Diagonal unitary and orthogonal symmetries in quantum theory II: Evolution operators

TL;DR

This paper explores bipartite unitary operators with diagonal symmetry, constructing new dual unitary gates, analyzing their entanglement properties, and establishing connections with complex Hadamard matrices for quantum information applications.

Contribution

It introduces new families of dual unitary gates invariant under diagonal symmetries and links their entangling power to complex Hadamard matrices, advancing quantum circuit models.

Findings

Constructed large families of dual unitary gates with diagonal symmetry.

Analyzed the entangling power and operator Schmidt rank of these operators.

Established a correspondence between maximally entangling operators and complex Hadamard matrices.

Abstract

We study bipartite unitary operators which stay invariant under the local actions of diagonal unitary and orthogonal groups. We investigate structural properties of these operators, arguing that the diagonal symmetry makes them suitable for analytical study. As a first application, we construct large new families of dual unitary gates in arbitrary finite dimensions, which are important toy models for entanglement spreading in quantum circuits. We then analyze the non-local nature of these invariant operators, both in discrete (operator Schmidt rank) and continuous (entangling power) settings. Our scrutiny reveals that these operators can be used to simulate any bipartite unitary gate via stochastic local operations and classical communication. Furthermore, we establish a one-to-one connection between the set of local diagonal unitary invariant dual unitary operators with maximum entangling power and the set of complex Hadamard matrices. Finally, we discuss distinguishability of unitary operators in the setting of the stated diagonal symmetry.