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 \bra{e,f}W\ket{e,f}=0 spans the corresponding Hilbert space, (ii) W is optimal, (iii) W=Q^Γ with Q denoting a positive operator supported on a completely entangled subspace (CES) and Γstanding for the partial transposition. While, implications $(i)\Rightarrow(ii)$ and $(ii)\Rightarrow(iii)$ 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 C^{2}\otimes C^{n} span after partial conjugation the whole space. On the other hand, already in the case of C^{3}\otimes C^{3} 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.
We disprove the conjecture that structural physical approximations to optimal entanglement witnesses are separable states. The conjecture holds true for extremal decomposable entanglement witnesses.
One of the unsolved problems in the characterization of the optimal entanglement witnesses is the existence of optimal witnesses acting on bipartite Hilbert spaces H_{m,n}=C^m\otimes C^n such that the product vectors obeying <e,f|W|e,f>=0 do not span H_{m,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 s…
It is well known that the support of an optimal decomposable entanglement witness is completely entangled. We add two more necessary conditions for the optimality: The orthogonal complement of the support must have a nonzero product vector; another one will be given in terms of related faces of a convex cone. With these necessary conditions, we show that there exist examples of non-optimal decomposable entanglement witnesses which are the partial transposes of positive semi-definite matrices supported on completely entangled spaces, whenever both of the local dimensions are greater than or equ…
The study of Entanglement of Formation of a mixed state of a bipartite system in high-dimensional Hilbert space is not easy in general. So, we focus on determining the amount of entanglement for a bipartite mixed state based on the concept of decomposable optimal entanglement witness (DOEW), that can be calculated as a minimum distance of an entangled state from the edge of positive partial transpose (PPT) states which has the most negative (positive) expectation value for non-PPT (bound) entangled states. We have constructed DOEWs based on the convex optimization method, then by using of it w…
We examine various notions related with the optimality for entanglement witnesses arising from Choi type positive linear maps. We found examples of optimal entanglement witnesses which are non-decomposable, but which are not `non-decomposable optimal entanglement witnesses' in the sense of [M. Lewenstein, B. Kraus, J. Cirac, and P. Horodecki, Phys. Rev. A 62, 052310 (2000)]. We suggest to use the term `PPTES witness' and `optimal PPTES witness' in the places of `non-decomposable entanglement witness' and `non-decomposable optimal entanglement witnesses' in order to avoid possible confusion. We…
We investigate the issue of finding common entanglement witness for certain class of states and extend this study to the case of Schmidt number witnesses. We also introduce the notion of common decomposable and non-decomposable witness operators which is specially useful for constructing a common witness where one of the entangled states is with a positive partial transpose. Our approach is illustrated with the help of suitable examples of qutrit systems.
It has been shown that finding generic Bell states diagonal entanglement witnesses (BDEW) for $d_{1}\otimes d_{2}\otimes ....\otimes d_{n}$ systems exactly reduces to a linear programming if the feasible region be a polygon by itself and approximately obtains via linear programming if the feasible region is not a polygon. Since solving linear programming for generic case is difficult, the multi-qubits, $2\otimes N$ and $3 \otimes 3$ systems for the special case of generic BDEW for some particular choice of their parameters have been considered. In the rest of this paper we obtain the optimal n…
It is shown that entanglement witnesses constructed via the family of spectral conditions are decomposable, i.e. cannot be used to detect bound entanglement. It supports several observations that bound entanglement reveals highly non-spectral features.
We address the problem of optimising entanglement witnesses when a limited fixed set of local measurements can be performed on a bipartite system, thus providing a procedure, feasible also for experiments, to detect entangled states using only the statistics of these local measurements. We completely characterize the class of entanglement witnesses of the form $W = P ^Γ$, where $Γ$ denotes partial transposition, that can be constructed from the measurements of the bipartite operators $σ_{x}\otimesσ_{x},$ $σ_{y}\otimesσ_{y}$ and $σ_{z}\otimesσ_{z}$ in the case of two-qubit systems. In particula…
The Gell-Mann $λ$ matrices for Lie algebra su(3) are the natural basis for the Hilbert space of Hermitian operators acting on the states of a three-level system(qutrit). So the construction of EWs for two-qutrit states by using these matrices may be an interesting problem. In this paper, several two-qutrit EWs are constructed based on the Gell-Mann matrices by using the linear programming (LP) method exactly or approximately. The decomposability and non-decomposability of constructed EWs are also discussed and it is shown that the $λ$-diagonal EWs presented in this paper are all decomposable b…
An entanglement witness (EW) is an operator that allows to detect 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 non-decomposable ones; the latter are those that can detect positive partial transpose entangled states (PPTES). We also present a method to systematically construct and optimize this last class of operators based on the existence of ``edge'' PPTES, i.e. states that violate the range separability criterio…
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.
Let $Φ_{t,π}: M_3({\mathbb C}) \rightarrow M_3({\mathbb C})$ be a linear map defined by $Φ_{t,π}(A)=(3-t)\sum_{i=1}^3E_{ii}AE_{ii}+t\sum_{i=1}^3E_{i,π(i)}AE_{i,π(i)}^†-A$, where $0\leq t\leq 3$ and $π$ is a permutation of $(1,2,3)$. We show that the Hermitian matrix $W_{Φ_{t,π}}$ induced by $Φ_{t,π}$ is an optimal entanglement witness if and only if $t=1$ and $π$ is cyclic.
In 2008, the conjecture that structural physical approximations to optimal entanglement witnesses are separable states (in general unnormalized) was posed. In an attempt to disprove it, in [K.-C. Ha and S.-H. Kye, Separable states with unique decompositions, arXiv:1210.1088v3], Ha and Kye proposed a decomposable entanglement witness whose SPA is entangled and argued that it is optimal. In this note, which is based on a comment to the latter work [R. Augusiak et al., Comment on "Separable states with unique decompositions", arXiv:1304.2040v1], we show, both analytically and numerically, that th…
Based on the general form of entanglement witnesses constructed from separable states, we first show a sufficient condition of violating the structural physical approximation (SPA) conjecture [Phys. Rev. A 78, 062105 (2008)]. Then we discuss the SPA conjecture for decomposable entanglement witnesses. Moreover, we make geometric illustrations of the connection between entanglement witnesses and the sets of quantum states, separable states, and entangled states comparing with planes and vectors in Euclidean space.
Recently, X. Qi and J. Hou [Phys. Rev. A 85, 022334 (2012)] provided optimal entanglement witnesses without the spanning property. These witnesses are associated to indecomposable positive linear maps, but it is not checked whether partial transposes of these witnesses are also optimal. We show that partial transposes of these entanglement witnesses have spanning property, and these witnesses are indeed optimal PPTES witnesses (non-decomposable optimal entanglement witnesses).
On the basis of linear programming, new sets of entanglement witnesses (EWs) for 3x3 and 4x4 systems are constructed. In both cases, the constructed EWs correspond to the hyperplanes contacting, without intersecting, the related feasible regions at line segments and restricted planes respectively. Due to the special property of the contacting area between the hyper-planes and the feasible regions, the corresponding hyper-planes can be turned around the contacting area throughout a bounded interval and hence create an infinite number of EWs. As these EWs are able to detect entanglement of some …
We propose a generalized form of optimal teleportation witness to demonstrate their importance in experimental detection of the larger set of entangled states useful for teleportation in higher dimensional systems. The interesting properties of our witness reveal that teleportation witness can be used to characterize mixed state entanglement using Schmidt numbers. Our results show that while every teleportation witness is also a entanglement witness, the converse is not true. Also, we show that a hermitian operator is a teleportation witness iff it is a decomposable entanglement witness. In ad…
Entanglement in symmetric quantum states and the theory of copositive matrices are intimately related concepts. For the simplest symmetric states, i.e., the diagonal symmetric (DS) states, it has been shown that there exists a correspondence between exceptional (non-exceptional) copositive matrices and non-decomposable (decomposable) Entanglement Witnesses (EWs). Here we show that EWs of symmetric, but not DS, states can also be constructed from extended copositive matrices, providing new examples of bound entangled symmetric states, together with their corresponding EWs, in arbitrary odd dime…
Two kinds of Bell-states diagonal (BSD) entanglement witnesses (EW) are constructed by using the algebra of Dirac $γ$ matrices in the space-time of arbitrary dimension $d$, where the first kind can detect some BSD relativistic and non-relativistic $m$-partite multispinor bound entangled states in Hilbert space of dimension $2^{m\lfloor d/2\rfloor}$, including the bipartite Bell-type and iso-concurrence type states in the four-dimensional space-time ($d=4$). By using the connection between Hilbert-Schmidt measure and the optimal EWs associated with states, it is shown that as far as the spin qu…
We construct optimal PPTES witnesses to detect $3\otimes 3$ PPT entangled edge states of type $(6,8)$ constructed recently \cite{kye_osaka}. To do this, we consider positive linear maps which are variants of the Choi type map involving complex numbers, and examine several notions related to optimality for those entanglement witnesses. Through the discussion, we suggest a method to check the optimality of entanglement witnesses without the spanning property.
A generic algorithm is developed to reduce the problem of obtaining linear and nonlinear entanglement witnesses of a given quantum system, to convex optimization problem. This approach is completely general and can be applied for the entanglement detection of any N-partite quantum system. For this purpose, a map from convex space of separable density matrices to a convex region called feasible region is defined, where by using exact convex optimization method, the linear entanglement witnesses can be obtained from polygonal shape feasible regions, while for curved shape feasible regions, envel…
We characterize a convex subset of entanglement witnesses for two qutrits. Equivalently, we provide a characterization of the set of positive maps in the matrix algebra of 3 x 3 complex matrices. It turns out that boundary of this set displays elegant representation in terms of SO(2) rotations. We conjecture that maps parameterized by rotations are optimal, i.e. they provide the strongest tool for detecting quantum entanglement. As a byproduct we found a new class of decomposable entanglement witnesses parameterized by improper rotations from the orthogonal group O(2).
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 $W$ is optimall if the set of product vectors $\{ϕ\otimes ψ: \langle ϕ\otimes ψ| W | ϕ\otimes ψ\rangle = 0\}$ spans the whole Hilbert…
Any two-qubit state can be represented geometrically by a steering ellipsoid inside the Bloch sphere. We extend this approach to represent any block positive two-qubit operator $B$. We derive a novel classification scheme based on the positivity of $\det B$ and $\det B^\mathrm{T_B}$; this shows that any ellipsoid inside the Bloch sphere must represent either a two-qubit state or a two-qubit entanglement witness. We focus on such witnesses and their corresponding ellipsoids, finding that properties such as witness optimality are naturally manifest in this geometric representation.
One of the problems concerning entanglement witnesses (EWs) is the construction of them by a given set of operators. Here several multi-qubit EWs called stabilizer EWs are constructed by using the stabilizer operators of some given multi-qubit states such as GHZ, cluster and exceptional states. The general approach to manipulate the multi-qubit stabilizer EWs by exact(approximate) linear programming (LP) method is described and it is shown that the Clifford group play a crucial role in finding the hyper-planes encircling the feasible region. The optimality, decomposability and non-decomposabil…
A new class of entanglement witnesses (EWs) called reduction type entanglement witnesses is introduced, which can detect some multi-qudit entangeled states including PPT ones with Hilbert space of dimension $d_{_{1}}\otimes d_{_{2}}\otimes...\otimes d_{_{n}}$. The novelty of this work comes from the fact that the feasible regions turn out to be convex polygons, hence the manipulation of these EWs reduces to linear programming which can be solved \emph{exactly} by using simplex method. The decomposability and non-decomposability of these EWs are studied and it is shown that it has a close conne…
In this expository note, we explain facial structures for the convex cones consisting of positive linear maps, completely positive linear maps, decomposable positive linear maps between matrix algebras, respectively. These will be applied to study the notions of entangled edge states with positive partial transposes and optimality of entanglement witnesses.
In this paper, we present a characterization of optimal entanglement witnesses in terms of positive maps and then provide a general method of checking optimality of entanglement witnesses. Applying it, we obtain new indecomposable optimal witnesses which have no spanning property. These also provide new examples which support a recent conjecture saying that the so-called structural physical approximations to optimal positive maps (optimal entanglement witnesses) give entanglement breaking maps (separable states).
The question under what conditions different witnesses may detect some entangled states simultaneously is answered for both finite- and infinite-dimensional bipartite systems. Finite many different witnesses can detect some entangled states simultaneously if and only if their convex combinations are still witnesses; they can not detect any entangled state simultaneously if and only if the set of their convex combinations contains a positive operator. For two witnesses $W_1$ and $W_2$, some more can be said: (1) $W_1$ and $W_2$ can detect the same set of entangled states if and only if they are…
Here we consider a class of $2\otimes2\otimes d$ chessboard density matrices starting with three-qubit ones which have positive partial transposes with respect to all subsystems. To investigate the entanglement of these density matrices, we use the entanglement witness approach. For constructing entanglement witnesses (EWs) detecting these density matrices, we attempt to convert the problem to an exact convex optimization problem. To this aim, we map the convex set of separable states into a convex region, named feasible region, and consider cases that the exact geometrical shape of feasible r…
We study extremal entanglement witnesses on a bipartite quantum system. We define the cone of witnesses as the dual of the set of separable density matrices, thus $\textrm{Tr}\,Ωρ\geq 0$ when $Ω$ is a witness and $ρ$ a pure product state, $ρ=ψψ^{\dagger}$ with $ψ=ϕ\otimesχ$. The set of witnesses of unit trace is a compact convex set, defined by its extremal points. The expectation value $f(ϕ,χ)=\mathrm{Tr}\,Ωρ$ as a function of $ϕ$ and $χ$ is a nonnegative biquadratic form. Every zero of $f(ϕ,χ)$ imposes real-linear constraints on $f$ and $Ω$. The Hessian matrix at the zero must be nonnegative…
We consider the characterization of entanglement depth in a quantum many-body system from the device-independent perspective; i.e., certifying how many particles are genuinely entangled without relying on assumptions on the system itself nor on the measurements performed. We obtain device-independent witnesses of entanglement depth (DIWEDs) using the Bell inequalities introduced in [J. Tura et al., Science 344, 1256 (2014)] and compute their $k$-producibility bounds. To this end, we present two complementary methods: First, a variational one, yielding a possibly optimal $k$-producible state. S…
Detecting Entanglement by State Preparation and Local Measurements
Aalborg University（奥尔堡大学）；Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院）；Birla Institute of Technology Mesra（比尔拉理工学院梅斯拉分校）；Nicolaus Copernicus University（尼古拉·哥白尼大学）
From the physical point of view entanglement witnesses define a universal tool for analysis and classification of quantum entangled states. From the mathematical point of view they provide highly nontrivial generalization of positive operators and they find elegant correspondence with the theory of positive maps in matrix algebras. We concentrate on theoretical analysis of various important notions like (in)decomposability, atomicity, optimality, extremality and exposedness. Several methods of construction are provided as well. Our discussion is illustrated by many examples enabling the reader…
