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arXiv:quant-ph/0503152 · Physical Review A

Quantifying entanglement with witness operators

Fernando G. S. L. Brandão

Published Aug 12, 2005• 241 citations

Abstract

We present a unifying approach to the quantification of entanglement based on entanglement witnesses, which includes several already established entanglement measures such as the negativity, the concurrence, and the robustness of entanglement. We then introduce an infinite family of new entanglement quantifiers, having as its limits the best separable approximation measure and the generalized robustness. Gaussian states, states with symmetry, states constrained to super-selection rules, and states composed of indistinguishable particles are studied under the view of the witnessed entanglement. We derive new bounds to the fidelity of teleportation ${d}_{\mathit{min}}$, for the distillable entanglement ${E}_{D}$ and for the entanglement of formation. A particular measure, the PPT-generalized robustness, stands out due to its easy calculability and provides sharper bounds to ${d}_{\mathit{min}}$ and ${E}_{D}$ than the negativity in most of the states. We illustrate our approach studying thermodynamical properties of entanglement in the Heisenberg XXX and dimerized models.

DOI: 10.1103/physreva.72.022310

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We present a unifying approach to the quantification of entanglement based on entanglement witnesses, which includes several already established entanglement measures such as the negativity, the concurrence, and the robustness of entanglement. We then introduce an infinite family of new entanglement quantifiers, having as its limits the best separable approximation measure and the generalized robustness. Gaussian states, states with symmetry, states constrained to super-selection rules, and states composed of indistinguishable particles are studied under the view of the witnessed entanglement. We derive new bounds to the fidelity of teleportation

{d}_{\mathit{min}}

, for the distillable entanglement

{E}_{D}

and for the entanglement of formation. A particular measure, the PPT-generalized robustness, stands out due to its easy calculability and provides sharper bounds to

{d}_{\mathit{min}}

and

{E}_{D}

than the negativity in most of the states. We illustrate our approach studying thermodynamical properties of entanglement in the Heisenberg XXX and dimerized models.

Entanglement measures quantify nonclassical correlations present in a quantum system, but can be extremely difficult to calculate, even more so, when information on its state is limited. Here, we consider broad families of entanglement criteria that are based on variances of arbitrary operators and analytically derive the lower bounds these criteria provide for two relevant entanglement measures: the best separable approximation and the generalized robustness. This yields a practical method for quantifying entanglement in realistic experimental situations, in particular, when only few measurements of simple observables are available. As a concrete application of this method, we quantify bipartite and multipartite entanglement in spin-squeezed Bose-Einstein condensates of ∼500 atoms, by lower bounding the best separable approximation and the generalized robustness only from measurements of first and second moments of the collective spin operator.

Abstract In this chapter the quantification of entanglement is discussed. Beginning with the pure-state entanglement measure called the entropy of entanglement, we discuss the paradigm of local operations and classical communication (LOCC) and its relation to majorization via Nielsen’s majorization theorem. We then turn to the asymptotic setting and discuss the formation and distillation of entanglement and the related entanglement measures: entanglement cost and distillable entanglement, and we examine the notions of distillability and bound entanglement. This brings us to a more general discussion of entanglement measures and monotones, and their desired properties, during which we present the entanglement of formation and concurrence, squashed entanglement, as well as the tangle and its relation to monogamy of entanglement, but also quantities like the relative entropy of entanglement, the Hilbert-Schmidt measure, and the (logarithmic) negativity. Finally we turn to the construction of entanglement witnesses and their geometric interpretation via the Bertlmann-Narnhofer-Thirring theorem

We discuss particle entanglement in systems of indistinguishable bosons and fermions, in finite Hilbert spaces, with focus on operational measures of the entanglement of particles. We show how to use von Neumann entropy, negativity, and entanglement witnesses in these cases, proving interesting relations. We obtain analytic expressions to quantify the entanglement of particles in homogeneous

D

-dimensional Hamiltonians with certain symmetries.

We present an approach to estimate the operational distinguishability between an entangled state and any separable state directly from measuring an entanglement witness. We show that this estimation also implies bounds on a variety of other well-known entanglement quantifiers. This approach for entanglement estimation is then extended to both the measurement-device-independent scenario and the fully device-independent scenario, where we obtain nontrivial but suboptimal bounds. The procedure requires no numerical optimization and is easy to compute. It offers ways for experimenters to not only detect, but also quantify, entanglement from the standard entanglement witness procedure.

The entanglement of superpositions [Linden et al., Phys. Rev. Lett. 97, 100502 (2006)]is generalized to the multipartite scenario: an upper bound to the multipartite entanglement of a superposition is given in terms of the entanglement of the superposed states and the superposition coefficients. This bound is proven to be tight for a class of states composed of an arbitrary number of qubits. We also extend the result to a large family of quantifiers, which includes the negativity, the robustness of entanglement, and the best separable approximation measure.