Four and a Half Axioms for Finite Dimensional Quantum Mechanics

TL;DR

This paper proposes four axioms that nearly derive finite-dimensional quantum mechanics, linking classical measurement, symmetry, and non-signaling states to the mathematical structure of quantum theory.

Contribution

It introduces a set of four strong axioms that lead to the derivation of the mathematical framework of finite-dimensional quantum mechanics.

Findings

Measurements represented by orthonormal subsets

States represented by vectors in an ordered real Hilbert space

Positive cone of the space is homogeneous and self-dual

Abstract

I discuss a set of strong, but probabilistically intelligible, axioms from which one can {\em almost} derive the appratus of finite dimensional quantum theory. Stated informally, these require that systems appear completely classical as restricted to a single measurement, that different measurements, and likewise different pure states, be equivalent (up to the action of a compact group of symmetries), and that every state be the marginal of a bipartite non-signaling state perfectly correlating two measurements. This much yields a mathematical representation of measurements and states that is already very suggestive of quantum mechanics. In particular, in any theory satisfying these axioms, measurements can be represented by orthonormal subsets of, and states, by vectors in, an ordered real Hilbert space -- in the quantum case, the space of Hermitian operators, with its usual tracial inner product. One final postulate (a simple minimization principle, still in need of a clear interpretation) forces the positive cone of this space to be homogeneous and self-dual and hence, to be the the state space of a formally real Jordan algebra. From here, the route to the standard framework of finite-dimensional quantum mechanics is quite short.

Equations

$${\omega }_{2|x}(y)=\omega (x,y)/{\omega }_1(x)\text{and}{\omega }_{1|y}(x)=\omega (x,y)/{\omega }_2(y)$$

$${\omega }_2(y)=\sum _{x\in E}{\omega }_{2|x}(y){\omega }_1(x),$$

$$q_x=x-⟨x,u⟩u,$$

$$⟨q_x,u⟩=0.$$

$$0=⟨q_x,0⟩=⟨q_x,\sum _{y\in E}{q}_y⟩=r^2+(n-1)s_q.$$

$$⟨v_x,v_y⟩=⟨x,y⟩-s.$$

$$⟨v_x,v_y⟩=⟨q_x,q_y⟩+c^2=⟨q_x,q_y⟩-s_q.$$

$$⟨q_x,q_y⟩=⟨x-mu,y-mu⟩=⟨x,y⟩-m⟨x,u⟩-m⟨u,y⟩+m^2=⟨x,y⟩-m^2.$$

$$s_q=s-m^2.$$

$$⟨v_x,v_y⟩=⟨x,y⟩-m^2-s_q=⟨x,y⟩-s,$$

$$\Vert x{\Vert }^2+2s=⟨x,u⟩=s+⟨x,b⟩+⟨x,y⟩.$$

$$⟨v_x,v⟩=\sum _{y\in E}⟨v_x,v_y⟩=⟨v_x,v_x⟩=1.\square$$

$${\omega }_1(x)=\sum _{y\in F}{\omega }_2(y){\omega }_{1|y}=\sum _{x\in E}{\omega }_2(f(x)){\omega }_{1|f(x)},$$

$$g\varphi (a)=\sum _{x\in E}b(gx)gx=\sum _{y\in F}b(y)y=b.\square$$

$$0\le \Vert x{\Vert }^2=⟨a_{+}-b_{+},a_{-}-b_{-}⟩=-(⟨b_{+},a_{-}⟩+⟨a_{+},b_{-}⟩).$$

$$⟨a_{+},b_{-}⟩=⟨a_{-},b_{+}⟩=0,$$

$$v_f=\sum _{x\in E}f(x)v_x=\sum _{x\in E}f(x)(1-⟨x,u⟩+nc)x.$$

$$⟨s1+a_o,t1+b_o⟩=st+\frac{\lambda }{n}\text{Tr}(a_o{b}_o).$$

$$⟨P_x,P_y⟩$$

Full text

This much yields a mathematical representation of measurements and states that is already very suggestive of quantum mechanics. In particular, in any theory satisfying these axioms, measurements can be represented by orthonormal subsets of, and states, by vectors in, an ordered real Hilbert space – in the quantum case, the space of Hermitian operators, with its usual tracial inner product. One final postulate (a simple minimization principle, still in need of a clear interpretation) forces the positive cone of this space to be homogeneous and self-dual and hence, to be the the state space of a formally real Jordan algebra. From here, the route to the standard framework of finite-dimensional quantum mechanics is quite short.

1 Introduction

It is an old idea that quantum mechanics – or, a bit more precisely, its probabilistic apparatus – might be derivable from a small number of well-motivated axioms having clear physical, operational or probabilistic interpretations. This goal is all but explicit in von Neumann’s book [27], and is made both explicit and programmatic in Mackey’s work in the late 1950s [20]. There is a small literature of attempts at such a derivation, including the seminal papers of von-Neumann and Birkhoff [9], Zierler [31], and Piron [22], framing the quantum-logical approach to the problem, and the work of Ludwig [19], Gunson [13], Mielnik [21], Araki [2] and others, approaching the problem in terms of ordered linear spaces. More recently, and with a new impetus from quantum information theory, there has been a resurgence of interest in the problem. Whereas most of the earlier work cited above focussed on the structure of single systems, and aimed to obtain the full apparatus of infinite-dimensional (non-relativistic) quantum mechanics, the newer work [3, 4, 12, 15, 10, 24] has focussed on deriving finite-dimensional quantum mechanics, and has a distinctive emphasis on properties of composite systems. Many of these last-cited works (notably those of Hardy [15], Rau [24] and D’Ariano [10] derive, or come very close to deriving, finite-dimensional QM on the basis of simple and operationally meaningful postulates.

This paper explores, in a preliminary way, a slightly different – and possibly less arduous – route towards such an axiomatization of finite-dimensional quantum theory, or of something reasonably close to it. The main ideas are that (i) both classical and quantum systems are very symmetrical; (ii) irreducible finite-dimensional systems with homogeneous, self-dual cones are pretty close to being quantum, thanks to the Koecher-Vinberg Theorem. Therefore, (iii) if we can somehow use symmetry assumptions to ground homogeneity and self-duality, we’ll be heading in the right direction.

In a bit more detail: the Koecher-Vinberg Theorem [17, 26] classifies homogeneous self-dual cones in finite dimensions as the positive cones of formally real Jordan algebras. It follows (from the Jordan-von Neumann-Wigner classification of such algebras) that, with the single exceptional example of the cone of positive $3\times 3$ matrices over the octonions, all physical systems having having an irreducible, homogeneous, self-dual cone of (un-normalized) states, are either either quantum-mechanical111allowing here real or quaternionic cases as “quantum”, or arise as so-called spin factors, i.e., their normalized state spaces are $n$-dimensional balls. Evidently, then, one path the deriving the mathematical framework of QM from first principles goes by way of supplying an operational motivation for homogeneity and self-duality. It will then remain either to dismiss, or to make room for, spin factors and the exceptional octonionic example as physical models.222In fact, there is a fairly direct route from Jordan algebras to complex Quantum Mechanics, at least in finite dimensions. A theorem of Hanche-Olsen [14] shows that the only Jordan algebras having a Jordan-algebraic tensor product with $M_{2}({\mathbb{C}})$ – that is, with a qbit – are the Jordan parts of $C^{\ast}$-algebras. Since the structure of qbits can be reasonably well-motivated on directly operational grounds, the only irreducible systems in a Jordan-algebraic theory supporting a reasonable tensor product, will be full matrix algebras. The condition that bipartite states be uniquely determined by the joint probabilities they assign to the two component systems – a condition sometimes called local tomography – then dictates that the scalar field be ${\mathbb{C}}$ [2, 16, 7].

Working in a framework in which a physical system is described by specifying a set of basic observables and a finite-dimensional compact, convex set of states [3, 30], I propose four axioms that, taken together, may be glossed as saying that (i) a system appears completely classical as restricted to any single basic observable, (ii) all basic observables are equivalent (up to the action of a compact group of physical symmetries), and (iii) all states arise as marginals of bipartite states that perfectly correlate some pair of basic observables. These assumptions suffice to obtain a representation of measurements as orthonormal subsets of a finite-dimensional ordered real Hilbert space, of states, as vectors therein, and of symmetries, as unitaries acting thereon. In the quantum case, the space in question is the space of Hermitian operators with its usual tracial inner product. A single additional postulate – a simple and natural minimization principle, for which one hopes to find an interpretation – forces the positive cone of this ordered Hilbert space to be homomgeneous and self-dual with respect to the given inner product.

Two disclaimers are in order before proceeding, one historical, and the other programmatic. First, the general line of attack taken here is not entirely new. The possibility of using the Koecher-Vinberg theorem is mentioned by Gunson [13] as long ago as 1967, but the suggestion seems not to have been followed up (perhaps owing to the then-prevailing focuss on infinite dimensional systems). An exception is the work of Kummer [18], which, in the context of a quite different set of axioms, also exploits the Koecher-Vingberg Theroem. What is novel about the approach taken here is the emphasis on symmetry considerations, and the use of an axiom involving composite systems.

Secondly, it should be stressed that the postulates discussed below are not advanced as possible “laws of thought”: the aim here is not to derive QM as the uniquely reasonable non-classical probability theory (though that would of course be very nice!), but only to find simple and transparent characterizations of quantum probability theory in autonomously probabilistic or information-theoretic terms, even if still as a theory with contingent elements.

The balance of this paper is organized as follows. In section 2, I review the general framework of abstract probabilistic models elaborated in [3, 4, 5, 29] (and deriving from the older tradition mentioend above, e.g, [20, 19, 11]). This leads very naturally to a representation of a finite-dimensional physical system in terms of an ordered linear space $V$, generated by the system’s normalized states, and its dual space $V^{\ast}$, housing the system’s “effects”. In section 3, I introduce a strong, but physically interpretable, symmetry postulate, namely, that all basic observables, and likewise all pure states, are equivalent up to the action of a compact group. This already allows one to introduce an inner product on the space $V^{\ast}$, in terms of which measurements can be represented as orthonormal sets (not necessarily bases) of vectors. With the addition of one purely mathematical regularity requirement (the “half axiom” of my title) — a simple extremality condition, satisfied in quantum mechanics, and for which one hopes that an interpretation can be adduced333As observed by D’Ariano [10], every attempt to date at an axiomatic reconstruction of QM has included at least one such axiom! — I obtain the self-duality of the positive cone of $V^{\ast}$. In section 4, I introduce two additional axioms, each again having a reasonably clear physical meaning, and from these, together with the assumptions of Section 2, deduce that this cone must also be homogeneous. By the Koecher-Vinberg Theorem, it now follows that $V^{\ast}$ can be represented as a formally real Jordan algebra. Section 5 collects some final thoughts and poses some questions for further study. Several appendices consider alternative approaches to some of the material in sections 3 and 4.

2 Preliminaries

A test space [11, 28] is a collection $\mathfrak{A}$ of non-empty sets $E,F,...$, each considered as the outcome-set of some experiment, measurement, or test. Subsets of tests are termed events. We allow the possibility that distinct tests may overlap, so that one outcome may belong to several tests. We write $X=X({\mathfrak{A}})$ for the total outcome space of $\mathfrak{A}$, i.e., $X=\bigcup{\mathfrak{A}}$. Outcomes $x,y\in X$ are termed orthogonal, and we write $x\perp y$, if they are distinct, but belong to a single test. Note that, at present, there is no linear structure, let alone an inner product, in view. Note, too, that we do not assume that every pairwise-orthogonal set is an event. A state on a test space ${\mathfrak{A}}$ is a mapping $\alpha:X\rightarrow[0,1]$ such that $\sum_{x\in E}\alpha(x)=1$ for every test $E\in{\mathfrak{A}}$. (In other language: a state is a “non-contextual” assignment of a probability to every outcome of every test.)

The simplest case is that in which ${\mathfrak{A}}$ comprises just a single test, say ${\mathfrak{A}}=\{E\}$. In this case, states on ${\mathfrak{A}}$ are simply probability weights on $E$, and we recover discrete classical probability theory.444Measure-theoretic classical probability theory is also subsumed by this framework: if $(S,\Sigma)$ is a measurable space, then the collection ${\mathfrak{B}}(S,\Sigma)$ of measurable partitions of $S$ by non-empty measurable sets in $\Sigma$ is a test space, and the states on ${\mathfrak{B}}(S,\Sigma)$ are exactly probability measures on $(S,\Sigma)$. Discrete quantum probability theory arises as the special case in which the test space is the collection ${\mathfrak{F}}({\bf H})$ of all frames, or unordered orthonormal bases, of a Hilbert space ${\bf H}$. Note that the outcome-set $X$ of ${\mathfrak{F}}$ is the unit sphere of ${\bf H}$. Gleason’s Theorem tells us that states on $\mathfrak{A}$ are all of the form $\alpha(x)=\langle\rho x,x\rangle=\text{Tr}(\rho p_{x})$ where $\rho$ is a density operator on ${\bf H}$ and $p_{x}$ is the rank-one orthogonal projection operator $p_{x}(y)=\langle y,x\rangle x$.

In many contexts, it is natural to endow the outcome-set $X$ of a test space with a Hausdorff topology for which which the relation $\perp$ is closed in $X\times X$ [28]. In this case, I shall speak of $X$ as the outcome space of ${\mathfrak{A}}$. Moreover, I shall assume – in the main, tacitly – that all states are continuous in some given topology on $X$. This will be important only for the proof of Lemma 1 below.555It is a consequence of Gleason’s Theorem that all states on ${\mathfrak{F}}({\bf H})$ are continuous in the usual topology on the unit sphere $X$ of ${\bf H}$, but in general, a test space will admit discontinuous states as well.

In what follows, a physical system is modeled by a pair $({\mathfrak{A}},\Omega)$, where $\mathfrak{A}$ is a test space with outcome-space $X$ and $\Omega$ is a closed, convex, outcome-separating set of continuous states thereon (possibly a proper subset of the full state space). We write $V$ for the cone-base space this generates (that, is, the linear hull of $\Omega$ in ${\mathbb{R}}^{X}$, ordered pointwise on $X$), and $V^{\ast}$ for the order-unit space dual to $V$. It will be convenient, if a trifle sloppy, to identify each outcome $x\in X$ with the corresponding evaluation functional, so that if $\alpha\in{\bf V}({\mathfrak{A}})$, we may write $x(\alpha)$ for $\alpha(x)$. Writing $u$ for the order unit in $V^{\ast}({\mathfrak{A}})$, we have $\sum_{x\in E}x=u$ for every test $E\in{\mathfrak{A}}$. Note that if $\mathfrak{A}$ is the frame manual of a Hilbert space ${\bf H}$, then we have $V^{\ast}({\mathfrak{A}})\simeq V({\mathfrak{A}})\simeq{\cal L}_{h}({\bf H})$, the space of Hermitian operators on ${\bf H}$.

If $\mathfrak{A}$ is any test space, ${\mathfrak{A}}\times{\mathfrak{A}}$ denotes the space of product tests $E\times F$ with $E,F\in{\mathfrak{A}}$. A state on ${\mathfrak{A}}\times{\mathfrak{A}}$ is non-signaling iff the marginal states $\omega_{1}(x)=\sum_{y\in F}\omega(x,y)$ and $\omega_{2}(y)=\sum_{x\in E}\omega(x,y)$ are well-defined, i.e., independent of $E,F\in{\mathfrak{A}}$. In this case we have conditional states $\omega_{2|x},\omega_{1|y}$ defined by

[TABLE]

(where these make sense, and defined to be [math] if not). Note that we have laws of total probability: for any test $E\in{\mathfrak{A}}$,

[TABLE]

and similarly for $\omega_{1}(y)$.

In everything that follows, I assume that $V$ is finite-dimensional, and that ${\mathfrak{A}}$ is locally finite, meaning that all tests in ${\mathfrak{A}}$ are finite sets.

3 Symmetry and Self-Duality

In this section, I introduce two operationally transparent axioms, and one simple (but much less transparent) minimization condition, and from these deduce that $V^{\ast}_{+}$ is self-dual.

The first axiom requires that systems be highly symmetrical in that (i) all outcomes of any given test look alike; (ii) all tests look alike; (iii) all pure states look alike. This axiom is satisfied by both (discrete) classical and pure quantum systems, and, as we’ll see in a moment, already leads to some surprisingly strong consequences.

To make this precise, let us agree that a symmetry of a system $({\mathfrak{A}},\Omega)$ with outcome space $X$ is a homeomorphism $g:X\rightarrow X$ such that (i) $gE\in{\mathfrak{A}}$ for every test $E\in{\mathfrak{A}}$, and (ii) $g^{\ast}(\alpha)=\alpha\circ g^{-1}$ belongs to $\Omega$ for every state $\omega\in\Omega$. An action of a group on $({\mathfrak{A}},\Omega)$ is an action by symmetries. We say that $\mathfrak{A}$ is fully symmetric [28] under such an action if (i) all tests have the same cardinality, and (ii) for any bijection $f:E\rightarrow F$ between tests $E,F\in{\mathfrak{A}}$, there exists some $g\in G$ with $f(x)=gx$ for all $x\in E$.

Notice that any symmetry $g$ of ${\mathfrak{A}}$ also determines an affine automorphism of $\Omega({\mathfrak{A}})$ by $(g\alpha)(x)=\alpha(g^{-1}x)$. We say that $g$ is a symmetry of the model $({\mathfrak{A}},\Omega)$ iff $g\alpha\in\Omega$ for all $\alpha\in\Omega$ and all $g\in G$. It is easy to see that $g$ takes extreme points of $\Omega$ to extreme points of $\Omega$. We shall say that an action of $G$ on $({\mathfrak{A}},\Omega)$ is continuous iff, for every $\alpha\in\Omega$ and every outcome $x\in X$, $g\mapsto\alpha(g^{-1}x)$ is continuous as a function of $G$.

Axiom 1 (Symmetry): There is a compact group $G$ acting continuously on $({\mathfrak{A}},\Omega)$, in such a way that (i) $G$ acts fully symmetrically on $\mathfrak{A}$, and (ii) $G$ acts transitively on $\Omega_{\mbox{ext}}$.

A classical test space ${\mathfrak{A}}=\{E\}$ satisfies Axiom 1 trivially with $G=S(E)$, the symmetric group on $E$. A quantum test space $({\mathfrak{F}}({\bf H}),\Omega_{{\bf H}})$ satisfies Axiom 1 with $G=U({\bf H})$, the unitary group of ${\bf H}$.

Call an inner product on $V^{\ast}$ positive iff $\langle a,b\rangle\geq 0$ for all $a,b\in V^{\ast}_{+}$. Note that the trace inner product on $V^{\ast}={\cal L}_{h}({\bf H})$ is positive in this sense.

Lemma 1: Subject to Axiom 1, there exists a positive, $G$-invariant inner product on $V^{\ast}$.

Proof: Represent $\Omega_{\mbox{ext}}$ as $G/K$ where $K=G_{\alpha}$, the stabilizer of some (any) pure state $\alpha_{o}$. Any $f\in V^{\ast}$ gives rise to a function $\hat{f}:G\rightarrow{\mathbb{R}}$, defined by $\hat{f}(g)=f(g\alpha_{o})$ for all $g\in G$. This is continuous, so we have an embedding of $V^{\ast}$ as a $G$-invariant real subspace of the algebra $C(G)$ of continuous complex-valued functions on $G$. The restriction of the natural inner product on the latter to $V^{\ast}$ is a real, $G$-invariant inner product, and is positive, simply because the convolution of positive functions on $G$ is positive. $\Box$

Let’s agree to call the specific inner product arising from $C(G)$ the canonical inner product. When we need to differentiate this from other choices of such an inner product, let’s denote it by $\langle\ ,\ \rangle_{G}$.

Remarks:

(1) The canonical inner product of Lemma 1 satisfies $\langle u,u\rangle=1$, and hence defines a symmetric non-signaling state on ${\mathfrak{A}}\times{\mathfrak{A}}$.

(2) Another way to obtain a positive, inner product on $V^{\ast}$ is just to declare some minimal informationally complete observable orthogonal (as D’Ariano notes in [8]). If this observable is $G$-invariant, so will be the resulting inner product.

(3) Appendix D classifies all unitarily invariant positive inner products on the space $V^{\ast}={\cal L}_{h}({\bf H})$ with $\langle{\bf 1},{\bf 1}\rangle=1$. These are found to depend on a single real parameter $\lambda\in(0,1]$, with $\lambda=1$ corresponding to the normalized trace inner product.

Lemma 2 [23]: Let $\langle,\rangle$ be any positive, $G$-invariant inner product on $V^{\ast}$. There is an embedding $x\mapsto v_{x}$ of the outcome-space $X$ into the unit sphere of $V^{\ast}$ with $x\perp y$ implying $\langle v_{x},v_{y}\rangle=0$.666I remind the reader that here, $x\perp y$ means only that the outcomes $x,y\in X$ are distinct and belong to a common test; this does not (yet) imply that $\langle x,y\rangle=0$.

Proof: For each $x\in X$, set

[TABLE]

so that

[TABLE]

Notice that $L^{\ast}_{\alpha}q_{x}=q_{\alpha x}$ for all $\alpha\in G$ and all $x\in X$. Since $L$ is unitary and $G$ acts transitively on $X$, the vectors $q_{x}$ have a constant norm $\|q_{x}\|=r$. Moreover, since $G$ takes any orthogonal pair of outcomes to any other, $\langle q_{x},q_{y}\rangle$ is constant for any pair $x\perp y$ in $X$. Call this value $s_{q}$. If $s_{q}=0$, we are done: simply set $v_{x}=q_{x}/\|q_{x}\|$. If not, we have

[TABLE]

In particular, $s_{q}=-\frac{r^{2}}{n-1}<0$. In this case, set $v_{x}=q_{x}+cu$ where $c=r/\sqrt{n-1}$ (so that $s_{q}=-c^{2}$). Then, using 2, we have $\langle v_{x},v_{y}\rangle=0$. Normalizing if necessary, we can take each $v_{x}$ to be a unit vector. Obviously, the mapping $x\mapsto q_{x}$ is injective $\langle x,u\rangle$ is constant on $X$; hence, so is $x\mapsto v_{x}$. $\Box$

In order to get maximum mileage out of this, we impose a very simple, but very strong condition. To set the stage, we need the following observation.

Lemma 3: Let $s$ denote the constant value of $\langle x,y\rangle$ where $x\perp y$. With notation as in Lemma 3, we have, for all outcomes $x$ and $y$, that

[TABLE]

Proof: Letting $m$ denote the (constant) value of $\langle x,u\rangle$, we set $q_{x}=x-mu$ as in the proof of Lemma 3, so that $\langle q_{x},u\rangle=0$ for all $x$. Recall that $v_{x}=q_{x}+cu$ where $-c^{2}=s_{q}$, the constant value of $\langle q_{x},q_{y}\rangle$ when $x\perp y$. Thus, we have

[TABLE]

Now

[TABLE]

Considering the case where $x\perp y$, this yields

[TABLE]

Hence,

[TABLE]

as promised. $\Box$

Definition: Call a $G$-invariant, positive inner product on $V^{\ast}$ minimizing iff the constant $s$ of Lemma 3 is in fact the minimum value of $\langle x,y\rangle$ on $X\times X$.

Note that this is certainly the case for the trace inner product on ${\cal L}_{h}({\bf H})$, where $s=0$!

Lemma 4: For a minimizing inner product, the vectors $v_{x}$ of Lemma 2 lie in the positive cone of $V^{\ast}$.

Proof: Immediate from Lemma 3. $\Box$

Provisional Axiom 2 (Minimization): There exists a minimizing $G$-invariant, positive inner product on $V^{\ast}$.

As we’ll see in section 3, all positive inner products on $V^{\ast}={\cal L}_{h}({\bf H})$ invariant under the unitary group of ${\bf H}$, are in fact minimizing. Thus, it is not out of the question that Axiom 4 is actually a theorem. In any case, one would like to have an operational interpretation for minimization. What such an interpretation would look like, I’m not sure – but I’m betting there is one.

There is one important class of examples in which the existence of a minimizing inner product does follow from the previous axioms. Call a test space $\mathfrak{A}$ $2$-connected iff every pair of outcomes $x,y\in X$ there exist tests $E,F\in{\mathfrak{A}}$ with $x\in E,y\in F$ and $E\cap F\not=\emptyset$. Equivalently, $\mathfrak{A}$ is $2$-connected iff, for all outcomes $x,y$ there exists an outcome $z$ with $x\perp z\perp y$. Example: the frame manual of a Hilbert space.

Lemma 5: If $\mathfrak{A}$ is a fully-symmetric, rank-three, $2$-connected test space, then any invariant, positive inner product is minimizing.

Proof: Let $x\not\perp y$. By $2$-connectedness, we can find an outcome $z$ with $x\perp z\perp y$. As $\mathfrak{A}$ has rank three, we have tests $E=\{x,a,z\}$ and $F=\{z,b,y\}$. Now, as all outcomes have the same norm in $V^{\ast}$ (here, I conflate an outcome with the corresponding evaluation functional in $V^{\ast}$), we see that $\langle x,t\rangle$ is maximized over outcomes $t$ by $t=x$. Let $s$ be the common value of $\langle e,f\rangle$ where $e$ and $f$ are outcomes with $e\perp f$. Noting that $x+a+z=u=z+b+y$, we have

[TABLE]

This yields $\langle x,b\rangle+\langle x,y\rangle=s+\|x\|^{2}$. Since $s<\|x\|^{2}$ and neither $\langle x,b\rangle$ and $\langle x,y\rangle$ can exceed $\|x\|^{2}$, it follows that both must exceed $s$. $\Box$

Remark: 2-connectivity seems close to requiring the sets $x^{\perp}$, $x$ ranging over outcomes, to form a projective geometry. See [23] for more on this.

Lemma 6: Subject to Axiom 1 and Provisional Axiom 2, For every $x\in X$, $\alpha_{x}(y):=\langle v_{x}|v_{y}\rangle$ defines a state on $\mathfrak{A}$.

Proof: By Lemma 4, $\langle v_{x},v_{y}\rangle\geq 0$ for all $y$. Since $v_{x}$ and $v_{y}$ are unit vectors, we also have $\langle v_{x},v_{y}\rangle\leq 1$ for all $y$. Finally, letting $x\in E\in{\mathfrak{A}}$, we have, by Lemma 2, and with $v:=\sum_{y\in E}v_{y}$, a multiple of $u$777since $v_{y}=q_{y}+cu$, and $\sum_{y\in E}q_{y}=0$, we have $\sum_{y\in E}v_{y}=ncu$, where $n=|E|$ is independent of $E$ by virtue of $\mathfrak{A}$’s being fully symmetric, that

[TABLE]

We now impose another axiom that, while decidely strong, has a clear physical meaning: it says that if we know for certain that a particular outcome will occur, then we know the system’s state.

Axiom 3 (Sharpness): To every outcome $x\in X$, there corresponds a unique state $\epsilon_{x}\in\Omega$ with $\epsilon_{x}(x)=1$.

Note that $\epsilon_{x}$ is necessarily a pure state. Note, too, that both (discrete) classical and non-relativistic QM satisfy this postulate. For some further discussion of (and motivation for) Axiom 3, see Appendix A.

Proposition 1: Subject to Axioms 1-3, $V^{\ast}_{+}$ is self-dual.

Proof: Let $\langle,\rangle$ be a minimizing, $G$-invariant positive inner product. Positivity gives us $V^{\ast}_{+}\subseteq V^{\ast+}\simeq V_{+}$. Letting $v_{x}$ be defined as in Lemma 2, Lemma 7 tells us that $\alpha_{x}(y):=\langle v_{x},v_{y}\rangle$ defines a state making $x$ certain (since $\langle v_{x},v_{x}\rangle=\|v_{x}\|=1$). By Axiom 3, there is but one such state, which, by virtue of its uniqueness, is pure. It follows from Axiom 2 that every pure state has the form $g\alpha_{x}=\alpha_{gx}$ for some $g\in G$. Thus, every pure state is represented in the cone $V^{\ast}_{+}$, so that $V^{\ast+}\subseteq V^{\ast}_{+}$. $\Box$

An alternative argument, based on slightly different assumptions, is presented in Appendix C.

4 Correlation, Filtering and Homogeneity

Having secured the self-duality of $V^{\ast}_{+}$, the next order of business is to secure its homogeneity. This will follow from two further axioms. The frst of these tells us that all states of a single system are consistent with that system’s being part of a larger composite in a state of perfect correlation between some pair of observables. To be more precise, call a bipartite non-signaling state correlating iff, for some tests $E,F\in{\mathfrak{A}}$, and some bijection $f:E\rightarrow F$, $\omega(xy)=0$ for all $(x,y)\in E\times F$ with $y\not=f(x)$.

Axiom 4 (Correlation): Every state is the marginal of a correlating non-signaling state.

Again, this is satisfied by both classical and quantum systems: trivially in the first case, and not-so-trivially (i.e., by the Schmidt decomposition) in the second.

Lemma 6 [29]: Subject to Axioms 3 and 4, for every $\mu\in V_{+}$ there exists a test $E$ such that $\sum_{x\in E}\mu(x)\epsilon_{x}$.

Proof: Suppose first that $\alpha$ is a normalized state on ${\mathfrak{A}}$. By Axiom 4, there exists a test space ${\mathfrak{B}}$ and a correlating, non-signaling state $\omega\in\Omega({\mathfrak{A}}\times{\mathfrak{B}})$ with $\alpha=\omega_{1}$. Suppose $\omega$ correlates $E\in{\mathfrak{A}}$ with $F\in{\mathfrak{B}}$ along a bijection $f:E\rightarrow F$. The bipartite law of total probability (1) tells us that

[TABLE]

where $\omega_{1|y}=\omega_{1|f(x)}$ is the conditional state on $\mathfrak{A}$ given outcome $y=f(x)\in F$. Since $\omega(x,y)=0$ for $y\not=f(x)$, we have $\omega_{1|f(x)}(x)=1$ if $y=f(x)\in F$; thus, $\omega_{2|f(x)}=\epsilon_{x}$, and $\alpha=\sum_{x\in E}\alpha(x)\epsilon_{x}$ as promised.

Now suppose $\mu\in V_{+}$. Then $\mu=r\alpha$ for some $\alpha\in\Omega$ and real constant $r\geq 0$. Expanding $\alpha$ as above, we have $\mu=\sum_{x\in E}r\alpha(x)\epsilon_{x}=\sum_{x\in E}\mu(x)\epsilon_{x}$. $\Box$

The following postulate completes the set.

Axiom 5 (Filtering): For every test $E$ and every $f:E\rightarrow(0,1]$, there exists an order-isomorphism $\phi:V^{\ast}\rightarrow V^{\ast}$ with $\phi(x)=f(x)x$.

This says that the outcomes of a test can simultaneously and independently be attenuated by any (non-zero) factors we like by a reversible physical process. This is equivalent to saying that, for any test $E$, any outcome $x\in E$, and any $0<c\leq 1$, there exists an order-automorphism $\phi$ such that $\phi(x)=cx$ and $\phi(y)=y$ for all $y\in E\setminus\{x\}$. This is clearly the case in both classical and quantum probability theory, and corresponds to the operationally natural idea that an outcome is always represented by a physical process, which can be subjected to a filter reducing its intensity by any specified factor. Possibly one could make this intuition more precise by considering two-stage tests (what D’Ariano calls cascades in [10]).

Proposition 2: Subject to Axioms 1-5, the cone $V^{\ast}_{+}$ is homogeneous.888A different route to homogeneity, via slightly different axioms, is discussed in appendix A.

Proof: Let $a$, $b$ be interior points of $V^{\ast}_{+}$. By Proposition 1, $V^{\ast}_{+}$ is self-dual; hence, $\langle a|$ and $\langle b|$ are (un-normalized) states. Let us write $a(x)$ for $\langle a,x\rangle$ and similarly for $b$. By Lemma 6, $\langle a|$ and $\langle b|$ have decompositions $\langle a|=\sum_{x\in E}a(x)\epsilon_{x}$ and $\langle b|=\sum_{y\in F}b(y)\epsilon_{y}$ for some pair of tests $E,F\in{\mathfrak{A}}$. As $\epsilon_{x}=\langle x|$, we have $\langle a|=\sum_{x\in E}a(x)\langle x|$, or, more simply, $a=\sum_{x\in E}a(x)x$, and similarly for $b$. Since $a$ and $b$ are interior points, $a(x)$ and $b(y)$ are non-zero for all $x,y$. Let $g$ be a bijection matching $E$ with $F$ (courtesy of Axiom 4), and set $t(x)=b(gx)/a(x)$. Then, by Axiom 5, there is an order-automorphism $\phi$ of $V^{\ast}$ taking $x$ to $t(x)x$ for every $x\in E$. Hence, $\phi(a)=\sum_{x\in E}a(x)\phi(x)=\sum_{x\in E}a(x)t(x)x=\sum_{x\in E}b(gx)x$. Applying $g$, we have

[TABLE]

It now follows from the Koecher-Vinberg theorem that the positive cone $V({\mathfrak{A}})_{+}$ is the set of positive elements in a formally real Jordan algebra. It is possible that there is a more direct route to this conclusion – certainly, I have not made use of all of the available structure. For example, the full power of the assumption that $\mathfrak{A}$ is fully $G$-symmetric (as opposed to merely $2$-symmetric) is not really exploited. It may also be useful that $V({\mathfrak{A}})$, regarded as a subspace of the group algebra ${\mathbb{C}}[G]$, is closed under convolution, hence, an algebra – though the connection between this structure and the $C^{\ast}$-algebraic structure in QM is not obvious.

Neither is it at all obvious that every self-dual homogeneous cone has a representation as $V({\mathfrak{A}})$ with $\mathfrak{A}$ satisfying all of the foregoing axioms – again, full symmetry seems rather strong, as does Axiom 4 on correlation. It is possible that these axioms constrain the set of models much more severely.

5 Summary and open questions

Axioms 1, 3 and 4 seem natural, or at any rate, intelligible: one understands what they say about a system. Although strong (and certainly, not “laws of thought”), they do identifying a natural class of especially simple and tractable systems that we might expect to find well represented “in nature”. Axiom 5 seems natural in a slightly more restricted context, in which the measurements we make involve sending systems through filters that they may or may not pass, with probabilities that can be attenuated at our discretion. (The idea of a filter also shows up prominently in the work of Ludwig [19] and others following in the same path, e.g., [13, 21, 18].) In a broader sense, Axiom 5 captures, at least in part, the idea that a system should look completely classical, as restricted to a single measurement. In particular, a process allowable in classical probability theory should be implementable by a “physical” process acting on $V({\mathfrak{A}})$.

Provisional Axiom 2 is obviously more problematic, but on the evidence, seems likely to be satisfied by a wide range of systems. In order better to understand the scope and significance of this postulate, one would like to endow an invariant positive inner product on $V^{\ast}$ with some operational, perhaps information-theoretic meaning.

Problem 1: Find an information-theoretic meaning for the canonical inner product $\langle\ ,\ \rangle_{G}$ arising from the group algebra.

In this connection, it would probably be useful to know more about the non-tracial positive, invariant inner products on ${\cal L}_{h}({\bf H})$. In Appendix D, it is shown that the positive, unitarily invariant inner products on ${\cal L}_{h}({\bf H})$ form a one-parameter family $\langle a,b\rangle_{\lambda}$, with $0<\lambda\leq 1$ and $\lambda=1$ corresponding to the trace inner product. It would be interesting to know what value of $\lambda$ corresponds to $\langle\ ,\ \rangle_{G}$ where $G$ is the unitary group of ${\bf H}$.

It would of course be nice if something like the result of Appendix D were true in more generality – say, for irreducible cones:

Problem 2: Classify the invariant, positive inner products on (i) any $V^{\ast}$ arising from a model $({\mathfrak{A}},\Omega)$ satisfying Axioms 1, 3, 4 and 5; (ii) any irreducible, homogeneous self-dual cone.

In this connection, it would also be interesting to know whether some reduction theory is available for systems satisfying Axioms 1, 3, 4 and perhaps Axiom 5. Thus, we have

Problem 3: Is a system satisfying Axioms 1, 3, 4 and 5 a direct sum (in some sense!) of irreducible such systems? Is the cone of an irreducible such system irreducible as a convex cone?

If such a reduction theory is available, there is perhaps a chance that, using the same line of argument as that of Appendix C, one may be able to eliminate the troublesome Axiom 2 altogether.

There are likely many connections between the approach sketched above and that of D’Ariano [10], based on sequential tests and conditioning operators. (The earlier work of Kummer, cited above, also depends heavily on ideas involving sequential measurements and conditioning.) There are probably also connections with the approach of Rau [24], which depends upon symmetry considerations similar to, but in some ways stronger than, those of our Axiom 2. In particular, Rau assumes that the group $G$ of physical symmetries is a compact Lie group.

Problem 4: Clarify these connections. In particular, (i) clarify the connection between Axiom 5 and the closure of $\mathfrak{A}$ under formation of compound experiments (“cascades”, in the language of [10]), together with the closure of $\Omega$ under conditioning. (ii) What is the relationship between the existence of a positive $G$-invariant inner product and D’Ariano’s “Choi-Jamiolkowski” axiom? [10] (iii) What additional leverage, if any, do we get from the foregoing axioms if $G$ is a compact Lie group and $X$ is an homogeneous space for $G$?

Another, particularly important, issue is that of how one can construct, by hand as it were, tensor products compatible with the foregoing axioms.

Problem 5: Under what conditions do systems satisfying axioms 1-5, or any subset of these, have non-signalling tensor products (containing all product states!) that also satisfy these axioms?

Where this desideratum is met, we would seem to come within hailing distance of Hardy’s axioms [15]. See [30] for some further discussion of the problems involved in constructing a class of test spaces closed under such a tensor product.

Acknowledgement: I wish to thank Howard Barnum for reading and commenting on an earlier draft of this paper, and, more especially, for introducing me to the papers of Koecher and Vinberg, on which the present exercise depends. Thanks also to C. M. Edwards for pointing out the paper [14] of Hanche-Olsen.