Strong NP-Hardness of the Quantum Separability Problem

Given the density matrix rho of a bipartite quantum state, the quantum separability problem asks whether rho is entangled or separable. In 2003, Gurvits showed that this problem is NP-hard if rho is located within an inverse exponential (with respect to dimension) distance from the border of the set of separable quantum states. In this paper, we extend this NP-hardness to an inverse polynomial distance from the separable set. The result follows from a simple combination of works by Gurvits, Ioannou, and Liu. We apply our result to show (1) an immediate lower bound on the maximum distance between a bound entangled state and the separable set (assuming P != NP), and (2) NP-hardness for the problem of determining whether a completely positive trace-preserving linear map is entanglement-breaking.Comment: 18 pages, 1 figure. v5: Updated version to appear in Quantum Information & Computation. Includes additional details in proof of NP-hardness of determining whether a quantum channel is entanglement-breaking, as well as minor updates to improve readability throughout. Thank you to anonymous referees for their comment

On the NP-hardness of checking matrix polytope stability and continuous-time switching stability

Motivated by questions in robust control and switched linear dynamical systems, we consider the problem checking whether all convex combinations of k matrices in R[superscript ntimesn] are stable. In particular, we are interested whether there exist algorithms which can solve this problem in time polynomial in n and k. We show that if k=n[superscript d] for any fixed real d > 0, then the problem is NP-hard, meaning that no polynomial-time algorithm in n exists provided that P ne NP, a widely believed conjecture in computer science. On the other hand, when k is a constant independent of n, then it is known that the problem may be solved in polynomial time in n. Using these results and the method of measurable switching rules, we prove our main statement: verifying the absolute asymptotic stability of a continuous-time switched linear system with more than n[superscript d] matrices A[subscript i] isin R[superscript ntimesn] satisfying 0 ges A[subscript i] + A[subscript i] [superscript T] is NP-hard