arXiv:1809.07366v3 [quant-ph] 29 Apr 2019 hc eetatifrainaottesystem. the about information extract we which o emn’ BNs hoe [ theorem (BvN’s) Neumann’s von aet oewt rbblt itiuinoe the over distribution [ probability outcomes possible a of with quantum set cope a to rather on we measurement have deterministically; predicted a be of cannot result system the that sense t etr h e fsc bet scne,adan Birkhoff- and by convex, given is is objects it such of of probabil- a characterisation set is The important row and vector. column ity each matrices, that stochastic such theory. doubly matrices quantum at aim of we terms specifically, in vec- More implications probability their of and features tors standard of correspondent entries. increas- the by of dimension obtained the former, are ing the latter the of Namely, operator-version operator. associ- identity an the the to with 1 of together ation semi- operators ana- positive Hermitian the and definite numbers Hence, real connections non-negative natural the between given identity. direct, more even the is measure- ments to quantum oper- and vectors up semi-definite probability between logy positive sum of that collection ators a by scribed steqatmaaou fapoaiiyvco,from vector, probability a interpreted normal- of be analogue and can quantum positivity matrix the this density as to the Due by features, given isation trace. are unit oper- states of semi-definite its positive ators are and which space matrices, Hilbert density a by scribed tcatcmtie hr rbblte r substi- are doubly call probabilities we which measurements, where quantum by tuted matrices stochastic convex a is matrices. permutation matrix Con- of stochastic combination matrices. doubly permutation every the sequently, are points extremal its ∗ † [email protected] [email protected] unu hoyi neetypoaiitc nthe in probabilistic, inherently is theory Quantum nti okw xlr hsprle,ivsiaigthe investigating parallel, this explore we work this In de- are measurements quantum hand, other the On eitouea prtrvrino doubly of operator-version an introduce We 2 nttt eMtmtc saítc nvrsaeFeder Univerisdade - Estatística e Matemática de Instituto DT) n omlt orsodn eso fBirkhoff- of version corresponding a formulate and (DNTs), napriua ntneo on esrblt rbe,r problem, measurability joint theory. operator a general of Con instance particular Neumann’s. a Birkhoff-von in to o similar feature characterisation mathematical doubl a a of as set naturally the arises of measurability points extremal the are permutations that oyt en eeaiaino obysohsi matrice stochastic doubly of generalisation a semi-de define positive to by logy replaced are numbers real non-negative unu esrmnscnb nepee sageneralisatio a as interpreted be can measurements Quantum 1 nentoa etefrTertclPyis-SuhAmeri South - Physics Theoretical for Centre International .INTRODUCTION I. nttt eFsc erc NS,R r et ebloF Teobaldo Bento Dr. R. 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R . c r components n n n p p ; ( = ( = nn 1 . . . p n i    ≥ p p (2a) ; nj in 0, ) ) i j 5 ∑ ∈ ∈ i .Nevertheless, ]. p S S p n sgvnby given is n i (2b) , (2c) . = ,Brazil e, .(1) 1. 3 ], 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 AI 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. To address this question we need to intro- symmetric post-processing functions, in the sense that duce the well-known notion of joint measurability [3]. µ(j R(i), l) = µ(i C(j), l). We now show that these con- | | Definition 3. A set of m quantum measurements ditions are also sufficient for ensuring such decomposi- (1) (m) tion. A ,..., A Pn is said to be jointly measurable if { } ⊂ M there exists a so-called mother measurement , from which Proposition 3. Let = [Aij] be an n n DNT and = we can recover each measurement of the set by post-processing A × R R(i) =(A ) , = C(j) = (A ) its sets of row- and it, i.e., ij j i ij i j column-measurements,{ } C satisfying{ } (i) A(i) Aj = ∑ µ(j , k)Mk. (15) (i) is jointly measurable; and k | R∪C (ii) the post-processing map is symmetric, µ(j R(i), k) = where µ is a probability distribution conditioned on the meas- | (i) µ(i C(j), k), for all i, j, k. urement A we wish to obtain and on the operator Mk of M, | and therefore satisfies µ(j A(i), k) 0 and ∑ µ(j A(i), k) = P | ≥ j | Then there exists a coefficient-measurement Q n! such 1, i, k. ∈ ∀ that This expresses the fact that for any given quantum sys- Π (i) = ∑ l Ql. (17) tem, one can determine an outcome for each A by per- A l ⊗ forming M and, depending on the outcome k obtained, flip a coin µ( A(i), k). Importantly, the joint measurabil- Proof. Let M be a mother measurement from which the ity of a finite·| set of d-dimensional measurements can be symmetric post-processing map µ yields computationally decided in an efficient way by means (i) (i) (j) (j) of semi-definite programming (SDP) [9]. Aij = Rj = ∑ µ(j R , k)Mk = ∑ µ(i C , k)Mk = Ci . We can now present the following proposition, which k | k | shows that any combination of permutation tensors has (18) jointly measurable rows and columns. Notice that any asymmetric µ satisfies the relation above, but the assumed symmetry says that the two n! Π Proposition 2. Let = ∑l=1 l Ql be a combina- sums in (18) match term by term. This ensures that tion of permutation tensors.B Then ⊗is a DNT, and the B set R(1),..., R(n), C(1),..., C(n) of all row- and column- ∑ µ(j R(i), k) = ∑ µ(i C(j), k) = 1, j. (19) | | ∀ measurements{ is jointly measurable.} i i Proof. is a DNT for Proposition 1, hence we need only Also, by definition, for all k we have to proveB that its rows/columns are both jointly measur- able. ∑ µ(j R(i), k) = 1, i. (20) | ∀ Defining the operators Bab as in (13), the row- j (i) measurements are given by R =(Bij)j. Then the coef- (i) ficient measurement Q is also a mother measurement for Hence, for each k we see that the matrix (µ(j R , k))ij = (i) | R , since for any i, j ∑ µ(j R(i), k)E is doubly stochastic, which according { } ij | ij to Birkhoff-von Neumann’s theorem has a decomposi- B = ∑ Q = ∑ δ Π Q , (16) ij l i, l(j) l tion into permutation matrices, l:i=Πl(j) l n so we can recover the j-th element of the row- (i) (k) ∑ µ(j R , k)Eij = ∑ r Πl, (21) (i) l measurement R by post-processing Q with i,j=1 | l 4

(k) (k) S for some mother measurement M = (M ) having lin- where r =(rl )l n! is a probability vector, for any k k. ∈ early independent elements. Since the columns of A Therefore, using (18)and (21) we obtain also sum to the identity, we have

= ∑ Eij Aij (22a) A ⊗ (i) ij I = ∑ Aij = ∑ ∑ µ(j R , k) Mk. (25) i k i | ! = ∑ E ∑ µ(j R(i), k)M (22b) ij k ∑ ij ⊗ k | ! Since k Mk = I by the normalisation of quantum meas- urements, linear independence implies that (i) = ∑ ∑ µ(j R , k)Eij Mk (22c) (i) k ij | ! ⊗ ∑ µ(j R , l) = 1. (26) i | ( ) = ∑ ∑ r k Π M (22d) l l ⊗ k Thus, the extended post-processing map µ(i C(j), k) := k l ! | µ(j R(i), k) (initially conditioned only on the rows) is | = Π (k) well defined and yields the column-measurements of , ∑ l ∑ rl Mk . (22e) A l ⊗ k ! which therefore are jointly measurable with . Then M and µ satisfy conditions (i) (ii) and we canR apply Prop. ( ) Thus, defining Q := ∑ r k M we see that Q 0 3. − l k l k l ≥ and ∑l Ql = I. Therefore, Q = (Ql) is the coefficient- measurement that concludes the proof. IV. RELAXING JOINT MEASURABILITY Notice that if the response functions µ(j R(i), k) are deterministic, say, probability measures whose| support In spite of characterising the set of DNTs that are contains a unique point (as the ones in the proof of Prop. decomposable into permutation tensors, Theorem 4 2), then the left-hand side of (21) is already a permuta- does not describe general DNTs. For that, we would tion matrix and the coefficient measurement Q equals need to show that all DNTs have jointly measurable the mother measurement M. row-measurements, besides the condition on the post- Putting together Propositions 2 and 3 we obtain the processing map. Perhaps surprisingly, the next example following characterisation. shows that this is not the case, and not even the strong Theorem 4. A doubly normalised tensor of positive semi- correlations between the rows of a DNT (given by the definite operators admits a decomposition into permutation normalisation of its columns) are sufficient to enforce tensors if and only if its columns and rows are jointly measur- joint measurability. able and admit symmetric post-processing functions. Example 1. For d = 2, consider the 3-outcome measurement A simpler hypothesis that can replace conditions (i) − A = (A1, A2, A3) whose elements are vertices of an equilat- (ii) in Proposition 3 refers to the linear independence of eral triangle in the Bloch sphere representation, the operators of the mother measurement. This allows us to assume joint measurability of only the rows (or, I + σ A = x , (27a) equivalently, of only the columns) of the DNT, relaxing 1 3 (i) but further specifying (ii). I (σx √3σz)/2 Corollary 5. Let =[A ] be an n n DNT satisfying A2 = − − , (27b) A ij × 3 √ R(i) I (σx + 3σz)/2 (i’) the set of rows = = (Aij)j i of is jointly A3 = − , (27c) measurable; andR { } A 3

(ii’) admits a mother measurement with linearly inde- where σx and σz are Pauli matrices. Writing A′ = σx/3 and R 1 pendent elements. A1′′ = I/3, we have A1 = A1′ + A1′′ and the following DNT, Then there exists a coefficient measurement Q P such n! A + A A A that ∈ 1′ 1′′ 2 3 = A A + A A , A  2 1′ 3 1′′  = ∑ Π Q . (23) A3 A′′ A′ + A2 A l ⊗ l 1 1 l   where each entry is positive semi-definite. However, one Proof. We will show that (i ) (ii ) imply (i) (ii) of ′ ′ can check via semi-definite programming that these row- Prop. 3. Assume that for each−i, j we have − measurements are not jointly measurable [9], and use Propos- (i) ition 2 to conclude that is not a combination of permutation Aij = ∑ µ(j R , k)Mk, (24) A k | tensors. 5

Nevertheless, we can still write V. DNTS ARISING FROM A JOINT MEASURABILITY PROBLEM 1 0 0 1 0 0 = 0 1 0 A′ + 0 0 1 A′′ (28a) A   ⊗ 1   ⊗ 1 Throughout this work, we presented our motivations 0 0 1 0 1 0 to define DNTs and a generalisation of Birkhoff-von 0 1 0 0 0 1 Neumann’s theorem as purely mathematical, namely to further extend the clear parallel between probability vec- + 1 0 0 A2 + 0 1 0 A3, (28b) 0 0 1 ⊗ 1 0 0 ⊗ tors and POVMs. We now show that the set of DNTs     can be reached also by a quantum-theoretical path, more which is a combination of permutation matrices where not specifically in terms of joint measurability. all coefficient-operators are positive semi-definite, given that The property of joint measurability is based on the existence of a mother measurement, but another nat- A1′  0. ural question refers to the uniqueness of such object Despite the fact that the rows of the DNT in the above [7]. Restricting the post-processing map to be determ- example are not jointly measurable, they still can be re- inistic, and therefore a marginalisation (which can be constructed by applying a post-processing map to the done without loss of generality [10]), we can display the tuple of coefficients (A1′ , A1′′, A2, A3), as the proof of mother measurement M = (Mab) for a pair of POVMs A =(A ,..., A ), B =(B ,..., B ) as a table, Prop. 2 shows. Since A1′  0, this tuple is not a mother 1 n 1 n measurement, but it plays the same role as one. There- M11 M1n A1 fore, we will call it a pseudo-mother. . ···. . . (1) (n) . . . . Indeed, any set of measurements ,..., ad- . . . . , (30) {B B } M M A mits a pseudo-mother like that; if we no longer im- n1 ··· nn n pose positive semi-definitiveness, we can simply con- B1 Bn (1) (n) ··· sider the products Mb ...b = B ... B and check 1 n b1 bn emphasising the marginalisations ∑b Mab = Aa and (i) ∑ M = B . that µ(j B , b1 ... bn) = δbi,j is an appropriate post- a ab b processing| map for it,e since it satisfies Once we decide to study the plurality of mother measurements for a given pair, it is reasonable to start (i) from the most basic specimen. Taking A = B = B = ∑ Mb ...b δb ,j. (29) j 1 n i (I/n,..., I/n), we guarantee that the pair is trivially b1,...,bn joint measurable, for being both copies of the same e Notice that M is still normalised, and many other such POVM, and, on top of that, a trivial one. However, this pseudo-mothers can be constructed. For example, the trivial case allows to see that the general mother meas- urement for this pair, upon rescaling all the operators by order of thee operators in its defining product can be ar- bitrary, as long as it is the same for each element of the a factor of n (the number of outcomes), yields a table pseudo-mother. nM nM I Hence, by dropping positive semi-definitiveness from 11 ··· 1n . .. . . the results in the last section, it is straightforward to ob- . . . . , (31) tain the following result. nM nM I n1 ··· nn I I Theorem 6. Let = [Aij] be an n n DNT and = ··· (i) A (j) × R R = (Aij)j i, = C =(Aij)i j its sets of row- and which has exactly the DNT features, i.e., it can be column-measurements.{ } C Then{ the following} are equivalent: provided with a tensor structure, and its rows and columns form POVMs. (a) admits a pseudo-mother measurement M with Thus, we see that a generalisation of BvN’s theorem R∪C a symmetric post-processing map, µ(j R(i), k) = emerges not only from an operator-version of the ori- | µ(i C(j), k) for all i, j, k; e ginal result, but also as the description of the set of | mother measurements for perhaps the most trivial pair of POVMs that accepts multiple mothers. (b) = ∑l Πl Ql, where the operators Ql are normal- ised,A but are not⊗ necessarily positive semi-definite. e e Although every set of measurements admits a pseudo- VI. DISCUSSION mother, we were not able to show that one with a symmetric post-processing can always be found, nor to In this work we explored the analogy between prob- present any counter-example to it. Therefore, we leave ability vectors and POVMs and extrapolated it to invest- it as an open question whether every DNT satisfies item igate a generalised version of Birkhoff-von Neumann’s (a) of Theorem 6, and consequently can be characterised theorem. The richer structure of non-negative oper- by the decomposition presented in item (b). ators yields a discrepancy between the set of doubly 6 normalised tensors (that generalise doubly stochastic ferent decompositions exist, the problem of computing matrices) and decompositions into permutation tensors the optimal one (with the minimal number of terms) (that correspond to convex combinations of permuta- is NP-complete [14]. Here, Theorem 4 establishes that tions). We showed that the latter can be perfectly de- any jointly measurable, symmetric DNT can be decom- scribed as the DNTs composed of jointly measurable posed into permutations matrices, with coefficient oper- POVMs with symmetric post-processing maps, remark- ators that form a POVM. Hence, an extremal DNT with ing joint measurability – a quantum-theoretical concept these properties must correspond to an extremal coeffi- with a strongly operational motivation – as an relevant cient POVM. Extremal POVMs on dimension d have at mathematical property by itself. most d2 non-null elements, thus by fixing the dimension On the other hand, the general set of DNTs remains of the underlying Hilbert space (which is independent to be characterised. Our Theorem 6 states that the DNTs of the size n n of the tensor), we arrive at an uniform arising from a symmetric post-processing can be written upper bound× for the number of terms in the decomposi- as affine combinations of permutation tensors; it is not tion of extremal DNT’s. clear whether such symmetry imposes a non-trivial con- Finally, we recall that doubly stochastic matrices go straint or, on the contrary, arbitrary DNTs satisfy this re- together with the concept of majorisation of real vec- quirement. This result is similar to a characterisation of tors and Hermitian matrices [15, 16], a celebrated con- unital quantum channels presented in Ref. [13], where nection with important applications to quantum inform- it was proved that these objects are affine combinations ation theory [17]. It would be interesting to understand of unitary channels. The similarity is curious, given whether DNTs are associated to a similar notion of ma- that unital quantum channels are also generalisations of jorisation for POVMs, together with its consequences for doubly stochastic matrices in some sense. In a broader joint measurability. context, the need for quasi-POVMs in our description of incompatible DNTs dialogues with the need for quasi- probability representations in quantum theory [11, 12]. ACKNOWLEDGEMENTS As mentioned along the text, in the literature on BvN’s theorem is also posed the question of how many The authors are thankful T. Perche for fruitful dis- permutations are needed to describe an arbitrary doubly cussions. LG is supported by the São Paulo Research stochastic matrix. It was shown that (n 1)2 + 1 per- Foundation (FAPESP) under grants 2016/01343-7 and mutations were enough [6], and that although− many dif- 2018/04208-9. AB is partially supported by CNPq.

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