Dual unitaries as maximizers of the distance to local product gates

The problem of finding the resource-free, closest local unitary, to any bipartite unitary gate U is addressed. Previously discussed as a measure of nonlocality, the distance K₃ (U) to the nearest product unitary has implications for circuit complexity and related quantities. Dual unitaries, currently of great interest in models of complex quantum many-body systems, are shown to have a preferred role as these are maximally and equally away from the set of local unitaries. This is proved here for the case of qubits, and we present strong numerical and analytical evidence that it is true in general. An analytical evaluation of K₃ (U) is presented for general two-qubit gates. For arbitrary local dimensions, that K₃ (U) is largest for dual unitaries is substantiated by its analytical evaluations for an important family of dual-unitary and for certain nondual gates. A closely allied result concerns for any bipartite unitary the existence of a pair of maximally entangled states that it connects. We give efficient numerical algorithms to find such states and to find K₃ (U) in general.