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.

In von Neumann’s1 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? Gleason2 gave an affirmative answer to this question in the case of factors of type In 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 theorem2 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 author3 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.

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

H=H1H2Hn

with dim Hi ≥ 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 Hi are of dimension 2 and all of the rest have dimensions > 2. Thus

H=kC2Hk+1Hn

with dim Hi > 2 and if n = k, then by convention the last factor is C. H is given the tensor product Hilbert structure, |. A vector in H is called unentangled (a product vector) if it is a tensor product of unit vectors, one from each Hi factor. Two such vectors v1v2 ⊗ ⋯ ⊗ vn and w1wn ⊗ ⋯ ⊗ wn are orthogonal,

v1v2vn|w1w2wn=0,

if and only if there is at least one i with vi|wi=0. An Unentangled Orthonormal Basis (UOB) {ui} 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.
An unentangled frame function is a map
f:ΣR0
such that for every UOB {ui},
if(ui)=1.
(1)

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=nC2 be the space of n qubits. A UOB is then a basis {u0,u1,,u2n1} of Hn consisting of orthogonal (unit norm) unentangled vectors.

Let σ:C2C2 be defined by v=(x,y)v^=(ȳ,x¯). We note that v|σv=0 and if v is a state, then up to phase σv 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

P1(C)C

given by

(x,y)xy.

We note that under this identification as a map from C to itself,

σz=1z¯.

In particular, on S,1 it is given by zz.

Here is a simple fundamental domain for σ,

F={zC|z|<1}{zC|z|=1,Im z>0}{1}.

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 aF, and since the sum of the values of f over orthogonal vectors is 1, f(a) = 1 − ϕ(σa) if aF.

Let Hn denote n-qubit space. We choose a fundamental domain F for the map zσ(z) = z^ in one qubit. We set Fn = FF ⊗ ⋯ ⊗ F (n copies). Let Ωn = {1, …, n}. If J ⊂ Ωn is a subset, we define σJ = T1T2 ⊗ ⋯ ⊗ Tn with Tj = σ if jJ and Tj=I (identity operator) if jJ. We set Jc = ΩnJ. We note

Σ=JΩnσJ(Fn)

is a disjoint union. If z = z1 ⊗ ⋯ ⊗ znFn, if J ⊂ Ωn, and if Jc = {i1, …, ik} with i1 < i2 < ⋯ < ik, then we set τJ(z)=(zi1,,zik) if J ≠ Ωn; if J = Ωn, then use the symbol ω for the value.

Lemma IV.1.
If f is a function on the product states ofHnsuch that f sums to a fixed constant c on all UOB’s, then for each J ⊂ Ωnthere exists a function ϕJon Fn−|J|(direct product of n − |J| copies of F, F0={ω},ϕΩn(ω)=c) such that if zFn, then
LJf(σL(z))=ϕJ(τJ(z)).

Proof.
After permuting the factors, we may assume that J = {1, …, j}; thus, the sum on the left-hand side is
I{1,,j}f(σI(z1zjzj+1zn)),
which we call b, and note that σI acts only on z1 ⊗ ⋯ ⊗ zj as σ and as I everywhere else. Observing that if the set
Z={σI(z1zjzj+1zn)IJ}
is extended to a UOB by adjoining elements u1, …, ur with r = 2j(2nj − 1), then ∑f(ui) = 1 − b no matter how we found the extension. Also Z is an orthonormal basis of jC2zj+1zn which only depends on zj+1 ⊗ ⋯ ⊗ zn. This proves the result.

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

Lemma IV.2.
If zFnand if f is a function on the product states satisfying the hypotheses of Lemma IV.1, then
f(σJ(z))=LJ(1)|JL|ϕL(τL(z)).

Proof.
Inclusion exclusion says: Let α and β be functions from the set of all subsets of Ωn to C and such that if J ⊂ Ωn, then
LJα(L)=β(J),
then
α(J)=LJ(1)|JL|β(L)
(cf. Ref. 5). The lemma follows from this assertion by taking α(J) = f(σJ(z)) and β(J) = ϕJ(τJ(z)).

Theorem IV.3.
IfLΩn, let ϕLbe a real-valued function onFnL. Assume thatϕΩn(ω)=c; then, the functionf:HnRdefined by
f(σJ(z))=LJ(1)JLϕL(τL(z))
for J ⊂ Ωnand zFnsatisfies
i=12nf(zi)=c,
ifz1,,z2nis a UOB.

Proof.

We first note that if zFn−1 and J ⊂ {2, …, n}, then

  1. f(aσJ(z)) + f(âσJ(z)) is independent of a.

Indeed,
f(aσJ(z))=LJ(1)J||LϕL(τL(az))
and (if 1 ∈ L, write L = {1} ∪ L′ with L′ ⊂ {2, …, n})
f(âσJ(z))=f(σJ{1}(az))=LJ{1}(1)|J|+1|L|ϕL(τL(az))
=LJ(1)J||LϕL(τL(az))+LJ(1)|J||L|ϕL{1}(τL{1}(az)).
We therefore have
f(aσJ(z))+f(âσJ(z))=LJ(1)|J||L|ϕL{1}(τL{1}(az)).
This proves (1) since τL′∪{1}(az) is independent of a.
We will now prove the theorem by induction on n. If n = 1, the assertion is that if aF, then f(a) + cf(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 aF, then define fa(z) = f(az). Then if JΩ={2,,n} and zFn−1, we have
fa(σJ(z))=LJ(1)JLϕL(τL(az))
and
fa(σΩ(z))=ϕΩ(a).
Thus the inductive hypothesis implies that fa satisfies
i=12n1fa(zi)=ϕΩ(a)
for any UOB {z1,,z2n1} of Hn1.
We are now ready to prove the theorem. Let B={z1,,z2n} be a UOB. Then Theorem 6 in Ref. 3 implies that there exist a1, …, arF, V1, …, Vr orthogonal subspaces of Hn1 such that
Hn1=V1Vr
and uij and vij, j = 1, …, di are the orthonormal basis of Vi consisting of product vectors such that
B={aiuiji=1,,r,j=1,,di}{âiviji=1,,r,j=1,,di}.
For each i, we apply the inductive hypothesis to fai and find that
j=1difai(ui,j)
depends only on ai and Vi and not on the particular orthonormal basis of Vi. Thus we can replace ui,j with vi,j without changing the sum. Now
f(aivi,j)+f(âivij)
is independent of aiby 1. Thus we can replace all of the ai with a fixed element aF without changing the sum. Thus the sum is given by
ijf(avij)+ijf(âvij).
We now observe that if we define
g(z)=f(az)+f(âz),
then [see the Proof of (1)]
g(σJ(z))=LJ(1)|J||L|ϕL{1}(τL{1}(az))
and
JΩg(σJ(z))=c,
for all zFn−1. Finally we can apply the inductive hypothesis to replace the basis {vij} by {σJ(z)∣J ⊂ Ω} with zFn−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.
B={aB1,âB2},

for an arbitrary aF, 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,,z2n}, we have
f(zi)=c,
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 ϕJ 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.

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

H=H1H2Hn

with dim Hi ≥ 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 {ui} is a UOB, then ∑ f(ui) = c.

If z is a product state in kC2, then setting fz(x) = f(zx) for x, a product state in Hk+1 ⊗ ⋯ ⊗ Hr, fz is an unentangled frame function on Hk+1 ⊗ ⋯ ⊗ Hr with

fz(ui)=c(z)

for {ui}, a UOB of Hk+1 ⊗ ⋯ ⊗ Hr (so c(z) = if(zui) depends only on z and f). The unentangled Gleason’s theorem3 implies that there exists A(z) a trace class self-adjoint non-negative operator on Hk+1 ⊗ ⋯ ⊗ Hr such that

  1. trA(z) = c(z).

  2. 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.

LetH=kC2Hk+1Hrbe as above. Then an unentangled frame function forHis the restriction of one forkC2Hwith H being the completion of Hk+1 ⊗ ⋯ ⊗ Hr.

In light of this lemma, we need to only classify the unentangled frame functions on Hilbert spaces of the form kC2H 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.

Theorem V.2.
Let for each J ⊂ {1, …, k}, ϕJ:Fk|J|T(H)(here as usual, we set F0 = {ω}). Let zFkand let u be a state in H,
f(σJ(z)u)=LJ(1)|LJ|u|ϕL(τL(z))|u.  (*)
Let
c=trϕΩk(ω).
If {wj} is a UOB ofkC2H, then
f(wj)=c.
If f is an unentangled frame function onkC2Hand if J ⊂ Ωk, we set for zFkand u a state in H
γJ,z(u)=LJf(σL(z)u),
thenγJ,z(u)=u|ϕJ(τJ(z))|uwithϕJ:Fk|J|T(H)and f is given by (*).

Proof.
If f is an unentangled frame function on kC2H and if z is a state in kC2, then, as above, fz(u) = f(zu) is a frame function on H. Thus there exists a trace class self-adjoint operator on H, α(z), such that fz(u)=u|α(z)|u for u a state in H. Similarly we see that for fixed u, a state in H,
z,u|α(z)|u
defines an unentangled frame function on kC2. Thus there exist for each u a state in H, and for J ⊂ Ωk, a function ξu,J defined by
ξu,J(τJz)=LJu|α(σLz)|u,
and for z, a state in kC2. Now for each fixed z, a state in kC2 and J ⊂ Ωk,
uξu,J(τJz)
is a frame function on H. Thus
ξu,J(τJz)=u|ϕJ(τJz)|u
and the rest of the argument is clear. We also note that we can run this argument backwards using the result with H=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. Furthermore, Definition II.1 imposes the obvious inequalities on the sums in (*) that they be non-negative.

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’s3 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 2(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.

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.

1.
J.
von Neumann
,
Mathematische Grundlagen der Quantenmechanik
(
Springer
,
Berlin
,
1932
).
2.
A. M.
Gleason
, “
Measures on the closed subspaces of a Hilbert space
,”
J. Math. Mech.
6
(
6
),
885
893
(
1957
).
3.
N. R.
Wallach
, “
An unentangled Gleason’s theorem
,”
Contemp. Math.
305
,
291
298
(
2002
).
4.
J.
Hamhalter
,
Quantum Measure Theory
(
Kluwer Academic Publishers
,
Dordrecht, Boston
,
2003
).
5.
G.
Rota
, “
On the foundations of combinatorial theory. I. Theory of Möbius functions
,”
Z. Wahrscheinlichkeitstheor. Geb.
2
,
340
368
(
1963
).
6.
J.
Lebl
,
A.
Shakeel
, and
N.
Wallach
, “
Local distinguishability of generic unentangled orthonormal bases
,”
Phys. Rev. A
93
,
012330-1
012330-6
(
2016
).