A unified approach to quantum resource theories and a new class of free operations

TL;DR

This paper proposes a unified algebraic framework for quantum resource theories (QRTs), identifying free operations as automorphisms of a preserved algebraic structure, and introduces new resource non-increasing operations for Lie-algebra based QRTs.

Contribution

It unifies the understanding of QRTs through algebraic structures and generalizes SLOCC to new operations, solving an open problem in the field.

Findings

Identified algebraic structures for various QRTs like entanglement and coherence.

Defined new resource non-increasing operations for Lie-algebra based QRTs.

Proved these operations map free states to free states and do not increase resources.

Abstract

In quantum resource theories (QRTs) certain quantum states and operations are deemed more valuable than others. While the determination of the ``free'' elements is usually guided by the constraints of some experimental setup, this can make it difficult to study similarities and differences between QRTs. In this work, we argue that QRTs follow from the choice of a preferred algebraic structure $\mathcal{E}$ to be preserved, thus setting the free operations as the automorphisms of $\mathcal{E}$. We illustrate our finding by determining $\mathcal{E}$ for the QRTs of entanglement, Clifford stabilizerness, purity, imaginarity, fermionic Gaussianity, reference frames, thermodynamics and coherence; showing instances where $\mathcal{E}$ is a Lie algebra, group, ring, or even a simple set. This unified understanding allows us to generalize the concept of stochastic local operations and classical communication (SLOCC) to identify novel resource non-increasing operations for Lie-algebra based QRTs, thus finding a new solution to an open problem in the literature. We showcase the sanity of our new set of operations by rigorously proving that they map free states to free states, as well as determine more general situations where these transformations strictly do not increase the resource of a state.