Phase-sensitive superposition of quantum states

TL;DR

This paper introduces a new way to quantify quantum superposition based on phase sensitivity, establishing conservation relations and exploring implications for quantum algorithms like Grover's search.

Contribution

It proposes a novel phase-sensitive superposition measure, links it to coherence, and analyzes its dynamics and properties in quantum processes.

Findings

Phase-sensitive superposition quantifiers are introduced.

A conservation relation similar to wave-particle duality is established.

Superposition dynamics are analyzed in Grover's search algorithm.

Abstract

Although the principle of superposition lies at the heart of quantum mechanics and is the root of almost all quantum phenomena such as coherence and entanglement, its quantification, except for that related to the resource theory of coherence and interference, remains relatively less studied. In this work, we address quantification of superposition from an information-theoretic perspective. We introduce a family of quantifiers of superposition, the phase-sensitive superposition, by taking into account the phases of amplitudes in the superposition of a fixed basis states (e.g., computational basis states). We establish a conservation relation for the phase-sensitive superposition, which is a kind of complementary relation and is reminiscent of wave-particle duality. We evaluate explicitly the second moment of phase-sensitive superposition and show that it is intrinsically related to the $l^2$-norm coherence. We characterize the dephasing channel induced by the maximally superposed states. We investigate the minimum and maximum superpositions, reveal their basic properties, and illustrate them through various examples. We further explore the dynamics of superposition in the Grover search algorithm, and demonstrate a complementary relation between superposition and success probability of the search algorithm. These results and quantifiers offer tools for analyzing structural features and implications of quantum superposition.