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.
Highlights
A clutch of quantities such as state entanglement, operator entanglement, operator scrambling, out-of-time-ordered correlators, and various measures of mutual information are currently being actively pursued as a means to understand information transport in complex quantum systems and to characterize quantum chaos [1,2,3,4,5,6,7,8]
We characterize the boundaries of the set K2 representing all two-qubit gates projected onto the plane showing that the fractional powers of the SWAP operator form a parabolic boundary of K2, while the other bounds are formed by two straight lines
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
Summary
A clutch of quantities such as state entanglement, operator entanglement, operator scrambling, out-of-time-ordered correlators, and various measures of mutual information are currently being actively pursued as a means to understand information transport in complex quantum systems and to characterize quantum chaos [1,2,3,4,5,6,7,8]. Such a bipartite structure could form a building block for more general random quantum circuits with fixed, possibly atypical, nonlocal gates and random or generic local interaction Another setting in which products of nonlocal unitaries interspersed with local ones arise naturally is in kicked systems, which are being used extensively. In a central result in this context, we demonstrate the exponentially fast thermalization of the average entangling power of U (n) = nj=1U j with time to that of a typical unitary operator. The second part of this work shows the thermalization of the entangling power of U under time evolution with nonautonomous local evolutions Such exponential saturation seems to provide excellent approximations in the case of autonomous Floquet systems [44] whose dimensions are not very small, the circumstances under which this holds need further investigation.
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