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Physical Review A

Determining the existence of the common entanglement witnesses for some entangled states

Yu-Chun Wu, Guang‐Can Guo

Published May 24, 2007• 13 citations
Publisher

Abstract

Entanglement witnesses (EWs) are nonpositive Hermitian operators which can detect the presence of entanglement. In this paper, we prove that for a given pair of entangled states ${\ensuremath{\rho}}_{1}$ and ${\ensuremath{\rho}}_{2}$ an EW, $W$, common to both of them exists if and only if for all $0\ensuremath{\leqslant}\ensuremath{\lambda}\ensuremath{\leqslant}1$, $\ensuremath{\lambda}{\ensuremath{\rho}}_{1}+(1\ensuremath{-}\ensuremath{\lambda}){\ensuremath{\rho}}_{2}$ is an entangled state. Based on this result, we find a sufficient condition for the presence of entanglement. We then show that an EW which can detect all bell states simultaneously does not exist.

DOI
10.1103/physreva.75.052333

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Entanglement witnesses (EWs) are nonpositive Hermitian operators which can detect the presence of entanglement. In this paper, we prove that for a given pair of entangled states \ensuremathρ1{\ensuremath{\rho}}_{1}\ensuremathρ1​ and \ensuremathρ2{\ensuremath{\rho}}_{2}\ensuremathρ2​ an EW, WWW, common to both of them exists if and only if for all 0\ensuremath⩽\ensuremathλ\ensuremath⩽10\ensuremath{\leqslant}\ensuremath{\lambda}\ensuremath{\leqslant}10\ensuremath⩽\ensuremathλ\ensuremath⩽1, \ensuremathλ\ensuremathρ1+(1\ensuremath−\ensuremathλ)\ensuremathρ2\ensuremath{\lambda}{\ensuremath{\rho}}_{1}+(1\ensuremath{-}\ensuremath{\lambda}){\ensuremath{\rho}}_{2}\ensuremathλ\ensuremathρ1​+(1\ensuremath−\ensuremathλ)\ensuremathρ2​ is an entangled state. Based on this result, we find a sufficient condition for the presence of entanglement. We then show that an EW which can detect all bell states simultaneously does not exist.
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.
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.
Quantum entanglement is a core resource for quantum information processing, and entanglement witnesses are indispensable tools for detection and verification. This paper first clarifies some already-known properties of entanglement witnesses and the Bell State. By applying mathematical tools such as Kronecker product and matrix algebra, this paper analyzes elements in the Hermitian matrix and discusses the conditions for it to form an entanglement witness. This paper also considers both necessary and sufficient conditions of entanglement witness to impose restrictions on elements and investigated several inequality properties of two-qubit entanglement witness to detect Bell states.
A necessary precondition for secure quantum key distribution is that sender and receiver can prove the presence of entanglement in a quantum state that is effectively distributed between them. In order to deliver this entanglement proof one can use the class of entanglement witness (EW) operators that can be constructed from the available measurements results. This class of EWs can be used to provide a necessary and sufficient condition for the existence of quantum correlations even when a quantum state cannot be completely reconstructed. The set of optimal EWs for two well-known entanglement-based (EB) schemes, the six-state and the four-state EB protocols, has been obtained recently [M. Curty et al., Phys. Rev. Lett. 92, 217903 (2004).] Here we complete these results, now showing specifically the analysis for the case of prepare and measure (PM) schemes. For this, we investigate the signal states and detection methods of the four-state and the two-state PM schemes. For each of these protocols we obtain a reduced set of EWs. More importantly, each set of EWs can be used to derive a necessary and sufficient condition to prove that quantum correlations are present in these protocols.
The question of under what conditions different witnesses (e.g., W1,W2{W}_{1},{W}_{2}W1​,W2​) may detect some common entangled states [i.e., there exists some state $\ensuremath{\rho}$ so that $\mathrm{Tr}({W}_{1}\ensuremath{\rho})<0$ and $\mathrm{Tr}({W}_{2}\ensuremath{\rho})<0$] is answered for both finite-dimensional and infinite-dimensional bipartite systems. Finitely many different witnesses W1,W2,…,Wn{W}_{1},{W}_{2},\dots{},{W}_{n}W1​,W2​,…,Wn​ can detect some common entangled states if and only if ${\ensuremath{\sum}}_{i=1}^{n}{d}_{i}{W}_{i}$ is still a witness for any nonnegative numbers d1,d2,…,dn{d}_{1},{d}_{2},\dots{},{d}_{n}d1​,d2​,…,dn​ with ${\ensuremath{\sum}}_{i=1}^{n}{d}_{i}=1$; they cannot detect any common entangled state if and only if ${\ensuremath{\sum}}_{i=1}^{n}{c}_{i}{W}_{i}$ is a positive operator for some nonnegative numbers c1,c2,…,cn{c}_{1},{c}_{2},\dots{},{c}_{n}c1​,c2​,…,cn​ with ${\ensuremath{\sum}}_{i=1}^{n}{c}_{i}=1$. For two witnesses W1{W}_{1}W1​ and W2{W}_{2}W2​ more can be said. First, W1{W}_{1}W1​ and W2{W}_{2}W2​ can detect the same set of entangled states if and only if W1=aW2{W}_{1}=a{W}_{2}W1​=aW2​ for some number a>0a>0a>0. Second, W2{W}_{2}W2​ can detect more entangled states than W1{W}_{1}W1​ can if and only if W1=aW2+D{W}_{1}=a{W}_{2}+DW1​=aW2​+D for some number a>0a>0a>0 and a positive operator DDD. As an application, some characterizations of the optimal witnesses are given and some structural properties of the decomposable optimal witnesses are presented.
We present a systematic investigation of the spectral properties of entanglement witnesses (EWs). Specifically, we analyze the infimum and supremum of the largest eigenvalue, the smallest eigenvalue, the negativity (defined as the absolute value of the sum of negative eigenvalues), and the squared Frobenius norm of a unit-trace (normalized) entanglement witness, along with the conditions under which these values are attained. Our study provides distinct characterizations for decomposable (DEWs) and nondecomposable entanglement witnesses (NDEWs). While these two classes share many spectral similarities, we reveal a fundamental divergence by proving that the infimum of the smallest eigenvalue can be attained by DEWs, yet remains strictly unattainable for all NDEWs. We apply the results to provide necessary conditions for an EW to possess a mirrored EW. Furthermore, we demonstrate the superior detection capability of NDEWs by proving that any non-positive-partial-transpose state beyond the two-qubit and qubit-qutrit systems can be detected by an NDEW.