Note on the Correspondence Between Quantum Correlations and the Completely Positive Semidefinite Cone Laura Manˇcinska

David E. Roberson

October 8, 2014

Introduction The purpose of this note is to point out a connection between the set of (twoparty) quantum correlations and the completely positive semidefinite cone recently introduced by Laurent and Piovesan [1]. A correlation with input sets X and Y and output sets A and B is a joint conditional probability distribution P (a, b|x, y), a ∈ A, b ∈ B, x ∈ X, y ∈ Y . In other words, P is a function from A × B × X × Y to the interval [0, 1] such that X P (a, b|x, y) = 1 for all x ∈ X, y ∈ Y. a∈A,b∈B

Such a correlation is quantum if it can be implemented by two separated parties by measuring a shared entangled state. In mathematical terms, P is a quantum correlation if P (a, b|x, y) = ψ † (Exa ⊗ Fyb ) ψP for some unit P vector ψ and positive semidefinite operators Exa , Fyb such that a Exa = b Fyb = I. In [1], Laurent and Piovesan define a completely positive semidefinite matrix to be one which is the Gram matrix of some set of positive semidefinite matrices. In other words, an n × n matrix M is completely positive semidefinite if there exist positive semidefinite matrices ρ1 , . . . , ρn such that Mij = Tr(ρi ρj ). The set of n × n completely positive semidefinite matrices is a convex cone and we denote it by CS n+ . If we do not wish to specify the matrix dimensions we will simply write CS + . We will see that CS + is intimately related to the set of quantum correlations. In this note we will often use matrices whose rows and columns are indexed by ordered pairs (z, c), for instance. To make things less cluttered, we will opt to use Mzcz0 c0 to indicate the entry of M in its (z, c) row and (z 0 , c0 ) column, as opposed to M(z,c),(z0 ,c0 ) which might be expected.

Main Result Before we prove our main result, we need the following lemma concerning completely positive semidefinite matrices.

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Lemma 0.1. Let C and Z be finite sets and let M be the Gram matrix of positive semidefinite matrices {ρzc for c ∈ C, z ∈ Z}. Then X Mzcz0 c0 = 1 for all z, z 0 ∈ Z c,c0

if and only if there exists ρ  0 such that Tr(ρ2 ) = 1 and X ρzc = ρ for all z ∈ Z. c

Proof. Define ρz = to

P

c

ρzc for all z ∈ Z. The first condition above is equivalent !

1=

X

Tr(ρzc ρz0 c0 ) = Tr

c,c0

X c

ρzc

X

ρz 0 c 0

= Tr(ρz ρz0 ) for all z, z 0 ∈ Z.

c0

It is easy to see that this holds if and only if ρz is the same for all z ∈ Z. We now prove our main result showing that quantum correlations correspond to certain entries of completely positive semidefinite matrices Theorem 0.2. Let A, B, X, and Y be finite disjoint sets and let C = A ∪ B and Z = X ∪ Y . A function P : A × B × X × Y → [0, 1] is a quantum correlation if and only if there exists a completely positive semidefinite matrix M with rows and columns indexed by Z × C satisfying the following: X Mzcz0 c0 = 1 for all z, z 0 ∈ Z c,c0

Mzcz0 c0 = 0 if z ∈ X, z 0 ∈ Y and (c ∈ B or c0 ∈ A) Mxayb = P (a, b|x, y) for all a ∈ A, b ∈ B, x ∈ X, y ∈ Y. Proof. Suppose that P is a quantum Then there exist positive P correlation. P semidefinite Exa and Fyb satisyfing a Exa = b Fyb = I, and a unit vector ψ such that P (a, b|x, y) = ψ † (Exa ⊗ Fyb ) ψ. We can write the vector ψ as vec(R) where R is an arbitrary square matrix and vec is the linear operator which maps uv | to u ⊗ v. In other words, vec maps a matrix to the vector obtained by stacking the rows on top of each other. Using singular value decomposition, we have that R = U DV † where U and V are unitary and D is a nonnegative diagonal matrix such that Tr(D2 ) = 1. Using the identity E ⊗ F vec(N ) = vec(EN F | ), we have | ψ † (Exa ⊗ Fyb ) ψ = vec† (U DV † ) vec(Exa U DV † Fyb ) | = Tr(V DU † Exa U DV † Fyb )   | = Tr (D1/2 U † Exa U D1/2 )(D1/2 V † Fyb V D1/2 ) .

For c ∈ C and z ∈ Z, define positive semidefinite operators ρzc as follows:  1/2 †  D U Exa U D1/2 if c ∈ A, z ∈ X | D1/2 V † Fyb V D1/2 if c ∈ B, z ∈ Y ρzc =  0 o.w. 2

It is now easy to see that if M is the Gram matrix of the ρzc , then Mxayb = ψ † (Exa ⊗ Fyb ) ψ = P (a, b|x, y) for all a ∈ A, b ∈ B, x ∈ X, y ∈ Y . Also, if z ∈ X and c ∈ B, then ρzc = 0 and thus Mzcz0 c0 = 0, and similarly for z 0 ∈ Y and c0 ∈ A. Furthermore, if z ∈ X, then X X ρzc = D1/2 U † Exa U D1/2 = D1/2 U † IU D1/2 = D, c∈C

a∈A

and similarly for z ∈ Y . By Lemma 0.1, this implies that X Mzcz0 c0 = 1 for all z, z 0 ∈ Z, c,c0

and thus we are done with this direction of the theorem. Conversely, suppose that a completely positive semidefinite matrix M exists satisfying the conditions of the theorem. There exist positive semidefinite operators ρzc such that Mzcz0 c0 = Tr(ρzc ρz0 c0 ) for all c, c0 ∈ C, z, z 0 ∈ Z. Furthermore, by Lemma 0.1, there exists ρ  0 such that Tr(ρ2 ) = 1 and X ρzc = ρ for all z ∈ Z. c

Lastly, we can assume that ρ is of full rank and is thus invertible. Now suppose that z ∈ X, z 0 ∈ Y and c ∈ B. Then Mzcz0 c0 = 0 and so we have X Tr(ρzc ρz0 c0 ) = Tr(ρzc ρ). 0= c0

Therefore ρzc = 0 for z ∈ X and c ∈ B. Similarly ρzc = 0 for z ∈ Y and c ∈ A. For a ∈ A and x ∈ X, let Exa = ρ−1/2 ρxa ρ−1/2 , and for b ∈ B, y ∈ Y let | Fyb = ρ−1/2 ρyb ρ−1/2 .

Obviously, the Exa and Fyb are positive semidefinite. Moreover, X X Exa = ρ−1/2 ρxa ρ−1/2 = ρ−1/2 ρρ−1/2 = I for all x ∈ X, a

a

and similarly for the Fyb . Letting ψ = vec(ρ), we see that ψ † ψ = Tr(ρ2 ) = 1, and thus ψ is a unit vector. Furthermore, again using the identity E ⊗ F vec(N ) = vec(EN F | ), we have ψ † (Exa ⊗ Fyb )ψ = Tr(ρExa ρFyb ) = Tr(ρρ−1/2 ρxa ρ−1/2 ρρ−1/2 ρyb ρ−1/2 ) = Tr(ρxa ρyb ) = Mxayb = P (a, b|x, y). Therefore, P is a quantum correlation. 3

Closedness Let A, B, X, and Y be disjoint finite sets and let n = |A| · |B| · |X| · |Y |. We will denote by QC(A, B, X, Y ) the set of quantum correlations with input sets X and Y and respective output sets A and B. Though above we defined correlations as being functions, here we think of them as being subsets of Rn , which is clearly equivalent. Theorem 0.3. Let A, B, X, and Y be disjoint finite sets and let n = (|A| + |B|)(|X| + |Y |). If CS n+ is closed, then so is QC(A, B, X, Y ) Proof. Let C = A ∪ B and Z = X ∪ Y . Suppose that CS n+ is closed, and that P 1 , P 2 , . . . is a sequence of quantum correlations which converges to some point P ∈ R|A|·|B|·|X|·|Y | . By Theorem 0.2, for each i ∈ N there exist matrices M i ∈ CS n+ with rows and columns indexed by Z × C such that X

i 0 Mzcz 0 c0 = 1 for all z, z ∈ Z

c,c0 i 0 0 Mzcz 0 c0 = 0 if z ∈ X, z ∈ Y and (c ∈ B or c ∈ A) i Mxayb = P i (a, b|x, y) for all a ∈ A, b ∈ B, x ∈ X, y ∈ Y.

The sequence M 1 , M 2 , . . . may not converge since there are entries of M i not determine by P i . However, by the first condition above and the fact that completely positive semidefinite matrices are entrywise nonnegative, we have that i 0 0 ≤ Mzcz ∈ Z, c, c0 ∈ C. Therefore the M i are 0 c0 ≤ 1 for all i ∈ N, z, z contained in a compact set and thus there exists a convergent subsequence (M j(i) )i∈N . Let M = limi→∞ M j(i) . It is easy to see that Mxayb = P (a, b|x, y) for all a ∈ A, b ∈ B, x ∈ X, y ∈ Y and that M satisfies the other two conditions of Theorem 0.2. Therefore, P is indeed a quantum correlation with input sets X and Y and respective output sets A and B, i.e. P ∈ QC(A, B, X, Y ). Since every convergent suequence of QC(A, B, X, Y ) converges to a point in QC(A, B, X, Y ), this set is closed.

References [1] Teresa Piovesan and Monique Laurent. Conic approach to quantum graph parameters using linear optimization over the completely positive semidefinite cone. 2014.

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