Extremal entanglement witnesses

Abstract

We study extremal entanglement witnesses on a bipartite quantum system. We define the cone of witnesses as the dual of the set of separable density matrices, thus $\text{Tr}\Omega \rho \ge 0$ when $\Omega$ is a witness and $\rho$ a pure product state, $\rho =\psi {\psi }^{†}$ with $\psi =\varphi \otimes \chi$. The set of witnesses of unit trace is a compact convex set, defined by its extremal points. The expectation value $f(\varphi ,\chi )=\mathrm{Tr}\Omega \rho$ as a function of $\varphi$ and $\chi$ is a nonnegative biquadratic form. Every zero of $f(\varphi ,\chi )$ imposes real-linear constraints on $f$ and $\Omega$. The Hessian matrix at the zero must be nonnegative. Its eigenvectors with zero eigenvalue, if any, we call Hessian zeros. A zero of $f(\varphi ,\chi )$ is quadratic if it has no Hessian zeros, otherwise it is quartic. We call a witness quadratic if it has only quadratic zeros, and quartic otherwise. We prove that a witness is extremal if and only if no other witness has the same, or a larger, set of zeros and Hessian zeros. A quadratic extremal witness has a minimum number of isolated zeros depending on dimensions. If a witness is not extremal, the constraints defined by its zeros and Hessian zeros determine all directions in which to search for witnesses having more zeros or Hessian zeros. A finite number of iterated searches in random directions lead to an extremal witness which is usually quadratic with the minimum number of zeros. We discuss some topics related to extremal witnesses, in particular the relation between the facial structures of the dual sets of witnesses and separable states. We discuss the relation between extremality and optimality of witnesses, and a conjecture of separability of the structural physical approximation (SPA) of an optimal witness. We discuss how to treat the entanglement witnesses on a complex Hilbert space as witnesses on a real Hilbert space.

61 pages, 7 figures, 4 tables. Updated version accepted for publication in International Journal of Quantum Information

References in corpus (13)

• Detecting Genuine Multipartite Entanglement with Two Local Measurements
• Quantifying Entanglement with Witness Operators
• Optimal entanglement criterion for mixed quantum states
• All bipartite entangled states display some hidden nonlocality
• Structural approximations to positive maps and entanglement breaking channels
• Extreme points of the set of density matrices with positive partial transpose
• Optimality for indecomposable entanglement witnesses
• Numerical studies of entangled PPT states in composite quantum systems
• Quantum Separability and Entanglement Detection via Entanglement-Witness Search and Global Optimization
• A class of exposed indecomposable positive maps
• The structural physical approximations and optimal entanglement witnesses
• Entanglement witnesses arising from Choi type positive linear maps
• Disproving the conjecture on structural physical approximation to optimal decomposable entanglement witnesses

Cited by in corpus (3)

• Entanglement witnesses: construction, analysis and classification
• Checking the optimality of entanglement witnesses: an application to structural physical approximations
• Visualizing extremal positive maps in unital and trace preserving form