Thermalization of entangling power with arbitrarily weak interactions
Abstract
The change of the entangling power of $n$ fixed bipartite unitary gates, describing interactions, when interlaced with local unitary operators describing monopartite evolutions, is studied as a model of the entangling power of generic Hamiltonian dynamics. A generalization of the local unitary averaged entangling power for arbitrary subsystem dimensions is derived. This quantity shows an exponential saturation to the random matrix theory (RMT) average of the bipartite space, indicating thermalization of quantum gates that could otherwise be very nongeneric and have arbitrarily small, but nonzero, entanglement. The rate of approach is determined by the entangling power of the fixed bipartite unitary, which is invariant with respect to local unitaries. The thermalization is also studied numerically via the spectrum of the reshuffled and partially transposed unitary matrices, which is shown to tend to the Girko circle law expected for random Ginibre matrices. As a prelude, the entangling power $e_p$ is analyzed along with the gate typicality $g_t$ for bipartite unitary gates acting on two qubits and some higher dimensional systems. We study the structure of the set representing all unitaries projected into the plane $(e_p, g_t)$ and characterize its boundaries which contains distinguished gates including Fourier gate, CNOT and its generalizations, swap and its fractional powers. In this way, a family of gates with extreme properties is identified and analyzed. We remark on the use of these operators as building blocks for manybody quantum systems.
 Publication:

arXiv eprints
 Pub Date:
 September 2019
 arXiv:
 arXiv:1909.08139
 Bibcode:
 2019arXiv190908139J
 Keywords:

 Quantum Physics;
 Condensed Matter  Disordered Systems and Neural Networks
 EPrint:
 16 pages, 9 figures