# Complete hierarchies of efficient approximations to problems in entanglement theory

@article{Eisert2004CompleteHO, title={Complete hierarchies of efficient approximations to problems in entanglement theory}, author={Jens Eisert and Philipp Hyllus and Otfried Guehne and Marcos Curty}, journal={Physical Review A}, year={2004}, volume={70}, pages={062317} }

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…

## 9 Citations

Quantum separability and entanglement detection via entanglement-witness search and global optimization

- Computer Science
- 2006

This paper focuses on determining the separability of an unknown bipartite quantum state {rho} by invoking a sufficiently large subset of all possible entanglement witnesses given the expected value of each element of a set of mutually orthogonal observables, and discusses in more generality such algorithms, which, in this case, assume a subroutine that computes the global maximum of a real function of several variables.

Quantifying entanglement with witness operators

- Computer Science
- 2005

An infinite family of new entanglement quantifiers is introduced, having as its limits the best separable approximation measure and the generalized robustness, and provides sharper bounds to d_(min) and E_D than the negativity in most the states.

Entanglement and the shareability of quantum states

- Physics
- 2014

This brief review discusses the problem of determining whether a given quantum state is separable or entangled. I describe an established approach to this problem that is based on the monogamy of…

Investigating a class of 2⊗2⊗d bound entangled density matrices via linear and nonlinear entanglement witnesses constructed by exact convex optimization

- Computer Science
- 2008

This work considers a class of chessboard density matrices starting with three-qubit ones which have positive partial transposes with respect to all subsystems, and attempts to convert the problem to an exact convex optimization problem.

The distillability problem revisited

- MathematicsQuantum Inf. Comput.
- 2006

The distillability problem can be reformulated as a special instance of the separability problem, for which a large number of tools and techniques are available, and the results suggest that bound entanglement is primarily a phenomenon found in low dimensional quantum systems.

Computing finite-dimensional bipartite quantum separability

- Computer ScienceArXiv
- 2005

This work motivates a new interior-point algorithm which, given the expected values of a subset of an orthogonal basis of observables of an otherwise unknown quantum state, searches for an entanglement witness in the span of the subset of Observables, and solves the separability problem.

Computational complexity of the quantum separability problem

- Computer ScienceQuantum Inf. Comput.
- 2007

This paper gives the first systematic and comprehensive treatment of this (bipartite) quantum separability problem, focusing on its deterministic computational complexity.

Relaxations of separability in multipartite systems: Semidefinite programs, witnesses and volumes

- Computer Science
- 2015

An operational witness construction arising from such relaxations that is capable of detecting every entangled state is described and it is proved that simple semi-definite relaxations in the multiparticle case cannot be an equally good approximation for any scenario.

Entanglement of genuinely entangled subspaces and states: Exact, approximate, and numerical results

- PhysicsPhysical Review A
- 2019

Genuinely entangled subspaces (GESs) are those subspaces of multipartite Hilbert spaces that consist only of genuinely multiparty entangled pure states. They are natural generalizations of the…

A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations

- Computer Science
- 2008

It is shown that in some cases it is possible to conclude that a given set of correlations is quantum after performing only a finite number of tests, and used in particular to bound the quantum violation of various Bell inequalities.