A class of stochastic products on the convex set of quantum states
Paolo Aniello
TL;DR
This paper introduces stochastic products on quantum states, explores their properties, especially group-covariant and associative types, and demonstrates how they form Banach algebras, with implications for quantum symmetry and algebraic structures.
Contribution
It develops a new class of group-covariant, associative stochastic products called twirled stochastic products, linking them to projective representations and Banach algebra structures.
Findings
Twirled stochastic products form Banach algebras extending to trace class operators.
These products are commutative when based on abelian groups.
Explicit treatment of Weyl system-generated stochastic products.
Abstract
We introduce the notion of stochastic product as a binary operation on the convex set of quantum states (the density operators) that preserves the convex structure, and we investigate its main consequences. We consider, in particular, stochastic products that are covariant wrt a symmetry action of a locally compact group. We then construct an interesting class of group-covariant, associative stochastic products, the so-called twirled stochastic products. Every binary operation in this class is generated by a triple formed by a square integrable projective representation of a locally compact group, by a probability measure on that group and by a fiducial density operator acting in the carrier Hilbert space of the representation. The salient properties of such a product are studied. It is argued, in particular, that, extending this binary operation from the density operators to the whole…
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A class of stochastic products on the convex set of
quantum states
Paolo Aniello1,2
1*Dipartimento di Fisica “Ettore Pancini”, Università di Napoli “Federico II”,
Complesso Universitario di Monte S. Angelo, via Cintia, I-80126 Napoli, Italy
2Istituto Nazionale di Fisica Nucleare, Sezione di Napoli,
Complesso Universitario di Monte S. Angelo, via Cintia, I-80126 Napoli, Italy*
Abstract
We introduce the notion of stochastic product as a binary operation on the convex set of quantum states (the density operators) that preserves the convex structure, and we investigate its main consequences. We consider, in particular, stochastic products that are covariant wrt a symmetry action of a locally compact group. We then construct an interesting class of group-covariant, associative stochastic products, the so-called twirled stochastic products. Every binary operation in this class is generated by a triple formed by a square integrable projective representation of a locally compact group, by a probability measure on that group and by a fiducial density operator acting in the carrier Hilbert space of the representation. The salient properties of such a product are studied. It is argued, in particular, that, extending this binary operation from the density operators to the whole Banach space of trace class operators, this space becomes a Banach algebra, a so-called twirled stochastic algebra. This algebra is shown to be commutative in the case where the relevant group is abelian. In particular, the commutative stochastic products generated by the Weyl system are treated in detail. Finally, the physical interpretation of twirled stochastic products and various interesting connections with the literature are discussed.
1 Introduction
Operator algebras and various related structures turn out to play a remarkable role in quantum mechanics [1, 2, 3, 4, 5]. Moreover, essentially the same algebraic structures and tools are fundamental in the context of quantum field theory [6], in quantum statistical mechanics [7] and in non-commutative geometry [8]. In particular, quantum states can be defined as normalized positive functionals on the -algebra of (bounded) observables [1, 2, 4, 6].
In several contexts, e.g., in quantum information science [9, 10] and in quantum measurement theory [11, 12, 13], one actually restricts to a distinct class of states — the so-called normal or completely additive states [1, 4] (in the finite-dimensional setting, all states are of this kind) — that can be realized as Hilbert space operators: von Neumann’s statistical operators, or density operators; namely, as normalized, positive trace class operators. These operators — that in the following will be simply referred to as (quantum) states — form a closed convex subset of the complex Banach space of trace class operators, which, endowed with the ordinary product of operators (i.e., composition) and with the involution (the adjoining map), becomes a Banach -algebra. This algebra is embedded in the -algebra of bounded operators as a two-sided ideal, and the pairing between states and observables — that form the selfadjoint part of this -algebra — is realized by means of the trace functional.
One should not forget, however, that the ordinary product of operators defines the algebraic structure of quantum observables, and states are only indirectly involved in this structure as positive functionals. Actually, the product of two positive operators is, in general, not positive itself; in particular, the product of two density operators is not, in general, a density operator. Endowing the Banach space of trace class operators with, e.g., the Jordan or with the Lie product , one obtains algebraic structures that preserve selfadjointness; i.e., the — Jordan or Lie — product of two selfadjoint operators is selfadjoint too. But the Jordan and the Lie products are not associative, and the composition or the Jordan product of two density operators is a quantum state if and only if the two factors are equal and pure — i.e., trivially, we have the idempotent product of a rank-one projection operator by itself — whereas the Lie product of two density operators is never a quantum state.
On the one hand, as previously stressed, these facts are not surprising, because all the mentioned products pertain to the canonical structure of the -algebra of quantum observables, whose selfadjoint part can in fact be regarded as a Jordan-Lie Banach algebra [3, 14]. On the other hand, it is quite natural to wonder whether one can endow the Banach space of trace class operators with the structure of an algebra enjoying the property of being state-preserving; namely, such that the product of two quantum states is again a quantum state. Of course, in order to avoid trivial examples, one should require that this product be a genuinely binary operation; i.e., the binary operation should effectively involve both its arguments. Moreover, it would be desirable to have an associative algebra, whose product be continuous wrt some suitable topology. Also, due to the central importance of symmetry transformations in quantum mechanics [15, 16, 17, 18, 19, 20], it would be interesting to achieve a state-preserving product enjoying some suitable covariance property wrt a symmetry action of a certain abstract group. Finally, a physical interpretation of such a product would be in order.
The analogy with classical (statistical) physics is often a powerful guide when dealing with new aspects of quantum theory. In this regard, it is worth recalling that classical states — namely, Borel probability measures on phase space (say, ) — are embedded in the Banach algebra of complex (finite) Borel measures on , where the algebra product is realized by convolution [21, 22]. In particular, the convolution of two probability measures is still a probability measure. Otherwise stated, convolution is state-preserving: The convolution of two (classical) states is a state too. Notice that this structure extends immediately to the Banach algebra of complex Radon measures on a locally compact group [23]. It is then natural to wonder whether a similar structure may be envisaged in the quantum setting as well.
Convolution is a group-theoretical notion; hence, we will consider a general group-theoretical framework where — regarded as the group of translations on phase space — is replaced by some abstract topological group . In spite of this great generality, however, one expects that the case of the group of translations on phase space be a meaningful example. We will actually show that it is possible to define a whole class of associative state-preserving products on the space of trace class operators, for both finite-dimensional and infinite-dimensional quantum systems. Each product is generated by a square integrable (in general, projective) representation [24, 25, 26, 27, 28] of a locally compact group , by a Borel probability measure on and by a fiducial state , i.e., a density operator acting in the carrier Hilbert space of the representation. Our construction will yield a state-preserving product on the Banach space of trace class operators in , which, endowed with this binary operation, will be shown to be a Banach algebra. As in the case of convolution, this algebra turns out to be commutative if the group is abelian. We have first briefly sketched this kind of product in [28], in the simplest case where (the Dirac measure at the identity of ). We will now consider the general framework, studying the main properties of this new class of products and providing complete proofs. Moreover, we will work out two remarkable examples: the case where is compact and the case of the abelian group . In particular, in the latter case, for , we get a ‘quantum convolution’. It is worth observing that, unlike the ‘classical’ convolution, the quantum convolution possesses a degree of freedom; namely, it depends on the choice of a fiducial state. An interpretation of this fact in terms of Wigner functions and of the associated quantum characteristic functions will be provided. We will also argue that the quantum convolution is intimately related to Werner’s remarkable approach to quantum harmonic analysis on phase space [29].
The first part of the paper will actually be devoted to set the basic rules of the game; i.e., to outline the general algebraic background of our work. We will introduce the abstract notion of stochastic product as a binary operation on the convex set of quantum states that preserves the convex structure, and we will study its salient properties. In particular, we will argue that a stochastic product can be extended to a state-preserving bilinear map on . Such an algebra structure on will be called, in the associative case, a stochastic algebra. In this context, our group-theoretical construction of a stochastic product will yield the class of twirled stochastic algebras.
The paper is organized as follows. In sect. 2, we establish the relevant mathematical background. Next, in sect. 3, we introduce the abstract notions of stochastic product and stochastic algebra, and derive their main consequences. We will then define and study the class of twirled products of trace class operators, that, under suitable assumptions, give rise to stochastic products and algebras, the aforementioned twirled stochastic algebras; see sect. 4. In sect. 5, we will show by means of examples that twirled stochastic products can be constructed for every Hilbert space dimension. Eventually, in sect. 6, a physical interpretation of these stochastic products will be proposed by discussing some interesting connections with quantum measurement theory, with a final glance at some possible developments of our work.
2 Preliminaries: state-preserving linear and bilinear maps
In this section, we fix our main terminology and notations, and collect some basic facts and results that will be fundamental in the rest of the paper.
Let be a separable complex Hilbert space. We will suppose that (we neglect the trivial one-dimensional case). We assume the scalar product in to be linear in its second argument, and the symbol will be adopted for the identity operator. We denote by the complex Banach space of trace class operators in , by \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} the real Banach space of selfadjoint trace class operators, by \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny+}}\subset\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} the convex cone of all positive trace class operators operators and by \mathcal{S}(\mathcal{H})\subset\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} the convex set of all density operators (unit trace, positive trace class operators). will be regarded as the set of (completely additive) states of a quantum system. The extreme points of the convex set form the set of pure states (the rank-one projectors in ). We denote by the (trace) norm in both and \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}. The Banach space dual of is identified — via the pairing — with , the complex Banach space of all bounded operators in , endowed with the standard operator norm . Similarly, the dual space of \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} is \mathcal{B}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}, the real Banach space of (bounded) quantum observables. The symbols , will denote the Banach spaces of bounded linear operators in and , respectively, with norms and . Finally, we denote by the unitary group of and by the unitary-antiunitary group.
2.1 Some basic facts concerning state-preserving linear maps
Consider a map . is called positive if \Phi(\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny+}})\subset\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny+}}; it is called trace-preserving (alternatively, adjoint-preserving) if, for every , (respectively, ). It is clear that, if is adjoint-preserving, then it is also selfadjoint-preserving; namely, \Phi(\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}})\subset\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}.
If , for all , is said to be contractive. In the case where is linear, this condition amounts to requiring that and .
We say that preserves the set of states — in short, that it is state-preserving — if ; we say that is stochastic if it is linear and state-preserving. Clearly, a stochastic map in is the quantum analogue of a stochastic map (or matrix) of classical probability theory [10]. A (quantum) stochastic map which is also completely positive [12, 13, 30] is often called a quantum channel in quantum information science.
Example 1**.**
A stochastic map is, in general, neither injective nor surjective. Consider, e.g., the collapse channel , where is a fixed density operator. We will call the image rank of the collapse channel. The collapse channel is said to be pure if . Collapse channels in form a convex subset of whose extreme points are the pure collapse channels.
Obviously, if is positive and trace-preserving, then, in particular, it is state-preserving. Conversely, in the case where is linear, is state-preserving (if and) only if it is both positive and trace-preserving; i.e., if is stochastic, then it is a trace-preserving positive linear map. Indeed, for every , we can write
[TABLE]
where A_{\mbox{\tiny\Re}}, A_{\mbox{\tiny\Im}} are uniquely determined, and
[TABLE]
where:
[TABLE]
The four positive operators A_{\mbox{\tiny\Re};\mbox{\tiny+}},\ldots,A_{\mbox{\tiny\Im};\mbox{\tiny-}} are uniquely determined if we impose the condition
[TABLE]
whose solution is: 2\hskip 1.70717ptA_{\mbox{\tiny\Re};\mbox{\tiny+}}=|A_{\mbox{\tiny\Re}}|+A_{\mbox{\tiny\Re}},\ldots,2\hskip 1.70717ptA_{\mbox{\tiny\Im};\mbox{\tiny-}}=|A_{\mbox{\tiny\Im}}|-A_{\mbox{\tiny\Im}}. We then set
[TABLE]
Notice that, by this definition and by relation (4), \check{A}_{\mbox{\tiny\Re};\mbox{\tiny+}}\hskip 0.85358pt\check{A}_{\mbox{\tiny\Re};\mbox{\tiny-}}=0=\check{A}_{\mbox{\tiny\Im};\mbox{\tiny+}}\hskip 0.85358pt\check{A}_{\mbox{\tiny\Im};\mbox{\tiny-}}.
Therefore, if is linear and state-preserving, then it is positive and
[TABLE]
By an analogous reasoning, a selfadjoint-preserving linear map in is adjoint-preserving.
With the obvious exception of the notion of adjoint-preserving map, analogous definitions and facts can be established for a map from \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} into itself. Clearly, every selfadjoint-preserving map in can be restricted to \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}. In particular, if is a positive linear map, then, setting
[TABLE]
we obtain a positive (real-)linear map \Phi_{\mbox{\tiny\mathbb{R}}}\colon\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}\rightarrow\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}, because, for every A\in\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}},
[TABLE]
Namely, is selfadjoint-preserving, and, being linear, also adjoint-preserving: . We will call \Phi_{\mbox{\tiny\mathbb{R}}} the restriction of the (selfadjoint-preserving) map to \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}. Obviously, the restriction \Phi_{\mbox{\tiny\mathbb{R}}} is trace-preserving or contractive if is. By linearity and by decomposition (1), it is easy to see that, conversely, if \Phi_{\mbox{\tiny\mathbb{R}}} is trace-preserving, then is trace-preserving too.
On the other hand, if \Psi\colon\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}\rightarrow\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} is a linear map, then setting
[TABLE]
we obtain an adjoint-preserving (complex-)linear map \Psi_{\mbox{\tiny\mathbb{C}}}\colon\mathcal{B}_{1}(\mathcal{H})\rightarrow\mathcal{B}_{1}(\mathcal{H}). Note that \Psi_{\mbox{\tiny\mathbb{C}}} — the canonical extension of the linear map to , or, simply, the complexification of — is the unique (adjoint-preserving) linear map in whose restriction to \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} is . It is clear that \Psi_{\mbox{\tiny\mathbb{C}}} is positive or trace-preserving if (and only if) is.
Proposition 1**.**
Let be a linear map. If is positive, then it is bounded: ; if is also trace-preserving, then it is mildly contractive, i.e., . If is trace-preserving and contractive, then it is positive (thus, actually, mildly contractive: ). Analogous results hold for a linear map from \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} into itself.
Proof.
It is known that every positive linear map in is bounded; see, e.g., Proposition 6.2 of [12]. Let be such a map and let be its Banach space adjoint (wrt the pairing ), which is a positive map too. Moreover, if is also trace-preserving, then is unital; i.e., . Therefore, if is positive and trace-preserving, then, by the Russo-Dye theorem (Corollary 2.9 of [31]), ; which is equivalent to . Now, let be a trace-preserving bounded linear map. If , then the unital map satisfies . Therefore, by Proposition 2.11 of [31], — hence, — is positive.
Next, if \Psi\colon\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}\rightarrow\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} is a positive linear map, then its extension \Psi_{\mbox{\tiny\mathbb{C}}} to is a positive linear map too. Therefore, \Psi_{\mbox{\tiny\mathbb{C}}} — hence, — is bounded. If, in addition, is trace-preserving, then \Psi_{\mbox{\tiny\mathbb{C}}} is trace-preserving as well, so that \|\Psi_{\mbox{\tiny\mathbb{C}}}\|_{[1]}=1; hence: \|\Psi\|_{[1]}\leq\|\Psi_{\mbox{\tiny\mathbb{C}}}\|_{[1]}=1. On the other hand, for every ,
[TABLE]
and thus . Finally, if \Psi\colon\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}\rightarrow\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} is a contractive, trace-preserving linear map, for every , , we have:
[TABLE]
Hence, \Psi(A)_{\mbox{\tiny-}}=0 and is a positive map. ∎
Remark 1*.*
Notice that a trace-preserving and adjoint-preserving linear map , which is also contractive on \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}, is positive. Indeed, its restriction \Phi_{\mbox{\tiny\mathbb{R}}} to \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} is trace-preserving and contractive. Hence, by Proposition 1, \Phi_{\mbox{\tiny\mathbb{R}}} is positive, so that is positive too.
By the previous discussion and by Proposition 1, one immediately concludes that
Proposition 2**.**
For every linear map in (alternatively, in \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}) the following facts are equivalent:
- •
* is stochastic.*
- •
* is the complexification (respectively, the restriction to \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}) of a stochastic map in \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} (respectively, in ).*
- •
* is positive and trace-preserving.*
- •
* is trace-preserving and contractive.*
- •
* is trace-preserving, bounded and mildly contractive .*
2.2 Adjoint-preserving isometries, symmetries and pureness-preserving maps
Later on, we will also exploit some useful facts related to Wigner’s theorem on symmetry transformations that will be now discussed.
Theorem 1**.**
Let be an isometric, adjoint-preserving, surjective linear map in . Then, is either of the form , or of the form , where the (constant) factor , and , are a unitary and an antiunitary operator in , respectively, that are uniquely defined up to multiplication by a phase factor.
Proof.
By a classical result due to Russo [32, 33] — also see Theorem 5 and Remark 2 of [20] — a surjective linear isometry is either of the form , or of the form , where and are pairs, respectively, of unitary and antiunitary operators in . In particular, if is adjoint-preserving, then we must have that and . Indeed, if were such that — or — then the rank-one operator , or , would not be selfadjoint; in fact, we would have: and , or and . Hence, we actually have: — or — for all . We stress that here the factor cannot depend on the choice of , because the unitary (linear) operators , — or the antiunitary (antilinear) operators , — generate the same symmetry transformation [16, 18, 19, 20], so that must be a constant factor; see, in particular, Theorem 2 of [16]. Therefore, an isometric, adjoint-preserving, surjective linear map in is — possibly up to a factor — the bonded linear extension of a symmetry transformation () where, by Wigner’s theorem [16, 18, 19, 20], the unitary operator (the antiunitary operator ) is uniquely defined up to a phase factor. ∎
Corollary 1**.**
Let be an isometric, positive linear map (hence, a stochastic map) in . If is surjective, then it is a symmetry transformation; i.e., either of the form , or of the form , where , are a unitary and an antiunitary operator in , respectively, uniquely defined up to a phase factor.
Remark 2*.*
According to Wigner [15], a symmetry transformation is a bijective map on the pure states preserving the transition probability. In our setting of linear maps in , a symmetry transformation will be simply a map in of the form specified in Corollary 1. This is of course coherent with Wigner’s classical theorem [15, 16, 17, 18, 19, 20], and with a linear version of this theorem [19, 20], where the assumption of preservation of the transition probability between pure states becomes immaterial.
Remark 3*.*
By Corollary 1, the group (wrt composition) of all positive, surjective linear isometries in is precisely the group of all symmetry transformations in , which is of course isomorphic to the projective unitary-antiunitary group , where is identified with the subgroup of the group of all unitary or antiunitary operators in .
Remark 4*.*
Let be a linear isometry in . By Proposition 1, is positive if (and only if) it is trace-preserving.
Remark 5*.*
Clearly, if , then a linear isometry in is automatically surjective.
Observe that, taking into account the previous remark and the contractivity of stochastic maps, from Corollary 1 one immediately derives the following:
Corollary 2**.**
Let be a stochastic map, with . Then, exactly one of the following two alternatives is realized:
* is a symmetry transformation.* 2. 2.
There exists a non-positive , with , such that .
Example 2**.**
In the case where , it is easy to show that the previous result does not hold by constructing an explicit example of an isometric stochastic map which is not a symmetry transformation. Indeed, for any , or for , let be a set of linear or antilinear isometries such that
[TABLE]
and let be a strictly positive probability distribution. Consider the linear map in defined by
[TABLE]
Here, in the case where , the sum converges wrt the norm of . is an isometric stochastic map — as the reader may easily check — which is a symmetry transformation if and only if and , with . Indeed, for , , for all , but , so that is a symmetry transformation if and only if ( is a unitary or antiunitary). For and , the state is not pure, because, by condition (12), ; thus, in this case is not a symmetry transformation. Accordingly, is surjective if and only if and . In fact — taking and , for and ; and , for — we have: , for all ().
Notice that, in the previous example, the case corresponds precisely to those stochastic maps of the form (13) with the property of mapping pure states into pure states, and this class of maps includes, in particular, the symmetry transformations. We will now describe the whole class of stochastic maps having the mentioned property.
Definition 1**.**
We say that a map is pureness-preserving if .
The structure of pureness-preserving linear maps is determined by the following refinement of Theorem 1 of [19] (in the proofs of these results essentially the same arguments are used):
Theorem 2**.**
Let be a densely defined linear operator in — with , the linear subspace of finite rank operators — mapping the set of pure states into itself. Then, is closable, and its closure is a bounded operator in of one of the following types:
an isometric stochastic map of the form
[TABLE]
where , are, respectively, a linear isometry and an antilinear isometry in , uniquely determined up to a phase factor; 2. 2.
a collapse channel of the form
[TABLE]
for some (fixed) pure state .
Proof.
By the spectral decomposition of a selfadjoint finite rank operator — with each spectral projection regarded as a sum of mutually orthogonal rank-one projections — and by the fact that is mapped by into itself, one concludes that is positive, trace-preserving and bounded on its dense domain , the linear subspace of all finite rank operators in . Thus, is closable and its closure is a bounded operator defined on . By the spectral decomposition of a selfadjoint trace class operator (converging in the trace norm), one argues as above that is a trace-preserving and positive too.
Therefore, is, in particular, a positive linear map, mapping the set of all pure elements of the positive cone \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny+}} of — the rank-one positive operators [34] — into itself, namely, a pure positive map. Then, by a classical result of Davies (Theorem 3.1 of [35]; also see Theorem 3.1, in chapt. 2 of [34]), is of one of the following forms:
- (i)
, for some bounded linear operator in , uniquely determined up to a phase factor; 2. (ii)
, for some bounded antilinear operator in , uniquely determined up to a phase factor; 3. (iii)
, for some positive operator and some pure state .
In the case (i), must be actually a linear isometry, since is also trace-preserving, so that
[TABLE]
Here, we have used the fact that is (isomorphic to) . Analogously, in the case (ii), , and must be an antilinear isometry. Finally, in the case (iii), we have: , for all ; hence, . ∎
Corollary 3**.**
Every pureness-preserving stochastic map is of the form (14) or (15).
Proof.
A stochastic map in is bounded, so that, if it is also pureness-preserving, it must coincide with the closure of a linear operator, with domain (the finite-rank operators), that maps into itself. Then, by Theorem 2 the statement follows. ∎
Remark 6*.*
Recalling Remark 5, observe that, in the case where () , the two possible forms (14) or (15) of a pureness-preserving stochastic map in realize, respectively, the two alternative forms of a stochastic map in Corollary 2.
Remark 7*.*
Notice that a pureness-preserving stochastic map preserves the purity of a state if and only if it is of the form (14) — ( linear isometry), ( antilinear isometry) — whereas a map of the form (15) strictly increases the purity of every non-pure state. Besides, a stochastic map that does not decrease the purity of states must map pure states into pure states.
By the previous remark, we have:
Corollary 4**.**
A stochastic map is purity-nondecreasing — , — if and only if it is pureness-preserving.
2.3 State-preserving bilinear maps
We will now extend some of the previous definitions and results to a binary operation on or on \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} (it is often useful to switch back and forth between the two spaces). We start with those cases where this extension is straightforward.
Definition 2**.**
We say that a map is positive if
[TABLE]
We say that preserves the set of states — in short, that it is state-preserving — if, for every pair of states , . We say that is stochastic if it is bilinear and state-preserving. Analogous definitions we set for a map from \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}\times\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} into \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}.
Definition 3**.**
We say that a map is selfadjoint-preserving if \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}\boxdot\hskip 0.85358pt\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}\subset\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}; we say that it is adjoint-preserving if
[TABLE]
Remark 8*.*
The definition of an adjoint-preserving binary operation on is already less obvious than the previous ones. Indeed, e.g., , endowed with the standard product of operators (composition) and with the adjoining map , becomes a Banach -algebra (in particular, we have: ). On the other hand, the adjoining map is an involution — in particular, (compare with (18)) — hence, the operator product is not adjoint-preserving.
Clearly, an adjoint-preserving binary operation on is selfadjoint-preserving; moreover:
Proposition 3**.**
Every selfadjoint-preserving bilinear map is adjoint-preserving. In particular, if a bilinear map on is positive, then it is (selfadjoint-preserving, hence) adjoint-preserving.
Proof.
If is bilinear and selfadjoint-preserving, for , writing A=A_{\mbox{\tiny\Re}}+\mathrm{i}\hskip 1.70717ptA_{\mbox{\tiny\Im}} and B=B_{\mbox{\tiny\Re}}+\mathrm{i}\hskip 1.70717ptB_{\mbox{\tiny\Im}}, with A_{\mbox{\tiny\Re}},\ldots,B_{\mbox{\tiny\Im}}\in\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}, and noting that A_{\mbox{\tiny\Re}}\boxdot B_{\mbox{\tiny\Re}},\ldots,A_{\mbox{\tiny\Im}}\boxdot B_{\mbox{\tiny\Re}}\in\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}, we have:
[TABLE]
i.e., is adjoint-preserving. If is a positive bilinear map, for every A=A_{\mbox{\tiny+}}-A_{\mbox{\tiny-}} and B=B_{\mbox{\tiny+}}-B_{\mbox{\tiny-}} in \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}, with A_{\mbox{\tiny+}},A_{\mbox{\tiny-}},B_{\mbox{\tiny+}},B_{\mbox{\tiny-}}\in\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny+}}, we see that is selfadjoint; i.e., the map is selfadjoint-preserving, hence, adjoint-preserving. ∎
An adjoint-preserving bilinear map — in particular, a positive bilinear map — on can be restricted to a bilinear map on the real Banach space \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}. Conversely, given a bilinear map (\hskip 0.85358pt\cdot\hskip 0.85358pt)\boxminus(\hskip 0.85358pt\cdot\hskip 0.85358pt)\colon\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}\times\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}\rightarrow\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}, for — with A=A_{\mbox{\tiny\Re}}+\mathrm{i}A_{\mbox{\tiny\Im}}, B=B_{\mbox{\tiny\Re}}+\mathrm{i}B_{\mbox{\tiny\Im}}, A_{\mbox{\tiny\Re}},A_{\mbox{\tiny\Im}},B_{\mbox{\tiny\Re}},B_{\mbox{\tiny\Im}}\in\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} — one can set
[TABLE]
so obtaining an adjoint-preserving (complex-)bilinear map on , which will be called the complexification of the bilinear map on \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}.
At this point, it is not immediately clear what a trace-preserving binary operation should be. For the moment, just notice that the following definition is compatible with the requirement that the binary operation be bilinear.
Definition 4**.**
We say that a map is trace-preserving if
[TABLE]
An analogous definition we set for a map from \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}\times\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} into \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}.
For a bilinear map and trace class operators — using notations (1)–(5) for , and analogous notations B_{\mbox{\tiny\Re}},B_{\mbox{\tiny\Im}}, b_{\mbox{\tiny\Re};\mbox{\tiny+}},\ldots,b_{\mbox{\tiny\Im};\mbox{\tiny-}}, \check{B}_{\mbox{\tiny\Re};\mbox{\tiny+}},\ldots,\check{B}_{\mbox{\tiny\Im};\mbox{\tiny-}} relative to — we have:
[TABLE]
and
[TABLE]
Hence:
Proposition 4**.**
A stochastic map is positive and trace-preserving. An analogous result holds for a stochastic map from \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}\times\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} into \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}.
Let us denote by , \mathscr{B}\mathscr{L}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} the Banach space of bonded bilinear maps on and on \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}, respectively, endowed with the norm
[TABLE]
Recall that there are two natural Banach space isomorphisms between (or \mathscr{B}\mathscr{L}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}) and the Banach space
[TABLE]
that are given by
[TABLE]
These isomorphisms justify our use, in the following, of the symbol to denote, as well, the standard operator norm in and in \mathcal{B}(\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}},\mathscr{L}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}).
Definition 5**.**
We say that a map is contractive on a set if
[TABLE]
We say that is mildly contractive on if it is contractive on and
[TABLE]
We say tout court that is contractive (alternatively, mildly contractive) if it is contractive (respectively, mildly contractive) on . Analogous definitions we set for a map from \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}\times\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} to \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}.
We have the following analogue of Proposition 2:
Proposition 5**.**
Let be a bilinear map on (alternatively, on \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}). Then, the following facts are equivalent:
(P1)
* is stochastic.*
(P1′)
The transpose (\hskip 0.85358pt\cdot\hskip 0.85358pt)\boxtimes^{{\mbox{\tiny\mathsf{T}}}}(\hskip 0.85358pt\cdot\hskip 0.85358pt) of — A\boxtimes^{{\mbox{\tiny\mathsf{T}}}}B\mathrel{\mathop{:}}=B\boxtimes A — is stochastic.
(P2)
* is the complexification (respectively, the restriction to a binary operation on \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}) of a stochastic map (\hskip 0.85358pt\cdot\hskip 0.85358pt)\boxminus(\hskip 0.85358pt\cdot\hskip 0.85358pt)\colon\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}\times\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}\rightarrow\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} (respectively, of a stochastic map ).*
(P3)
* is positive and trace-preserving.*
(P4)
For every , the linear map is trace-preserving, bounded and satisfies the condition .
(P5)
* is contractive on \mathcal{S}(\mathcal{H})\times\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} and trace-preserving.*
(P6)
* is mildly contractive on \mathcal{S}(\mathcal{H})\times\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} and trace-preserving.*
(P7)
* is contractive on \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}\times\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} and trace-preserving.*
(P8)
* is mildly contractive on \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}\times\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} and trace-preserving.*
Proof.
The equivalence between (P1) and (P1′), and between (P1) and (P2), is clear. By Proposition 4, (P1) and (P3) are equivalent too.
Moreover, (P3) implies (P4). Indeed, if (P3) holds, then, for every , the linear map is positive and trace-preserving. Hence, by the first assertion of Proposition 1, is bounded and we have that . Conversely, let (P4) hold. By linearity in its first argument, the map is trace-preserving. Moreover, by the second assertion of Proposition 1, the linear map is positive, for every ; hence, by linearity in its first argument, is positive and (P3) holds true.
Notice that (P4) implies (P6). Indeed, if (P4) holds, then — since (P4) (P3) — is trace-preserving and positive. Moreover,
[TABLE]
i.e., is mildly contractive on \mathcal{S}(\mathcal{H})\times\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}. Here, in the case where is regarded as a bilinear map on , we are using the fact that, for every , is positive and trace-preserving, so that
[TABLE]
Obviously, (P6) implies (P5). Also, (P5) implies (P3). In fact, if the map is trace-preserving and , for all and B\in\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}, then, for every , the linear map \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}\ni B\mapsto A\boxtimes B\in\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} is trace-preserving and contractive; hence positive, by the second assertion of Proposition 1. Thus, is trace-preserving and, by linearity in its first argument, positive.
(P3) implies (P7). Indeed, if the bilinear map is positive and trace-preserving — denoting by its restriction to a bilinear map on \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}, in the case where is regarded as a binary operation on , or, otherwise, the map itself — we have:
[TABLE]
Thus, is (trace-preserving and) contractive on \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}\times\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}.
Clearly, (P8) (P7). Let us prove that (P7) (P8), as well. If (P7) holds, then
[TABLE]
Moreover, (P7) (P5) (P3); hence, , for all , i.e., inequality (32) is actually saturated, and is mildly contractive on \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}\times\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}.
In conclusion, (P1′) (P1) (P2), (P1) (P3) (P4) (P6) (P5) (P3) (P7) (P8) and (P7) (P5); thus, the proof is complete. ∎
By the Principle of Uniform Boundedness [36], it turns out that the separate continuity of a bilinear map on or \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} is equivalent to its joint continuity — wrt the norm in — or, equivalently, to its boundedness. Besides, the boundedness of is implied by the property of preserving the set of all states; precisely:
Proposition 6**.**
Every stochastic map is bounded, contractive on the sets \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}\times\mathcal{B}_{1}(\mathcal{H}) and \mathcal{B}_{1}(\mathcal{H})\times\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}, and its norm satisfies ; whereas, its restriction to a bilinear map on \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} is such that .
Proof.
Exploiting decomposition (1)–(5) for , and, next, the implication (P1) (P4) in Proposition 5, we have:
[TABLE]
In particular, setting A_{\mbox{\tiny\Im}}=0, we conclude that is contractive on \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}\times\mathcal{B}_{1}(\mathcal{H}). Replacing with its transpose, we see that , for (A,B)\in\mathcal{B}_{1}(\mathcal{H})\times\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}. Moreover, by the fact that (P1) implies (P8) in Proposition 5, is mildly contractive. ∎
3 Stochastic products and algebras
In this section, we will introduce the notion of stochastic product and study its main consequences. In particular, it turns out that an associative stochastic product is always associated with a ‘stochastic algebra’. We will also consider some special classes of points in the domain of a stochastic product and a natural property of group-covariance. As in the previous section, we will suppose that .
3.1 Definition and basic facts
We now define a stochastic product as a binary operation on preserving the natural convex structure of this set. The sets and , endowed with the distances and , respectively, become metric spaces. It would be natural to require that a stochastic product be continuous wrt the associated topologies, but, as it will be clear soon, this property is automatically satisfied.
Remark 9*.*
The weak and the strong operator topologies on , and the topology induced on by any Schatten -norm , , all coincide; we will call this topology the standard topology on . Indeed, applying Theorem 2.20 of [37] with , we conclude that, if a sequence in converges wrt the weak operator topology, then it converges wrt the trace norm topology as well; hence, a fortiori, wrt any other of the previously mentioned topologies. Thus, the topology induced on by the distance coincides with the product topology associated with the standard topology on .
Definition 6**.**
A stochastic product on is a map that is convex-linear in both its arguments, i.e.,
[TABLE]
for all and .
A fundamental connection of the previous definition with the state-preserving bilinear maps on is provided by the following fact:
Proposition 7**.**
Every stochastic product can be obtained as a suitable restriction of a unique state-preserving bilinear map; namely, for every stochastic product , there is a uniquely determined stochastic map — the so-called canonical extension of — such that
[TABLE]
Proof.
Let us define a bilinear map by extending the stochastic product , which will be now regarded as a map from to , convex-linear in both its arguments. We will extend this map from to \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny+}}\times\mathcal{S}(\mathcal{H}), then to \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}\times\mathcal{S}(\mathcal{H}) and , and finally to . We denote by \mathbb{R}_{\ast}^{\phantom{\mbox{\tinyi}}} the set of nonzero real numbers and by \mathbb{R}_{\hskip 0.85358pt\ast}^{\mbox{\tiny+}} the set of strictly positive real numbers.
Let us first fix some arbitrary state and set . For every A\in\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny+}}, , we define
[TABLE]
Then, for every r\in\mathbb{R}_{\hskip 0.85358pt\ast}^{\mbox{\tiny+}} and 0\neq A\in\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny+}},
[TABLE]
and, by the convex-linearity of in its first argument, for A,B\in\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny+}}, ,
[TABLE]
Let us further extend the map to \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}\times\mathcal{S}(\mathcal{H}). To this aim, as usual it is convenient to write
[TABLE]
where A_{\mbox{\tiny+}},A_{\mbox{\tiny-}}\in\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny+}}. We then set
[TABLE]
By the previous definition and by (36), for every r\in\mathbb{R}_{\ast}^{\phantom{\mbox{\tinyi}}} and A\in\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}, we have:
[TABLE]
moreover, . Next, notice that, given A,B\in\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}, we have:
[TABLE]
Therefore, (A+B)_{\mbox{\tiny+}}+A_{\mbox{\tiny-}}+B_{\mbox{\tiny-}}=(A+B)_{\mbox{\tiny-}}+A_{\mbox{\tiny+}}+B_{\mbox{\tiny+}}, and from this relation we get
[TABLE]
By this equation, we obtain:
[TABLE]
We have therefore constructed a (real-)linear map in \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}.
We then extend the map to , by setting, for every ,
[TABLE]
Then, for , , and , we have:
[TABLE]
Moreover, (A+B)_{\mbox{\tiny\Re}}=A_{\mbox{\tiny\Re}}+B_{\mbox{\tiny\Re}} and (A+B)_{\mbox{\tiny\Im}}=A_{\mbox{\tiny\Im}}+B_{\mbox{\tiny\Im}}; hence, , and we get a (complex-)linear map in .
Observe that the mapping constructed so far is such that, for every , the map is convex-linear.
Thus, arguing as above for extending the second argument of , we finally obtain a bilinear map from to , which is of course stochastic. Notice that, if a stochastic product is the restriction of a bilinear map , then the previous procedure — without the arguments proving linearity — can be regarded as a reconstruction of by linearity; hence, this bilinear map is uniquely determined. ∎
Remark 10*.*
By the same kind of reasoning that one uses in the proof of the previous result, we may have called stochastic a convex-linear map in , and then argued that such a map is the restriction to of a state-preserving linear map in .
Corollary 5**.**
Every stochastic product on is jointly continuous.
Proof.
By Proposition 7, every stochastic product is the restriction of a state-preserving bilinear map, which, by Proposition 6, is bounded; thus, (jointly) continuous. ∎
Let us denote by the convex set of all trace-preserving, positive linear maps in . Proposition 7 also implies the following:
Corollary 6**.**
For every stochastic product , there exist two bounded linear maps
[TABLE]
— respectively, the left partial map and the right partial map associated with — such that , uniquely determined by
[TABLE]
Moreover, the sets \mathsf{L}(\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}), \mathsf{R}(\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}) consist of adjoint-preserving bounded maps, and
[TABLE]
where is the canonical extension of the stochastic product , is the restriction of to a bilinear map on \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} and the bounded linear map \mathsf{L}_{\mbox{\tiny\mathbb{R}}}\in\mathcal{B}(\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}},\mathscr{L}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}) is defined by \mathsf{L}_{\mbox{\tiny\mathbb{R}}}\colon\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}\ni A\mapsto\mathsf{L}(A)_{\mbox{\tiny\mathbb{R}}}=A\boxminus(\hskip 0.85358pt\cdot\hskip 0.85358pt)\in\mathscr{L}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}.
Proof.
We only need to observe that the map is bounded — — and to justify relations (49). To this aim, let us recall the norm estimates in Proposition 6, and the Banach space isomorphisms (26) between (or \mathscr{B}\mathscr{L}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}) and (\mathcal{B}(\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}},\mathscr{L}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}})). ∎
We will say that a stochastic product is left-constant (alternatively, right-constant) on if
[TABLE]
i.e., if the left partial map (the right partial map ) is constant on . Clearly, is left-constant (right-constant) tout court when .
Remark 11*.*
Taking into account the fact that an element of can be expressed as a linear combination of (at most) four density operators, it is clear that is left-constant (right-constant) if and only if the associated linear map () is of the form (), where is arbitrary; i.e., a collapse channel.
A (nonempty) subset of of the form
[TABLE]
is called a left level set (a right level set) for the stochastic product . These level sets are convex subsets of . Clearly, two states , belong to the same left (right) level set if and only if (); thus: is injective on . Actually:
Proposition 8**.**
The linear map is injective on if and only if ; i.e., if and only if it is injective.
Proof.
We need to prove the ‘only if part’. For , using notations (1)–(5), we have that
[TABLE]
because \mathsf{L}(A_{\mbox{\tiny\Re}}),\mathsf{L}(A_{\mbox{\tiny\Im}})\in\mathscr{L}_{1}(\mathcal{H}) are adjoint-preserving maps. Moreover:
[TABLE]
because \mathsf{L}(\check{A}_{\mbox{\tiny\Re};\mbox{\tiny+}}),\mathsf{L}(\check{A}_{\mbox{\tiny\Re};\mbox{\tiny-}})\in\mathscr{P}\mathscr{T}(\mathcal{H})\cup\{0\} (since \check{A}_{\mbox{\tiny\Re};\mbox{\tiny+}},\check{A}_{\mbox{\tiny\Re};\mbox{\tiny-}}\in\mathcal{S}(\mathcal{H})\cup\{0\}). Similarly, \mathsf{L}(A_{\mbox{\tiny\Im}})=0 implies that \mathsf{L}(\check{A}_{\mbox{\tiny\Im};\mbox{\tiny+}})=\mathsf{L}(\check{A}_{\mbox{\tiny\Im};\mbox{\tiny-}}). If , then A_{\mbox{\tiny\Re}}\neq 0 and/or A_{\mbox{\tiny\Im}}\neq 0. Therefore, if (\Rightarrow\mathsf{L}(A_{\mbox{\tiny\Re}})=0=\mathsf{L}(A_{\mbox{\tiny\Im}})), with , supposing without loss of generality that A_{\mbox{\tiny\Re}}\neq 0, we have:
[TABLE]
Indeed, A_{\mbox{\tiny\Re}}=a_{\mbox{\tiny\Re};\mbox{\tiny+}}\hskip 0.85358pt\check{A}_{\mbox{\tiny\Re};\mbox{\tiny+}}-a_{\mbox{\tiny\Re};\mbox{\tiny-}}\hskip 0.85358pt\check{A}_{\mbox{\tiny\Re};\mbox{\tiny-}}\neq 0 cannot be positive or negative (otherwise, \mathsf{L}(A_{\mbox{\tiny\Re}}) would be a strictly positive or negative multiple of a trace-preserving, positive linear map) — thus, \check{A}_{\mbox{\tiny\Re};\mbox{\tiny+}},\check{A}_{\mbox{\tiny\Re};\mbox{\tiny-}}\neq 0 are both density operators — and we recall that these density operators are mutually orthogonal, i.e., \check{A}_{\mbox{\tiny\Re};\mbox{\tiny+}}\hskip 0.85358pt\check{A}_{\mbox{\tiny\Re};\mbox{\tiny-}}=0. In conclusion, we find: \check{A}_{\mbox{\tiny\Re};\mbox{\tiny+}},\check{A}_{\mbox{\tiny\Re};\mbox{\tiny-}}\in\mathcal{S}(\mathcal{H}), \check{A}_{\mbox{\tiny\Re};\mbox{\tiny+}}\neq\check{A}_{\mbox{\tiny\Re};\mbox{\tiny-}} and \mathsf{L}(\check{A}_{\mbox{\tiny\Re};\mbox{\tiny+}})=\mathsf{L}(\check{A}_{\mbox{\tiny\Re};\mbox{\tiny-}}). Therefore, if , then is not injective on . ∎
Example 3**.**
It is clear that there is a one-to-one correspondence between the set and the left-constant stochastic products on , i.e.,
[TABLE]
and an analogous correspondence holds, of course, for the right-constant products. However, these products are ‘trivial’, in the sense that they actually ‘involve’ just one of their arguments. Moreover, they are, in general, non-associative. The class of stochastic products that are simultaneously left-constant and right-constant consists of all collapse products; i.e., the stochastic products of the form
[TABLE]
where is a fixed state in .
Example 4**.**
There is a special class of left-constant stochastic products. Consider the stochastic product defined by
[TABLE]
for all , where is a unitary operator in . We call such a product a left-constant unitary product in . More generally, given a subset of , we will say that is left-constant unitary on if (57) holds for all and , and for some unitary operator . An analogous definition can be given for a left-constant antiunitary product, associated with an antiunitary operator in , and, of course, for the right-constant counterparts of these products.
Remark 12*.*
The (constant on ) left partial map associated with a left-constant antiunitary product on is of the form
[TABLE]
where is, of course, an antiunitary operator in .
The two previous examples motivate the following:
Definition 7**.**
We say that a stochastic product is genuinely binary if it is neither left-constant nor right-constant.
Example 5**.**
Is is easy to construct a genuinely binary (but, in general, non-associative) stochastic product on . Indeed, for every pair of maps , and for every , let us set
[TABLE]
This is a genuinely binary stochastic product if and only if neither nor are constant on ; i.e., of the form , where is some fixed state (collapse channel). The left and right partial maps associated with this product are given by
[TABLE]
Denoting by the stochastic bilinear map extending this product, we have that
[TABLE]
where we have used the inequality — see Theorem 3.1 of [37] — and the fact that (hence: ). Thus, in this case, the inequality (Proposition 6) is not saturated.
Example 6**.**
Let E\in\mathcal{B}(\mathcal{H})_{\mbox{\tiny+}} be an effect [12, 13] — i.e., — and let . We set:
[TABLE]
We then obtain a stochastic product that admits a straightforward generalization. Denoting by the convex set of all effects in , let be a discrete POVM [12, 13]; i.e., . Let, moreover, be linear maps in . A stochastic product in is defined by setting
[TABLE]
Example 7**.**
Let be a trace-preserving positive linear map, where is an ‘auxiliary’ (separable complex) Hilbert space. Denoting by the partial trace, we set
[TABLE]
We then obtain a stochastic product whose left and right partial maps are given by
[TABLE]
3.2 Stochastic algebras
The notion of stochastic product is related, in a natural way, to a notion of ‘stochastic algebra’. We will require that such an algebra be associative.
Definition 8**.**
The Banach space , endowed with a map that is both stochastic and an associative binary operation, is called a stochastic algebra.
Remark 13*.*
Since, by Proposition 7, every stochastic product on is the restriction of a unique stochastic bilinear map on , which is easily seen to be an associative operation if is associative, one may equivalently define a stochastic algebra as a Banach space of trace class operators together with an associative stochastic product on .
Remark 14*.*
Restricting the binary operation of a stochastic algebra to a bilinear map on \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}}, by Proposition 6 one obtains a (real) Banach algebra (\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}},(\hskip 0.85358pt\cdot\hskip 0.85358pt)\boxminus(\hskip 0.85358pt\cdot\hskip 0.85358pt)); in particular, . Taking into account the same proposition, it is clear that on may ‘renormalize’ by a factor the binary operation in such a way to obtain a complex Banach algebra; but the renormalized product would not be state-preserving. Besides, as Example 5 shows, a stochastic algebra may well be a (complex) Banach algebra.
3.3 Bijective and pureness-preserving points for a stochastic product
It is natural to distinguish certain classes of points in , relatively to a given stochastic product.
Definition 9**.**
We say that is a left injective point (a right injective point) for a stochastic product if () maps injectively into itself. Left (and right) surjective and bijective points for are defined analogously. Denoting by () the set of left (right) bijective points for , we say that this product is left bijective (right bijective) if (). A point in () is called a left non-bijective point (a right non-bijective point) for .
Remark 15*.*
The term ‘left bijective product’, which is coherent with the definition of a ‘left bijective point’ for the product, should not confuse the reader; a stochastic product which is left bijective is actually bijective wrt its right entry (say ‘bijective on the right’), for each point in its left entry (kept fixed).
Proposition 9**.**
* is a left (right) injective point for the stochastic product if and only if ; i.e., if and only if the linear map is injective. Moreover, if is a left (right) surjective point, then is surjective.*
Proof.
To prove the first assertion, we use an argument similar to the proof of Proposition 8. Again, we prove the ‘only if part’, the converse implication being obvious. For , since is adjoint-preserving, we have that
[TABLE]
Moreover, if 0=\big{(}\mathsf{L}(\rho)\big{)}(A_{\mbox{\tiny\Re}})=a_{\mbox{\tiny\Re};\mbox{\tiny+}}\hskip 0.85358pt\big{(}\mathsf{L}(\rho)\big{)}(\check{A}_{\mbox{\tiny\Re};\mbox{\tiny+}})-a_{\mbox{\tiny\Re};\mbox{\tiny-}}\hskip 0.85358pt\big{(}\mathsf{L}(\rho)\big{)}(\check{A}_{\mbox{\tiny\Re};\mbox{\tiny-}}), then (a_{\mbox{\tiny\Re};\mbox{\tiny+}}=a_{\mbox{\tiny\Re};\mbox{\tiny-}} and)
[TABLE]
because \check{A}_{\mbox{\tiny\Re};\mbox{\tiny+}},\check{A}_{\mbox{\tiny\Re};\mbox{\tiny-}}\in\mathcal{S}(\mathcal{H})\cup\{0\} and . Analogously, \big{(}\mathsf{L}(\rho)\big{)}(A_{\mbox{\tiny\Im}})=0 implies that \big{(}\mathsf{L}(\rho)\big{)}(\check{A}_{\mbox{\tiny\Im};\mbox{\tiny+}})=\big{(}\mathsf{L}(\rho)\big{)}(\check{A}_{\mbox{\tiny\Im};\mbox{\tiny-}}). If , then A_{\mbox{\tiny\Re}}\neq 0 and/or A_{\mbox{\tiny\Im}}\neq 0. Therefore, if — hence, \big{(}\mathsf{L}(\rho)\big{)}(A_{\mbox{\tiny\Re}})=0=\big{(}\mathsf{L}(\rho)\big{)}(A_{\mbox{\tiny\Im}}) — with , supposing that, say, A_{\mbox{\tiny\Re}}\neq 0, we have:
[TABLE]
Indeed, A_{\mbox{\tiny\Re}}=a_{\mbox{\tiny\Re};\mbox{\tiny+}}\hskip 0.85358pt\check{A}_{\mbox{\tiny\Re};\mbox{\tiny+}}-a_{\mbox{\tiny\Re};\mbox{\tiny-}}\hskip 0.85358pt\check{A}_{\mbox{\tiny\Re};\mbox{\tiny-}}\neq 0 cannot be positive or negative (otherwise, \big{(}\mathsf{L}(\rho)\big{)}(A_{\mbox{\tiny\Re}}) would be a strictly positive or negative multiple of a state) — thus, \check{A}_{\mbox{\tiny\Re};\mbox{\tiny+}},\check{A}_{\mbox{\tiny\Re};\mbox{\tiny-}}\neq 0 are both density operators — and we recall that these density operators are mutually orthogonal, i.e., \check{A}_{\mbox{\tiny\Re};\mbox{\tiny+}}\hskip 0.85358pt\check{A}_{\mbox{\tiny\Re};\mbox{\tiny-}}=0. In conclusion: \check{A}_{\mbox{\tiny\Re};\mbox{\tiny+}},\check{A}_{\mbox{\tiny\Re};\mbox{\tiny-}}\in\mathcal{S}(\mathcal{H}), \check{A}_{\mbox{\tiny\Re};\mbox{\tiny+}}\neq\check{A}_{\mbox{\tiny\Re};\mbox{\tiny-}} and \big{(}\mathsf{L}(\rho)\big{)}(\check{A}_{\mbox{\tiny\Re};\mbox{\tiny+}})=\big{(}\mathsf{L}(\rho)\big{)}(\check{A}_{\mbox{\tiny\Re};\mbox{\tiny-}}). Therefore, if , then is not injective on .
The second assertion is an immediate consequence of the fact that every trace class operator is a linear combination of (at most) four density operators and of the linearity of . ∎
Another natural notion is the following:
Definition 10**.**
We say that is a left (right) pureness-preserving point for a stochastic product if () is a pureness-preserving map (Definition 1). The subset of formed by all such points will be denoted by (). If (), the product is called left (right) pureness-preserving.
A characterization of the bijective points is provided by the following result:
Theorem 3**.**
Given and a stochastic product on , and denoting by the left (right) partial map associated with this product, the following facts are equivalent:
- (i)
* is a left (right) bijective point for .* 2. (ii)
* is isometric — i.e., \|\big{(}\mathsf{L}(\rho)\big{)}(A)\|_{1}=1, for all — and surjective.* 3. (iii)
* is a left (right) pureness-preserving point for such that \big{(}\mathsf{L}(\rho)\big{)}(\mathcal{P}(\mathcal{H}))=\mathcal{P}(\mathcal{H}) (\big{(}\mathsf{R}(\rho)\big{)}(\mathcal{P}(\mathcal{H}))=\mathcal{P}(\mathcal{H})).* 4. (iv)
* is such that*
[TABLE]
where is a unitary or antiunitary operator in , uniquely defined up to a phase factor; equivalently, is a symmetry transformation in .
Proof.
We prove the equivalence between each of properties (i)–(iii) and (iv). Clearly, (iv) implies (i)–(iii). If maps bijectively onto itself, the map \mathcal{S}(\mathcal{H})\ni\sigma\mapsto\big{(}\mathsf{L}(\rho)\big{)}(\sigma)=\rho\odot\sigma\in\mathcal{S}(\mathcal{H}) is a ‘Kadison automorphism’ [18] (i.e., a convex-linear map from onto itself), so that, by a variant of Wigner’s theorem on symmetry transformations [17, 18, 19, 20], relation (69) is verified (and, by linearity, is a symmetry transformation); therefore, (i) (iv). Moreover, if the positive linear map is a surjective isometry, then, by Proposition 1, is a symmetry transformation in and relation (69) is again satisfied; namely, (ii) (iv). Finally, (iii) implies (iv) because, if maps onto itself, then, by a linear version of Wigner’s theorem [19, 20], is once again a symmetry transformation. ∎
By the previous result, ; we will see that, actually, . Moreover:
Corollary 7**.**
If , is a left non-bijective (a right non-bijective) point for a stochastic product on if and only if there is a non-positive operator , with , such that \|\big{(}\mathsf{L}(\rho)\big{)}(A)\|_{1}<1 (\|\big{(}\mathsf{R}(\rho)\big{)}(A)\|_{1}<1). This result does not hold if .
Proof.
Taking into account the equivalence of properties (i) and (iv) in Theorem 3, just recall the dichotomy in Corollary 2, for the first assertion. The second one follows from Example 2. ∎
Let us now consider the pureness-preserving points.
Theorem 4**.**
Let be a left (right) pureness-preserving point for . Then, is of one of the following types:
an isometric stochastic map of the form , or , where , are, respectively, a linear isometry and an antilinear isometry in , uniquely determined up to a phase factor; 2. 2.
a collapse channel of the form , for some (fixed) pure state .
Therefore, , and is an injective point if and only if is not a collapse channel. Moreover, a left (right) pureness-preserving point is left (right) bijective if and only if it is left (right) surjective.
Proof.
The first assertion follows immediately from the definition of a pureness-preserving point and from Corollary 3. By the assumption that , a pureness-preserving point , such that is collapse channel, belongs to , and all non-injective points in are of this type. Moreover, if is a left pureness-preserving point, the stochastic map is bijective if and only if it is of the first kind in the dichotomy of the first assertion of the theorem and surjective; i.e., a symmetry transformation in . ∎
By the second assertion of the previous theorem, the pureness preserving points fall in two classes.
Definition 11**.**
is called a left collapsing point (a right collapsing point) for if () is a collapse channel. In particular, the set () can be partitioned into the subset of all collapsing and the subset of all injective left (right) pureness-preserving points.
Notice that, in the finite-dimensional case, every linear or antilinear isometry in is a unitary or an antiunitary operator; thus, in this case, coincides with the set of all injective left pureness-preserving points.
Let us now derive a few further consequences of Theorem 4.
Corollary 8**.**
A stochastic product on is left-constant (right-constant) on every convex set consisting of left (right) pureness-preserving points. Therefore, is either empty or partitioned into maximal convex subsets that are left (right) level sets for the stochastic product. In particular, a left (right) pureness-preserving stochastic product cannot be genuinely binary, because it must be left-constant (right-constant).
Proof.
The statement is trivial if the convex set of pureness-preserving points is a singleton. Then, let be two states belonging to a convex set of left pureness-preserving points for , with . Let us first suppose that these states are both injective points for the stochastic product. Then, for every , and, supposing that is an injective point too, by Theorem 4 we have that
[TABLE]
where are linear or antilinear isometries in , uniquely defined up to a phase factor. This relation implies that, actually, generate the same stochastic map in ; i.e., that they coincide, up to an irrelevant phase factor. This is easily shown by taking , where ranges over the pure states . Besides, it is also clear that cannot be a collapsing point, if both and are not. By an analogous reasoning, one concludes that the pair cannot be formed by a collapsing and a non-collapsing (i.e., an injective pureness-preserving) point. The only other possibility is that they are both collapsing points, as well as their convex combinations, and they give rise to the same collapse channel. Hence, every convex subset of formed by left pureness-preserving points is contained in a left level set of , which is a convex set itself and, then, a maximal convex subset of . ∎
We can also establish some constraints concerning the injective pureness-preserving points and the collapsing points.
Corollary 9**.**
If a stochastic product possesses both left and right collapsing points, then all the associated collapse channels must coincide. If the product possesses both left and right injective pureness-preserving points, then all these points are states of the same rank. If the product admits both left collapsing points and right injective pureness-preserving points, or vice versa, then each collapsing point has rank equal to the image rank of the associated collapse channel.
Proof.
Let be a pair formed by a left and a right collapsing point, respectively. Then, \big{(}\mathsf{L}(\rho)\big{)}(\tau)=\rho_{0} and \big{(}\mathsf{R}(\sigma)\big{)}(\upsilon)=\sigma_{0} — for all , and for some fixed states and — so that, actually, \rho_{0}=\big{(}\mathsf{L}(\rho)\big{)}(\sigma)=\rho\odot\sigma=\big{(}\mathsf{R}(\sigma)\big{)}(\rho)=\sigma_{0}. Next, let be a pair formed by a left and a right injective pureness-preserving point, respectively. Then, by Theorem 4 we have: U\sigma\hskip 0.85358ptU^{\ast}=\big{(}\mathsf{L}(\rho)\big{)}(\sigma)=\rho\odot\sigma=\big{(}\mathsf{R}(\sigma)\big{)}(\rho)=V\rho\hskip 0.85358ptV^{\ast}, for some linear or antilinear isometries and ; hence, . Finally, if, say, is a left collapsing point whereas is a non-collapsing right pureness-preserving point, then \rho_{0}=\big{(}\mathsf{L}(\rho)\big{)}(\sigma)=\rho\odot\sigma=\big{(}\mathsf{R}(\sigma)\big{)}(\rho)=V\rho\hskip 0.85358ptV^{\ast}, with a linear or antilinear isometry; hence, the image rank of the collapsing channel () is equal to the rank of the collapsing point . The reader will easily complete this reasoning. ∎
Finally, something can be also said about the points that are not pureness-preserving.
Proposition 10**.**
* belongs to if and only if, for some state , .*
Proof.
The statement follows from Corollary 4. ∎
Example 8**.**
We now reconsider the stochastic product (62) in Example 6. Let us assume that and . Suppose, moreover, that is a symmetry transformation, and let be the (possibly empty) convex subset of defined by . Notice that . Moreover and, by the equivalence of the properties (i) and (iv) in Theorem 3, we have:
[TABLE]
Therefore, the product is left constant — unitary or antiunitary — on the convex set , which is (either empty or) a left level set for the stochastic product; moreover, if and only if is not a symmetry transformation and/or . In the case where is a symmetry transformation too (with ) and , we have that , with the partition formed by two maximal convex subsets of . Consider next the more general stochastic product (63). If the product is right injective, then the discrete POVM must be informationally complete [12, 13]; i.e., for every , with , we must have that , for some . Hence, by a well known elementary result — see Proposition 3.49 of [13] — if , the constraint must be satisfied.
3.4 Group-covariant stochastic products and equivariant families of products
Let be a locally compact, second countable, Hausdorff topological group; in short, a l.c.s.c. group, and let , pair of projective representations [38] of this group in a (separable complex) Hilbert space . As usual, these representations are supposed to be weakly Borel maps [38].
Let \mathscr{SI}_{1}(\mathcal{H})_{\mbox{\tiny+}}\simeq\mathrm{P}\overline{\mathcal{U}}{(\mathcal{H})} be the group of positive, surjective linear isometries in — see Remark 3 — endowed with the strong operator topology of bounded linear operators in . The map
[TABLE]
associated with the projective representation , is a continuous group homomorphism (see [39], Proposition 4.1). This map gives rise to a (symmetry) action of the group on the space of trace class operators where the quantum states live. We stress that, although is (in general) projective,
[TABLE]
behaves like an ordinary group representation (i.e., as already mentioned, homomorphically):
[TABLE]
Let us first introduce a natural notion of covariance for a stochastic product.
Definition 12**.**
We say that a stochastic product is left (right) covariant wrt the pair of projective representations if
[TABLE]
for all and all . We say that the product is left (right) covariant wrt if (74) holds with . An analogous definition we set for any algebra product (i.e., any bilinear map) on .
Let us deduce a few immediate consequences of covariance of a stochastic product. Observe that, in the case where , admits a maximally mixed state .
Lemma 1**.**
Let the projective representation be irreducible, and let . Then:
[TABLE]
Proof.
Note that , for all , if and only if belongs to the commutant of . In the case where is irreducible, by Schur’s lemma (that holds also for genuinely projective representations), ; hence, in this case, if and only if and is the maximally mixed state. ∎
Proposition 11**.**
Suppose that is finite-dimensional and is irreducible, and let be a stochastic product, left (right) covariant wrt the pair . Then: .
Proof.
Indeed, for every , we have that . Therefore, by Lemma 1, for every , . ∎
If the stochastic product is commutative, then all collapsing points are both left and right collapsing. Then, by the first assertion of Corollary 9, all the associated collapse channels must coincide. This conclusion is coherent with the following result.
Proposition 12**.**
Let be irreducible, and let be a commutative stochastic product, (left and right) covariant wrt the pair . If is a (left and right) collapsing point, then and
[TABLE]
Proof.
Let be a collapsing point for the commutative stochastic product ; i.e., , for all . Since is both left and right covariant wrt the pair , we have: , for all . Hence, by Lemma 1, for every , . ∎
In addition to covariance, there are two further group-theoretical notions concerning stochastic products, but this time involving a family of products: invariance and equivariance. Precisely, let be a -space [38] wrt to a (left) group action . Suppose that the points of label a family of stochastic products, all defined on the same set of states .
Definition 13**.**
We say that the family of stochastic products is invariant wrt the -action if
[TABLE]
We say that is left (right) inner equivariant wrt the pair if
[TABLE]
Moreover, we say that is outer equivariant wrt the pair if
[TABLE]
Analogous definitions we set for any algebra product in .
Example 9**.**
One can easily construct examples of equivariant families of stochastic products from a given stochastic product and a projective representation . Indeed, consider the family of stochastic products defined by
[TABLE]
This family of products is left inner equivariant wrt the pair , where is the left action of on itself; i.e.,
[TABLE]
In an analogous way, one can construct an outer equivariant family of stochastic products. It is clear that, conversely, every family of stochastic products labeled by the points of the group , and left inner equivariant wrt to the pair , where is the left action of on itself, is of the previously specified form. In fact, denoting by the identity of , it is sufficient to set
[TABLE]
4 Constructing a class of stochastic products: twirled products
We will now explicitly construct a class of group-covariant, associative stochastic products. As clarified in the previous section, every associative stochastic product is embedded in a suitable algebra on the Banach space of trace class operators, a so-called stochastic algebra. It will be convenient to develop some technical tools first, then to construct such an algebra structure — we will actually achieve a larger class of algebras, the ‘twirled algebras’ — next to prove the covariance properties of these algebras wrt the relevant group actions and, finally, to obtain our stochastic products by imposing suitable conditions and by restricting to density operators.
4.1 First step: technical tools
The basic ingredients of our construction are the following:
We consider a pair of projective representations , of a l.c.s.c. group in a separable complex Hilbert space . We will further assume that the group is unimodular — i.e., , where is the modular function [23, 38] on — and that the representation is (irreducible and) square integrable [24, 25, 26, 27, 28]. 2. 2.
We fix a complex Borel measure [40] on . We will denote by the Banach space of all complex Borel measures on . Notice that, since is a l.c.s.c. topological space, every measure in is regular, hence, a Radon measure; see, e.g., Theorem 7.8 of [40]. We will denote by \mathcal{M}\hskip 0.85358pt(G)_{\mbox{\tiny+}}\subset\mathcal{M}\hskip 0.85358pt(G) the convex cone of all finite, positive Borel measures on and by \mathcal{P}\mathcal{M}\hskip 0.85358pt(G)\subset\mathcal{M}\hskip 0.85358pt(G)_{\mbox{\tiny+}} the convex set of Borel probability measures. Recall that, for every \nu\in\mathcal{M}\hskip 0.85358pt(G)_{\mbox{\tiny+}}, the union of all -null open subsets of is a -null open set too, whose complement is the support of the Radon measure (equivalently, is the intersection of all closed subsets of of full -measure); see [40], chapt. 7, sect. 1, or [41], chapt. 7, sect. 2. Let us also point out that, since the topological group is second countable, for every pair , of Radon measures on , the standard product measure is a Radon measure too; see Theorem 7.20 of [40]. 3. 3.
We also fix a fiducial operator in . The twirled algebra that we are going to define will depend on the choice of this trace class operator, as it will be clear soon, as well as on the choice of the previously mentioned complex measure .
The fact that the representation is square integrable entails that, for every pair of vectors , the Borel function is square integrable wrt (a normalization of) the Haar measure [38, 23] on , which, in this case ( unimodular), is both left and right invariant. Moreover, the so-called orthogonality relations hold; i.e.,
[TABLE]
where is a (strictly) positive constant, depending only on and on the normalization of .
Remark 16*.*
In the general (i.e., not necessarily unimodular) case, the orthogonality relations for a square integrable representation involve a positive selfadjoint linear operator in — the so-called Duflo-Moore operator [25, 26, 27, 28] — which is bounded if and only if is unimodular, and, in such case, this operator is a multiple of the identity: , . In particular, every irreducible unitary representation of a compact group is square integrable, since the Haar measure of such a group is finite. Moreover, if is normalized as a probability measure, then (according to the Peter-Weyl theorem [23]).
Recall that the projective representations and give rise to (strongly) continuous isometric representations
[TABLE]
of the l.c.s.c. group , acting in the Banach space ; see (71). Notice that, if is the multiplier [38] of , then ; hence, by the cyclic property of the trace,
[TABLE]
We will now establish two fundamental technical facts.
Lemma 2**.**
For every pair of trace class operators , the bounded continuous function G\ni g\mapsto\mathrm{tr}\big{(}A\hskip 0.85358pt({\mathsf{U}\vee\mathsf{U}\hskip 0.56905pt}(g)F)\big{)}=\mathrm{tr}(A\hskip 0.85358pt\hskip 0.85358pt\mathsf{U}\hskip 0.56905pt(g)\hskip 0.85358ptF\hskip 0.85358pt\hskip 0.85358pt\mathsf{U}\hskip 0.56905pt(g)^{\ast})\in\mathbb{C} is -integrable and
[TABLE]
where is the constant appearing in the orthogonality relations (83) for the square integrable representation . Therefore, we can define a complex measure \nu_{\mbox{\tinyA,F}}\in\mathcal{M}\hskip 0.85358pt(G) by setting
[TABLE]
Proof.
The bounded function G\ni g\mapsto\mathrm{tr}\big{(}A\hskip 0.85358pt({\mathsf{U}\vee\mathsf{U}\hskip 0.56905pt}(g)F)\big{)}\in\mathbb{C} is also continuous, because the representation is strongly continuous. Relation (86) is the ‘second trace formula for square integrable representations’; see [42], Proposition 7. ∎
Remark 17*.*
By the previous lemma, it is clear that, if and are density operators, then \nu_{\mbox{\tinyA,F}} is a Borel probability measure on .
Lemma 3**.**
For every trace class operator and every complex measure , the vector-valued function is Bochner-integrable wrt to . Moreover, the trace class operator , defined as a Bochner integral
[TABLE]
has trace equal to , and, in the case where \nu\in\mathcal{M}\hskip 0.85358pt(G)_{\mbox{\tiny+}} and B\in\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny+}}, it is a positive element of . Finally, if , then is contained in the closed convex hull
[TABLE]
Proof.
Taking into account that the Banach space is separable ( being separable), by Pettis’ measurability theorem [43], the continuous function is -measurable. Moreover, this vector-valued function is bounded (wrt the norm ). By these facts, it is Bochner-integrable wrt . Observe now that, exchanging the trace with the Bochner integral (so getting an ordinary integral of a -valued function),
[TABLE]
and, for every ,
[TABLE]
Thus, if, in particular, is a positive measure and B\in\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny+}}, then {\nu[\hskip 1.13809pt\mathsf{V}]}\hskip 1.70717ptB\in\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny+}}. The final assertion relies on the fact that we can restrict the integral in (88) to and on a well known property of Bochner’s integral; see [43], chapt. II, Corollary 8. ∎
By the two previous lemmas, for every tern of operators , we can consistently define a trace class operator by setting
[TABLE]
where the integral on the rhs is, as in (88), a Bochner integral, and the measure \nu_{\mbox{\tinyA,F}}\circledcirc\nu\in\mathcal{M}\hskip 0.85358pt(G) is the convolution [23] of the complex Borel measures \nu_{\mbox{\tinyA,F}} and .
Proposition 13**.**
For every , we have that
[TABLE]
where \nu_{\mbox{\tinyA,F}}\otimes\nu\hskip 0.56905pt is the standard product measure, all integrals are in the sense of Bochner and the iterated integrals can be interchanged.
Proof.
The function is continuous — hence, measurable wrt the product measure \nu_{\mbox{\tinyA,F}}\otimes\nu\hskip 0.56905pt ( being separable) — and norm bounded. Therefore, it is Bochner-integrable. By Fubini’s theorem for Bochner integrals (see, e.g., Theorem 3.7.13 of [44]),
[TABLE]
where the iterated Bochner integrals on the rhs can be interchanged. Moreover, for every , we have that
[TABLE]
by the definition of the convolution of measures (notice that this relation holds even if the integrand functions are only assumed to be continuous and bounded; see Remark 2 of [45]). Thus, by (94),
[TABLE]
where, again, Bochner integrals are understood. ∎
Note that — for (Dirac measure at the identity of ) — \nu_{\mbox{\tinyA,F}}\circledcirc\nu=\nu_{\mbox{\tinyA,F}}\circledcirc\delta=\nu_{\mbox{\tinyA,F}} and we have:
[TABLE]
In the following, we will also need a further technical result. Denoting by |\nu_{\mbox{\tinyA,F}}|\in\mathcal{M}\hskip 0.85358pt(G)_{\mbox{\tiny+}} the total variation measure [40] associated with the complex measure \nu_{\mbox{\tinyA,F}}, we have:
Lemma 4**.**
For every pair of trace class operators , the -norm \|\nu_{\mbox{\tinyA,F}}\|\mathrel{\mathop{:}}=|\nu_{\mbox{\tinyA,F}}|(G) of the associated complex measure \nu_{\mbox{\tinyA,F}} satisfies the inequality
[TABLE]
Proof.
Let us consider the singular value decomposition of (see, e.g., [4], chapt. 4):
[TABLE]
Here, , are the singular values of , — thus: and — , , , are orthonormal systems in and convergence wrt the trace norm is understood. We then have:
[TABLE]
for all (absolute convergence on the rhs), where , (). Since is a square integrable representation, a_{kl},b_{kl}\in\mathrm{L}^{\mbox{\tiny2}}(G)\equiv\mathrm{L}^{\mbox{\tiny2}}(G,\mu_{G};\mathbb{C}) and, by the orthogonality relations (83),
[TABLE]
Moreover, the function belongs to \mathrm{L}^{\mbox{\tiny1}}(G). Hence, by (100), we have a series of functions in \mathrm{L}^{\mbox{\tiny1}}(G) converging point-wise to the function \mathrm{tr}\big{(}A\hskip 0.85358pt({\mathsf{U}\vee\mathsf{U}\hskip 0.56905pt}(\hskip 0.85358pt\cdot\hskip 0.85358pt)F)\big{)}\in\mathrm{L}^{\mbox{\tiny1}}(G) (Lemma 2). By a well known result (an immediate consequence, e.g., of Corollary 2.32 of [40]), it follows that the series — which converges absolutely in \mathrm{L}^{\mbox{\tiny1}}(G), since — must converge to \mathrm{tr}\big{(}A\hskip 0.85358pt({\mathsf{U}\vee\mathsf{U}\hskip 0.56905pt}(\hskip 0.85358pt\cdot\hskip 0.85358pt)F)\big{)} in \mathrm{L}^{\mbox{\tiny1}}(G), as well; therefore:
[TABLE]
At this point, it is sufficient to notice that
[TABLE]
and the proof is complete. ∎
4.2 Second step: constructing twirled algebras
Let us now study the main properties of the trace class operator . It will be convenient, from this point onwards, to fix the normalization of the Haar measure in such a way that .
It is clear that the map
[TABLE]
is linear wrt each of its four arguments; besides:
Proposition 14**.**
For every and every , we have that
[TABLE]
where is the complex conjugate of the measure . Hence, if the operators are selfadjoint and is real-valued (i.e., a finite signed Borel measure), then is selfadjoint too. Furthermore,
[TABLE]
Proof.
Observe that we have and, by the basic properties of the trace,
[TABLE]
Then, by (93), relation (105) holds. Relation (106) is a consequence of the second assertion of Lemma 3 and of the fact that, for , \nu_{\mbox{\tinyA,F}} is a positive measure. ∎
Another remarkable property of the map is the following:
Proposition 15**.**
For every and every , we have that
[TABLE]
and
[TABLE]
Proof.
Recalling the expression of the trace class operator appearing in the last line of (93) (with ), observe that, exchanging the trace with the Bochner integrals, we have:
[TABLE]
Then, by the first assertion of Lemma 2, we conclude that \mathrm{tr}\big{(}[[A,F,B]]_{\nu}\big{)}=\nu(G)\hskip 1.70717pt\mathrm{tr}(A)\hskip 1.70717pt\mathrm{tr}(F)\hskip 1.70717pt\mathrm{tr}(B). Next, by a well known property of the Bochner integral (see, e.g., property (c) in Proposition 4.5 of [12]), and taking into account the norm estimate \|\nu_{\mbox{\tinyA,F}}\|\leq\|A\|_{1}\hskip 0.85358pt\|F\|_{1} in Lemma 4, we find:
[TABLE]
for all and . ∎
The associativity of the stochastic algebras that we are going to define is ensured by the following:
Proposition 16**.**
If the representation coincides with the square integrable representation , then, for every and every , we have that
[TABLE]
Proof.
Recalling relation (93), setting (with square integrable) and exchanging the trace with the Bochner integrals, we have:
[TABLE]
Here, all the Bochner integrals can be interchanged (not only the first one with the second, and the remaining two, by the final claim of Proposition 13). Indeed — setting etc. — the function is integrable wrt the product measure . This fact is easily checked by taking , and in \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny+}}, and in \mathcal{M}\hskip 0.85358pt(G)_{\mbox{\tiny+}}, at first, so that, by Tonelli’s theorem and by the first assertion of Lemma 2,
[TABLE]
By linearity, one extends this conclusion to all and (recall [40] that admits a decomposition \nu=\nu_{\mbox{\tiny\Re}}+\mathrm{i}\hskip 1.70717pt\nu_{\mbox{\tiny\Im}}=\nu_{\mbox{\tiny\Re};\mbox{\tiny+}}-\nu_{\mbox{\tiny\Re};\mbox{\tiny-}}+\mathrm{i}\hskip 1.70717pt(\nu_{\mbox{\tiny\Im};\mbox{\tiny+}}-\nu_{\mbox{\tiny\Im};\mbox{\tiny-}}), \nu_{\mbox{\tiny\Re};\mbox{\tiny+}},\nu_{\mbox{\tiny\Re};\mbox{\tiny-}},\nu_{\mbox{\tiny\Im};\mbox{\tiny+}},\nu_{\mbox{\tiny\Im};\mbox{\tiny-}}\in\mathcal{M}\hskip 0.85358pt(G)_{\mbox{\tiny+}}, analogous to (1)–(3)). Therefore, the integrals in (113) can be freely interchanged by Fubini’s theorem for Bochner integrals, because the function is continuous and norm bounded, hence, Bochner-integrable wrt .
Besides, since the bounded linear operator in and the relevant Bochner integrals can be interchanged, and taking into account the final claim of Proposition 13, we have:
[TABLE]
Here, for obtaining the second equality, we have exploited the change of variables and the invariance of the Haar measure , while the third equality follows from relation (85). Finally, comparing the expressions obtained in (113) and (115), where the order of the iterated integrals is irrelevant, we see that relation (112) is verified. ∎
Finally, we focus on the case where the l.c.s.c. group is abelian.
Proposition 17**.**
If is abelian and coincides with the square integrable representation , then, for every and every , we have that
[TABLE]
Proof.
As in the proof of Proposition 16, we set etc. Notice that , and, since is abelian, . Let us assume that , and are in \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny+}}, and in \mathcal{M}\hskip 0.85358pt(G)_{\mbox{\tiny+}}, at first, and take any positive bounded operator , so that we deal with positive functions in the following argument and we can freely exchange integrals, by Tonelli’s theorem. By (93) and, next, by the first assertion of Lemma 2, we have:
[TABLE]
At this point, let us observe that
[TABLE]
By this relation, by suitably exchanging the integrals and, next, again by the first assertion of Lemma 2, we find that
[TABLE]
Note that the third and the fourth of the above equalities are obtained by relation (85) (since ) and by the change of variables , respectively. Hence, by linearity, we conclude that , without any assumption of positivity of the integrand functions; i.e., for all , and . Finally, by the arbitrariness of the operator in , it follows that relation (116) is verified. ∎
Summarizing all the previous facts, we obtain the following result:
Theorem 5**.**
Let , be projective representations of a unimodular l.c.s.c. group in a separable complex Hilbert space , with square integrable, and, for every and , let us consider the bilinear map
[TABLE]
If , this map is trace-preserving. If F\in\mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} and is real-valued, then the bilinear map is adjoint-preserving; in particular, it is positive, if . If the representation coincides with the square integrable representation , the product (120) is associative, hence if, moreover, and are such that , then the Banach space , endowed with the product (120), is a Banach algebra; in particular,
[TABLE]
Finally, if is abelian and coincides with the square integrable representation , then the product (120) is commutative.
There is a particular case of the previous result which deserves a special attention:
Corollary 10**.**
If is a square integrable projective representation, a density operator in and a Borel probability measure on , then the Banach space , endowed with the binary operation (120), is both a stochastic algebra and a Banach algebra. This algebra is also commutative in the case where is abelian.
Remark 18*.*
Under the assumptions of the previous corollary, in the Banach algebra the inequality (121) is actually saturated by every pair of positive operators and .
We will call a pair of the form
[TABLE]
the twirled algebra on generated by the tetrad .
4.3 Third step: proving covariance, equivariance and invariance properties of twirled algebras
Given a complex measure , let us first define the associated left and right -translate measures — denoted by and , respectively — as
[TABLE]
where is a Borel subset of ; namely, for every Borel function ,
[TABLE]
Observe that the maps
[TABLE]
are (left) group actions, because
[TABLE]
Lemma 5**.**
For every and every , we have:
[TABLE]
[TABLE]
[TABLE]
Proof.
Let us prove relation (127). In fact, , being a bounded operator in , commutes with the Bochner integrals and we have:
[TABLE]
Regarding the first of relations (128), observe that, by the right-invariance of the Haar measure , we find:
[TABLE]
The proofs of the second of relations (128) and of relation (129) are similar. Relation (129) can also be regarded as a consequence of the two relations (128) and of the fact that . ∎
Proposition 18**.**
Let
[TABLE]
be the twirled algebra generated by the tetrad . Then, the algebra product is left-covariant wrt the pair . Moreover, the family of products
[TABLE]
is right inner equivariant wrt the pair , where is the (left) group action
[TABLE]
it is invariant wrt the group action
[TABLE]
Finally, setting , the family of products
[TABLE]
is right inner equivariant wrt the pair , where is the group action
[TABLE]
Proof.
The first assertion follows directly from (127). The right inner equivariance of the families of products (133) and (136) is a consequence of the second of relations (128) and of relation (129), respectively. Exploiting the first of relations (128), one obtains the invariance of the family of products (133) wrt the action (135). ∎
4.4 Fourth step: defining twirled stochastic products
We now complete our program. Given a square integrable projective representation , for every fiducial state and every probability measure , we define
[TABLE]
where, as above, the integrals are in the Bochner sense and the Haar measure is normalized in such a way that . For the sake of conciseness, we set . Recall, moreover, that and denote, respectively, the left and the right -translates of the Borel probability measure .
Taking into account Proposition 13, and applying Corollary 10 and Proposition 18, we obtain the following result:
Theorem 6**.**
With the previous notations and assumptions, the pair
[TABLE]
is an associative stochastic product that is left-covariant wrt the representation , namely,
[TABLE]
Moreover, the family of stochastic products
[TABLE]
is right inner equivariant wrt the pair , where is the group action
[TABLE]
namely,
[TABLE]
it is invariant wrt the group action
[TABLE]
namely,
[TABLE]
Setting , the family of stochastic products is right inner equivariant wrt the pair , where is the group action , , namely,
[TABLE]
In the case where the l.c.s.c. group is abelian, the stochastic product (138) is commutative.
Remark 19*.*
In the case where the l.c.s.c. group is abelian, due to the commutativity of the stochastic product (138) and to the fact that , by combining the previous covariance, equivariance and invariance properties, we find various further symmetry relations:
[TABLE]
[TABLE]
[TABLE]
We call the (associative) stochastic product (138) the twirled stochastic product associated with the triple . The algebra on that is obtained as the canonical extension of this stochastic product will be called the twirled stochastic algebra associated with the triple .
5 Twirled stochastic products for every Hilbert space dimension
We now show by means of examples that there exist twirled stochastic products for every Hilbert space dimension (as usual, we neglect the trivial case where ).
5.1 Finite-dimensional twirled stochastic products
Suppose that the group is compact (hence, unimodular) and is an irreducible unitary representation; thus, in this case: (the multiplier is trivial) and . Then, is square integrable, because the Haar measure on is finite; see [27, 46]. Moreover, by the classical Peter-Weyl theorem [23], assuming that the Haar measure is normalized as a probability measure — — we have that ; hence:
[TABLE]
for all and . In particular, for the maximally mixed state , the following noteworthy relations hold:
[TABLE]
The first of these relations is clear considering the expression (150) of the twirled product and recalling Lemma 2 (, with \mathrm{d}\hskip 0.56905pt\mu(g)=n\hskip 1.70717pt\mathrm{tr}\big{(}\rho\hskip 1.70717pt({\mathsf{U}\vee\mathsf{U}\hskip 0.56905pt}(g)\hskip 0.85358pt\upsilon)\big{)}\hskip 2.56073pt\mathrm{d}\hskip 0.56905pt\mu_{G}(g), is a probability measure). The other two are a consequence of the fact that \mathrm{tr}\big{(}\rho\hskip 1.70717pt({\mathsf{U}\vee\mathsf{U}\hskip 0.56905pt}(g)\hskip 0.85358pt\upsilon)\big{)}=n^{-1}, for or , and of the relation
[TABLE]
see Proposition 6 of [42] (the ‘first trace formula for square integrable representations’; just make the substitution in this formula, for unimodular). Indeed, by the previous facts and by Lemma 3, we have:
[TABLE]
Notice that the second of relations (151) also descends from Proposition 11, by virtue of the left covariance of the twirled stochastic product. Therefore, the maximally mixed state is a left and right collapsing point for the twirled product — i.e., \big{(}\mathsf{L}(\Omega)\big{)}(A)=\big{(}\mathsf{R}(\Omega)\big{)}(A)=\mathrm{tr}(A)\hskip 1.70717pt\Omega, for all — and, choosing as a fiducial state, we get a collapse product. Similarly, setting , one finds a collapse product too:
[TABLE]
This fact is readily verified using the invariance of the measure and, once again, relation (152).
Observe that exploiting, e.g., the irreducible unitary representations of the group , one is able to construct twirled products for every finite Hilbert space dimension. Moreover, composing the twirled stochastic product (associated with an irreducible representation of a compact group) with suitable stochastic maps — regarded as convex-linear maps in , see Remark 10 — one can provide examples of stochastic products having collapsing points different from and giving rise to different collapse channels.
5.2 Infinite-dimensional products and the quantum convolution
We now consider an infinite-dimensional example. Let be the group of translations on phase space — — \mathcal{H}=\mathrm{L}^{\mbox{\tiny2}}(\mathbb{R}^{n}) and the Weyl system [27, 42, 46]:
[TABLE]
i.e., , where , are the (vector) position and momentum operators in \mathrm{L}^{\mbox{\tiny2}}(\mathbb{R}^{n}). This is a (strongly continuous) irreducible projective representation, with multiplier
[TABLE]
Moreover, the Weyl system is square integrable and, setting \mathrm{L}^{\mbox{\tiny2}}(G)=\mathrm{L}^{\mbox{\tiny2}}(\mathbb{R}^{n}\times\mathbb{R}^{n},(2\pi)^{-n}\mathrm{d}\hskip 0.56905pt^{n}q\hskip 2.27621pt\mathrm{d}\hskip 0.56905pt^{n}p\hskip 0.85358pt;\mathbb{C}), we have that . Therefore, in this case the twirled product associated with the triple is of the form
[TABLE]
We will call this product the phase-space stochastic product. Although this is not immediately clear from the previous expression, the phase-space stochastic product is commutative, since it stems from a representation of an abelian group. It follows, by Proposition 12, that this stochastic product does not admit collapsing points, because the Hilbert space \mathrm{L}^{\mbox{\tiny2}}(\mathbb{R}^{n}) is infinite-dimensional.
In particular, for , we have:
[TABLE]
We will call this stochastic product the quantum convolution. In order to justify this term, it is worth expressing this binary operation in terms of the Wigner (quasi-probability) distributions — see [21, 22, 27, 28, 42, 46, 48, 49] and references therein — , , and associated with the density operators , , and , respectively. It can be shown [47] that, setting , ,
[TABLE]
Therefore, we actually find a double convolution of Wigner functions, where the function plays a sort of pivotal role and has no analogue in the classical setting, where the convolution of two probability distributions on phase space is a probability distribution too. We will further comment on this crucial point in the next section.
The function
[TABLE]
is a probability distribution wrt the Lebesgue measure on . E.g., setting , and choosing the pure state — — as the fiducial state , whose Wigner function is
[TABLE]
we obtain the probability distribution
[TABLE]
the so-called Husimi-Kano function [48, 49] associated with the state . Considering now a generic pure state , one obtains a class of probability distributions associated with a quantum state that admit a remarkable operational interpretation in the context of quantum optics [50, 51, 52].
Hence, in this case, the quantum convolution, expressed in terms of phase-space functions, is of the form
[TABLE]
where we have used the commutativity of the product (or the associativity and commutativity of convolution in relation (159)). Note that here we have the convolution of an integrable function, the Husimi-Kano distribution, with a square integrable function, which gives rise to a square integrable function as it should (all Wigner functions are square integrable [27, 28, 42, 46]); see, e.g., Proposition 2.40 of [23].
6 Final remarks, conclusions and perspectives
In this paper, we have introduced the notion of stochastic product defined as a binary operation on the convex set of quantum states — the density operators — that preserves the convex structure. Such a product is automatically (jointly) continuous wrt the natural product topology on (Remark 9, Corollary 5), and can be extended to an algebra on the Banach space of trace class operators, the so-called canonical extension of the stochastic product (Proposition 7). We have also defined a stochastic algebra as an algebra on arising from an associative stochastic product; by restricting to a binary operation on the real Banach space \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} of selfadjoint trace class operators, one obtains a real Banach algebra (Remark 14). We have then shown — see sect. 4 — that one can explicitly construct a class of stochastic algebras, the so-called twirled stochastic algebras, by means of a square integrable projective representation of a (unimodular) locally compact group , and by choosing a fiducial state and a Borel probability measure on ; in the simplest case, we set (the Dirac measure at the identity). This is actually a special case of a more general class of algebras — the twirled algebras (tout court) — associated with a pair of projective group representations (where , as above, is supposed to be square integrable), with a fiducial trace class operator and with a complex Borel measure . It turns out that the twirled stochastic algebras are, as well, complex Banach algebras (i.e., restricting to an algebra on \mathcal{B}_{1}(\mathcal{H})_{\mbox{\tiny\mathbb{R}}} is not necessary, in this case). Moreover, these stochastic algebras enjoy nice covariance, equivariance and invariance properties, and, in the case where the relevant group is abelian, our construction yields a commutative algebra (Theorem 6). One then easily shows, by means of examples, the existence of twirled stochastic algebras for every Hilbert space dimension (sect. 5).
Not surprisingly, our group-theoretical construction of a stochastic product involves some important tools typical of various applications of abstract harmonic analysis to the foundations of quantum theory (quantization, phase-space quantum mechanics, quantum measurement theory).
In this regard, let us first recall that, using our previous notations, the mapping
[TABLE]
where is a Borel set and a density operator, is a covariant quantum observable; i.e., a POVM covariant wrt the irreducible representation [53]: , for every Borel subset of and every . Actually, formula (164) provides the general expression of such a POVM; namely, if a POVM is covariant wrt to the irreducible projective representation , then this representation must be square integrable and the POVM must be of the form (164), for exactly one state . Hence, the function g\mapsto\mathrm{tr}\big{(}\rho\hskip 1.70717pt({\mathsf{U}\vee\mathsf{U}\hskip 0.56905pt}(g)\hskip 0.85358pt\upsilon)\big{)} used in our construction of a stochastic product can be regarded as the probability density on — regarded as a sample space — of the covariant observable , relative to the observed state ; namely, the Radon-Nikodym derivative of the probability measure \nu_{\mbox{\tiny\rho,\upsilon}}\colon\mathscr{E}\mapsto\int_{\mathscr{E}}\mathrm{d}\hskip 0.56905pt\mu_{G}(g)\hskip 2.56073pt\mathrm{tr}\big{(}\rho\hskip 1.70717pt({\mathsf{U}\vee\mathsf{U}\hskip 0.56905pt}(g)\hskip 0.85358pt\upsilon)\big{)} wrt the (suitably normalized) Haar measure . This function can also be regarded as the restriction, to the diagonal subgroup of the direct product group , of the frame transform [42] of , generated by the square integrable representation , with analyzing operator .
Moreover, for every Borel probability measure on , the mapping
[TABLE]
where this time is a generic projective representation, defines a quantum dynamical map (or quantum channel), a so-called twirling operator [39]. Therefore, if is square integrable, then the stochastic twirled product can be thought of as (recall (92))
[TABLE]
where is the left partial map associated with this product (Corollary 6).
As a further observation within the same circle of ideas, it is worth mentioning that, for every square integrable representation ( unimodular) and every pair of states , the mapping
[TABLE]
— where is a Borel set and a quantum operation [13] (in particular, a quantum channel for ) — is a quantum instrument [13, 54, 55, 56]; more specifically, a -covariant quantum instrument based on [56], because
[TABLE]
(The reader may notice, however, that the covariant quantum instrument , as defined in (167), is not expressed in the form displayed in Corollary 12 of [56]; in order to properly compare the two expressions of the quantum instrument, one has to suitably re-elaborate formula (167).) Clearly, a connection with the twirled stochastic product associated with the triple , and with the related twirling operator, is obtained by setting ; i.e.,
[TABLE]
Therefore, the linear map from into that extends the convex-linear application coincides with the right partial map associated with the given stochastic product. Besides, since , from relation (168) we recover the left-covariance of the twirled stochastic product. Also note that the associativity of this product translates into the following relation for the quantum channels :
[TABLE]
Observe, moreover, that the condition
[TABLE]
for every and every Borel subset of , uniquely determines a (-covariant) POVM; see subsect. 5.1.3 of [13]. Precisely, recalling the expression (164), one finds out that . We stress that the class of covariant quantum instruments of the form (167) — with and (an arbitrary state) kept fixed, and ranging over — gives rise to the whole class of POVMs that are covariant wrt the representation . Precisely, for every , the mapping \mathsf{E}_{\upsilon}\mapsto\mathcal{I}_{\mbox{\tiny(\hskip 0.85358pt\cdot\hskip 0.85358pt)}}^{\hskip 0.85358pt\upsilon,\hskip 0.28453pt\sigma}, from the set of all -covariant POVMs to the set of all -covariant quantum instruments, can be regarded as a cross section wrt to the partition of the latter set into equivalence classes of instruments compatible with the same observable.
The twirled stochastic product admits a remarkable expression in terms of the covariant symbols associated with the density operators [47]. Given a square integrable projective representation ( unimodular) and a trace class operator , the (covariant) symbol of is the complex function on defined by
[TABLE]
where . Moreover, , where \mathfrak{D}\colon\mathcal{B}_{2}(\mathcal{H})\rightarrow\mathrm{L}^{\mbox{\tiny2}}(G)\equiv\mathrm{L}^{\mbox{\tiny2}}(G,\mu_{G};\mathbb{C}) denotes a linear isometry mapping the Hilbert space of Hilbert-Schmidt operators into the Hilbert space of square integrable functions \mathrm{L}^{\mbox{\tiny2}}(G). This isometry can be regarded as a dequantization map, which is directly related to the Wigner transform in the case where is the group of translations on phase space [42, 46, 27, 28]. The operator an be explicitly re-constructed from its symbol via the quantization map [46].
In the case where is the group of phase-space translations and is the Weyl system (see subsect. 5.2), for the symbol is also called the quantum characteristic function of the state — essentially, the Fourier transform of the Wigner function — in analogy with the classical characteristic function of a probability measure on a locally compact abelian group [21, 22, 27, 28]. Recall, indeed, that, for every , , we can identify the characteristic function of — i.e., its Fourier-Stieltjes transform , where is the Pontryagin dual [23] of — with the function defined by
[TABLE]
It turns out [47] that, setting \mathrm{L}^{\mbox{\tiny2}}(G)=\mathrm{L}^{\mbox{\tiny2}}(\mathbb{R}^{n}\times\mathbb{R}^{n},(2\pi)^{-n}\mathrm{d}\hskip 0.56905pt^{n}q\hskip 2.27621pt\mathrm{d}\hskip 0.56905pt^{n}p\hskip 0.85358pt;\mathbb{C}) (hence, ), the phase-space stochastic product associated with the triple — expressed in terms of the characteristic function and of the symbols , , of the states , , — assumes the simple form
[TABLE]
from which it is evident that this product is commutative.
Considering formula (174) for the phase-space stochastic product, one may regard this nice expression as a straightforward way to achieve a commutative stochastic product. Indeed, whereas the pointwise product of two quantum characteristic functions is not, in general, a function of the same kind, the product of a ‘classical’ characteristic function on phase space — the Fourier-Stieltjes transform of a probability measure on — by a quantum characteristic function, is a function of the latter type [21, 22]. Moreover, for every , is a classical characteristic function, the (symplectic) Fourier-Stieltjes transform of the probability measure \nu_{\mbox{\tiny\rho,\upsilon}}, \mathrm{d}\hskip 0.56905pt\nu_{\mbox{\tiny\rho,\upsilon}}{(q,p)}=(2\pi)^{-n}\hskip 0.85358pt\mathrm{tr}\big{(}\rho\hskip 1.70717pt({\mathsf{U}\vee\mathsf{U}\hskip 0.56905pt}{(q,p)}\hskip 0.85358pt\upsilon)\big{)}\hskip 1.70717pt\mathrm{d}\hskip 0.56905pt^{n}q\hskip 2.27621pt\mathrm{d}\hskip 0.56905pt^{n}p\hskip 0.85358pt; namely, recalling (160) and denoting by the convolution of the functions and (),
[TABLE]
Therefore, the function can be regarded as the pointwise product of two classical characteristic functions — and — which is again a function of the same kind (the Fourier-Stieltjes transform of the convolution of two probability measures), multiplied by the quantum characteristic function , so achieving a function of the latter type. Interestingly, the fact that is a classical characteristic function can also be proved ‘intrinsically’ using the properties of classical and quantum positive definite functions [21].
Analyzing the previous example, various intriguing links connecting the notion of quantum convolution with Werner’s seminal work on quantum harmonic analysis on phase space [29] come to light. In Werner’s remarkable approach, beside the ordinary convolution of two integrable functions and on phase space, one can also construct the convolution of a function f\in\mathrm{L}^{\mbox{\tiny1}}(\mathbb{R}^{n}\times\mathbb{R}^{n}) with an operator (and vice versa) — where \mathcal{H}=\mathrm{L}^{\mbox{\tiny2}}(\mathbb{R}^{n}) — and the convolution of two operators . Precisely,
[TABLE]
where is the Weyl system — i.e., — and, denoting by the parity operator in \mathrm{L}^{\mbox{\tiny2}}(\mathbb{R}^{n}) (, ),
[TABLE]
It turns out that associativity is satisfied; in particular, . Therefore, in this language, the quantum convolution (the phase-space stochastic product with ) can be written, unambiguously, in the following form:
[TABLE]
We stress that, here, the commutativity of the quantum convolution may seem, at first sight, an immediate consequence of the fact that in definition (176) and of the easily verified relation ; but it actually also involves the non-trivial associativity of Werner’s convolution of operators.
Beside the aforementioned formulation of twirled products in terms of covariant symbols, there are several aspects of stochastic products that are not treated in this paper and that we plan to study; in particular:
- •
For the sake of simplicity, we have considered the twirled product associated with a square integrable representation of a unimodular locally compact group. Taking into account some mathematical intricacy related to an unbounded Duflo-Moore operator (Remark 16), one can work out the non-unimodular case, as well.
- •
There is a natural notion of complete positivity for a stochastic product.
- •
The classification of covariant stochastic products is an interesting open problem.
- •
The behaviour of a quantum entropy [57, 58, 59] — not necessarily the von Neumann entropy — wrt to a stochastic product is another interesting issue. In particular, it is natural to wonder whether the twirled stochastic product is, say, entropy-nondecreasing; i.e., whether the entropy of the product of two states is not smaller than the entropy of each of these states. Observe that such a property would provide a nice physical interpretation of the fact that the maximally mixed state is both a left and a right collapsing point for the twirled stochastic product associated with an irreducible unitary representation of a compact group; see subsect. 5.1.
- •
An abstract setting for a stochastic algebra may give rise to useful tools and new insights.
In connection with the last point, we suggest that the following algebraic structure might be appropriate for our purposes. Let us consider a mathematical object, consisting of two sets and four maps, of the form
[TABLE]
where:
is a proper -algebra [46, 60, 61], equipped with the product and the involution (let us denote by the scalar product, assumed to be linear in its second argument, in the complex Hilbert space ). 2. 2.
is the trace class [62] of the proper -algebra , and an associative algebra wrt the binary operation . 3. 3.
the bilinear map preserves the convex cone
[TABLE]
of all positive elements of ; i.e., \mathcal{T}_{\mbox{\tiny+}}\boxdot\mathcal{T}_{\mbox{\tiny+}}\subset\mathcal{T}_{\mbox{\tiny+}}. 4. 4.
is the the canonical (positive) trace functional of the trace class ; therefore, , for all . 5. 5.
The trace is multiplicative wrt binary operation : .
We propose to call such an object a (abstract) stochastic -algebra. A stochastic algebra on , as defined in subsect. 3.2, fits in this abstract scheme with the following identifications: the -algebra is the Hilbert-Schmidt space , endowed with the standard product of operators (composition) and with the natural involution (adjoining map); the associated trace class is, of course, the space , endowed with the trace functional and with an associative, state-preserving bilinear map .
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