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Joint Measurability Meets Birkhoff-Von Neumann's Theorem

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Joint Measurability Meets Birkhoff-Von Neumann's Theorem Joint measurability meets Birkhoff-von Neumann’s theorem 1, 2, † Leonardo Guerini ∗ and Alexandre Baraviera 1International Centre for Theoretical Physics - South American Institute for Fundamental Research & Instituto de Física Teórica - UNESP, R. Dr. Bento Teobaldo Ferraz 271, São Paulo, Brazil 2Instituto de Matemática e Estatística - Univerisdade Federal do Rio Grande do Sul, Av. Bento Gonçalves 9500, Porto Alegre, Brazil Quantum measurements can be interpreted as a generalisation of probability vectors, in which non-negative real numbers are replaced by positive semi-definite operators. We extrapolate this ana- logy to define a generalisation of doubly stochastic matrices that we call doubly normalised tensors (DNTs), and formulate a corresponding version of Birkhoff-von Neumann’s theorem, which states that permutations are the extremal points of the set of doubly stochastic matrices. We prove that joint measurability arises naturally as a mathematical feature of DNTs in this context, needed to establish a characterisation similar to Birkhoff-von Neumann’s. Conversely, we also show that DNTs appear in a particular instance of a joint measurability problem, remarking the relevance of this property in general operator theory. I. INTRODUCTION normalised tensors. Following BvN’s theorem, our goal is to study the extremal points of such a set. We Quantum theory is inherently probabilistic, in the extend the analogy between real numbers and operat- sense that the result of a measurement on a quantum ors to convex combinations, and find that a necessary system cannot be predicted deterministically; we rather condition for achieving a decomposition in terms of have to cope with a probability distribution over the permutations given in terms of joint measurability [3], set of possible outcomes [1]. A quantum system is de- a property o measurements that plays a central role in scribed by a Hilbert space and its states are given by many quantum information topics, such as Bell non- density matrices, which are positive semi-definite oper- locality [4] and uncertainty relations [5]. Nevertheless, ators of unit trace. Due to this positivity and normal- we also show that not all doubly normalised tensors isation features, the density matrix can be interpreted present this property, and hence a complete description as the quantum analogue of a probability vector, from of this set and its relations to permutations remains which we extract information about the system. open. We finish by presenting yet another connection On the other hand, quantum measurements are de- to joint measurability: Starting from quantum theory scribed by a collection of positive semi-definite oper- and studying the plurality of joint measurements, we ators that sum up to the identity. Hence, the ana- see the emergence of our generalised BvN’s theorem logy between probability vectors and quantum measure- arising from this context. ments is even more direct, given the natural connections between non-negative real numbers and positive semi- definite Hermitian operators together with the associ- II. PRELIMINARIES ation of 1 to the identity operator. Namely, the latter are an operator-version of the former, obtained by increas- A probability vector of n components p is given by ing the dimension of the entries. In this work we explore this parallel, investigating the Rn p =(p1,..., pn) ; pi 0, ∑ pi = 1. (1) correspondent of standard features of probability vec- ∈ ≥ i tors and their implications in terms of quantum theory. arXiv:1809.07366v3 [quant-ph] 29 Apr 2019 More specifically, we aim at doubly stochastic matrices, We denote the set of n-component probability vectors matrices such that each column and row is a probabil- by Sn. An n n doubly stochastic matrix is a matrix n n × ity vector. The set of such objects is convex, and an D R × such that each column and each row is a prob- important characterisation of it is given by Birkhoff- ability∈ vector, von Neumann’s (BvN’s) theorem [2], which states that its extremal points are the permutation matrices. Con- p11 ... p1n sequently, every doubly stochastic matrix is a convex D = . .. ; (2a) combination of permutation matrices. . p ... p We introduce an operator-version of doubly n1 nn stochastic matrices where probabilities are substi- 1 n S c =(pi1)i,..., c =(pin)i n, (2b) tuted by quantum measurements, which we call doubly ∈ r1 =(p ) ,..., rn =(p ) S . (2c) 1j j nj j ∈ n The set of doubly stochastic matrices is convex. Among ∗ [email protected] the most important results on this topic lies Birkhoff-von † [email protected] Neumann’s theorem [2], which states that the extremal 2 points of this set are the permutation matrices. This im- points of the set of DNTs. Associating a DNT to plies that every doubly stochastic matrix D can be writ- the tuple of its row-measurements, it is knownA that ten as a convex combination of permutation matrices Πl, this tuple is extremal if and only if each measurement is extremal [7]. Thus, applying the same argument to n! the columns, is extremal if and only each row and A (i) D = ∑ qlΠl, (3) column is an extremal POVM. But since each row R (j) l=1 and column C share a common element (namely, Aij), there are more correlations in that should allow for a q = (q ) S where l l n!. Although this decomposition more refined characterisation ofA its extremality. ∈ n involves in principle ! terms, it was shown in Ref. [6] A first (naive) conjecture would be that the extremal that only (n 1)2 + 1 terms are sufficient. − points of the set of DNTs are the operator-versions of In analogy to (1), a quantum measurement of n out- permutations, such as comes acting on a Hilbert space is modelled by a H positive-operator valued measure (POVM) A, described 0 1 0 I . (8) by 1 0 7→ I 0 n A =(A ,..., An) ( ) ; A 0, ∑ A = I, (4) 1 ∈L H i ≥ i Nevertheless, convex combinations of DNTs like the i one in the right-hand side above yield DNTs where all where ( ) is the space of linear operators acting in , entries are proportional to I. Clearly this does not re- L H H cover the whole set, as for any operator 0 A I we is the partial order that define positive semi-definite ≤ ≤ operators≥ and I is the identity operator. Hence, a POVM can construct the DNT is an operator-version of a probability vector, obtained A I A by enlarging the dimension of the entries. We denote . (9) I A− A the set of n-outcome quantum measurements on by − P H n. A second attempt is to extend the correspondence to We can consider now the main object of this work. operators also to convex combinations. In other words, Definition 1. A doubly normalised tensor of positive we can consider combinations of permutation matrices semi-definite operators (DNT, for short) is a tensor in which the convex weights are associated to operat- n n A ∈ ( ) × which, in analogy to (2), each element is positive ors that form a quantum measurement, attached to each semi-definite,L H and each column and each row sums up to the term via tensor product. We formalise this idea in the identity, following way: Definition 2. Consider a set of operators Q = (Q ) P A11 ... A1n l n! that form a quantum measurement. We call ∈ = . .. (5) A . n! An1 ... Ann ∑ Π Q , (10) l ⊗ l This is to say that each row R(i) and each column C(j) of is l=1 A a quantum measurement, a decomposition into permutation tensors. C1 =(A ) ,..., Cn =(A ) P , (6a) i1 i in i ∈ n We prove now that every combination of permutation R1 =(A ) ,..., Rn =(A ) P . (6b) tensors of the form (10) is a DNT. 1j j nj j ∈ n n! Π Notice that we can write Proposition 1. If = ∑l=1 l Ql, where Q = (Ql) is a POVM and Π Bis the set of n-dimensional⊗ permutation n { l} = E A , (7) matrices, then is a DNT. ∑ i,j ij B A i,j=1 ⊗ Proof. Notice that the permutation Πl acting on the ca- where Ei,j = i j is the n n matrix with entries nonical basis can be written as[8] | ih | × Rn n eab = δa,iδb,j, that belongs to the canonical basis of × . Π = ∑ EΠ . (11) A DNT can be taken as an operator-version of doubly l l (i),i i stochastic matrices, and will be denoted by =[A ]. A ij We have III. AN OPERATOR-VERSION OF BIRKHOFF-VON = E Q (12a) NEUMANN’S THEOREM ∑ ∑ Πl (i),i l B l i ! ⊗ In view of Birkhoff-von Neumann’s theorem and the = ∑ Eab ∑ Ql. (12b) ⊗ Π decomposition (3), we turn our attention to the extremal ab l:a= l(b) 3 (i) Thus, defining µ(j R , l) = δ Π . Therefore the rows of are | i, l(j) B jointly measurable. Bab := ∑ Ql, (13) Extending the post-processing function for the l:a=Π (b) l columns as µ(i C(j), l) = δ , we can interpret (16) i,Πl(j) | (j) it satisfies = [Bab] and Bab 0. Also, for all a, b we as obtaining the i-th element of C , and hence Q is also B ≥ have a mother for the jointly measurable columns of . B ∑ Bab = ∑ Bab = ∑ Ql = I, (14) Proposition 2 establishes that a necessary condition b a l for a DNT to possess a decomposition into permutation matrices is that the row- and column-measurements are hence each row and column of is in Pn. B jointly measurable. Crucially, in the proof we see that The question left is whether tensors that admit a de- both the rows and the columns of the DNT not only can composition into permutation tensorsB are the only pos- be obtained from a single measurement, but also admit sible DNTs.
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