Epsilon-Net Method for Optimizations over Separable States

We give algorithms for the optimization problem: maxρ 〈Q, ρ〉, where Q is a Hermitian matrix, and the variable ρ is a bipartite separable quantum state. This problem lies at the heart of several problems in quantum computation and information, such as the complexity of QMA(2). While the problem is NPhard, our algorithms are better than brute force for several instances of interest. In particular, they give PSPACE upper bounds on promise problems admitting a QMA(2) protocol in which the verifier performs only logarithmic number of elementary gates that act on both proofs, as well as the promise problem of deciding if a bipartite local Hamiltonian’s ground energy is large or small. For Q ≥ 0, our algorithm runs in time exponential in ‖Q‖F. While the existence of such an algorithm was first proved recently by Brandão, Christandl and Yard [Proceedings of the 43rd annual ACM Symposium on Theory of Computation, 343–352, 2011], our algorithm is conceptually simpler. Entanglement is an essential ingredient in many ingenious applications of quantum information processing. Understanding and exploiting entanglement remains a central theme in quantum information processing research [HHH+09]. Denote by SepD (A1 ⊗A2) the set of separable (i.e, unentangled) density operators over the spaceA1⊗A2. A fundamental question known as the weak membership problem for separability is to decide, given the classical description of a quantum state ρ over A1 ⊗A2, whether ρ is inside or e far away in trace distance from SepD (A1 ⊗A2). Unfortunately, this basic problem turns out to be intractable. In 2003, Gurvits [Gur03] proved the NP-hardness of the problem when e is inverse exponential in the dimension of A1 ⊗A2. The dependence on e was later improved to inverse polynomial [Ioa07, Gha10]. In this paper we study a closely related problem to the weak membership problem discussed above. More precisely, we consider the linear optimization problem over separable states. Problem 1. Given a Hermitian matrix Q over A1 ⊗A2 (of dimension d× d), compute the optimum value, denoted by OptSep(Q), of the optimization problem max 〈Q,X〉 subject to X ∈ SepD (A1 ⊗A2) . It is well-known in convex optimization [GLS93, Ioa07] that the weak membership problem and the weak linear optimization, a special case of Problem 1, over certain convex set, such as SepD (A1 ⊗A2), are equivalent up to polynomial loss in precision and polynomial-time overhead. Thus the hardness result on the weak membership problem for separability passes directly to Problem 1. Besides the connection with the weak membership problem for separability, Problem 1 can also be understood from many other aspects. Firstly, as the objective function is the inner-product of a Hermitian matrix and a quantum state, which represents the average value of some physical observable, the optimal value of Problem 1 inherently possesses certain physical meaning. Secondly, in the study of the tensor product space [DF92], the value OptSep(Q) is precisely the injective norm of Q in L(A1)⊗ L(A2), where L(A) denote the Banach space of operators on A with the operator norm. Finally, one may be equally motivated from the study in operations research. Problem 1 appeared in an equivalent form in [LQNY09] as “bi-quadratic optimization over unit spheres”. Subsequent works [HLZ10, So11] demonstrated that Problem 1 is just a special case of a more general class of optimization problems called homogenous polynomial optimization with quadratic constraints, which is currently an active research topic in operation research. Another motivation to study Problem 1 is the recent interest on the complexity class QMA(2). The class QMA [KSV02] is the quantum counterpart of the classical complexity class NP (or more precisely, MA).