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arXiv:quant-ph/0508043 · Physical Review A

Optimal entanglement witnesses for qubits and qutrits

Reinhold A. Bertlmann, Katharina Durstberger-Rennhofer, Beatrix C. Hiesmayr, Philipp Krammer

Published Nov 28, 2005• 74 citations
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Abstract

We study the connection between the Hilbert-Schmidt measure of entanglement (that is the minimal distance of an entangled state to the set of separable states) and entanglement witness in terms of a generalized Bell inequality which distinguishes between entangled and separable states. A method for checking the nearest separable state to a given entangled one is presented. We illustrate the general results by considering isotropic states, in particular two-qubit and two-qutrit states---and their generalizations to arbitrary dimensions---where we calculate the optimal entanglement witnesses explicitly.

DOI
10.1103/physreva.72.052331

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We study the connection between the Hilbert-Schmidt measure of entanglement (that is the minimal distance of an entangled state to the set of separable states) and entanglement witness in terms of a generalized Bell inequality which distinguishes between entangled and separable states. A method for checking the nearest separable state to a given entangled one is presented. We illustrate the general results by considering isotropic states, in particular two-qubit and two-qutrit states---and their generalizations to arbitrary dimensions---where we calculate the optimal entanglement witnesses explicitly.
A new method is developed for obtaining entanglement witnesses (EW) and Bell operators for n-qubit state. Observing that both EW and Bell operators must have nonzero overlap with the given state we suggest constructing them using some of the terms that appear in the Hilbert-Schmidt (HS) decomposition of the state (not necessarily with the same coefficients). We illustrate the method for several 3 and 4 qubits states, and show improvement of the entanglement region compared with known results. Since the Bell operator and the EW may be restricted to few HS terms, separability/nonseparability of the quantum state can be checked by measuring only these terms.
The use of entanglement witness (EW) for non-full separability and the Bell operator for non-local hidden-variables (LHV) model are analyzed by relating them to the Hilbert-Schmidt (HS) decomposition of n-qubits states and these methods are applied explicitly to some 3 and 4 qubits states. EW for non-full separability (fs) is given by fs parameter minus operator G where the choice of G in the HS decomposition leads to the fs parameter and to the condition for non-separability by using criterions which are different from those used for genuine entanglement. We analyze especially entangled states with probability p mixed with white noise with probability 1-p and find the critical value p(crit.) above which the system is not fully separable. As the choice of EW might not be optimal we add to the analysis of EW explicit construction of fully separable density matrix and find the critical value p below which the system is fully separable. If the two values for p coincide we conclude that this parameter gives the optimal result. Such optimal results are obtained in the present work for some 3 and 4 qubits entangled states mixed with white noise. The use of partial-transpose (PT ) (say relative to qubit A) gives also p value above which the system is not fully separable. The use of EW gives better results (or at least equal) than those obtained by PT .
The basic question that is addressed in this paper is finding the closest separable state for a given entangled state, measured with the Hilbert-Schmidt distance. While this problem is in general very difficult, we show that the following strongly related problem can be solved: find the Hilbert-Schmidt distance of an entangled state to the set of all partially transposed states. We prove that this latter distance can be expressed as a function of the negative eigenvalues of the partial transpose of the entangled state, and show how it is related to the distance of a state to the set of positive, partially transposed (PPT) states. We illustrate this by calculating the closest biseparable state to the W state and give a simple and very general proof for the fact that the set of W-type states is not of measure zero. Next we show that all surfaces with states whose partial transposes have constant minimal negative eigenvalue are similar to the boundary of PPT states. We illustrate this with some examples on bipartite qubit states, where contours of constant negativity are plotted on two-dimensional intersections of the complete state space.
An entanglement witness (EW) is an operator that allows the detection of entangled states. We give necessary and sufficient conditions for such operators to be optimal, i.e., to detect entangled states in an optimal way. We show how to optimize general EW, and then we particularize our results to the nondecomposable ones; the latter are those that can detect positive partial transpose entangled states (PPTES's). We also present a method to systematically construct and optimize this last class of operators based on the existence of ``edge'' PPTES's, i.e., states that violate the range separability criterion [Phys. Lett. A 232, 333 (1997)] in an extreme manner. This method also permits a systematic construction of nondecomposable positive maps (PM's). Our results lead to a sufficient condition for entanglement in terms of nondecomposable EW's and PM's. Finally, we illustrate our results by constructing optimal EW acting on $H={C}^{2}\ensuremath{\bigotimes}{C}^{4}.$ The corresponding PM's constitute examples of PM's with minimal ``qubit'' domains, or---equivalently---minimal Hermitian conjugate codomains.
We show how to detect entangled, bound entangled and separable bipartite quantum states of arbitrary dimension and mixedness using geometric entanglement witnesses. These witnesses are constructed using properties of the Hilbert–Schmidt geometry and can be shifted along parameterized lines. The involved conditions are simplified using Bloch decompositions of operators and states. As an example we determine the three different types of states for a family of two-qutrit states that is part of the 'magic simplex', i.e. the set of Bell-state mixtures of arbitrary dimension.
Abstract Entanglement witnesses (EWs) are a versatile tool to detect entangled states and characterize related properties of entanglement in quantum information theory. The verification of entangled states via EWs relies on the fact that separable states form a convex set; this also means that the framework presented by EWs generally applies to other quantum resources where free resources contain the convexity. A witness W corresponds to an observable satisfying Tr [ W σ sep ] ⩾ 0 for all separable states σ sep ; entangled states are detected once the inequality is violated. Recently, mirrored EWs have been introduced by showing that there exist non-trivial upper bounds to EWs, u W ⩾ Tr [ W σ sep ] ⩾ 0. An upper bound to a witness W signifies the existence of the other one M , called a mirrored EW, such that W + M = u W I ⊗ I . The framework of mirrored EWs shows that a single EW can be even more useful, as it can detect a larger set of entangled states by lower and upper bounds. In this work, we develop and investigate mirrored EWs for multipartite qubit states and also for high-dimensional systems, to find the efficiency and effectiveness of mirrored EWs in detecting entangled states. We provide mirrored EWs for n -partite GHZ states, graph states such as two-colorable states and tripartite bound entangled states. We also show that optimal EWs can be reflected with each other. For bipartite systems, we present mirrored EWs for existing optimal EWs and also construct a mirrored pair of optimal EWs in dimension three. Finally, we generalize mirrored EWs such that a pair of EWs can be connected by another EW, i.e. W + M = K is also an EW. Our results enhance the capability of EWs to detect a larger set of entangled states in multipartite and high-dimensional quantum systems.