Coffman-Kundu-Wootters inequality for fermions

Gábor Sárosi and Péter Lévay
Phys. Rev. A 90, 052303 – Published 4 November 2014

Abstract

We derive an inequality for three fermions with six single-particle states which reduces to the sum of the famous Coffman-Kundu-Wootters inequalities when an embedded three-qubit system is considered. We identify the quantities which are playing the role of the concurrence, the three-tangle and the invariant detρA+detρB+detρC for this tripartite system. We show that this latter one is almost interchangeable with the von Neumann entropy and conjecture that it measures the entanglement of one fermion with the rest of the system. We prove that the vanishing of the fermionic “concurrence” implies that the two-particle reduced-density matrix is a mixture of separable states. Also, the vanishing of this quantity is only possible in the Greenberger-Horne-Zeilinger class, where some genuine tripartite entanglement is present and in the separable class. Based on this, we conjecture that this “concurrence” measures the amount of entanglement between pairs of fermions. We identify the well-known “spin-flipped” density matrix in the fermionic context as the reduced-density matrix of a special particle-hole dual state. We show that, in general, this dual state is always canonically defined by the Hermitian inner product of the fermionic Fock space and that it can be used to calculate covariants under stochastic local operations and classical communication (SLOCC). We show that Fierz identities known from the theory of spinors relate SLOCC covariants with reduced-density-matrix elements of the state and its spin-flipped dual.

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  • Received 3 September 2014

DOI:https://doi.org/10.1103/PhysRevA.90.052303

©2014 American Physical Society

Authors & Affiliations

Gábor Sárosi* and Péter Lévay

  • Department of Theoretical Physics, Institute of Physics, Budapest University of Technology, H-1521 Budapest, Hungary

  • *sarosi@phy.bme.hu
  • levay@phy.bme.hu

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Issue

Vol. 90, Iss. 5 — November 2014

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