Unentangled Measurements and Frame Functions

TL;DR

This paper extends Gleason's theorem to unentangled frame functions in multi-partite quantum systems, revealing their structure across various dimensions and uncovering a combinatorial pattern with potential foundational significance.

Contribution

It classifies unentangled frame functions for multi-qubit and mixed-dimension systems, including infinite-dimensional spaces, generalizing previous results and revealing new structural insights.

Findings

Classified unentangled frame functions for multi-qubit systems.

Extended the classification to systems with varying and infinite dimensions.

Discovered a combinatorial structure suggesting fundamental interpretations.

Abstract

Gleason's theorem asserts the equivalence of von Neumann's density operator formalism of quantum mechanics and frame functions, which are functions on the pure states that sum to 1 on any orthonormal basis of Hilbert space of dimension at least 3. The unentangled frame functions are initially only defined on unentangled (that is, product) states in a multi-partite system. The third author's Unentangled Gleason's Theorem shows that unentangled frame functions determine unique density operators if and only if each subsystem is at least 3-dimensional. In this paper, we determine the structure of unentangled frame functions in general. We first classify them for multi-qubit systems, and then extend the results to factors of varying dimensions including countably infinite dimensions (separable Hilbert spaces). A remarkable combinatorial structure emerges, suggesting possible fundamental interpretations.

Equations

$$\mathcal{H}=H_1\otimes H_2\otimes \cdots \otimes H_n$$

$$\mathcal{H}={\otimes }^k{\mathbb{C}}^2\otimes H_{k+1}\otimes \cdots \otimes H_n$$

$$⟨v_1\otimes v_2\otimes \cdots \otimes v_n|w_1\otimes w_2\otimes \cdots \otimes w_n⟩=0$$

$$f:\Sigma \to {\mathbb{R}}_{\ge 0},$$

$$\sum _{i}f(u_i)=1.$$

$${\mathbb{P}}^1(\mathbb{C})\to \mathbb{C}\cup \infty$$

$$(x,y)⟼\frac{x}{y}.$$

$$\sigma z=-\frac{1}{\bar{z}}.$$

$$F=\{z\in \mathbb{C}\mid |z|<1\}\cup \{z\in \mathbb{C}\mid |z|=1,\mathrm{Im}z>0\}\cup \{1\}.$$

$$\Sigma =\bigcup _{J\subset {\Omega }_n}{\sigma }_J(F_n)$$

$$\sum _{L\subset J}f({\sigma }_L(z))={\varphi }_J({\tau }_J(z)).$$

$$\sum _{I\subset \{1,\dots ,j\}}f({\sigma }_I(z_1\otimes \cdots \otimes z_j\otimes z_{j+1}\otimes \cdots \otimes z_n))$$

$$Z=\{{\sigma }_I(z_1\otimes \cdots \otimes z_j\otimes z_{j+1}\otimes \cdots \otimes z_n)\mid I\subset J\}$$

$$f({\sigma }_J(z))=\sum _{L\subset J}(-1{)}^{|J-L|}{\varphi }_L({\tau }_L(z)).$$

$$\sum _{L\subset J}\alpha (L)=\beta (J),$$

$$\alpha (J)=\sum _{L\subset J}(-1{)}^{|J-L|}\beta (L).$$

$$f({\sigma }_J(z))=\sum _{L\subset J}(-1{)}^{|J-L|}{\varphi }_L({\tau }_L(z))$$

$$\sum _{i=1}^{2^n}f(z_i)=c\text{.}$$

$$f(a\otimes {\sigma }_J(z))=\sum _{L\subset J}(-1{)}^{|J|-|L|}{\varphi }_L({\tau }_L(a\otimes z))$$

$$f(\hat{a}\otimes {\sigma }_J(z))=f({\sigma }_{J\cup \{1\}}(a\otimes z))=\sum _{L\subset J\cup \{1\}}(-1{)}^{|J|+1-|L|}{\varphi }_L({\tau }_L(a\otimes z))=$$

$$-\sum _{L\subset J}(-1{)}^{|J|-|L|}{\varphi }_L({\tau }_L(a\otimes z))+\sum _{L^{\prime }\subset J}(-1{)}^{|J|-|L^{\prime }|}{\varphi }_{L^{\prime }\cup \{1\}}({\tau }_{L^{\prime }\cup \{1\}}(a\otimes z)).$$

$$f(a\otimes {\sigma }_J(z))+f(\hat{a}\otimes {\sigma }_J(z))=\sum _{L^{\prime }\subset J}(-1{)}^{|J|-|L^{\prime }|}{\varphi }_{L^{\prime }\cup \{1\}}({\tau }_{L^{\prime }\cup \{1\}}(a\otimes z)).$$

$$f_a({\sigma }_J(z))=\sum _{L\subset J}(-1{)}^{|J-L|}{\varphi }_L({\tau }_L(a\otimes z))$$

$$f_a({\sigma }_{\Omega }(z))={\varphi }_{\Omega }(a).$$

$$\sum _{i=1}^{2^n-1}{f}_a(z_i)={\varphi }_{\Omega }(a)\text{.}$$

$${\mathcal{H}}_{n-1}=V_1\oplus \cdots \oplus V_r$$

$$\mathcal{B}=\{a_i\otimes u_{ij}\mid i=1,\dots ,r,j=1,\dots ,d_i\}\cup \{{\hat{a}}_i\otimes v_{ij}\mid i=1,\dots ,r,j=1,\dots ,d_i\}.$$

$$\sum _{j=1}^{d_i}{f}_{a_i}(u_{i,j})$$

$$f(a_i\otimes v_{i,j})+f({\hat{a}}_i\otimes v_{ij})$$

$$\sum _{ij}f(a\otimes v_{ij})+\sum _{ij}f(\hat{a}\otimes v_{ij}).$$

Full text

Quantum physics, Gleason’s Theorem, Frame function, Product state, Mixed state, Entanglement, Quantum measurements

I Introduction

In von Neumann’s vn:mgdq approach to quantum mechanics, the formalism assumes a factor algebra with an endowed trace function. The mixed states in this set up are the self-adjoint, positive elements of trace $1$ looked upon as defining a measure on the set of self-adjoint, idempotent elements. This led Mackey to ask the natural question: Is every such measure given by a mixed state? Gleason gave an affirmative answer to this question in the case of factors of type $\text{I}_{n}$ with $n>2$ (i.e. the bounded operators on a separable Hilbert space). In his approach to his proof, he introduced the notion of frame function (a non-negative function on the pure states that sums to $1$ on an orthonormal basis of the Hilbert space) which is easily seen to be equivalent to that of measure on the set of self-adjoint idempotents. Gleason’s theorem shows that such a function, $f$, must be of the form $f(v)=\left\langle v|A|v\right\rangle$ for $v$ any unit norm vector in the Hilbert space and $A$ a non-negative, self-adjoint operator of trace $1$ if the dimension of the Hilbert space is at least $3$. In many contexts of quantum information theory that deal with state spaces of many independent particles, the only pure states that are sampled are product (or unentangled) states. This led the last named author W:Gleason to ask if such a sampling of only unentangled states would allow for more general frame functions. In this paper we classify all the unentangled frame functions, for an arbitrary but finite number of tensor factors, including those of countably infinite dimension (separable Hilbert spaces), completing the treatment of the unentangled frame functions. A thorough account of quantum measurement theory in the setting of operator algebras, including Gealson’s theorem and its variants, is in h:qmt (J. Hamhalter).

The organization of the rest of the paper is as follows. We first introduce the idea of an unentangled frame function in Section II. In Section III, we choose the fundamental domain in $\mathbb{P}^{1}(\mathbb{C})$ that we use to identify a vector with its orthogonal. In Section IV we classify all multi-qubit frame functions. Section V generalizes this classification further to include the case when some of the tensor factors are of dimensions at least $3$ and at most countable (separable Hilbert spaces). Section VI concludes our discussion with some remarks.

II Unentangled frame functions

Let us take a brief look at unentangled Gleason setup as in W:Gleason , and make more precise the ideas from the Introduction. Let

[TABLE]

with $\dim H_{i}\geq 2$. Technically we should be looking at the completed tensor product if two or more of the factors are infinite dimensional. However, since we will only be looking at product vectors this will not be necessary. Applying a permutation of the factors we may assume that the first $k$ of the $H_{i}$ are of dimension $2$ and all of the rest have dimension $>2$. Thus

[TABLE]

with $\dim H_{i}>2$ and if $n=k$ then by convention the last factor is $\mathbb{C}$. $\mathcal{H}$ is given the tensor product Hilbert structure, $\left\langle\ldots|\ldots\right\rangle$. A vector in $\mathcal{H}$ is called unentangled (a product vector) if it is a tensor product of unit vectors, one from each $H_{i}$ factor. Two such vectors $v_{1}\otimes v_{2}\otimes\cdots\otimes v_{n}$ and $w_{1}\otimes w_{n}\otimes\cdots\otimes w_{n}$ are orthogonal

[TABLE]

if and only if there is at least one $i$ with $\left\langle v_{i}|w_{i}\right\rangle=0$. An Unentangled Othonormal Basis (UOB) $\{u_{i}\}$ is a basis of $\mathcal{H}$ consisting of orthogonal (unit norm) unentangled vectors. Let $\Sigma$ be the set of all unentangled vectors in $\mathcal{H}$.

Definition II.1**.**

An unentangled frame function is a map

[TABLE]

such that for every UOB $\{u_{i}\}$,

[TABLE]

In the next two sections we will be dealing with the multi-qubit case, so we specialize the above to prepare for it. Let $\mathcal{H}_{n}={\otimes}^{n}\mathbb{C}^{2}$ be the space of $n$ qubits. A UOB is then a basis $\{u_{0},u_{1},\ldots,u_{2^{n}-1}\}$ of $\mathcal{H}_{n}$ consisting of orthogonal (unit norm) unentangled vectors.

III A fundamental domain

Let $\sigma:\mathbb{C}^{2}\rightarrow\mathbb{C}^{2}$ be defined by $v=(x,y)\longmapsto\hat{v}=(-\bar{y},\bar{x})$. We note that $\left\langle v|\sigma v\right\rangle=0$ and if $v$ is a state then up to phase $\sigma v$ is the unique state perpendicular to $v$. The $\sigma$ induces a map of $\mathbb{P}^{1}(\mathbb{C})$ to itself which we also denote by $\sigma$. We have the standard map

[TABLE]

given by

[TABLE]

We note that under this identification as a map from $\mathbb{C\cup\infty}$ to itself

[TABLE]

In particular, on $S^{1}$ it is given by $z\longmapsto-z$.

Here is a simple fundamental domain for $\sigma$:

[TABLE]

A frame function $f$ in a single qubit space corresponds to an arbitrary function $\phi\colon F\to[0,1]$ by letting $f(a)=\phi(a)$ if $a\in F$ and since the sum of the values of $f$ over orthogonal vectors is $1$, $f(a)=1-\phi(\sigma a)$ if $a\notin F$.

IV Unentangled frame functions in $n$ qubits

Let $\mathcal{H}_{n}$ denote $n$–qubit space. We choose a fundamental domain $F$ for the map $[z]\rightarrow\sigma([z])=[\hat{z}]$ in one qubit. We set $F_{n}=F\otimes F\otimes\cdots\otimes F$ ($n$ copies). Let $\Omega_{n}=\{1,\ldots,n\}$. If $J\subset\Omega_{n}$ is a subset we define $\sigma_{J}=T_{1}\otimes T_{2}\otimes\cdots\otimes T_{n}$ with $T_{j}=\sigma$ if $j\in J$ and $T_{j}=\mathbb{I}$ (identity operator), if $j\notin J$. We set $J^{c}=\Omega_{n}-J$. We note

[TABLE]

is a disjoint union. If $z=z_{1}\otimes\cdots\otimes z_{n}\in F_{n}$, if $J\subset\Omega_{n}$ and if $J^{c}=\{i_{1},\ldots,i_{k}\}$ with $i_{1}<i_{2}<\cdots<i_{k}$ then we set $\tau_{J}(z)=(z_{i_{1}},\ldots,z_{i_{k}})$ if $J\neq\Omega_{n}$, if $J=\Omega_{n}$ then use the symbol $\omega$ for the value.

Lemma IV.1**.**

If $f$ is a function on the product states of $\mathcal{H}_{n}$ such that $f$ sums to a fixed constant $c$ on all UOB’s then for each J\subset\Omega_{n}\there exists a function $\phi_{J}$ on $F^{n-|J|}$ (direct product of $n-|J|$ copies of $F$, $F^{0}=\{\omega\},\phi_{\Omega_{n}}(\omega)=c$) such that if $z\in F_{n}$ then

[TABLE]

Proof.

After permuting the factors we may assume that $J=\{1,\ldots,j\}$, thus the sum on the left hand side is

[TABLE]

call it $b$, and note that $\sigma_{I}$ acts only on $z_{1}\otimes\cdots\otimes z_{j}$ as $\sigma$ and as $\mathbb{I}$ everywhere else. Observing that if the set

[TABLE]

is extended to a $UOB$ by adjoining elements $u_{1},\ldots,u_{r}$ with $r=2^{j}(2^{n-j}-1)$ then $\sum f(u_{i})=1-b$ no matter how we found the extension. Also $Z$ is an orthonormal basis of $\otimes^{j}\mathbb{C}^{2}\otimes z_{j+1}\otimes\cdots\otimes z_{n}$ which only depends on $z_{j+1}\otimes\cdots\otimes z_{n}$. This proves the result. ∎

Now applying inclusion exclusion we have in the notation of the previous lemma

Lemma IV.2**.**

If $z\in F_{n}$ and if $f$ is a function on the product states satisfying the hypotheses of Lemma IV.1 then

[TABLE]

Proof.

Inclusion exclusion says: Let $\alpha$ and $\beta$ be functions from the set of all subsets of $\Omega_{n}$ to $\mathbb{C}$ and such that if $J\subset\Omega_{n}$ then

[TABLE]

then

[TABLE]

(c.f. Rota r:mobius ). The lemma follows from this assertion by taking $\alpha(J)=f(\sigma_{J}(z))$ and $\beta(J)=\phi_{J}(\tau_{J}(z))$. ∎

Theorem IV.3**.**

If $L\subsetneqq\Omega_{n}$ let $\phi_{L}$ be a real valued function on $F_{n-\left|L\right|}$. Assume that $\phi_{\Omega_{n}}(\omega)=c$ then the function $f:\mathcal{H}_{n}\rightarrow\mathbb{R}$ defined by

[TABLE]

for $J\subset\Omega_{n}$ and $z\in F_{n}$ satisfies

[TABLE]

if $z_{1},\ldots,z_{2^{n}}$ is a UOB.

Proof.

We first note that if $z\in F_{n-1}$ and $J\subset\{2,\ldots,n\}$ then

1. $f(a\otimes\sigma_{J}(z))+f(\hat{a}\otimes\sigma_{J}(z))$ is independent of $a$.

Indeed,

[TABLE]

and (if $1\in L$ write $L=\{1\}\cup L^{\prime}$ with $L^{\prime}\subset\{2,\ldots,n\}$)

[TABLE]

[TABLE]

We therefore have

[TABLE]

This proves 1. since $\tau_{L^{\prime}\cup\{1\}}(a\otimes z)$ is independent of $a$.

We will now prove the theorem by induction on $n$. If $n=1$ the assertion is that if $a\in F$ then $f(a)+c-f(a)=c$. So the result is true for $n=1$. Now assume the result for $n-1\geq 1$. We now prove it for $n.$ If $a\in F$ then define $f_{a}(z)=f(a\otimes z)$. Then if $J\subsetneqq\Omega=\{2,\ldots,n\}$ and $z\in F_{n-1}$ we have

[TABLE]

and

[TABLE]

Thus the inductive hypothesis implies that $f_{a}$ satisfies

[TABLE]

for any UOB $\mathcal{\{}z_{1},\ldots,z_{2^{n-1}}\}$ of $\mathcal{H}_{n-1}$.

We are now ready to prove the theorem. Let $\mathcal{B}=\{z_{1},\ldots,z_{2^{n}}\}$ be a UOB. Then Theorem $6$ in W:Gleason implies that there exist $a_{1},\ldots,a_{r}\in F$, $V_{1},\ldots,V_{r}$ orthogonal subspaces of $\mathcal{H}_{n-1}$ such that

[TABLE]

and $u_{ij}$ and $v_{ij},j=1,\ldots,d_{i}$ orthonormal basis of $V_{i}$ consisting of product vectors such that

[TABLE]

For each $i$ we apply the inductive hypothesis to $f_{a_{i}}$ and find that

[TABLE]

depends only on $a_{i}$ and $V_{i}$ and not on the particular orthonormal basis of $V_{i}$. Thus we can replace $u_{i,j}$ with $v_{i,j}$ without changing the sum. Now

[TABLE]

is independent of $a_{i}$ $by$ $1.$ Thus we can replace all of the $a_{i}$ with a fixed element $a\in F$ without changing the sum. Thus the sum is given by

[TABLE]

We now observe that if we define

[TABLE]

then (see the proof of 1.)

[TABLE]

and

[TABLE]

for all $z\in F_{n-1}$. Finally we can apply the inductive hypothesis to replace the basis $\{v_{ij}\}$ by $\{\sigma_{J}(z)\mid J\subset\Omega\}$ with $z\in F_{n-1}$ and the theorem is proved. ∎

A consequence of the proof of this theorem is that it suffices to specify a frame function on the highest dimensional component of the space of UOBs. This is described inlsw:lduobag . A generic UOB in this component is recursively defined as follows.

Definition IV.4**.**

[TABLE]

for an arbitrary $a\in F$, and where $\mathcal{B}_{i},i=1,2$ are again defined in the same manner as $\mathcal{B}$ for one less qubit.

Corollary IV.5**.**

If $f$ is a function on the product state such that there exists a constant $c$ and for every generic UOB, $\{z_{1},\ldots,z_{2^{n}}\}$we have

[TABLE]

then the same is true for every UOB.

Proof.

The characterization of the generic UOB in lsw:lduobag makes it clear that if $f$ sums to $c$ on all generic UOB then $f$ and the functions $\phi_{J}$ have the property in Lemma IV.1. Lemma IV.2 and Theorem IV.3 now complete the proof. ∎

Notice that the corollary is interesting only when $n\geq 3$, since when $n<3$, every UOB is part of a maximal dimensional family.

V General unentangled frame functions

In this section we will give a complete description of unentangled frame functions for separable Hilbert spaces of the form

[TABLE]

with $\dim H_{i}\geq 2$.

Let $f$ be an unentangled frame function on $\mathcal{H}$. Thus $f$ is a real valued function product states such that there exists a scalar $c$ such that if $\{u_{i}\}$ is a $UOB$ then $\sum f(u_{i})=c$.

If $z$ is a product state in $\mathcal{\otimes}^{k}\mathbb{C}^{2}$ then setting $f_{z}(x)=f(z\otimes x)$ for $x$ a product state in $H_{k+1}\otimes\cdots\otimes H_{r}$, $f_{z}$ is an unentangled frame function on $H_{k+1}\otimes\cdots\otimes H_{r}$ with

[TABLE]

for $\{u_{i}\}$ a $UOB$ of $H_{k+1}\otimes\cdots\otimes H_{r}$ (so $c(z)=\sum_{i}f(z\otimes u_{i})$ depends only on $z$ and $f$). The unentangled Gleason’s theorem W:Gleason implies that there exists $A(z)$ a trace class, self-adjoint non-negative operator on $H_{k+1}\otimes\cdots\otimes H_{r}$ such that

1. $\mathrm{tr}A(z)=c(z)$.

2. $f_{z}(u)=\left\langle u|A(z)|u\right\rangle$.

Note that the proof of the unentangled Gleason theorem given in W:Gleason does not use finite dimensionality and therefore applies to the context of this paper.

This proves the following reduction of the problem.

Lemma V.1**.**

Let $\mathcal{H}=\mathcal{\otimes}^{k}\mathbb{C}^{2}\otimes H_{k+1}\otimes\cdots\otimes H_{r}$ be as above. Then an unentangled frame function for $\mathcal{H}$ is the restriction of one for $\mathcal{\otimes}^{k}\mathbb{C}^{2}\otimes H$ with $H$ the completion of $H_{k+1}\otimes\cdots\otimes H_{r}$.

In light of this lemma we need only classify the unentangled frame functions on Hilbert spaces of the form $\mathcal{\otimes}^{k}\mathbb{C}^{2}\otimes H$ with $H$ a separable Hilbert space of whose dimension is not $2$. We note that if $\dim H=1$ then a self-adjoint operator on $H$ is a real scalar. Let $\mathcal{T}(H)$ be the space of trace class, self-adjoint operators on $H$.

Theorem V.2**.**

Let for each $J\subset\{1,\ldots,k\}$, $\phi_{J}:F^{k-|J|}\rightarrow\mathcal{T}(H)$ (here as usual, we set $F^{0}=\{\omega\}$). Let $z\in F^{k}$ and let $u$ be a state in $H$

[TABLE]

Let

[TABLE]

If $\{w_{j}\}$ is a $UOB$ of $\mathcal{\otimes}^{k}\mathbb{C}^{2}\otimes H$ then

[TABLE]

If $f$ is an unentangled frame function on $\mathcal{\otimes}^{k}\mathbb{C}^{2}\otimes H$ and if $J\subset\Omega_{k}$ we set for $z\in F^{k}$ and $u$ a state in $H$

[TABLE]

then $\gamma_{J,z}(u)=\left\langle u|\phi_{J}(\tau_{J}(z))|u\right\rangle$ with $\phi_{J}:F^{k-|J|}\rightarrow\mathcal{T}(H)$ and $f$ is given by $(\ast)$.

Proof.

If $f$ is an unentangled frame function on $\mathcal{\otimes}^{k}\mathbb{C}^{2}\otimes H$ and if $z$ is a state in $\mathcal{\otimes}^{k}\mathbb{C}^{2}$ then, as above, $f_{z}(u)=f(z\otimes u)$ is a frame function on $H$. Thus there exists a trace class self-adjoint operator on $H$, $\alpha(z)$, such that $f_{z}(u)=\left\langle u|\alpha(z)|u\right\rangle$ for $u$ a state in $H$. Similarly we see that for fixed $u$ a state in $H$,

[TABLE]

defines an unentangled frame function on $\mathcal{\otimes}^{k}\mathbb{C}^{2}$. Thus there exist for each $u$ a state in $H$ and for $J\subset\Omega_{k}$ we have a function $\xi_{u,J}$ defined by

[TABLE]

for $z$ a state in $\mathcal{\otimes}^{k}\mathbb{C}^{2}$. Now for each fixed $z$ a state in $\mathcal{\otimes}^{k}\mathbb{C}^{2}$ and $J\subset\Omega_{k}$

[TABLE]

is a frame function on $H$. Thus

[TABLE]

and the rest of the argument is clear. We also note that we can run this argument backwards using the result with $H=\mathbb{C}$ to prove the converse (second part of the theorem). ∎

Remark V.3*.*

In the above, the frame functions can sum up to an arbitrary constant $c$ on a UOB. To be a generalized version of a mixed state in quantum mechanics, as required by eq. (1), we need to set $c=1$. Further, the definition II.1 imposes the obvious inequalities on the sums in $\overset{}{(\ast)}$ that they be non-negative.

VI Conclusion

We have classified the unentangled frame functions, first for the multi-qubit system and then generally when tensor factors are of different dimensions, including separable Hilbert spaces. The proofs use the theory of Möbius functions to explicitly show the combinatorial nature of the multi-qubit UOB encountered in lsw:lduobag , and the multi-qubit unentangled frame functions quantify the result of measurements via such UOB. The structure of the frame functions thus revealed is sufficiently elegant that we surmise it points to interesting physical interpretations within the fundamentals of quantum mechanics. Indeed, qubit is the most basic quantum system, and it is common to see it as a subsystem in quantum algorithms, and it is also not unusual to use spatio-temporally separated measurements, which are by very definition, unentangled. Thus the information gleaned by such measurements falls within the context of analysis in this paper. On the other hand, if all the systems being measured have dimensions at least $3$, then the conclusion of the unentangled Gleason’s W:Gleason theorem applies, which agree with the original Gleason’s theorem. Another area where we often encounter a mix of systems is the Hydrogen atom. Its phase space is a spin $\frac{1}{2}$ space tensored with $\ell^{2}(\mathbb{R}^{3})$, so it is precisely a sub-case of the general case we discuss in Section V. These are two of the more obvious examples, but underline a need to understand the significance of unentangled measurements.

Acknowledgements.

The authors would like to thank David Meyer for productive discussions.