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arXiv:1802.02404 · Physical Review Letters

Monogamy of Particle Statistics in Tripartite Systems Simulating Bosons and Fermions

Marcin Karczewski, Dagomir Kaszlikowski, Paweł Kurzyński

Published Aug 31, 2018• 14 citations
arXivHTMLPDF

Abstract

In the quantum world, correlations can take the form of entanglement which is known to be monogamous. In this Letter we show that another type of correlation, indistinguishability, is also restricted by some form of monogamy. Namely, if particles A and B simulate bosons, then A and C cannot perfectly imitate fermions. Our main result consists in demonstrating to what extent it is possible.

DOI
10.1103/physrevlett.121.090403

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In the quantum world, correlations can take the form of entanglement which is known to be monogamous. In this Letter we show that another type of correlation, indistinguishability, is also restricted by some form of monogamy. Namely, if particles A and B simulate bosons, then A and C cannot perfectly imitate fermions. Our main result consists in demonstrating to what extent it is possible.
It is well known that a particle cannot freely share entanglement with two or more particles. This restriction is generally called monogamy. However the formal quantification of such restriction is only known for some measures of entanglement and for two-level systems. The first and broadly known monogamy relation was established by Coffman, Kundu, and Wootters for the square of the concurrence. Since then, it is usually said that the entanglement of formation is not monogamous, as it does not obey the same relation. We show here that despite that, the entanglement of formation cannot be freely shared and therefore should be said to be monogamous. Furthermore, the square of the entanglement of formation does obey the same relation of the squared concurrence, a fact recently noted for three particles and extended here for N particles. Therefore the entanglement of formation is as monogamous as the concurrence. We also numerically study how the entanglement is distributed in pure states of three qubits and the relation between the sum of the bipartite entanglement and the classical correlation.
Two important results of quantum physics are the no-cloning theorem and the monogamy of entanglement. The former forbids the creation of an independent and identical copy of an arbitrary unknown quantum state and the latter restricts the shareability of quantum entanglement among multiple quantum systems. For distinguishable particles, one of these results implies the other. In this Letter, we show that in qubit systems with indistinguishable particles (where each particle cannot be addressed individually) a maximum violation of the monogamy of entanglement is possible by the measures that are monogamous for distinguishable particles. To derive this result, we formulate the degree of freedom trace-out rule for indistinguishable particles corresponding to a spatial location where each degree of freedom might be entangled with the other degrees of freedom. Our result removes the restriction on the shareability of quantum entanglement for indistinguishable particles, without contradicting the no-cloning theorem.
We study the monogamy of arbitrary quantum entanglement measures EEE for tripartite quantum systems. Both the sufficient and necessary conditions for EEE to be monogamous in terms of the $\ensuremath{\alpha}\mathrm{th}$ power of EEE are explicitly derived. It is shown that such monogamy of an entanglement measure EEE only depends on the boundedness of the solution set of certain equations. Moreover, the monogamy conditions are also obtained with respect to certain subsets of quantum states for a given quantum correlation. Detailed examples are given to illustrate our results.
Quantum entanglement and quantum nonlocality are known to exhibit monogamy; that is, they obey strong constraints on how they can be distributed among multipartite systems. Quantum correlations that comprise and go beyond entanglement are quantified by, e.g., quantum discord. It was observed recently that for some states quantum discord is not monogamous. We prove, in general, that any measure of correlations that is monogamous for all states and satisfies reasonable basic properties must vanish for all separable states: only entanglement measures can be strictly monogamous. Monogamy of other than entanglement measures can still be satisfied for special, restricted cases: we prove that the geometric measure of discord satisfies the monogamy inequality on all pure states of three qubits.
We strengthen the set of Bell-type inequalities presented by Sun and Fei [Phys. Rev. A 74, 032335 (2006)] that give a classification for biseparable correlations and entanglement in tripartite quantum systems. We will furthermore consider the restriction to local orthogonal spin observables and show that this strengthens all previously known such tripartite inequalities. The quadratic inequalities we find indicate a type of monogamy of maximal biseparable tripartite quantum correlations, although the nonmaximal ones can be shared. This is contrasted to recently found monogamy inequalities for bipartite Bell correlations in tripartite systems.
Abstract One of the fundamental differences between classical and quantum mechanics is in the ways correlations can be distributed among the many parties that compose a system. While classical correlations can be shared among many subsystems, in general quantum correlations cannot be freely shared. This unique property is known as monogamy of quantum correlations. In this work, we study the monogamy properties of the correlated coherence for the l 1 -norm and relative entropy measures of coherence. For the l 1 -norm the correlated coherence is monogamous for a particular class of quantum states. For the relative entropy of coherence, and using maximally mixed state as the reference incoherent state, we show that the correlated coherence is monogamous for tripartite pure quantum systems.