Optimal Entanglement Witnesses Research Articles

Entanglement measure based on optimal entanglement witness

Entanglement measure based on optimal entanglement witness

On the structure of mirrored operators obtained from optimal entanglement witnesses

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.

Probing quantum entanglement from magnetic-sublevels populations: beyond spin squeezing inequalities

Spin squeezing inequalities (SSI) represent a major tool to probe quantum entanglement among a collection of few-level atoms, and are based on collective spin measurements and their fluctuations. Yet, for atomic ensembles of spin-j atoms and ultracold spinor gases, many experiments can image the populations in all Zeeman sublevels s=−j,−j+1,…,j, potentially revealing finer features of quantum entanglement not captured by SSI. Here we present a systematic approach which exploits Zeeman-sublevel population measurements in order to construct novel entanglement criteria, and illustrate our approach on ground states of spin-1 and spin-2 Bose-Einstein condensates. Beyond these specific examples, our approach allows one to infer, in a systematic manner, the optimal permutationally-invariant entanglement witness for any given set of collective measurements in an ensemble of d-level quantum systems.

Class of Bell-diagonal entanglement witnesses in C4⊗C4 : Optimization and the spanning property

Two classes of Bell-diagonal indecomposable entanglement witnesses in ${\mathbb{C}}^{4}\ensuremath{\bigotimes}{\mathbb{C}}^{4}$ are considered. Within the first class, we find a generalization of the well-known Choi witness from ${\mathbb{C}}^{3}\ensuremath{\bigotimes}{\mathbb{C}}^{3}$, while the second one contains the reduction map. Interestingly, contrary to the ${\mathbb{C}}^{3}\ensuremath{\bigotimes}{\mathbb{C}}^{3}$ case, the generalized Choi witnesses are no longer optimal. We perform an optimization procedure of finding spanning vectors that eventually gives rise to optimal witnesses. Operators from the second class turn out to be optimal, however, without the spanning property. This analysis sheds light onto the intricate structure of optimal entanglement witnesses.

Optimal Entanglement Witnesses: A Scalable Data-Driven Approach.

Multipartite entanglement is a key resource allowing quantum devices to outperform their classical counterparts, and entanglement certification is fundamental to assess any quantum advantage. The only scalable certification scheme relies on entanglement witnessing, typically effective only for special entangled states. Here, we focus on finite sets of measurements on quantum states (hereafter called quantum data), and we propose an approach which, given a particular spatial partitioning of the system of interest, can effectively ascertain whether or not the dataset is compatible with a separable state. When compatibility is disproven, the approach produces the optimal entanglement witness for the quantum data at hand. Our approach is based on mapping separable states onto equilibrium classical field theories on a lattice and on mapping the compatibility problem onto an inverse statistical problem, whose solution is reached in polynomial time whenever the classical field theory does not describe a glassy system. Our results pave the way for systematic entanglement certification in quantum devices, optimized with respect to the accessible observables.

Detecting entanglement of unknown continuous variable states with random measurements

We develop a scheme for the detection of entanglement in any continuous variable system, by constructing an optimal entanglement witness from random homodyne measurements. To this end, we introduce a set of linear constraints that guarantee the necessary properties of a witness and allow for its optimisation via a semidefinite program. We test our method on the class of squeezed vacuum states and study the efficiency of entanglement detection in general unknown covariance matrices. The results show that we can detect entanglement, including bound entanglement, in arbitrary continuous variable states with fewer measurements than in full tomography. The statistical analysis of our method shows a good robustness to statistical errors in experiments.

Distance between Bound Entangled States from Unextendible Product Bases and Separable States

We discuss the use of the Gilbert algorithm to tailor entanglement witnesses for unextendible product basis bound entangled states (UPB BE states). The method relies on the fact that an optimal entanglement witness is given by a plane perpendicular to a line between the reference state, entanglement of which is to be witnessed, and its closest separable state (CSS). The Gilbert algorithm finds an approximation of CSS. In this article, we investigate if this approximation can be good enough to yield a valid entanglement witness. We compare witnesses found with Gilbert algorithm and those given by Bandyopadhyay–Ghosh–Roychowdhury (BGR) construction. This comparison allows us to learn about the amount of entanglement and we find a relationship between it and a feature of the construction of UPBBE states, namely the size of their central tile. We show that in most studied cases, witnesses found with the Gilbert algorithm in this work are more optimal than ones obtained by Bandyopadhyay, Ghosh, and Roychowdhury. This result implies the increased tolerance to experimental imperfections in a realization of the state.

Nonlinear improvement of qubit-qudit entanglement witnesses

The entanglement witness is an important and experimentally applicable tool for entanglement detection. In this paper, we provide a nonlinear improvement of any entanglement witness for $2\otimes d$ quantum systems. Compared with any existing entanglement witness, the improved separability criterion only needs two more measurements on local observables. Detailed examples are employed to illustrate the efficiency of the nonlinear improvement for general, optimal and non-decomposable entanglement witnesses.

Numerical Construction of Multipartite Entanglement Witnesses

Entanglement in multipartite systems is a key resource for quantum information and communication protocols, making its verification in complex systems a necessity. Because an exact calculation of arbitrary entanglement probes is impossible, we derive and implement a numerical method to construct multipartite witnesses to uncover entanglement in arbitrary systems. Our technique is based on a substantial generalization of the power iteration, an essential tool for computing eigenvalues, and it is a solver for the separability eigenvalue equations, enabling the general formulation of optimal entanglement witnesses. Beyond our rigorous derivation and direct implementation of this method, we also apply our approach to several examples of complexly quantum-correlated states and benchmark its general performance. Consequently, we provide an generally applicable numerical tool for the identification of multipartite entanglement.

Entanglement witnesses for families of the bound entangled three-qubit states

Entanglement witnesses are of special interest for distinguishing between separable and entangled states. For three-qubit states, a family of optimal entanglement witnesses has been introduced by Jafarizadeh et al. [Eur. Phys. J. D 50 (2008) 107]. We give an alternative demonstration for some elements of this family and show that they are able to detect bound entangled states. To illustrate this fact, we consider two families of bound entangled three-qubit states, introduced by Kay [Phys. Rev. A 83 (2011) 020303] and Kye [J. Phys. A, Math. Theor. 48 (2015) 235303], and show that these witnesses detect the entanglement of all states of these two families.

The structural physical approximation conjecture

It was conjectured that the structural physical approximation (SPA) of an optimal entanglement witness is separable (or equivalently, that the SPA of an optimal positive map is entanglement breaking). This conjecture was disproved, first for indecomposable maps and more recently for decomposable maps. The arguments in both cases are sketched along with important related results. This review includes background material on topics including entanglement witnesses, optimality, duality of cones, decomposability, and the statement and motivation for the SPA conjecture so that it should be accessible for a broad audience.

Characterization and properties of weakly optimal entanglement witnesses

We present an analysis of the properties and characteristics of weakly optimal entanglement witnesses, that is witnesses whose expectation value vanishes on at least one product vector. Any weakly optimal entanglement witness can be written as the form of $W^{wopt}=\sigma-c_{\sigma}^{max} I$, where $c_{\sigma}^{max}$ is a non-negative number and $I$ is the identity matrix. We show the relation between the weakly optimal witness $W^{wopt}$ and the eigenvalues of the separable states $\sigma$. Further we give an application of weakly optimal witnesses for constructing entanglement witnesses in a larger Hilbert space by extending the result of [P. Badzi\c{a}g {\it et al}, Phys. Rev. A {\bf 88}, 010301(R) (2013)], and we examine their geometric properties.

Boundary of the set of separable states

Motivated by the separability problem in quantum systems 2⊗4, 3⊗3 and 2⊗2⊗2, we study the maximal (proper) faces of the convex body, S 1 , of normalized separable states in an arbitrary quantum system with finite-dimensional Hilbert space H = H 1 ⊗ H 2 ⊗ ⋯ ⊗ H n . To any subspace V ⊆ H , we associate a face F V of S 1 consisting of all states ρ ∈ S 1 whose range is contained in V . We prove that F V is a maximal face if and only if V is a hyperplane. If V =| ψ 〉 ⊥ , where | ψ 〉 is a product vector, we prove that Dim F V = d 2 − 1 − ∏ ( 2 d i − 1 ) , where d i = Dim H i and d = ∏ d i . We classify the maximal faces of S 1 in the cases 2⊗2 and 2⊗3. In particular, we show that the minimum and the maximum dimension of maximal faces is 6 and 8 for 2⊗2, and 20 and 24 for 2⊗3. The boundary, ∂ S 1 , of S 1 is the union of all maximal faces. When d >6, it is easy to show that there exist full states on ∂ S 1 , i.e. states ρ ∈ ∂ S 1 such that all partial transposes of ρ (including ρ itself) have rank d . Ha and Kye have recently constructed explicit such states in 2⊗4 and 3⊗3. In the latter case, they have also constructed a remarkable family of faces, depending on a real parameter b >0, b ≠1. Each face in the family is a nine-dimensional simplex, and any interior point of the face is a full state. We construct suitable optimal entanglement witnesses for these faces and analyse the three limiting cases b = 0 , 1 , ∞ .

Entanglement witnesses for indistinguishable particles

We study the problem of witnessing entanglement among indistinguishable particles. For this purpose, we derive a set of equations which results in necessary and sufficient conditions for probing multipartite entanglement between arbitrary systems of Bosons or Fermions. The solution of these equations yields the construction of optimal entanglement witnesses for partial and full entanglement in discrete and continuous variable systems. Our approach unifies the verification of entanglement for distinguishable and indistinguishable particles. We provide general solutions for certain observables to study quantum entanglement in systems with different quantum statistics in noisy environments.

Experimental detection of entanglement with optimal-witness families.

We report an experiment in which one determines, with least tomographic effort, whether an unknown two-photon polarization state is entangled or separable. The method measures whole families of optimal entanglement witnesses. We introduce adaptive measurement schemes that greatly speed up the entanglement detection. The experiments are performed on states of different ranks, and we find good agreement with results from computer simulations.

Entanglement detection via continuous quantum channels

For any $N\ensuremath{\bigotimes}N$-type bipartite system with even $Ng2$, two-parameter continuous families of unital block-dephasing quantum channels and of weakly optimal entanglement witnesses are constructed from special forms of a parametrized positive map. The parameter domains of the families are complementary subsets of a convex set and this fact reveals unusual combination rules of quantum channels and entanglement witnesses. These rules make it possible to develop many physically implementable direct entanglement-detection methods that provide a unified framework to accomplish both information-processing and entanglement-detection tasks by means of smooth quantum channels, without any reference to entanglement witnesses.

Recurrent construction of optimal entanglement witnesses for2N-qubit systems

We provide a recurrent construction of entanglement witnesses for a bipartite systems living in a Hilbert space corresponding to $2N$ qubits ($N$ qubits in each subsystem). Our construction provides a method of generalization of the Robertson map that naturally meshes with $2N$-qubit systems, i.e., its structure respects the ${2}^{2N}$ growth of the state space. We prove that for $N>1$ these witnesses are indecomposable and optimal. As a byproduct we provide a family of PPT (Positive Partial Transpose) entangled states.

Disproving the conjecture on the structural physical approximation to optimal decomposable entanglement witnesses

We disprove the conjecture that structural physical approximations to optimal entanglement witnesses are separable states. The conjecture holds true for extremal decomposable entanglement witnesses.

Checking the optimality of entanglement witnesses: an application to structural physical approximations

In 2008, the conjecture that structural physical approximations (SPAs) to optimal entanglement witnesses are separable states (in general unnormalized) was posed. In an attempt to disprove it, Ha and Kye (2012 arXiv:1210.1088v3) proposed a decomposable entanglement witness, whose SPA is entangled, and argued that it is optimal. In this paper, which is based on a comment to the latter work (Augusiak et al 2013 arXiv:1304.2040v1), we show, both analytically and numerically, that this entanglement witness is not optimal, and as such it is not a counterexample to the conjecture. To this end, we make use of a method for checking the optimality of entanglement witnesses developed already in Lewenstein et al (2000 Phys. Rev. A 62 052310), however, hardly exploited so far in the literature.

Optimality of a Class of Entanglement Witnesses for 3⊗3 Systems

Let $\Phi_{t,\pi}: M_3({\mathbb C}) \rightarrow M_3({\mathbb C})$ be a linear map defined by $\Phi_{t,\pi}(A)=(3-t)\sum_{i=1}^3E_{ii}AE_{ii}+t\sum_{i=1}^3E_{i,\pi(i)}AE_{i,\pi(i)}^\dag-A$, where $0\leq t\leq 3$ and $\pi$ is a permutation of $(1,2,3)$. We show that the Hermitian matrix $W_{\Phi_{t,\pi}}$ induced by $\Phi_{t,\pi}$ is an optimal entanglement witness if and only if $t=1$ and $\pi$ is cyclic.