Complete hierarchies of efficient approximations to problems in entanglement theory

Abstract

We investigate several problems in entanglement theory from the perspective of convex optimization. This list of problems comprises (A) the decision whether a state is multi-party entangled, (B) the minimization of expectation values of entanglement witnesses with respect to pure product states, (C) the closely related evaluation of the geometric measure of entanglement to quantify pure multi-party entanglement, (D) the test whether states are multi-party entangled on the basis of witnesses based on second moments and on the basis of linear entropic criteria, and (E) the evaluation of instances of maximal output purities of quantum channels. We show that these problems can be formulated as certain optimization problems: as polynomially constrained problems employing polynomials of degree three or less. We then apply very recently established known methods from the theory of semi-definite relaxations to the formulated optimization problems. By this construction we arrive at a hierarchy of efficiently solvable approximations to the solution, approximating the exact solution as closely as desired, in a way that is asymptotically complete. For example, this results in a hierarchy of novel, efficiently decidable sufficient criteria for multi-particle entanglement, such that every entangled state will necessarily be detected in some step of the hierarchy. Finally, we present numerical examples to demonstrate the practical accessibility of this approach.

Separability criterion of tripartite qubit systems

“…If a standard numerical process is needed, we suggest that the approach introduced in Ref. [17] be preferred.…”

“…[8] also presents an alternative method to minimize the convex hull for mixed states. As to multipartite quantum systems, several separability criteria have been proposed [12,13,14,15,16,17]. The most notable one is 3-tangle for three qubits [13].…”

Unifying several separability conditions using the covariance matrix criterion

“…Proof: Without any loss of generality we assume ρ = |1 1| and take as observables the ones of the standard basis (14). Calculating directly and reordering of the matrix elements afterwards gives a block structure [28] …”

“…There are also other criteria that turned out to be directly related to the PPT criterion: The majorization criterion [9] and also entropic criteria [10] have been shown to be weaker than the PPT criterion [11,12]. Moreover, one can extend the PPT condition to a test based on a complete hierarchy of symmetric extensions, where each step constitutes a semidefinite program [13] (another complete family of semidefinite tests has been described in [14]). In such a hierarchy, every entangled state is necessarily detected as such in some step of the hierarchy.…”

Single-photon quantum key distribution in the presence of loss

“…(See also Ref. [21].) Also, one would like to include in the analysis the attenuation introduced by the quantum channel, not considered in Refs.…”