Showing papers by "Hamza Fawzi published in 2017"
TL;DR: In this paper, the authors show how to approximate the matrix logarithm with functions that preserve operator concavity and can be described using the feasible regions of semidefinite optimization problems of fairly small size.
Abstract: The matrix logarithm, when applied to Hermitian positive definite matrices, is concave with respect to the positive semidefinite order. This operator concavity property leads to numerous concavity and convexity results for other matrix functions, many of which are of importance in quantum information theory. In this paper we show how to approximate the matrix logarithm with functions that preserve operator concavity and can be described using the feasible regions of semidefinite optimization problems of fairly small size. Such approximations allow us to use off-the-shelf semidefinite optimization solvers for convex optimization problems involving the matrix logarithm and related functions, such as the quantum relative entropy. The basic ingredients of our approach apply, beyond the matrix logarithm, to functions that are operator concave and operator monotone. As such, we introduce strategies for constructing semidefinite approximations that we expect will be useful, more generally, for studying the approximation power of functions with small semidefinite representations.
52 citations
TL;DR: In this article, it was shown that Lieb's function admits an explicit semidefinite programming formulation for any rational t ∈ [ 0, 1 ] and that this formulation makes use of a semidefinite formulation of weighted matrix geometric means.
16 citations
TL;DR: In this article, the authors give an explicit construction of an equivariant psd lift of the regular 2n-gon of size 2n − 1, and prove that their construction is essentially optimal.
Abstract: Given a polytope P ⊂ ℝn, we say that P has a positive semidefinite lift (psd lift) of size d if one can express P as the projection of an affine slice of the d × d positive semidefinite cone. Such a representation allows us to solve linear optimization problems over P using a semidefinite program of size d and can be useful in practice when d is much smaller than the number of facets of P. If a polytope P has symmetry, we can consider equivariant psd lifts, i.e., those psd lifts that respect the symmetries of P. One of the simplest families of polytopes with interesting symmetries is regular polygons in the plane. In this paper, we give tight lower and upper bounds on the size of equivariant psd lifts for regular polygons. We give an explicit construction of an equivariant psd lift of the regular 2n-gon of size 2n − 1, and we prove that our construction is essentially optimal by proving a lower bound on the size of any equivariant psd lift of the regular N-gon that is logarithmic in N. Our construction is...
13 citations
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TL;DR: It is shown that the positive semidefinite rank of any convex body $C$ is at least $\sqrt{\log d}$, where $d$ is the smallest degree of a polynomial that vanishes on the boundary of the polar of $C$.
Abstract: The positive semidefinite rank of a convex body $C$ is the size of its smallest positive semidefinite formulation. We show that the positive semidefinite rank of any convex body $C$ is at least $\sqrt{\log d}$ where $d$ is the smallest degree of a polynomial that vanishes on the boundary of the polar of $C$. This improves on the existing bound which relies on results from quantifier elimination. The proof relies on the Bezout bound applied to the Karush-Kuhn-Tucker conditions of optimality. We discuss the connection with the algebraic degree of semidefinite programming and show that the bound is tight (up to constant factor) for random spectrahedra of suitable dimension.
10 citations
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18 May 2017
TL;DR: This work uses the approximation method proposed in [Fawzi, Saunderson, Parrilo, Semidefinite approximations of the matrix logarithm, arXiv:1705.00812] to provide numerical counterexamples for a proposed lower bound on the quantum conditional mutual information in terms of the relative entropy of recovery.
Abstract: Many quantum information measures can be written as an optimization of the quantum relative entropy between sets of states. For example, the relative entropy of entanglement of a state is the minimum relative entropy to the set of separable states. The various capacities of quantum channels can also be written in this way. We propose a unified framework to numerically compute these quantities using off-the-shelf semidefinite programming solvers, exploiting the approximation method proposed in [Fawzi, Saunderson, Parrilo, Semidefinite approximations of the matrix logarithm, arXiv:1705.00812]. As a notable application, this method allows us to provide numerical counterexamples for a proposed lower bound on the quantum conditional mutual information in terms of the relative entropy of recovery.
10 citations
TL;DR: A unified framework to numerically compute the quantum conditional mutual information in terms of the relative entropy of recovery using off-the-shelf semidefinite programming solvers, exploiting the approximation method proposed in Fawzi, Saunderson and Parrilo.
Abstract: Many quantum information measures can be written as an optimization of the quantum relative entropy between sets of states. For example, the relative entropy of entanglement of a state is the minimum relative entropy to the set of separable states. The various capacities of quantum channels can also be written in this way. We propose a unified framework to numerically compute these quantities using off-the-shelf semidefinite programming solvers, exploiting the approximation method proposed in [Fawzi, Saunderson, Parrilo, Semidefinite approximations of the matrix logarithm, arXiv:1705.00812]. As a notable application, this method allows us to provide numerical counterexamples for a proposed lower bound on the quantum conditional mutual information in terms of the relative entropy of recovery.
5 citations