Correspondence between entangled states and entangled bases under local transformations

TL;DR

This paper explores the relationship between entangled states and entangled bases under local transformations, revealing that such correspondences exist for certain dimensions and states, but not universally, highlighting complex dependencies on system size and dimension.

Contribution

It proves the existence of entangled bases corresponding to pure states for specific bipartite dimensions and characterizes when state-independent transformations are possible.

Findings

Every bipartite state with local dimension 2, 4, or 8 corresponds to an entangled basis.

Numerical evidence suggests similar correspondence for two qutrits and three qubits.

Not all four-qubit states admit such a basis, indicating limitations.

Abstract

We investigate whether pure entangled states can be associated to a measurement basis in which all vectors are local unitary transformations of the original state. We prove that for bipartite states with a local dimension that is either $2, 4$ or $8$, every state corresponds to a basis. Via numerics we strongly evidence the same conclusion also for two qutrits and three qubits. However, for some states of four qubits we are unable to find a basis, leading us to conjecture that not all quantum states admit a corresponding measurement. Furthermore, we investigate whether there can exist a set of local unitaries that transform \textit{any} state into a basis. While we show that such a state-independent construction cannot exist for general quantum states, we prove that it does exist for real-valued $n$-qubit states if and only if $n=2,3$, and that such constructions are impossible for any multipartite system of an odd local dimension. Our results suggest a rich relationship between entangled states and iso-entangled measurements with a strong dependence on both particle numbers and dimension.