UNENTANGLED MEASUREMENTS AND FRAME FUNCTIONS JIRˇ´I LEBL, ASIF SHAKEEL, AND NOLAN WALLACH Abstract. Gleason’s theorem shows the equivalence of von Neumann’s density operator formalism of quantum mechanics and frame functions that arise from associating measure- 7 mentprobabilitiestovectorsonasphere. Wallach’sUnentangledGleason’sTheoremextends 1 ittotheregimeofmeasurementsbyunentangledorthogonalbases(UOB)inamulti-partite 0 system in which each subsystem is at least 3-dimensional. In this paper, we determine 2 the structure of unentangled frame functions in general. We first classify the unentangled n frame functions for multi-qubit systems, and then extend it to factors of varying dimen- a J sions including countably infinite dimensions (separable Hilbert spaces). The unentangled 1 frame functions havearemarkablecombinatorialstructure suggestingpossible fundamental 2 interpretations. ] h p - 1. Introduction t n a In von Neumann’s [1] approach to quantum mechanics, the formalism assumes a factor u algebra with an endowed trace function. The mixed states in this set up are the self-adjoint, q [ positive elements of trace 1 looked upon as defining a measure on the set of self-adjoint, 1 idempotent elements. This led Mackey to ask the natural question: Is every such measure v given by a mixed state? Gleason gave an affirmative answer to this question in the case of 9 factorsoftype I withn > 2(i.e. theboundedoperatorsonaseparable Hilbert space). Inhis 6 n 0 approach to his proof he introduced the notion of frame function (a non-negative function on 6 the pure states that sums to 1 on an orthonormal basis of the Hilbert space) which is easily 0 seen to be equivalent to that of measure on the set of self-adjoint idempotents. Gleason’s . 1 theorem shows that such a function, f, must be of the form f(v) = hv|A|vi for v any unit 0 7 norm vector in the Hilbert space and A a non-negative, self-adjoint operator of trace 1 if 1 the dimension of the Hilbert space is at least 3. In many contexts of quantum information : v theory that deal with state spaces of many independent particles, the only pure states that i X are sampled are product (or unentangled) states. This led the last named author [3] to ask if r such a sampling of only unentangled states would allow for more general frame functions. In a 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. The organization of the rest of the paper is as follows. We first introduce the idea of an unentangled frame function in Section 2. In Section 3, we choose the fundamental domain in P1(C) that we use to identify a vector with its orthogonal. In Section 4 we classify all multi-qubit frame functions. Section 5 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 6 concludes our discussion with some remarks. Date: January 24, 2017. The first author was in part supported by NSF grant DMS-1362337 and Oklahoma State University’s DIG and ASR grants. 1 2 JIRˇ´I LEBL, ASIFSHAKEEL,ANDNOLANWALLACH 2. Unentangled frame functions Let us take a brief look at unentangled Gleason setup as in [3], and make more precise the ideas from the Introduction. Let H = H ⊗H ⊗···⊗H 1 2 n with dimH ≥ 2. Technically we should be looking at the completed tensor product if two i 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 are of dimension 2 and all of the rest have dimension > 2. i Thus H = ⊗kC2 ⊗Hk+1 ⊗···⊗Hn with dimHi > 2 and if n = k then by convention the last factor is C. Itisgiventhetensor productHilbertstructure, h...|...i. Avector inHiscalledunentangled (a product vector) if it is a tensor product of unit vectors, one from each H factor. Two i such vectors v ⊗v ⊗···⊗v and w ⊗w ⊗···⊗w are orthogonal 1 2 n 1 n n hv ⊗v ⊗···⊗v |w ⊗w ⊗···⊗w i = 0 1 2 n 1 2 n if and only if there is at least one i with hv |w i = 0. A UOB {u } is a basis of H consisting i i i of orthogonal (unit norm) unentangled vectors. Let Σ be the set of all unentangled vectors in H. Definition 2.1. An unentangled frame function is a map f: Σ → R≥0, (1) such that for every UOB {u }, i Xf(ui) = 1. (2) i In the next few sections 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,...,u2n−1} of Hn consisting of orthogonal (unit norm) unentangled vectors. 3. A fundamental domain Let σ : C2 → C2 be defined by (x,y) 7−→ (−y¯,x¯). We note that hx|σxi = 0 and if x is a state then up to phase σx is the unique state perpendicular to x. The σ induces a map of P1(C) to itself which we also denote by σ. We have the standard map 1 P (C) → C∪∞ given by x (x,y) 7−→ . y We note that under this identification as a map from C∪∞ to itself 1 σz = − . z¯ In particular, on S1 it is given by z 7−→ −z. Here is a simple fundamental domain for σ: F = {z ∈ C||z| < 1}∪{z ∈ C||z| = 1,Imz > 0}∪{0}. UNENTANGLED MEASUREMENTS AND FRAME FUNCTIONS 3 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 f(a) = φ(σa) if a ∈/ F. 4. Unentangled frame functions in n qubits Let H denote n–qubit space. We choose a fundamental domain F for the map [z] → n σ([z]) = [zˆ] in one qubit. We set F = F ⊗F ⊗···⊗F (n copies). Let Ω = {1,...,n}. If n n J ⊂ Ω is a subset we define σ = T ⊗T ⊗···⊗T with T = σ if j ∈ J and T = Id, if n J 1 2 n j j j ∈/ J. We set Jc = Ω −J. We note n Σ = ∪ σ (F ) J⊂Ωn J n is a disjoint union. If z = z ⊗ ··· ⊗ z ∈ F , if J ⊂ Ω and if Jc = {i ,...,i } with 1 n n n 1 k i < i < .... < i then we set τ (z) = (z ,...,z ) if J 6= Ω , if J = Ω then use the symbol 1 2 k J i1 ik n n ω for the value. Lemma 4.1. If f is a function on the product states of H such that f sums to a fixed n constant c on all UOB’s then for each J ⊂ Ω there exists a function φ on Fn−|J| (direct n J product of n−|J| copies of F, F0 = {ω},φ (ω) = c) such that if z ∈ F then Ωn n Xf(σL(z)) = φJ(τJ(z)). L⊂J Proof. After permuting the factors we may assume that J = {1,...,j}, thus the sum on the left hand side is X f(σI(z1 ⊗···⊗zj)⊗zj+1 ⊗···⊗zn) I⊂{1,...,j} call it b. Observing that if the set Z = {(σ (z ⊗···⊗z )⊗z ⊗···⊗z |I ⊂ J} I 1 j j+1 n is extended to a UOB by adjoining elements u ,...,u with r = 2j(2n−j−1) then f(u ) = 1 r P i 1 − b no matter how we found the extension. Also Z is an orthonormal basis of ⊗jC2 ⊗ z ⊗···⊗z which only depends on z ⊗···⊗z . This proves the result. (cid:3) j+1 n j+1 n Now applying inclusion exclusion we have in the notation of the previous lemma Lemma 4.2. If z ∈ F and if f is a function on the product states satisfying the hypotheses n of Lemma 4.1 then f(σJ(z)) = X(−1)|J−L|φL(τL(z)). L⊂J Proof. Inclusion exclusion says: Let α and β be functions from the set of all subsets of Ω n to C and such that if J ⊂ Ωn then Xα(L) = β(J), L⊂J then α(J) = X(−1)|J−L|β(L). L⊂J (c.f. Rota [5]). The lemma follows from this assertion by taking α(J) = f(σ (z)) and J β(J) = φ (τ (z)). (cid:3) J J 4 JIRˇ´I LEBL, ASIFSHAKEEL,ANDNOLANWALLACH Theorem 4.3. If L $ Ωn let φL be a real valued function on Fn−|L|. Assume that φΩn(ω) = c then the function f : Hn → R defined by f(σJ(z)) = X(−1)|J−L|φL(τL(z)) L⊂J for J ⊂ Ω and z ∈ F satisfies n n 2n Xf(zi) = c. i=1 if z1,...,z2n is a UOB. Proof. We first note that if z ∈ F and J ⊂ {2,...,n} then n−1 1. f(a⊗σ (z))+f(aˆ⊗σ (z)) is independent of a. J J Indeed, f(a⊗σJ(z)) = X(−1)|J|−|L|φL(τL(a⊗z)) L⊂J and (if 1 ∈ L write L = {1}∪L′ with L′ ⊂ {2,...,n}) f(aˆ⊗σJ(z)) = f(σJ∪{1}(a⊗z)) = X (−1)|J|+1−|L|φL(τL(a⊗z)) = L⊂J∪{1} −X(−1)|J|−|L|φL(τL(a⊗z))+ X(−1)|J|−|L′|φL′∪{1}(τL′∪{1}(a⊗z)). L⊂J L′⊂J We therefore have f(a⊗σJ(z))+f(aˆ⊗σJ(z)) = X(−1)|J|−|L′|φL′∪{1}(τL′∪{1}(a⊗z)). L′⊂J This proves 1. since τL′∪{1}(a⊗z) is independent of a. We will now prove the theorem by induction on 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 fa(z) = f(a⊗ z). Then if J $ Ω = {2,...,n} and z ∈ F we have n−1 fa(σJ(z)) = X(−1)|J−L|φL(τL(a⊗z)) L⊂J and f (σ (z)) = φ (a). a Ω Ω Thus the inductive hypothesis implies that f satisfies a 2n−1 X fa(zi) = φΩ(a). i=1 for any UOB {z1,...,z2n−1} of Hn−1. We are now ready to prove the theorem. Let B = {z1,...,z2n} be a UOB. Then Theorem 6 in [3] implies that there exist a ,...,a ∈ F, V ,...,V orthogonal subspaces of H such 1 r 1 r n−1 that H = V ⊕...⊕V n−1 1 r and u ,j = 1,...,d and v ,j = 1,...,d orthonormal basis of V consisting of product vectors ij i ij i i such that B = {a ⊗u |i = 1,...,r,j = 1,...,d }∪{aˆ ⊗v |i = 1,...,r,j = 1,...,d }. i ij i i ij i UNENTANGLED MEASUREMENTS AND FRAME FUNCTIONS 5 For each i we apply the inductive hypothesis to f and find that ai di Xfai(ui,j) j=1 depends only on a and V and not on the particular orthonormal basis of V . Thus we can i i i replace u with v without changing the sum. Now i,j i,j f(a ⊗v )+f(aˆ ⊗v ) i i,j i ij is independent of a by 1. Thus we can replace all of the a with a fixed element a ∈ F i i without changing the sum. Thus the sum is given by Xf(a⊗vij)+Xf(aˆ⊗vij). ij ij We now observe that if we define g(z) = f(a⊗z)+f(aˆ⊗z) then (see the proof of 1.) g(σJ(z)) = X(−1)|J|−|L′|φL′∪{1}(τL′∪{1}(a⊗z)). L′⊂J and Xg(σJ(z)) = c J⊂Ω for all z ∈ F . Finally we can apply the inductive hypothesis to replace the basis {v } by n−1 ij {σ (z)|J ⊂ Ω} with z ∈ F and the theorem is proved. (cid:3) J n−1 Corollary 4.4. 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 Xf(zi) = c then the same is true for every UOB. Proof. The characterization of the generic UOB in [4] makes it clear that if f sums to c on all generic UOB then f and the functions φ have the property in Lemma 4.1. Lemma 4.2 J (cid:3) and Theorem 4.3 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. 5. General unentangled frame functions In this section we will give a complete description of unentangled frame functions for separable Hilbert spaces of the form H = H ⊗H ⊗···⊗H 1 2 n with dimH ≥ 2. i Let f be an unentangled frame function on H. Thus f is a real valued function product states such that there exists a scalar c such that if {u } is a UOB then f(u ) = c. i P i 6 JIRˇ´I LEBL, ASIFSHAKEEL,ANDNOLANWALLACH If z is a product state in ⊗kC2 then setting fz(x) = f(z ⊗ x) for x a product state in H ⊗···⊗H , f is an unentangled frame function on H ⊗···⊗H with k+1 r z k+1 r Xfz(ui) = c(z) for {u } a UOB of H ⊗···⊗H (so c(z) = f(z ⊗u ) depends only on z and f). The i k+1 r Pi i unentangled Gleason theorem [3] implies that there exists A(z) a trace class, self-adjoint non-negative operator on H ⊗···⊗H such that k+1 r 1. trA(z) = c(z). 2. f (u) = hu|A(z)|ui. z Note that the proof of the unentangled Gleason theorem given in [3] does not use finite dimensionality and therefore applies to the context of this paper. This proves the following reduction of the problem. Lemma 5.1. Let H = ⊗kC2 ⊗ Hk+1 ⊗ ···⊗ Hr be as above. Then an unentangled frame function forH is the restriction of one for ⊗kC2⊗H with H the completionof Hk+1⊗···⊗Hr. In light of this lemma we need only classify the unentangled frame functions on Hilbert spaces of the form ⊗kC2⊗H with H a separable Hilbert space of whose dimension is not 2. We note that if dimH = 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 5.2. Let for each J ⊂ {1,...,k}, φ : Fk−|J| → T (H) (here as usual, we set J F0 = {ω}). Let for z ∈ Fk and let u be a state in H f(σJ(z)⊗u) = X(−1)|L−J|hu|φL(τL(z))|ui. (∗) L⊂J Let c = trφ (ω). Ωk If {wj} is a UOB of ⊗kC2 ⊗H then Xf(wj) = c. If f is an unentangled frame function on ⊗kC2 ⊗H and if J ⊂ Ωk we set for z ∈ Fk and u a state in H γJ(u) = Xf(σL(z)⊗u), L⊂J then γ (u) = hu|φ (τ (z))|ui with φ : Fk−|J| → T (H) and f is given by (∗). J J J J Proof. If f is an unentangled frame function on ⊗kC2⊗H and if z is a state in ⊗kC2 then, as above, f (u) = f(z ⊗ u) is a frame function on H. Thus there exists a trace class self z adjoint operator on H, α(z), such that f (u) = hu|α(z)|ui for u a state in H. Similarly we z see that for fixed u a state in H, z 7−→ hu|α(z)|ui defines an unentangled frame function on ⊗kC2. Thus there exist for each u a state in H and for J ⊂ Ω we have a function ξ defined by k u,J ξu,J(τJz) = Xhu|α(σLz)|ui L⊂J UNENTANGLED MEASUREMENTS AND FRAME FUNCTIONS 7 for z a state in ⊗kC2. Now for each fixed z a state in ⊗kC2 and J ⊂ Ωk u 7−→ ξ (τ z) u,J J is a frame function on H. Thus ξ (τ z) = hu|φ (τ z)|ui u,J J J J 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). (cid:3) Remark 5.3. 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. (2), we need to set c = 1. Further, the definition in eq. (2.1) imposes the obvious inequalities on the sums in (∗) that they be non-negative. 6. 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 M¨obius functions to explicitly show the combinatorial nature of the multi-qubit UOB encountered in [4], 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, andit iscommon tosee it asasubsystem inquantum algorithms, and it is also not unusual to use spatio-temporally separated measurements, which are by very definition, unentangled. Thus the informationgleanedby such measurements fallswithin 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 agree with the original Gleason’s theorem. Another area where we often encounter a mixofsystems istheHydrogenatom. Itsphasespaceisaspinbundleofdimension2tensored with ℓ2(R3), so it is precisely a sub-case of the general case we discuss in Section 5. 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. References [1] J. von Neumann, Mathematische Grundlagen der Quantenmechanik, J. Springer, Berlin, 1932. [2] A. M. Gleason, Measures on the closed subspaces of a Hilbert space, Journal of Mathematics and Me- chanics 6 (1957), no. 6, 885–893. [3] N. R. Wallach, An unentangled Gleason’s theorem, Contemporay Mathematics 305 (2002), 291–298. [4] J. Lebl, A. Shakeel, and N. Wallach, Local distinguishability of generic unentangled orthonormal bases, Physical Review A 93 (January 2016), no. 1, 012330/1–6. [5] G. Rota, On the foundations of combinatorial theory. I. Theory of Mbius functions, Zeitschrift fr Wahrscheinlichkeitstheorie und verwandte Gebiete 2 (1963), 340–368. 8 JIRˇ´I LEBL, ASIFSHAKEEL,ANDNOLANWALLACH Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA E-mail address: [email protected] E-mail address: [email protected] Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0112, USA E-mail address: [email protected]