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Phase-sensitive superposition of quantum states

Xiaotong Wang, Shunlong Luo, Yue Zhang

TL;DR

The paper develops a information-theoretic framework for quantum superposition by introducing phase-sensitive superposition $S_ heta( ho)$, the overlap with maximally superposed states in a fixed basis. It derives the first and second moments, revealing a conservation relation across phases and linking the second moment to the $l^2$-norm coherence $C_{l^2}( ho)$, while introducing a phase-ensemble channel $E( ho)$ related to complete decoherence. It then characterizes extreme values through $S_{ m min}( ho)$ and $S_{ m max}( ho)$, with analytical results for pure states and general bounds for mixed states, and demonstrates a complementary relationship between maximal superposition and Grover-search success probability. Together, these results provide concrete tools to quantify, compare, and exploit phase-dependent features of quantum superposition in computation and information processing.

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.

Phase-sensitive superposition of quantum states

TL;DR

The paper develops a information-theoretic framework for quantum superposition by introducing phase-sensitive superposition , the overlap with maximally superposed states in a fixed basis. It derives the first and second moments, revealing a conservation relation across phases and linking the second moment to the -norm coherence , while introducing a phase-ensemble channel related to complete decoherence. It then characterizes extreme values through and , with analytical results for pure states and general bounds for mixed states, and demonstrates a complementary relationship between maximal superposition and Grover-search success probability. Together, these results provide concrete tools to quantify, compare, and exploit phase-dependent features of quantum superposition in computation and information processing.

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 -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.
Paper Structure (6 sections, 112 equations, 1 figure)

This paper contains 6 sections, 112 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Quantifying phase-sensitive superposition
  3. Extreme values of phase-sensitive superposition
  4. Minimal superposition
  5. Maximal superposition
  6. Conclusion

Figures (1)

  • Figure 1: Dynamics of the maximal superposition $S_{\max}(|\varphi_t\rangle )$ and the success probability $P_{\max}(|\varphi_t\rangle )$ of the evolved state $|\varphi_t\rangle.$ Here $t$ denotes the number of iterations, $M=1,N=150.$ We see clearly that the superposition is complementary to the success probability, or equivalently, the depletion of superposition is proportional to the success probability: The Grover search algorithm consumes superposition.