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Quanutm-State Texture as a Resource: Measures and Nonclassical Interdependencies

Yuntao Cui, Zhaobing Fan, Sunho Kim

TL;DR

The paper treats quantum-state texture (QST) as a resource and defines free CPTP maps $\Lambda$ with $\Lambda(f)=f$. It introduces two new QST measures and a convex-roof framework for constructing additional measures, linking QST to the Hellinger distance via a special case. It derives the maximal stochastic transformation probabilities for both pure and mixed state conversions under free operations and provides explicit Kraus constructions to achieve these bounds. Finally, it analyzes relations between QST and other quantum resources, including coherence, imaginarity, and predictability, across $l_{1}$-, $l_{2}$-norms and skew information, offering a cohesive theoretical foundation for QST resource theory and its potential applications.

Abstract

Quantum-state texture is a newly recognized quantum resource that has garnered attention with the advancement of quantum theory. In this work, we address several key aspects of quantum-state texture resource theory, including the quantification of quantum-state texture, quantum state transformation under free operations, and the relationships between quantum resources. We first propose two new measures of quantum-state texture and introduce a specific functional form for constructing such measures via the convex roof method. Then, we determine the maximum probability of quantum state transformation under free operations. Finally, we establish connections between quantum-state texture and other prominent quantum resources, such as coherence, imaginarity, and predictability. Our research contributes to the measure theory of quantum-state texture and enriches the overall framework of quantum-state texture resource theory.

Quanutm-State Texture as a Resource: Measures and Nonclassical Interdependencies

TL;DR

The paper treats quantum-state texture (QST) as a resource and defines free CPTP maps with . It introduces two new QST measures and a convex-roof framework for constructing additional measures, linking QST to the Hellinger distance via a special case. It derives the maximal stochastic transformation probabilities for both pure and mixed state conversions under free operations and provides explicit Kraus constructions to achieve these bounds. Finally, it analyzes relations between QST and other quantum resources, including coherence, imaginarity, and predictability, across -, -norms and skew information, offering a cohesive theoretical foundation for QST resource theory and its potential applications.

Abstract

Quantum-state texture is a newly recognized quantum resource that has garnered attention with the advancement of quantum theory. In this work, we address several key aspects of quantum-state texture resource theory, including the quantification of quantum-state texture, quantum state transformation under free operations, and the relationships between quantum resources. We first propose two new measures of quantum-state texture and introduce a specific functional form for constructing such measures via the convex roof method. Then, we determine the maximum probability of quantum state transformation under free operations. Finally, we establish connections between quantum-state texture and other prominent quantum resources, such as coherence, imaginarity, and predictability. Our research contributes to the measure theory of quantum-state texture and enriches the overall framework of quantum-state texture resource theory.

Paper Structure

This paper contains 13 sections, 6 theorems, 90 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Quantum-state texture measures
  3. Quantum-state texture measurement based on affinity
  4. Quantum-state texture measurement based on Tsallis relative $\alpha$ entropy
  5. Quantum-state texture measurement based on function
  6. State transformations via free operations
  7. Stochastic transformations for pure states
  8. Stochastic transformations for mixed states
  9. Relationships between texture and other quantum properties
  10. Relationship under $l_{1}$-norm measure
  11. Relationship under $l_{2}$-norm measure
  12. Relationship under skew information
  13. Conclusion

Key Result

Theorem 1

$\Upsilon_{A}(\rho)=1-A_{\alpha}(\rho,f)$ is a QST measure.

Figures (2)

  • Figure 1: The purple vector (QST) can be decomposed into the vector sum of the yellow vector (Predictability), the green vector (Imaginarity) and the blue vector ($1-x$), serving as the intuitive representation on the Bloch sphere.
  • Figure 2: The vector sum of the red vector (Coherence) and the yellow vector (Predictability) constitutes the green vector ($C_{l_{1}}+P_{l_{1}}$), while the vector sum of the green vector ($C_{l_{1}}+P_{l_{1}}$) and the positive x-axis constitutes the purple vector (QST), serving as the intuitive representation on the Bloch sphere.

Theorems & Definitions (13)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Example 4
  • Theorem 5
  • proof
  • Theorem 6
  • ...and 3 more