Resource theory of superposition: State transformations

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 $d\ge 3$, scalar products of the pure superposition-free states have a greater place in seeking maximally resourceful states. Various explicit examples illustrate our findings.

arXiv comment14 pages, 1 figure and 1 table

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• Coherence and Imaginarity as Resources in Quantum Circuit Complexity
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• Coherence, superposition, and Löwdin symmetric orthogonalization
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• Methods of constructing superposition measures
• Generating quantum dissonance via local operations
• Golden states in resource theory of superposition
• Phase-sensitive superposition of quantum states