Approximate Completely Positive Semidefinite Rank

In this paper we provide an approximation for completely positive semidefinite (cpsd) matrices with cpsd-rank bounded above (almost) independently from the cpsd-rank of the initial matrix. This is particularly relevant since the cpsd-rank of a matrix cannot, in general, be upper bounded by a function only depending on its size. For this purpose, we make use of the Approximate Carath\'eodory Theorem in order to construct an approximate matrix with a low-rank Gram representation. We then employ the Johnson-Lindenstrauss Lemma to improve to a logarithmic dependence of the cpsd-rank on the size.Comment: v2: clarified and corrected some citation

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 nonnegative rank, the completely positive rank, and their symmetric analogues: the positive semidefinite rank and the completely positive semidefinite rank. We study the convergence properties of our hierarchies, compare them extensively to known lower bounds, and provide some (numerical) examples

La composición tipográfica de la Políglota

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 nonnegative rank, the positive semidefinite rank, and their symmetric analogs: the completely positive rank and the completely positive semidefinite rank. We study convergence properties of our hierarchies, compare them extensively to known lower bounds, and provide some (numerical) examples

Algorithms for Positive Semidefinite Factorization

This paper considers the problem of positive semidefinite factorization (PSD factorization), a generalization of exact nonnegative matrix factorization. Given an $m$-by-$n$ nonnegative matrix $X$ and an integer $k$, the PSD factorization problem consists in finding, if possible, symmetric $k$-by-$k$ positive semidefinite matrices $\{A^1,...,A^m\}$ and $\{B^1,...,B^n\}$ such that $X_{i,j}=\text{trace}(A^iB^j)$ for $i=1,...,m$, and $j=1,...n$. PSD factorization is NP-hard. In this work, we introduce several local optimization schemes to tackle this problem: a fast projected gradient method and two algorithms based on the coordinate descent framework. The main application of PSD factorization is the computation of semidefinite extensions, that is, the representations of polyhedrons as projections of spectrahedra, for which the matrix to be factorized is the slack matrix of the polyhedron. We compare the performance of our algorithms on this class of problems. In particular, we compute the PSD extensions of size $k=1+ \lceil \log_2(n) \rceil$ for the regular $n$-gons when $n=5$, $8$ and $10$. We also show how to generalize our algorithms to compute the square root rank (which is the size of the factors in a PSD factorization where all factor matrices $A^i$ and $B^j$ have rank one) and completely PSD factorizations (which is the special case where the input matrix is symmetric and equality $A^i=B^i$ is required for all $i$).Comment: 21 pages, 3 figures, 3 table

Atemporal diagrams for quantum circuits

A system of diagrams is introduced that allows the representation of various elements of a quantum circuit, including measurements, in a form which makes no reference to time (hence ``atemporal''). It can be used to relate quantum dynamical properties to those of entangled states (map-state duality), and suggests useful analogies, such as the inverse of an entangled ket. Diagrams clarify the role of channel kets, transition operators, dynamical operators (matrices), and Kraus rank for noisy quantum channels. Positive (semidefinite) operators are represented by diagrams with a symmetry that aids in understanding their connection with completely positive maps. The diagrams are used to analyze standard teleportation and dense coding, and for a careful study of unambiguous (conclusive) teleportation. A simple diagrammatic argument shows that a Kraus rank of 3 is impossible for a one-qubit channel modeled using a one-qubit environment in a mixed state.Comment: Minor changes in references. Latex 32 pages, 13 figures in text using PSTrick