Entanglement in mixed states of fermions

Deciding whether a bipartite mixed state is separable (unentangled) or not is a computationally intractable (NP-hard) problem. In the case of qubits, the partial transpose of density matrices is known as a good candidate to diagnose separable states. In particular, it can be used to define an entanglement measure called the (logarithmic) negativity. Surprisingly, the extension of the partial transpose (and so the negativity) to fermionic systems remained intractable even for the non-interacting Gaussian states. In this talk, I will introduce partial time-reversal transformation as an analog of partial transpose for fermionic density matrices. This definition naturally arises from the spacetime picture of density matrices in which partial transpose is equivalent to reversing the arrow of time for one subsystem relative to the other subsystem. I will show that the partial time-reversal of density matrix can be computed efficiently for fermionic Gaussian states and its norm satisfies the necessary quantum information theoretic properties to qualify as an entanglement measure. Hence, we call it the fermionic negativity.

 

Host: Mohammad Hafezi