matrix. Numerically we have ξ2psd(SH) ≈ 1.99 while ξpsd2 (DSH) ≈ 2.12, where D = Diag(2, 2, 1, 1, 1, 1). The bound ξpsd2 (DSH) is in fact tight (after rounding) for the complex positive semidefinite rank of DSH (and hence of SH): in [GGS17] it is shown that psd-rankC(SH) = 3.
5.7 Concluding remarks
We provide a Matlab implementation of all the lower bounds introduced in this chap-ter, at the arXiv submission of the paper on which this chapter is based [GdLL19].
The implementation uses the CVX package [GB14] and supports various semi-definite programming solvers; for our numerical examples we used Mosek [ApS17].
We now mention some corollaries of the results of this chapter and open prob-lems.
Testing membership in the completely positive cone and the completely positive semidefinite cone is another important problem to which our hierarchies can also be applied. It follows from the proof of Proposition 5.18 that if A is not completely positive then, for some order t, the program ξtcp(A) is infeasible or its optimum value is larger than the Carath´eodory bound on the cp-rank (which is similar to an earlier result in [Nie14a]). In the noncommutative setting the situation is more complicated: If ξ∗cpsd(A) is feasible, then A ∈ CS+, and if A 6∈ CSn+,vN, then ξcpsd∞ (A) is infeasible (Propositions 5.1 and 5.2). Here CSn+,vNis the cone defined in [BLP17]
consisting of the matrices admitting a factorization by positive elements in a von Neumann algebra with a trace. This cone can equivalently be characterized as the set of matrices of the form α (τ (aiaj)) where α ∈ R+, and τ is a tracial state on a C∗-algebra A and a1, . . . , an are positive elements in A.
Our lower bounds are on the complex version of the (completely) positive semi-definite rank. As far as we are aware, the existing generic lower bounds (except for the dimension-counting rank lower bound) are also on the complex (completely) positive semidefinite rank. It would be interesting to find a lower bound on the real (completely) positive semidefinite rank that can go beyond the complex (com-pletely) positive semidefinite rank. In [LWdW17] an ad-hoc argument is given to separate the complex positive semidefinite rank from the real positive semidefinite rank for a specific matrix.
Finally we mention that our approach applies more generally, for instance to the nonnegative tensor rank; see [GdLL19, Sec. 6] for more details.
Chapter 6
Matrices with high completely positive semidefinite rank
This chapter is based on the paper “Matrices with high completely positive semi-definite rank”, by S. Gribling, D. de Laat, and M. Laurent [GdLL17].
As we have seen in Chapter 2, the nonnegative rank, the positive semidefinite rank, and the completely positive rank can all be upper bounded by a function of the matrix size. In fact, the square of the matrix size is a (loose) upper bound on all three. In this chapter we study the question of whether the completely positive semidefinite rank can be upper bounded in terms of the matrix size n. We give an explicit construction of completely positive semidefinite matrices of size 4k2+ 2k + 2 with complex completely positive semidefinite rank 2k for any positive integer k.
This shows that if such an upper bound would exist, it has to be at least exponential in the matrix size. For this we exploit connections to quantum information theory and we construct extremal bipartite correlation matrices of large rank.
The main motivation for the above question is to decide whether the completely positive semidefinite cone is closed. Indeed, if an upper bound on cpsd-rankC that only depends on the matrix size exists, then a compactness argument shows that the cone is closed. If the cone is closed, that would immediately imply that affine slices of the cone are closed. In Section 3.4 we have seen that an important affine slice of the completely positive semidefinite cone is the set of bipartite quantum correlations.
After completion of the work in this chapter, Slofstra [Slo19] showed that the set of bipartite quantum correaltions is not closed. Using the above mentioned connection, this implies that the cone CSn+ is not closed, for n large enough; see Section 3.2 for a discussion. Hence, no upper bound exists on cpsd-rankC that only depends on the matrix size. Nevertheless, it remains challenging to construct explicit classes of completely positive semidefinite matrices with large cpsd-rank.
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104 Chapter 6. Matrices with high completely positive semidefinite rank
6.1 Introduction
In this chapter we study the completely positive semidefinite rank, one of the two symmetric matrix factorization ranks that we have seen in the previous chapter.
Recall the inclusions
CPn⊆ CSn+⊆ Sn+∩ Rn×n+ ,
where Sn+ is the cone of (real) positive semidefinite n × n matrices. The three cones coincide for n ≤ 4 (since doubly nonnegative matrices of size n ≤ 4 are completely positive), but both inclusions are strict for n ≥ 5 (see [LP15] for details). By Carath´eodory’s theorem, the completely positive rank of a matrix in CPn is at most n+12 + 1. As we now know, due to Slofstra’s work [Slo19], there does not exist an upper bound on the cpsd-rank that only depends on the matrix size n.
It remains a challenging task to construct explicit families of completely positive semidefinite matrices whose cpsd-rank is large.
In this chapter we construct an explicit family of matrices whose cpsd-rank grows exponentially in the matrix size n. Our main result is the following:
Theorem 6.1. For each positive integer k, there exists a completely positive semi-definite matrix M of size 4k2+ 2k + 2 with cpsd-rankC(M ) = 2k.
The proof of this result relies on a connection with quantum information theory and geometric properties of (bipartite) correlation matrices. A first basic ingredient is the fact from [SV17] that a quantum correlation P can be realized in local dimen-sion d if and only if there exists a certain completely positive semidefinite matrix M with cpsd-rankC(M ) at most d (see Section 3.4). Then, the key idea is to construct a class of quantum correlations P that need large local dimension. In Chapter 7 we will revisit the topic of quantifying the amount of entanglement needed to realize a quantum correlation. There we will not focus on explicit examples, rather we will propose a new measure for the amount of entanglement of a quantum correla-tion. Specifically, our new measure will differ from the “minimal local dimension”
measure by assuming that access to shared randomness is free.
The papers [VP09, Slo11, Ji13] each use different techniques to show the exis-tence of different quantum correlations that require large local dimension. Our main contribution is to provide a unified, explicit construction of the quantum correlations from [VP09] and [Slo11], which uses the seminal work of Tsirelson [Tsi87, Tsi93]
combined with convex geometry and recent insights from rigidity theory. In addi-tion, we also give an explicit proof of Tsirelson’s bound (see Corollary 6.11) and we show examples where the bound is tight.
More specifically, we construct such quantum correlations from bipartite correla-tion matrices. For this we use the classical results of Tsirelson [Tsi87, Tsi93], which characterize bipartite correlation matrices in terms of operator representations and, using Clifford algebras, we relate the rank of extremal bipartite correlations to the local dimension of their operator representations. In this way we reduce the prob-lem to finding bipartite correlation matrices that are extreme points of the set of bipartite correlations and have large rank.
6.2. The set of bipartite correlations 105