Metric Structure of the Space of Two-Qubit Gates, Perfect Entanglers and Quantum Control

TL;DR

This paper derives a metric for the space of two-qubit gates, computes the volume of perfect entanglers, and analyzes their implications for quantum control, revealing that most two-qubit gates are perfect entanglers.

Contribution

It introduces a coordinate system and metric for SU(4), enabling volume calculations of entanglers and informing quantum control strategies.

Findings

Over 84% of the two-qubit gate space consists of perfect entanglers.

The metric helps determine effective target sizes in quantum control.

The work provides a geometric framework for understanding two-qubit gate entanglement.

Abstract

We derive expressions for the invariant length element and measure for the simple compact Lie group SU(4) in a coordinate system particularly suitable for treating entanglement in quantum information processing. Using this metric, we compute the invariant volume of the space of two-qubit perfect entanglers. We find that this volume corresponds to more than 84% of the total invariant volume of the space of two-qubit gates. This same metric is also used to determine the effective target sizes that selected gates will present in any quantum-control procedure designed to implement them.