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arXiv:1610.00104

A Note on the Entropy of Entanglement and Entanglement Swapping Bounds

Simon J. D. Phoenix

Published Oct 1, 2016• 0 citations
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Abstract

Using the information content of correlations between multipartite systems, together with the notion of partitioning, we show that some general results about the evolution of correlations in quantum systems can be derived with only elementary methods. In particular, we show that for 2 quantum systems A and B, each comprised of a number of sub-systems, in which a partition of A interacts unitarily with a partition of B, then the total correlation can only increase (or remain unchanged) and is given simply by the sum of the initial correlation and the correlation that develops as a result of the interaction. We then show that in a 4 qubit entanglement swapping process the transferred degree of entanglement is bounded by the lower of the initial degrees of entanglements of the qubits.

DOI
10.48550/arxiv.1610.00104

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Using the information content of correlations between multipartite systems, together with the notion of partitioning, we show that some general results about the evolution of correlations in quantum systems can be derived with only elementary methods. In particular, we show that for 2 quantum systems A and B, each comprised of a number of sub-systems, in which a partition of A interacts unitarily with a partition of B, then the total correlation can only increase (or remain unchanged) and is given simply by the sum of the initial correlation and the correlation that develops as a result of the interaction. We then show that in a 4 qubit entanglement swapping process the transferred degree of entanglement is bounded by the lower of the initial degrees of entanglements of the qubits.
Quantum entanglement of pure states of a bipartite system is defined as the amount of local or marginal (i.e. referring to the subsystems) entropy. For mixed states this identification vanishes, since the global loss of information about the state makes it impossible to distinguish between quantum and classical correlations. Here we show how the joint knowledge of the global and marginal degrees of information of a quantum state, quantified by the purities or, in general, by information entropies, provides an accurate characterization of its entanglement. In particular, for Gaussian states of continuous variable systems, we classify the entanglement of two-mode states according to their degree of total and partial mixedness, comparing the different roles played by the purity and the generalized p-entropies in quantifying the mixedness and bounding the entanglement. We prove the existence of strict upper and lower bounds on the entanglement and the existence of extremally (maximally and minimally) entangled states at fixed global and marginal degrees of information. This results allow for a powerful, operative method to measure mixed-state entanglement without the full tomographic reconstruction of the state. Finally, we briefly discuss the ongoing extension of our analysis to the quantification of multipartite entanglement in highly symmetric Gaussian states of arbitrary 1 × N-mode partitions.
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Abstract. Quantum entanglement of pure states of a bipartite system is defined as the amount of local or marginal (i.e. referring to the subsystems) entropy. For mixed states this identification vanishes, since the global loss of information about the state makes it impossible to distinguish between quantum and classical correlations. Here we show how the joint knowledge of the global and marginal degrees of information of a quantum state, quantified by the purities or in general by information entropies, provides an accurate characterization of its entanglement. In particular, for Gaussian states of continuous variable systems, we classify the entanglement of two–mode states according to their degree of total and partial mixedness, comparing the different roles played by the purity and the generalized p−entropies in quantifying the mixedness and bounding the entanglement. We prove the existence of strict upper and lower bounds on the entanglement and the existence of extremally (maximally and minimally) entangled states at fixed global and marginal degrees of information. This results allow for a powerful, operative method to measure mixedstate entanglement without the full tomographic reconstruction of the state. Finally, we briefly discuss the ongoing extension of our analysis to the quantification of multipartite entanglement in highly symmetric Gaussian states of arbitrary 1 × N-mode partitions. 1.
The field of quantum information processing seeks to transform modern information technology, with applications such as quantum computing and distributed quantum networks illustrating its potential. The key ingredient underlying these applications is entanglement, a purely quantum property that allows disjoint systems to become correlated beyond classical limitations. Entanglement enables core protocols including quantum teleportation and entanglement swapping, which are well studied in few-body settings. However, extending these concepts to many-body systems is non-trivial, and forms the primary focus of this thesis. Concretely, the first half of this thesis focuses on the characterization and quantification of entanglement in generic quantum settings. These ideas culminate in the formulation of a many-body entanglement swapping protocol, describing the redistribution of entanglement between extended systems that have no prior causal connection. To test this protocol, a classical simulation of up to 202020 qubits is carried out, and the resulting entanglement is analyzed using various quantities, most notably von Neumann entropy, mutual information, and observable spin correlation functions. The simulations realize entanglement swapping between causally disconnected many-body systems, induced by measurements on auxiliary subsystems. This is captured by an increase in post-measurement mutual information and spin correlation functions when spin-conserving systems are considered. When the spin conservation requirement is relaxed, observable spin correlations become suppressed while mutual information remains largely unchanged, highlighting the regimes where observable correlation measures remain informative probes of the underlying entanglement dynamics.
Motivated by recent experimental studies on coherent dynamics transfer in three interacting atoms or electron spins [Phys. Rev. Lett 114 (2015) 113002, Phys. Rev. Lett 120 (2018) 243604], here we study entanglement entropy transfer in three interacting qubits. We analytically calculate time evolutions of wave function, density matrix and entanglement of the system. We find that initially entangled two qubits may alternatively transfer their entanglement entropy to other two qubit pairs. Thus dynamical evolution of three interacting qubits may produce a genuine three-partite entangled state through entanglement entropy transfers. In particular, different pairwise interactions of the three qubits endow symmetric and asymmetric evolutions of the entanglement transfer, characterized by the quantum mutual information and concurrence. Finally, we discuss an experimental proposal of three Rydberg atoms for testing the entanglement dynamics transfer of this kind .