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Witnessing quantum entanglement

Vlatko Vedral

Published Sep 28, 2006• 0 citations
Publisher

Abstract

Abstract Quantum entanglement is a key resource in quantum information. Teleportation, dense coding, and many other quantum protocols rely on the existence of entanglement, and would not be possible with just classical correlations. Furthermore, using entanglement between qubits that support computation, quantum computers can solve problems faster than classical computers. Bell's inequalities can sometimes help in discriminating between entangled and separable states. However, these inequalities are not always reliable — this is true in the sense that they may fail to detect genuinely entangled states. This question is addressed in the present chapter by focusing on an entanglement witness, a Hermitian operator winch helps us to decide whether a state is entangled or not. The basic idea is that the expectation value of the witness will be different for separable and entangled states. This chapter discusses entanglement witnesses, the Jamiolkowski isomorphism, and the Peres Horodecki criterion. More examples of entanglement witnesses are given.

DOI
10.1093/acprof:oso/9780199215706.003.0008

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Abstract Quantum entanglement is a key resource in quantum information. Teleportation, dense coding, and many other quantum protocols rely on the existence of entanglement, and would not be possible with just classical correlations. Furthermore, using entanglement between qubits that support computation, quantum computers can solve problems faster than classical computers. Bell's inequalities can sometimes help in discriminating between entangled and separable states. However, these inequalities are not always reliable — this is true in the sense that they may fail to detect genuinely entangled states. This question is addressed in the present chapter by focusing on an entanglement witness, a Hermitian operator winch helps us to decide whether a state is entangled or not. The basic idea is that the expectation value of the witness will be different for separable and entangled states. This chapter discusses entanglement witnesses, the Jamiolkowski isomorphism, and the Peres Horodecki criterion. More examples of entanglement witnesses are given.
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Quantum entangled states play very important roles in quantum information processing,such as quantum teleportation,dense coding,error correction,cryptography,and computation.The theory of entanglement mainly concerns the characterization of quantum entanglement and the classification of quantum states,as well as its applications in quantum information processing.We present some basic concepts and results,including:the relationship between locality and reality in quantum mechanics and Bell inequalities,and that between Bell inequalities and the separability of quantum states;the definition of separability for pure and mixed quantum states,and some separability criteria such as the positive map approach,positive partial transpose,reduction criterion,realignment,entanglement witnesses,and the covariance matrix approach and local uncertainty relations;some entanglement measures such as the entanglement of formation,concurrence,relative entropy,negativity,tangle,entanglement of assistance,and their estimation;the change in entanglement when the physical system evolves or undergoes interaction with the environment.
Chapter devoted to the basic quantum properties of entanglement and separability. Introduces the Schmidt decomposition for pure states and the positive partial transpose criterion for mixed states as entanglement witnesses. Introduces the famous Einstein–Podolsky–Rosen paradox and its implementation in terms of qubits, then the Bell inequality, quickly reviewing the experimental demonstrations that quantum mechanics violates this inequality. Gives examples of the use of entanglement in a quantum algorithm to accelerate an information task, namely a database search (Grover algorithm) and the possibility of teleportation of a quantum state.
Entanglement, a critical resource for quantum information processing, needs to be witnessed in many practical scenarios. Theoretically, witnessing entanglement is by measuring a special Hermitian observable, called an entanglement witness (EW), which has non-negative expected outcomes for all separable states but can have negative expectations for certain entangled states. In practice, an EW implementation may suffer from two problems. The first one is reliability. Due to unreliable realization devices, a separable state could be falsely identified as an entangled one. The second problem relates to robustness. A witness may not be optimal for a target state and fail to identify its entanglement. To overcome the reliability problem, we employ a recently proposed measurement-device-independent entanglement witness scheme, in which the correctness of the conclusion is independent of the implemented measurement devices. In order to overcome the robustness problem, we optimize the EW to draw a better conclusion given certain experimental data. With the proposed EW scheme, where only data postprocessing needs to be modified compared to the original measurement-device-independent scheme, one can efficiently take advantage of the measurement results to maximally draw reliable conclusions.
Abstract This chapter explains how entanglement witnesses can be measured in practice. The main idea behind the Mach–Zehnder interferometer experiment described earlier is to test for and even measure quantum entanglement. The key idea is discussed along with its exact application. Partial transposition is not a physical operation because it is a positive map rather than a CP-map. Therefore, it cannot be implemented directly within the quantum formalism. However, an entanglement witness is the average of some Hermitian operator, and this average is a physically measurable quantity. Thus, it is possible to measure the effects of the partial transposition in some indirect way. This chapter discusses the implementation of the Peres Horodecki criterion using an interferometer. The important message is that a simple apparatus that measures quantum superpositions, such as a Mach–Zehnder interferometer, can also be used for much more complicated measurements.
Quantum entanglement is a core resource for quantum information processing, and entanglement witnesses are indispensable tools for detection and verification. This paper first clarifies some already-known properties of entanglement witnesses and the Bell State. By applying mathematical tools such as Kronecker product and matrix algebra, this paper analyzes elements in the Hermitian matrix and discusses the conditions for it to form an entanglement witness. This paper also considers both necessary and sufficient conditions of entanglement witness to impose restrictions on elements and investigated several inequality properties of two-qubit entanglement witness to detect Bell states.