All unitaries having operator Schmidt rank 2 are controlled unitaries

TL;DR

This paper proves that all unitaries with operator Schmidt rank 2 are equivalent to controlled unitaries via local unitaries, enabling their implementation with LOCC and entanglement, regardless of system dimensions.

Contribution

It establishes that unitaries with operator Schmidt rank 2 are always locally equivalent to controlled unitaries, extending understanding of their structure and implementation.

Findings

All such unitaries are diagonalizable by local unitaries.

They are locally equivalent to controlled unitaries with specific control properties.

They can be implemented using LOCC and two-qubit entanglement.

Abstract

We prove that every unitary acting on any multipartite system and having operator Schmidt rank equal to 2 can be diagonalized by local unitaries. This then implies that every such multipartite unitary is locally equivalent to a controlled unitary with every party but one controlling a set of unitaries on the last party. We also prove that any bipartite unitary of Schmidt rank 2 is locally equivalent to a controlled unitary where either party can be chosen as the control, and at least one party can control with two terms, which implies that each such unitary can be implemented using local operations and classical communication (LOCC) and a maximally entangled state on two qubits. These results hold regardless of the dimensions of the systems on which the unitary acts.