Abstract Entanglement witnesses (EWs) based on a restricted set of local measurements are experimentally more accessible. For two-qubit systems, we provide a complete characterization of extremal decomposable EWs constructed from local measurements with an anti-diagonal correlation structure. These witnesses can be employed for the detection of gravitational entanglement. We then extend this characterization to higher-dimensional quantum systems and explicitly construct the corresponding extremal decomposable witnesses using generalized Gell-Mann matrices.
Abstract Entanglement witnesses (EWs) based on a restricted set of local measurements are experimentally more accessible. For two-qubit systems, we provide a complete characterization of extremal decomposable EWs constructed from local measurements with an anti-diagonal correlation structure. These witnesses can be employed for the detection of gravitational entanglement. We then extend this characterization to higher-dimensional quantum systems and explicitly construct the corresponding extremal decomposable witnesses using generalized Gell-Mann matrices.
We address the problem of optimizing 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 $W={P}^{\mathrm{\ensuremath{\Gamma}}}$, where $\mathrm{\ensuremath{\Gamma}}$ denotes partial transposition, that can be constructed from the measurements of the bipartite operators ${\ensuremath{\sigma}}_{x}\ensuremath{\bigotimes}{\ensuremath{\sigma}}_{x}, {\ensuremath{\sigma}}_{y}\ensuremath{\bigotimes}{\ensuremath{\sigma}}_{y}$, and ${\ensuremath{\sigma}}_{z}\ensuremath{\bigotimes}{\ensuremath{\sigma}}_{z}$ in the case of two-qubit systems. In particular, we consider all possible extremal decomposable witnesses within the considered class that can be defined from this set of measurements. Finally, we discuss possible extensions to higher-dimension bipartite systems when the set of available measurements is characterized by the generalized Gell-Mann matrices. We provide several examples of entanglement witnesses, both decomposable and indecomposable, that can be constructed with these limited resources.
We experimentally demonstrate a method for detection of entanglement via construction of entanglement witnesses from a limited fixed set of local measurements ( M ). Such a method does not require a priori knowledge about the form of the entanglement witnesses. It is suitable for a scenario where a full state tomography is not available, but the only resource is a limited set of M . We demonstrate the method on pure two-qubit entangled states and mixed two-qubit entangled states, which emerge from photonic implementation of controllable quantum noisy channels. The states we select are motivated by realistic experimental conditions, and we confirm it works well for both cases. Furthermore, possible generalizations to higher-dimensional bipartite systems have been considered, which can potentially detect both decomposable and indecomposable entanglement witnesses. Our experimental results show perfect validity of the method, which indicates that even a limited set of local measurements can be used for quick entanglement detection and further provide a practical test bed for experiments with entanglement witnesses.
Entanglement witnesses (EWs) are a collection of observables that can characterize separable states and, experimentally, estimating EWs can verify entangled states. In this work, we show that a fixed measurement setting on a multipartite entangled state, which we introduce as a network state for the purpose, can estimate EWs. Namely, entangled states can be fully verified in a measurement-based manner, in which experimenters do not necessarily change measurement settings. We present a fixed measurement setting and network states for estimating decomposable EWs, equivalent to the partial transpose criteria. We also consider non-decomposable EWs that detect bound entangled states beyond the partial transpose criteria. The results can be extended to multipartite states such as graph states, a resource for measurement-based quantum computing, and readily applied to distributed settings such as quantum metrology or sensor networks where multipartite entangled states are resourceful.
Abstract We investigate the inertia (i.e., the array of numbers of negative, zero and positive eigenvalues of an Hermitian matrix) of decomposable entanglement witnesses (EWs). We show that the 2 × n and two-qutrit decomposable EWs have the same inertias as those of non-positive-transpose (NPT) EWs. We also show that if an m × n EW W has inertia (p, a p , mn − p − a p ) with p≥1, then for every integer b ∈ [0, a p ], then we can find an EW W b such that InW b = (p, b, mn − p − b). If W is a decomposable (resp. NPT) EW, then we can choose W b as also a decomposable (resp. NPT) EW. We further show that the m × n decomposable EW with the maximum number of negative eigenvalues can be chosen as an NPT EW. Then we explicitly characterize the 2 × 3 EWs, and decomposable EWs P Γ + Q with positive semidefinite matrices P of rank one and Q. We also show that a 2 × 4 non-decomposable EW has no inertia (3, 2, 3). Then we show some properties of a 2 × 4 non-decomposable EW of inertia (2, 3, 3), if it exists.
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 dimensions.
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.