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arXiv:1012.3786 · Journal of Physics A Mathematical and Theoretical

A note on the optimality of decomposable entanglement witnesses and completely entangled subspaces

Remigiusz Augusiak, Jordi Tura, Maciej Lewenstein

Published Apr 27, 2011• 39 citations
arXivHTMLPDF

Abstract

Entanglement witnesses (EWs) constitute one of the most important entanglement detectors in quantum systems. Nevertheless, their complete characterization, in particular with respect to the notion of optimality, is still missing, even in the decomposable case. Here we show that for any qubit–qunit decomposable EW (DEW) W , the three statements are equivalent: (i) the set of product vectors obeying ⟨ e , f | W | e , f ⟩ = 0 spans the corresponding Hilbert space, (ii) W is optimal, and (iii) W = Q Γ , with Q denoting a positive operator supported on a completely entangled subspace (CES) and Γ standing for the partial transposition. While implications (i)⇒(ii) and (ii)⇒(iii) are known, here we prove that (iii) implies (i). This is a consequence of a more general fact saying that product vectors orthogonal to any CES in span after partial conjugation the whole space. On the other hand, already in the case of the Hilbert space, there exist DEWs for which (iii) does not imply (i). Consequently, either (i) does not imply (ii) or (ii) does not imply (iii), and the above transparent characterization, obeyed by qubit–qunit DEWs, does not hold in general.

DOI
10.1088/1751-8113/44/21/212001

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Entanglement witnesses (EWs) constitute one of the most important entanglement detectors in quantum systems. Nevertheless, their complete characterization, in particular with respect to the notion of optimality, is still missing, even in the decomposable case. Here we show that for any qubit–qunit decomposable EW (DEW) W , the three statements are equivalent: (i) the set of product vectors obeying ⟨ e , f | W | e , f ⟩ = 0 spans the corresponding Hilbert space, (ii) W is optimal, and (iii) W = Q Γ , with Q denoting a positive operator supported on a completely entangled subspace (CES) and Γ standing for the partial transposition. While implications (i)⇒(ii) and (ii)⇒(iii) are known, here we prove that (iii) implies (i). This is a consequence of a more general fact saying that product vectors orthogonal to any CES in span after partial conjugation the whole space. On the other hand, already in the case of the Hilbert space, there exist DEWs for which (iii) does not imply (i). Consequently, either (i) does not imply (ii) or (ii) does not imply (iii), and the above transparent characterization, obeyed by qubit–qunit DEWs, does not hold in general.
One of the unsolved problems in the characterization of the optimal entanglement witnesses is the existence of optimal witnesses acting on bipartite Hilbert spaces ${\mathcal{H}}_{m,n}={\mathbb{C}}^{m}\ensuremath{\bigotimes}{\mathbb{C}}^{n}$ such that the product vectors obeying $\ensuremath{\langle}e,f|W|e,f\ensuremath{\rangle}=0$ do not span Hm,n{\mathcal{H}}_{m,n}Hm,n​. So far, the only known examples of such witnesses were found among indecomposable witnesses, one of them being the witness corresponding to the Choi map. However, it remains an open question whether decomposable witnesses exist without the property of spanning. Here we answer this question affirmatively, providing systematic examples of such witnesses. Then, we generalize some of the recently obtained results on the characterization of $2\ensuremath{\bigotimes}n$ optimal decomposable witnesses [R. Augusiak et al., J. Phys. A 44, 212001 (2011)] to finite-dimensional Hilbert spaces Hm,n{\mathcal{H}}_{m,n}Hm,n​ with $m,n\ensuremath{\ge}3$.
Entanglement witnesses (EWs) are a versatile tool in the verification of entangled states. The framework of mirrored EW doubles the power of a given EW by introducing its twin-a mirrored EW-whereby two EWs related by mirroring can bound the set of separable states more efficiently. In this work, we investigate the relation between the EWs and its mirrored ones, and present a conjecture which claims that the mirrored operator obtained from an optimal EW is either a positive operator or a decomposable EW, which implies that positive-partial-transpose entangled states, also known as the bound entangled states, cannot be detected. This conjecture is reached by studying numerous known examples of optimal EWs. However, the mirrored EWs obtained from the non-optimal ones can be non-decomposable as well. We also show that mirrored operators obtained from the extremal decomposable witnesses are positive semi-definite. Interestingly, the witnesses that violate the well known conjecture of Structural Physical Approximation, do satisfy our conjecture. The intricate relation between these two conjectures is discussed and it reveals a novel structure of the separability problem.
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 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.
We investigate various properties of multipartite block-positive operators, decomposable EWs (DEWs), and non-decomposable EWs. We provide a necessary and sufficient condition to construct a special DEW using the multipartite GHZ state. For multipartite DEW, we explicitly characterize the supremum and infimum of maximum and minimum eigenvalues, as well as the trace of its square. We also derive other results concerning NDEW, eigenvalues of 2×n2 \times n2×n EWs and the corresponding physical implications. Furthermore, we investigate the tightness of inequalities of these eigenvalues with examples.
A general formulation of the problem of detection for a pair of two cones is presented. The special case is the detection of entangled states by entanglement witnesses. Having defined what means "to detect", one can identify the subset of elements, which detect optimally. I will present the properties of this set for a general pair of cones. In particular, I prove the generalization of the theorem of Lewenstein, Krauss, Cirac, Horodecki. The entanglement witness WWW is optimall if the set of product vectors {ϕ⊗ψ:⟨ϕ⊗ψ∣W∣ϕ⊗ψ⟩=0}\{ϕ\otimes ψ: \langle ϕ\otimes ψ| W | ϕ\otimes ψ\rangle = 0\}{ϕ⊗ψ:⟨ϕ⊗ψ∣W∣ϕ⊗ψ⟩=0} spans the whole Hilbert space of a system. There exist optimall entangled witness, which do not fullfill this property. It is closely related to some geometrical properties of the boundary of the set of entanglement witnesses and it is possible to say something more about location of such extraordinary states.