Resource theory of superposition: State transformations

TL;DR

This paper develops a resource theory framework for superposition, establishing conditions for state transformations based on scalar products, and identifying maximally resourceful states in finite-dimensional quantum systems.

Contribution

It introduces a new resource theory of superposition, deriving transformation conditions dependent on scalar products and characterizing maximally resourceful states.

Findings

Transformation conditions depend on scalar products of basis states.

Maximal superposition states are identified for different scalar product ranges.

Scalar products of superposition-free states influence resourcefulness in higher dimensions.

Abstract

A combination of a finite number of linear independent states forms superposition in a way that cannot be conceived classically. Here, using the tools of resource theory of superposition, we give the conditions for a class of superposition state transformations. These conditions strictly depend on the scalar products of the basis states and reduce to the well-known majorization condition for quantum coherence in the limit of orthonormal basis. To further superposition-free transformations of $d$-dimensional systems, we provide superposition-free operators for a deterministic transformation of superposition states. The linear independence of a finite number of basis states requires a relation between the scalar products of these states. With this information in hand, we determine the maximal superposition states which are valid over a certain range of scalar products. Notably, we show that, for $d\geq3$, scalar products of the pure superposition-free states have a greater place in seeking maximally resourceful states. Various explicit examples illustrate our findings.