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.

## I. INTRODUCTION

In von Neumann’s^{1} approach to quantum mechanics, the formalism assumes a factor algebra with an endowed trace function. The mixed states in this setup 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^{2} gave an affirmative answer to this question in the case of factors of type 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 the 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 the measure on the set of self-adjoint idempotents. Gleason’s theorem^{2} shows that such a function, *f*, must be of the form $f(v)=v|A|v$ where $v$ is any unit norm vector in the Hilbert space and *A* is 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 deals 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^{3} 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 dimensions (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 Gleason’s theorem and its variants, is in Ref. 4.

The organization of the rest of the paper is as follows. We first introduce the idea of an *unentangled frame function* in Sec. II. In Sec. III, we choose the fundamental domain in $P1(C)$ that we use to identify a vector with its orthogonal. In Sec. 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 the unentangled Gleason setup as in Ref. 3 and make more precise the ideas from the Introduction. Let

with dim *H*_{i} ≥ 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 dimensions > 2. Thus

with dim *H*_{i} > 2 and if *n* = *k*, then by convention the last factor is $C$. $H$ is given the tensor product Hilbert structure, $\u2026|\u2026$. A vector in $H$ is called unentangled (a product vector) if it is a tensor product of unit vectors, one from each *H*_{i} factor. Two such vectors $v1$ ⊗ $v2$ ⊗ ⋯ ⊗ $vn$ and $w1$ ⊗ $wn$ ⊗ ⋯ ⊗ $wn$ are orthogonal,

if and only if there is at least one *i* with $vi|wi=0$. An Unentangled Orthonormal Basis (UOB) {*u*_{i}} is a basis of $H$ consisting of orthogonal (unit norm) unentangled vectors. Let Σ be the set of all unentangled vectors in $H$.

*Definition II.1*.

*u*

_{i}},

In Secs. III and IV, we will be dealing with the multi-qubit case, so we specialize the above to prepare for it. Let $Hn=\u2297nC2$ be the space of *n* qubits. A UOB is then a basis ${u0,u1,\u2026,u2n\u22121}$ of $Hn$ consisting of orthogonal (unit norm) unentangled vectors.

## III. A FUNDAMENTAL DOMAIN

Let $\sigma :C2\u2192C2$ be defined by $v=(x,y)\u27fcv^=(\u2212\u0233,x\xaf)$. We note that $v|\sigma v=0$ and if $v$ is a state, then up to phase *σ*$v$ is the unique state perpendicular to $v$. The *σ* induces a map of $P1(C)$ to itself which we also denote by *σ*. We have the standard map

given by

We note that under this identification as a map from $C\u222a\u221e$ to itself,

In particular, on *S*,^{1} it is given by $z\u27fc\u2212z$.

Here is a simple fundamental domain for *σ*,

A frame function *f* in a single qubit space corresponds to an arbitrary function *ϕ*: *F* → [0, 1] by letting *f*(*a*) = *ϕ*(*a*) if *a* ∈ *F*, and since the sum of the values of *f* over orthogonal vectors is 1, *f*(*a*) = 1 − *ϕ*(*σa*) if *a* ∉ *F*.

## IV. UNENTANGLED FRAME FUNCTIONS IN *n* QUBITS

Let $Hn$ denote *n*-qubit space. We choose a fundamental domain *F* for the map *z* → *σ*(*z*) = $z^$ in one qubit. We set *F*_{n} = *F* ⊗ *F* ⊗ ⋯ ⊗ *F* (*n* copies). Let Ω_{n} = {1, …, *n*}. If *J* ⊂ Ω_{n} is a subset, we define *σ*_{J} = *T*_{1} ⊗ *T*_{2} ⊗ ⋯ ⊗ *T*_{n} with *T*_{j} = *σ* if *j* ∈ *J* and $Tj=I$ (identity operator) if *j* ∉ *J*. We set *J*^{c} = Ω_{n} − *J*. We note

is a disjoint union. If *z* = *z*_{1} ⊗ ⋯ ⊗ *z*_{n} ∈ *F*_{n}, if *J* ⊂ Ω_{n}, and if *J*^{c} = {*i*_{1}, …, *i*_{k}} with *i*_{1} < *i*_{2} < ⋯ < *i*_{k}, then we set $\tau J(z)=(zi1,\u2026,zik)$ if *J* ≠ Ω_{n}; if *J* = Ω_{n}, then use the symbol *ω* for the value.

*Lemma IV.1*.

*If f is a function on the product states of*$Hn$

*such that f sums to a fixed constant c on all UOB’s, then for each J*⊂ Ω

_{n}

*there exists a function ϕ*

_{J}

*on F*

^{n−|J|}

*(direct product of n*− |

*J*|

*copies of F*, $F0={\omega},\varphi \Omega n(\omega )=c$

*) such that if z*∈

*F*

_{n},

*then*

*Proof*.

*J*= {1, …,

*j*}; thus, the sum on the left-hand side is

*b*, and note that

*σ*

_{I}acts only on

*z*

_{1}⊗ ⋯ ⊗

*z*

_{j}as

*σ*and as $I$ everywhere else. Observing that if the set

*UOB*by adjoining elements

*u*

_{1}, …,

*u*

_{r}with

*r*= 2

^{j}(2

^{n−j}− 1), then

*∑f*(

*u*

_{i}) = 1 −

*b*no matter how we found the extension. Also

*Z*is an orthonormal basis of $\u2297jC2\u2297zj+1\u2297\cdots \u2297zn$ which only depends on

*z*

_{j+1}⊗ ⋯ ⊗

*z*

_{n}. This proves the result.

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

*Lemma IV.2*.

*If z*∈

*F*

_{n}

*and if f is a function on the product states satisfying the hypotheses of Lemma IV.1, then*

*Proof*.

*α*and

*β*be functions from the set of all subsets of Ω

_{n}to $C$ and such that if

*J*⊂ Ω

_{n}, then

*α*(

*J*) =

*f*(

*σ*

_{J}(

*z*)) and

*β*(

*J*) =

*ϕ*

_{J}(

*τ*

_{J}(

*z*)).

*If*$L\u2acb\Omega n$,

*let ϕ*

_{L}

*be a real-valued function on*$Fn\u2212L$.

*Assume that*$\varphi \Omega n(\omega )=c$

*; then, the function*$f:Hn\u2192R$

*defined by*

*for J*⊂ Ω

_{n}

*and z*∈

*F*

_{n}

*satisfies*

*if*$z1,\u2026,z2n$

*is a UOB*.

*Proof*.

We first note that if *z* ∈ *F*_{n−1} and *J* ⊂ {2, …, *n*}, then

*f*(*a*⊗*σ*_{J}(*z*)) +*f*(*â*⊗*σ*_{J}(*z*)) is independent of*a*.

*L*, write

*L*= {1} ∪

*L*′ with

*L*′ ⊂ {2, …,

*n*})

*τ*

_{L′∪{1}}(

*a*⊗

*z*) is independent of

*a*.

*n*. If

*n*= 1, the assertion is that if

*a*∈

*F*, then

*f*(

*a*) +

*c*−

*f*(

*a*) =

*c*. So the result is true for

*n*= 1. Now assume the result for

*n*− 1 ≥ 1. We now prove it for

*n*. If

*a*∈

*F*, then define

*f*

_{a}(

*z*) =

*f*(

*a*⊗

*z*). Then if $J\u2acb\Omega ={2,\u2026,n}$ and

*z*∈

*F*

_{n−1}, we have

*f*

_{a}satisfies

*a*

_{1}, …,

*a*

_{r}∈

*F*,

*V*

_{1}, …,

*V*

_{r}orthogonal subspaces of $Hn\u22121$ such that

*u*

_{ij}and $vij$,

*j*= 1, …,

*d*

_{i}are the orthonormal basis of

*V*

_{i}consisting of product vectors such that

*i*, we apply the inductive hypothesis to $fai$ and find that

*a*

_{i}and

*V*

_{i}and not on the particular orthonormal basis of

*V*

_{i}. Thus we can replace

*u*

_{i,j}with $vi,j$ without changing the sum. Now

*a*

_{i}

*by*1. Thus we can replace all of the

*a*

_{i}with a fixed element

*a*∈

*F*without changing the sum. Thus the sum is given by

*z*∈

*F*

_{n−1}. Finally we can apply the inductive hypothesis to replace the basis {$vij$} by {

*σ*

_{J}(

*z*)∣

*J*⊂ Ω} with

*z*∈

*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 in Ref. 6. A generic UOB in this component is recursively defined as follows.

*Definition IV.4*.

for an arbitrary *a* ∈ *F*, and where $Bi,i=1,2$ are again defined in the same manner as $B$ 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*, ${z1,\u2026,z2n}$,

*we have*

*then the same is true for every UOB*.

*Proof*.

The characterization of the generic UOB in Ref. 6 makes it clear that if *f* sums to *c* on all generic UOB, then *f* and the functions *ϕ*_{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* ≥ 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

with dim *H*_{i} ≥ 2.

Let *f* be an unentangled frame function on $H$. Thus *f* is a real-valued function product state such that there exists a scalar *c* such that if {*u*_{i}} is a *UOB*, then *∑ f*(*u*_{i}) = *c*.

If *z* is a product state in $\u2297kC2$, then setting *f*_{z}(*x*) = *f*(*z* ⊗ *x*) for *x*, a product state in *H*_{k+1} ⊗ ⋯ ⊗ *H*_{r}, *f*_{z} is an unentangled frame function on *H*_{k+1} ⊗ ⋯ ⊗ *H*_{r} with

for {*u*_{i}}, a *UOB* of *H*_{k+1} ⊗ ⋯ ⊗ *H*_{r} (so *c*(*z*) = *∑*_{i}*f*(*z* ⊗ *u*_{i}) depends only on *z* and *f*). The unentangled Gleason’s theorem^{3} implies that there exists *A*(*z*) a trace class self-adjoint non-negative operator on *H*_{k+1} ⊗ ⋯ ⊗ *H*_{r} such that

tr

*A*(*z*) =*c*(*z*).$fz(u)=u|A(z)|u$.

Note that the proof of the unentangled Gleason theorem given in Ref. 3 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* $H=\u2297kC2\u2297Hk+1\u2297\cdots \u2297Hr$ *be as above*. *Then an unentangled frame function for* $H$ *is the restriction of one for* $\u2297kC2\u2297H$ *with H* being *the completion of H*_{k+1} ⊗ ⋯ ⊗ *H*_{r}.

In light of this lemma, we need to only classify the unentangled frame functions on Hilbert spaces of the form $\u2297kC2\u2297H$ with *H* being a separable Hilbert space whose dimension is not 2. We note that if dim *H* = 1, then a self-adjoint operator on *H* is a real scalar. Let $T(H)$ be the space of trace class self-adjoint operators on *H*.

*Let for each J*⊂ {1, …,

*k*}, $\varphi J:Fk\u2212|J|\u2192T(H)$

*(here as usual*,

*we set F*

^{0}= {

*ω*}

*)*.

*Let z*∈

*F*

^{k}

*and let u be a state in H*,

*Let*

*If*{$wj$}

*is a UOB of*$\u2297kC2\u2297H$,

*then*

*If f is an unentangled frame function on*$\u2297kC2\u2297H$

*and if J*⊂ Ω

_{k},

*we set for z*∈

*F*

^{k}

*and u a state in H*

*then*$\gamma J,z(u)=u|\varphi J(\tau J(z))|u$

*with*$\varphi J:Fk\u2212|J|\u2192T(H)$

*and f is given by*(*).

*Proof*.

*f*is an unentangled frame function on $\u2297kC2\u2297H$ and if

*z*is a state in $\u2297kC2$, then, as above,

*f*

_{z}(

*u*) =

*f*(

*z*⊗

*u*) is a frame function on

*H*. Thus there exists a trace class self-adjoint operator on

*H*,

*α*(

*z*), such that $fz(u)=u|\alpha (z)|u$ for

*u*a state in

*H*. Similarly we see that for fixed

*u*, a state in

*H*,

*u*a state in

*H*, and for

*J*⊂ Ω

_{k}, a function

*ξ*

_{u,J}defined by

*z*, a state in $\u2297kC2$. Now for each fixed

*z*, a state in $\u2297kC2$ and

*J*⊂ Ω

_{k},

*H*. Thus

*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. Furthermore, Definition II.1 imposes the obvious inequalities on the sums in (*) 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 Ref. 6, and the multi-qubit unentangled frame functions quantify the result of measurements via such a 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^{3} theorem applies, which agrees 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 $12$ space tensored with $\u21132(R3)$, so it is precisely a sub-case of the general case we discussed in Sec. V. These are two of the more obvious examples but underline a need to understand the significance of unentangled measurements.

## ACKNOWLEDGMENTS

The authors would like to thank David Meyer for productive discussions. The first author was supported in part by NSF Grant No. DMS-1362337 and Oklahoma State University DIG and ASR grants.