Characterize mixture-of-local-unitaries transformations for two-qubit mixed states

Characterize whether and when two arbitrary two-qubit mixed states are related by a mixture of local unitary transformations.

Background

The paper relates Hamiltonian engineering with two subensemble control to transformations of bipartite mixed states under convex mixtures of local-unitary conjugations. For qubits, this connection concerns determining whether one two-qubit mixed state can be obtained from another by averaging transformations of the form consisting of independent local unitaries on the two parties.

The paper notes that this transformation problem is currently unresolved and uses the connection to explain the computational and structural difficulty of the Hamiltonian-engineering problem. A general characterization would clarify the exact reachability conditions beyond the necessary majorization conditions established in the paper.