Unitary quantum gates, perfect entanglers and unistochastic maps

TL;DR

This paper investigates the non-local properties of quantum gates, analyzing their entanglement entropy, distribution of interaction content, and conditions for unistochastic maps, providing insights into their entangling capabilities and classifications.

Contribution

It introduces a detailed analysis of the entropy of entanglement for unitary gates, derives the distribution of interaction content for two-qubit gates, and establishes conditions for unistochastic maps.

Findings

Asymptotic behavior of entanglement entropy under Haar measure

Relative volume of perfect entanglers is approximately 0.85

Explicit conditions for unistochastic maps are established

Abstract

Non-local properties of ensembles of quantum gates induced by the Haar measure on the unitary group are investigated. We analyze the entropy of entanglement of a unitary matrix U equal to the Shannon entropy of the vector of singular values of the reshuffled matrix. Averaging the entropy over the Haar measure on U(N^2) we find its asymptotic behaviour. For two--qubit quantum gates we derive the induced probability distribution of the interaction content and show that the relative volume of the set of perfect entanglers reads 8/3 \pi \approx 0.85. We establish explicit conditions under which a given one-qubit bistochastic map is unistochastic, so it can be obtained by partial trace over a one--qubit environment initially prepared in the maximally mixed state.