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

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.