Entanglement negativity of fermions: Monotonicity, separability criterion, and classification of few-mode states

Abstract

We study quantum information aspects of the fermionic entanglement negativity recently introduced in [Phys. Rev. B 95, 165101 (2017)] based on the fermionic partial transpose. In particular, we show that it is an entanglement monotone under the action of local quantum operations and classical communications-which preserves the local fermion-number parity-and satisfies other common properties expected for an entanglement measure of mixed states. We present fermionic analogs of tripartite entangled states such as W and Greenberger-Horne-Zeilinger states and compare the results of bosonic and fermionic partial transpose in various fermionic states, where we explain why the bosonic partial transpose fails in distinguishing separable states of fermions. Finally, we explore a set of entanglement quantities which distinguish different classes of entangled states of a system with two and three fermionic modes. In doing so, we prove that vanishing entanglement negativity is a necessary and sufficient condition for separability of N ≥ 2 fermionic modes with respect to the bipartition into one mode and the rest. We further conjecture that the entanglement negativity of inseparable states which mix local fermion-number parity is always non-vanishing.

Finite-temperature entanglement negativity of free fermions

“…For a system of two fermionic site, we provide an explanation based on the modified separability criterion for fermionic density matrices. We expect (but not argue) that the absence of sudden death carries through all fermionic systems with generic hopping terms [78]. A rigorous proof of the absence of sudden death in the entanglement negativity of a generic extended fermionic system with many sites or a counter example in which sudden death is observed could in principle be very enlightening.…”

“…In Refs. [66] and [78], we have shown that these issues do not carry over to our definition of partial transpose.…”

Entanglement and classical fluctuations at finite-temperature critical points

“…This is not possible for free-fermion models [80][81][82][83][84][85][86][87]. An alternative entanglement measure, which is effectively calculable using free-fermion techniques, has been introduced [15,85,86,[88][89][90], and it is also an upper bound for the negativity [87]. Very recently, much attention has been focused to study the behaviour of the negativity at a finite-temperature phase transition.…”

Fermionic entanglement in the Lipkin model

“…The finite fermionic concurrence of the reduced state ρ pq warrants non-zero bipartite entanglement for any bipartition of the four dimensional sp space [35]. In particular, ρ pq will lead to a finite up-down entanglement, which can be quantified through the pertinent negativity [39,67,68]. This partition involves four distinct states at each side: |0 , c † p− c † q− |0 , c † p− |0 and c † q− |0 for Alice and similar states at the upper level for Bob (Fig.…”

“…2), leading to a two-qudit system with d = 4. P z symmetry implies that just states with the same local fermion number parity are connected in ρ pq , entailing that partial trasposition will not mix local states with different number parity and standard formulas can be applied [39]. The ensuing negativity is just minus the sum of the two negative eigenvalues of the partial trasposed matrix:…”