The theory of manipulations of pure state asymmetry: basic tools and equivalence classes of states under symmetric operations

TL;DR

This paper develops a comprehensive framework for understanding pure state asymmetry under symmetric operations, introducing characteristic functions as key tools for classifying and converting states within symmetry constraints.

Contribution

It introduces the characteristic function as a complete invariant for pure state asymmetry and characterizes state convertibility under symmetric operations for compact Lie groups.

Findings

Pure state asymmetry is fully characterized by the state's characteristic function.

Two pure states are reversibly interconvertible if their characteristic functions differ by a 1D representation.

Conditions for state transformation under symmetric operations are explicitly derived for various paradigms.

Abstract

If a system undergoes symmetric dynamics, then the final state of the system can only break the symmetry in ways in which it was broken by the initial state, and its measure of asymmetry can be no greater than that of the initial state. It follows that for the purpose of understanding the consequences of symmetries of dynamics, in particular, complicated and open-system dynamics, it is useful to introduce the notion of a state's asymmetry properties, which includes the type and measure of its asymmetry. We demonstrate and exploit the fact that the asymmetry properties of a state can also be understood in terms of information-theoretic concepts, for instance in terms of the state's ability to encode information about an element of the symmetry group. We show that the asymmetry properties of a pure state psi relative to the symmetry group G are completely specified by the characteristic function of the state, defined as chi_psi(g)=<psi|U(g)|psi> where g\in G and U is the unitary representation of interest. For a symmetry described by a compact Lie group G, we show that two pure states can be reversibly interconverted one to the other by symmetric operations if and only if their characteristic functions are equal up to a 1-dimensional representation of the group. Characteristic functions also allow us to easily identify the conditions for one pure state to be converted to another by symmetric operations (in general irreversibly) for the various paradigms of single-copy transformations: deterministic, state-to-ensemble, stochastic and catalyzed.