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arXiv:0705.2665 · Physical Review A

Multiqubit entanglement witness

Lin Chen, Yi-Xin Chen

Published Aug 24, 2007• 22 citations
arXivHTMLPDF

Abstract

We introduce a feasible method of constructing the entanglement witness that detects genuine multiqubit entanglement in the vicinity of a given pure multiqubit state. We illustrate our method in the scenario of constructing the witnesses for the multiqubit states that are most often investigated theoretically and experimentally. It is shown that our method can construct the effective witnesses for experiments. We also investigate the entanglement detection of symmetric states and mixed states.

DOI
10.1103/physreva.76.022330

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We introduce a feasible method of constructing the entanglement witness that detects genuine multiqubit entanglement in the vicinity of a given pure multiqubit state. We illustrate our method in the scenario of constructing the witnesses for the multiqubit states that are most often investigated theoretically and experimentally. It is shown that our method can construct the effective witnesses for experiments. We also investigate the entanglement detection of symmetric states and mixed states.
With an aim to detect the genuine entanglement of a mixed state at the nearby pure state, we reflect a practicable approach concerning the construction of witness and express expectations to detect separable, bipartite states, and multiqubit entangled states. We introduce the related concept of the density matrix, multiqubit state, entangled state entanglement witness, etc. In addition, we give unambiguous definitions of separable states, entangled states, and genuinely entangled states. Therefore, we are able to detect the genuine entanglement of a mixed state at the nearby pure state employing the witness that is constructed for the multiqubit states.
We present general numerical methods to construct witness operators for entanglement detection and estimation of the fidelity. Our methods are applied to detecting entanglement in the vicinity of a six-qubit Dicke state with three excitations and also to further entangled symmetric states. All our witnesses are designed to keep the measurement effort small. We also present general results on the efficient local decomposition of permutationally invariant operators, which makes it possible to measure projectors to symmetric states efficiently.
We present general numerical methods to construct witness operators for entanglement detection and estimation of the fidelity. Our methods are applied to detecting entanglement in the vicinity of a six-qubit Dicke state with three excitations and also to further entangled symmetric states. All our witnesses are designed to keep the measurement effort small. We also present general results on the efficient local decomposition of permutationally invariant operators, which makes it possible to measure projectors to symmetric states efficiently.
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.
Quantum entanglement, as an indispensable resource in quantum communication and quantum computation, is widely used in the field of quantum information. However, people's understanding on entanglement is quite limited both theoretically and experimentally. How to determine whether a given quantum state is entangled is still an important task. The entanglement witness is a kind of special self-adjoint operator, it can be used to determine whether a quantum state is an entangled state. This provides a new direction for the determination of entangled states. Entanglement witness has its own unique characteristics in various kinds of entanglement criterion. It is the most effective tool for detecting multipartite entanglement, and the most useful method to detect entanglement in experiments. In the background of quantum theory, we use theory of operators to make a thorough and systematic study of the construction of entanglement witness in this paper. First, from the definition of an entanglement witness, a general method is given to construct an entanglement witness. It is proved that when the maximal expectation CA of an observable A in the separable pure states is strictly less than its biggest eigenvalue max(A), the operator WC=CI-A is an entanglement witness provided that CA C max(A). Although the entanglement witness WCA can detect more entangled states than WC, but it is difficult to calculate the exact value of CA, and the estimate of the upper bound of CA is easier. Therefore, it is more convenient to construct entanglement witness WC than WCA. In quantum computation, a graph state is a special kind of multi-qubit state that can be represented by a graph. Each qubit is represented by a vertex of the graph, and there is an edge between every interacting pair of qubits. Graph states play a crucial role in many applications of quantum information theory, such as quantum error correcting codes, measurement-based quantum computation, and quantum simulation. Consequently, a significant effort is devoted to the creation and investigation of graph states. In the last part of this paper, as applications of our method, a series of methods for constructing an entanglement witness is obtained in the stabilizer formalism. It is also proved that how entanglement witnesses can be derived for a given graph state, provided some stabilizing operators of the graph state are known. Especially, when A is made up of some stabilizing operators of a graph state, entanglement witness WCA becomes one in literature.