Constructing monotones for quantum phase references in totally dephasing channels

TL;DR

This paper develops methods to quantify 'frameness' or asymmetry as a resource in quantum systems lacking shared reference frames, linking it to entanglement measures and providing analytical tools for qubits.

Contribution

It introduces a way to adapt entanglement monotones as frameness monotones for phase-invariant channels and derives analytical formulas for qubits.

Findings

Constructed a family of concurrence monotones for U(1) frameness.

Extended entanglement measures to frameness via convex-roof extension.

Derived an analytical expression for frameness of formation in qubits.

Abstract

Restrictions on quantum operations give rise to resource theories. Total lack of a shared reference frame for transformations associated with a group G between two parties is equivalent to having, in effect, an invariant channel between the parties and a corresponding superselection rule. The resource associated with the absence of the reference frame is known as "frameness" or "asymmetry." We show that any entanglement monotone for pure bipartite states can be adapted as a pure-state frameness monotone for phase-invariant channels [equivalently U(1) superselection rules] and extended to the case of mixed states via the convex-roof extension. As an application, we construct a family of concurrence monotones for U(1) frameness for general finite-dimensional Hilbert spaces. Furthermore, we study "frameness of formation" for mixed states analogous to entanglement of formation. In the case of a qubit, we show that it can be expressed as an analytical function of the concurrence analogously to the Wootters formula for entanglement of formation. Our results highlight deep links between entanglement and frameness resource theories.