Completely positive semidefinite rank

  title={Completely positive semidefinite rank},
  author={Anupam Prakash and Jamie Sikora and Antonios Varvitsiotis and Zhaohui Wei},
  journal={Mathematical Programming},
An $$n\times n$$n×n matrix X is called completely positive semidefinite (cpsd) if there exist $$d\times d$$d×d Hermitian positive semidefinite matrices $$\{P_i\}_{i=1}^n$${Pi}i=1n (for some $$d\ge 1$$d≥1) such that $$X_{ij}= \mathrm {Tr}(P_iP_j),$$Xij=Tr(PiPj), for all $$i,j \in \{ 1, \ldots , n \}$$i,j∈{1,…,n}. The cpsd-rank of a cpsd matrix is the smallest $$d\ge 1$$d≥1 for which such a representation is possible. In this work we initiate the study of the cpsd-rank which we motivate in two… 
Algorithms for positive semidefinite factorization
This work introduces several local optimization schemes to tackle the problem of positive semidefinite factorization, a generalization of exact nonnegative matrix factorization and introduces a fast projected gradient method and two algorithms based on the coordinate descent framework.
Correlation matrices, Clifford algebras, and completely positive semidefinite rank
ABSTRACT A symmetric matrix X is completely positive semidefinite (cpsd) if there exist positive semidefinite matrices (for some ) such that for all . The of a cpsd matrix is the smallest for which
Positively factorizable maps
Matrices With High Completely Positive Semidefinite Rank
Approximate Completely Positive Semidefinite Rank.
This paper makes use of the Approximate Carath\'eodory Theorem in order to construct an approximate matrix with a low-rank Gram representation and employs the Johnson-Lindenstrauss Lemma to improve to a logarithmic dependence of the cpsd-rank on the size.
Semidefinite games
We introduce and study the class of semidefinite games, which generalizes bimatrix games and finite N -person games, by replacing the simplex of the mixed strategies for each player by a slice of the
Quantum machine learning with subspace states
A new approach for quantum linear algebra based on quantum subspace states is introduced and three new quantum machine learning algorithms are presented that reduce exponentially the depth of circuits used in quantum topological data analysis from O(n) to O(log n).
Self-dual polyhedral cones and their slack matrices
We analyze self-dual polyhedral cones and prove several properties about their slack matrices. In particular, we show that self-duality is equivalent to the existence of a positive semidefinite (PSD)
Lower Bounds on Matrix Factorization Ranks via Noncommutative Polynomial Optimization
We use techniques from (tracial noncommutative) polynomial optimization to formulate hierarchies of semidefinite programming lower bounds on matrix factorization ranks. In particular, we consider the
Completely positive completely positive maps (and a resource theory for non-negativity of quantum amplitudes)
In this work we examine quantum states which have non-negative amplitudes (in a fixed basis) and the channels which preserve them. These states include the ground states of stoquastic Hamiltonians


Positive semidefinite rank
The main mathematical properties of psd rank are surveyed, including its geometry, relationships with other rank notions, and computational and algorithmic aspects.
New Lower Bounds and Asymptotics for the cp-Rank
This paper constructs counterexamples to the DJL conjecture for all $n\ge {12}$ and shows the largest possible cp-rank of an completely positive matrix, p_n, is defined.
New approximations for the cone of copositive matrices and its dual
  • J. Lasserre
  • Mathematics, Computer Science
    Math. Program.
  • 2014
This work provides convergent hierarchies for the convex cone of copositive matrices and its dual, the cone of completely positive matrices, and provides outer approximations for these hierarchies, which have a very simple interpretation.
Conic Approach to Quantum Graph Parameters Using Linear Optimization Over the Completely Positive Semidefinite Cone
This new cone is investigated, a new matrix cone consisting of all $n\times n$ matrices that admit a Gram representation by positive semidefinite matrices (of any size) and is used to model quantum analogues of the classical independence and chromatic graph parameters.
On the Closure of the Completely Positive Semidefinite Cone and Linear Approximations to Quantum Colorings
A hierarchy of polyhedral cones is constructed which covers the interior of the completely positive semidefinite cone $\mathcal{CS}_+^n$, which is used for computing some variants of the quantum chromatic number by way of a linear program.
The square root rank of the correlation polytope is exponential
This work shows that the square root rank of the slack matrix of the correlation polytope is exponential, and a way to lower bound the rank of certain matrices under arbitrary sign changes of the entries using properties of the roots of polynomials in number fields.
A Note on Extreme Correlation Matrices
An $n\times n$ complex Hermitian or real symmetric matrix is a correlation matrix if it is positive semidefinite and all its diagonal entries equal one. The collection of all $n\times n$ correlation
Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization
In the first part of this thesis, we introduce a specific class of Linear Matrix Inequalities (LMI) whose optimal solution can be characterized exactly. This family corresponds to the case where the