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

TL;DR

This paper analyzes the entanglement properties of bipartite quantum gates, characterizes their boundaries in entanglement measures, and demonstrates how repeated application of such gates under random local dynamics leads to thermalization of entanglement, regardless of initial entanglement levels.

Contribution

It introduces a geometric characterization of two-qubit gates in entanglement measure space and derives an explicit formula showing entanglement thermalization under random local dynamics for arbitrary dimensions.

Findings

Fractional powers of swap form a parabolic boundary in entanglement measure space.

Entanglement saturates exponentially to a RMT-predicted value under repeated gate application.

Thermalization of entanglement occurs even from initially low or zero entanglement states.

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 \textsc{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 \textsc{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 analyze the influence of interlacing local unitary operations, which model generic Hamiltonian dynamics. An explicit formula for the entangling power 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 (RMT), 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.