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

We study quantum information aspects of the fermionic entanglement negativity recently introduced [H. Shapourian et al., 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 $N\ge 2$ 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 nonvanishing.

Figure 1

Comparison of entanglement negativity associated with fermionic and bosonic partial transpose for a Werner state (82). Logarithm is to base 2.

Figure 2

Hamiltonian of two coupled Majorana modes (a) as in the reduced density matrix of two adjacent intervals on a single chain and (b) two coupled chains.

Figure 3

Representation of Hamiltonians with entangled ground states. (a) The $W$ state of complex fermions, where each blue circle indicates a complex fermionic mode. The GHZ state of complex fermions can be realized by (b) three-site Kitaev Majorana chain, or (c) three coupled Kitaev Majorana chains in the $\mathrm{\Delta }$ geometry. Tripartite entangled state of Majorana modes is obtained by (d) a subsystem of kagome lattice with three sites, or (e) three coupled Kitaev Majorana chains in the $Y$ geometry. In (c) and (e), each blue rod is a Kitaev Majorana chain and the smaller circles within each chain represent Majorana end modes. Arrows in (a), (c), and (e) and solid lines in (b) and (d) represent quadratic tunneling terms.

Figure 4

Comparison of various measures of tripartite entanglement in a one-parameter fermionic state $|{\mathrm{\Psi }}_p\rangle =\sqrt{p}{|\text{GHZ}\rangle }_f-\sqrt{1-p}{|W\rangle }_f$. All entanglements are normalized to give 1 for a symmetric ${|\text{GHZ}\rangle }_f$ state.

Figure 5

Schematic structure of separable and inseparable sets of states and values of the negativity in a bipartite system of one fermion ${\mathcal{H}}^A$ and a Fock space ${\mathcal{H}}^{\overline{A}}$ (discussed in Theorem t5). (a) Separable states occupy a part of the zero-negativity subspace. (b) A situation in which there exists a subset of inseparable states with zero negativity away from separable states, that results in a contradiction (see the text). Therefore, the correct structure is given by (c), which means zero negativity is a necessary and sufficient condition for separability in this setup.