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.
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.
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.
Entanglement finds wide application in the development of quantum information theory. After reviewing states in classical mechanics, this chapter discusses the fundamental aspects of characterization of quantum entanglement like the definition, the detection, the classification and quantification of entanglement. The density operator is used to describe the state of a quantum system in a more general way than provided by the state vector of the Hilbert space. Multipartite entanglement theory is much complex compared to the bipartite case. One cannot generalize the results of bipartite entanglement to multipartite entanglement. The Peres criterion known as positive partial transpose criterion is an important criterion to find the separability of mixed states at least for low-dimensional systems. B.W. Terhal introduced the term entanglement witnesses for operators detecting quantum entanglement states. It is found that any entangled state can be detected by some entanglement witnesses.
Summary: \nEntanglement is a strange feature contained in the quantum mechanical framework, first observed theoretically by Erwin Schrödinger and Albert Einstein. An inherent attribute of entanglement is non-locality, and in the continuation of this, several non-reductionist features, which are conceptually challenging and philosophically interesting. \n \nA considerable growth of the interest in quantum entanglement has occurred since the early 1990s, and much of the reason for this is that the tools of quantum optics and modern instrumentation have made it possible to measure and manipulate entangled quantum systems. Today, several applications of entanglement exist, including quantum computers and several areas in quantum communications. \n \nA quantum state is either separable or entangled, it cannot be none or both. A separable state do not exhibit the non-local features found in entangled states, and are in this sense comparable to classic states. An important issue here is the separability problem, which is the problem of determining whether a given state is entangled or not. \n \nWe study these quantum states as Hermitian matrices with positive eigenvalues in convex spaces. The PPT criterion is based on the partial transposition operation, which essentially transposes one of the two subsystems in the quantum system. For some states the result is another quantum (positive) state, and these are PPT (Positive Partial Transpose) states, while others that gain negative eigenvalues are called NPT states. \n \nFor systems of dimensions 2 × 2 and 2 × 3, the PPT states are identical to the separable states, and the NPT states are identical to the entangled states, so the PPT criterion is always conclusive. But for larger dimensions, such as for instance the 3 × 3 system, there always exist PPT states that are entangled, so for this case the PPT criterion can only be used in a contrapositive way. Since entangled PPT states in this sense seem to have great significance, their characteristics are worth studying. \nAs mentioned, the PPT criterion uses the transposition map on one of the two subsystems of a composite quantum system. Transposition is an example of a positive map, in that it maps all quantum states into quantum states. It is possible to use positive maps to formulate a much more general criterion for the separability problem.There is a link (an isomorphism) between positive maps and a subset of Hermitian operators called entanglement witnesses. \n \nThe so-called Horodecki criterion states that a quantum state is separable if and only if the expectation value of all entanglement witnesses in the state are non-negative. If the expectation value of one (or possibly several) entanglement witnesses in a given quantum state is negative, then the state is always entangled. This criterion is both necessary and sufficient, but demands that we can characterize the entire set of entanglement witnesses. \n \nThe approach in this thesis is through geometry. Of fundamental importance is the convex nature of the sets in question. The knowledge and understanding of the extremal points in these sets is very relevant, since they describe the sets completely. Undertaking completely analytic studies of the geometry at hand is in general very hard, especially as the dimension of the systems grow. The development and use of numerical tools is therefore essential.
A complete chapter on quantum entanglement measures and detection of entanglement is given because there are a wide variety of entanglement measures for pure and mixed states which are used very often. The interesting feature of these routines is the fact that, these operations can be done between any random set of qubits not necessarily contiguous qubits. Key recipes to perform operations like partial tracing and partial transpose are disscussed in detail. After that, it was decided to devote several routines to calculate the entanglement measures like concurrence and its variants, von Neumann entropy, block entropy, Renyi entropy, Q measure, negativity and log negativity, entanglement spectrum and residual entanglement (three tangle). Last but not least, certain entanglement detection procedures like Positive Partial Transpose (the Peres-Horodecki criteria) and the reduction criteria are also explained via examples.
Abstract Efficient certification and quantification of high dimensional entanglement of composite systems are challenging both theoretically as well as experimentally. Here, we demonstrate how to measure the linear entropy, negativity and the Schmidt number of bipartite systems from the visibility of Mach–Zehnder interferometer using single copies of the quantum state. Our result shows that for any two qubit pure bipartite state, the interference visibility is a direct measure of entanglement. We also propose how to measure the mutual predictability experimentally from the intensity patterns of the interferometric set-up without having to resort to local measurements of mutually unbiased bases. Furthermore, we show that the entanglement witness operator can be measured in a interference setup and the phase shift is sensitive to the separable or entangled nature of the state. Our proposal bring out the power of Interferometric set-up in entanglement detection of pure and several mixed states which paves the way towards design of entanglement meter.
Abstract The Mach–Zehnder interferometer experiment described earlier shows why and how quantum mechanics is different from classical mechanics. A photon sent through a beam splitter behaves like a particle when it is observed by only one of the two detectors. When two beam splitters are used, the photon “interferes with itself” and behaves like a wave. This is the so-called wave-particle duality of quantum mechanics which leads to quantum entanglement. This chapter discusses quantum superpositions when two or more particles are present. Understanding and analysing entanglement is one of the most interesting directions in the field of quantum information. First, a historical background of quantum entanglement is given, followed by a discussion on Bell's inequalities, separable states that do not violate Bell's inequalities, pure states that violate Bell's inequalities, mixed states that do not violate Bell's inequalities, and entanglement in second quantisation.