Symmetrizing Cost of Quantum States

TL;DR

This paper defines a symmetrization process for quantum states, showing that the minimum randomness needed for symmetrization equals the relative entropy of frameness, linking an operational cost to a quantum resource measure.

Contribution

It introduces the concept of symmetrizing cost and proves its equivalence to the relative entropy of frameness for quantum states.

Findings

Symmetrizing cost equals the relative entropy of frameness.

Provides an operational interpretation for the relative entropy of frameness.

Analyzes asymptotic limits of quantum state symmetrization.

Abstract

We introduce and analyze a task that we call symmetrization, in which a state of a quantum system, associated with a symmetry group, is transformed by a random unitary operation to a symmetric state. Each element of the unitary ensemble is required to be symmetry preserving, in the sense that it keeps the set of symmetric states invariant. We consider an asymptotic limit of infinitely many copies and vanishingly small error, and analyze the symmetrizing cost, that is, the minimum cost of randomness per copy required for symmetrization. We prove that the symmetrizing cost of an arbitrary quantum state is equal to the relative entropy of frameness, thereby providing it with a direct operational meaning.