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arXiv:2207.02049 · Physical Review A

Quantitative characterization of several entanglement detection criteria

Alexander Sauer, József Zsolt Bernád

Published Sep 20, 2022• 6 citations
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Abstract

Quantitative characterization of different entanglement detection criteria for bipartite systems is presented. We review the implication sequence of these criteria and then numerically estimate volume ratios between criteria nonviolating quantum states and all quantum states. The numerical approach is based on the hit-and-run algorithm, which is applied to the convex set of all quantum states embedded into a Euclidean vector space of the Hilbert-Schmidt inner product. We demonstrate that reduction, majorization, and the R\'enyi-entropy-based criteria are very ineffective compared to the positive partial transpose. In the case of the R\'enyi-entropy-based criterion, we show that the ratio of detectable entanglement increases with the order of the R\'enyi entropy.

DOI
10.1103/physreva.106.032423

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Quantitative characterization of different entanglement detection criteria for bipartite systems is presented. We review the implication sequence of these criteria and then numerically estimate volume ratios between criteria nonviolating quantum states and all quantum states. The numerical approach is based on the hit-and-run algorithm, which is applied to the convex set of all quantum states embedded into a Euclidean vector space of the Hilbert-Schmidt inner product. We demonstrate that reduction, majorization, and the R'enyi-entropy-based criteria are very ineffective compared to the positive partial transpose. In the case of the R'enyi-entropy-based criterion, we show that the ratio of detectable entanglement increases with the order of the R'enyi entropy.
Abstract Euclidean volume ratios between quantum states with positive partial transpose and all quantum states in bipartite systems are investigated. These ratios allow a quantitative exploration of the typicality of entanglement and of its detectability by Bell inequalities. For this purpose a new numerical approach is developed. It is based on the Peres–Horodecki criterion, on a characterization of the convex set of quantum states by inequalities resulting from Newton identities and from Descartes’ rule of signs, and on a numerical approach involving the multiphase Monte Carlo method and the hit-and-run algorithm. This approach confirms not only recent analytical and numerical results on two-qubit, qubit-qutrit, and qubit-four-level qudit states but also allows for a numerically reliable numerical treatment of so far unexplored qutrit–qutrit states. Based on this numerical approach with the help of the Clauser–Horne–Shimony–Holt inequality and the Collins–Gisin inequality the degree of detectability of entanglement is investigated for two-qubit quantum states. It is investigated quantitatively to which extent a combined test of both Bell inequalities can increase the detectability of entanglement beyond what is achievable by each of these inequalities separately.
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.
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.
We provide an analytical formula for the volume ratio between bipartite X-states with positive partial transpose and all bipartite X-states. The result applies to arbitrary m×nm \times nm×n-bipartite systems and the volume expressions are derived with respect to the Hilbert-Schmidt measure.