Entanglement measures of bipartite quantum gates and their thermalization under arbitrary interaction strength

Abstract

Entanglement properties of bipartite unitary operators are studied via their local invariants, namely the entangling power e_{p} and a complementary quantity, the gate typicality g_{t}. We characterize the boundaries of the set K_{2} representing all two-qubit gates projected onto the plane (e_{p},g_{t}) showing that the fractional powers of the swap operator form a parabolic boundary of K_{2}, while the other bounds are formed by two straight lines. In this way, a family of gates with extreme properties is identified and analyzed. We also show that the parabolic curve representing powers of swap persists in the set K_{N} for gates of higher dimensions (N>2). Furthermore, we study entanglement of bipartite quantum gates applied sequentially n times, and we analyze the influence of interlacing local unitary operations, which model generic Hamiltonian dynamics. An explicit formula for the entangling power of a gate applied n times averaged over random local unitary dynamics is derived for an arbitrary dimension of each subsystem. This quantity shows an exponential saturation to the value predicted by the random matrix theory, indicating “thermalization” in the entanglement properties of sequentially applied quantum gates that can have arbitrarily small, but nonzero, entanglement to begin with. The thermalization is further characterized by the spectral properties of the reshuffled and partially transposed unitary matrices.