The Local Orthogonality between Quantum States and Entanglement Decomposition

TL;DR

This paper explores how quantum states decomposed into locally orthogonal mixed states allow entanglement to be similarly decomposed, extending previous results and highlighting conditions for such decompositions.

Contribution

It generalizes a known entanglement decomposition result to mixed states and clarifies conditions needed for entanglement cost and distillability.

Findings

Entanglement can be decomposed along with the state under local orthogonality.

The generalization extends previous pure state results to mixed states.

Conditions for entanglement cost decomposition are identified.

Abstract

In the paper, we show that when a quantum state can be decomposed as a convex combination of locally orthogonal mixed states, its entanglement can be decomposed into the entanglement of these mixed states without losing them. The obtained result generalizes a corresponding one proved by Horodecki [Acta Phys. Slov. 48, 141 (1998).]. But, for the entanglement cost it requires certain conditions for holding the decomposition, and the distillable entanglement only has a week result as inequality. Finally, we presented an example to show that the conditions of our conclusions are existence.