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Construction and characterization of measures in block coherence resource theory

Xiangyu Chen, Qiang Lei

Abstract

Quantum coherence, as a direct manifestation of the quantum superposition principle, is a crucial resource in quantum information processing. Block coherence resource theory generalizes the traditional coherence framework by defining coherence via a set of orthogonal projectors. Within this framework, we investigates the construction and comparison of block coherence measures. First, we propose two universal methods for constructing coherence measures and introduce a two-parameter family of measures based on the $α$-$z$ Rényi relative entropy and a family of measures based on the Tsallis relative operator entropy. Second, through theoretical proofs and numerical counterexamples, we compares the ordering relations and numerical magnitudes among different block coherence measures and establishes a series of universal numerical inequalities to constrain their values. Besides, we also use $C_{α,1}$ to show the role of coherence in complex dynamic evolution of the Kominis master equation that includes recombination reactions.

Construction and characterization of measures in block coherence resource theory

Abstract

Quantum coherence, as a direct manifestation of the quantum superposition principle, is a crucial resource in quantum information processing. Block coherence resource theory generalizes the traditional coherence framework by defining coherence via a set of orthogonal projectors. Within this framework, we investigates the construction and comparison of block coherence measures. First, we propose two universal methods for constructing coherence measures and introduce a two-parameter family of measures based on the - Rényi relative entropy and a family of measures based on the Tsallis relative operator entropy. Second, through theoretical proofs and numerical counterexamples, we compares the ordering relations and numerical magnitudes among different block coherence measures and establishes a series of universal numerical inequalities to constrain their values. Besides, we also use to show the role of coherence in complex dynamic evolution of the Kominis master equation that includes recombination reactions.

Paper Structure

This paper contains 13 sections, 17 theorems, 111 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Block Coherence Resource Theory Framework and Construction of Block Coherence Measures
  3. Block Coherence Resource Theory Framework
  4. Construction of Block Coherence Measures via Convex Combination and Convex Roof Extension
  5. Construction of block coherence measures via alternative framework
  6. Block coherence measures induced by $\alpha$-$z$ Rényi Relative Entropy
  7. Block coherence measures induced by Tsallis relative operator entropy
  8. Relations among Different Block Coherence Measures
  9. Review of Existing Measures
  10. Ordering Relations among Measures
  11. Numerical Magnitude Relations among Measures
  12. Dynamics of S-T Coherence Measures under the Kominis Master Equation
  13. Conclusion

Key Result

Theorem 1

For any set of block coherence measures $\left\lbrace C_j(\rho,\mathbf{P}) \right\rbrace_j$ associated with the projector set $\mathbf{P}=\left\lbrace P_k \right\rbrace_k$, and any set of constants $\left\lbrace q_j \right\rbrace_j$ satisfying $q_j\geqslant 0$ and $\sum_jq_j=1$, $C(\rho,\mathbf{P}):

Figures (8)

  • Figure 1: Relationships among various coherence resource theories.
  • Figure 2: The block coherence measure $C_{\alpha,z}$ and the difference function $DIS(\alpha)$.
  • Figure 3: Coherence measure $C_{0.5,1}$ and particle number (initial state: $\ket{S}$)
  • Figure 4: Singlet and Triplet Particle Number Change Rates (initial state: $\ket{S}$)
  • Figure 5: Coherence measure $C_{0.5,1}$ and particle number (initial state: $\ket{S}$)
  • ...and 3 more figures

Theorems & Definitions (35)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 1
  • Theorem 3
  • proof
  • Example 1
  • Example 2
  • Proposition 1
  • ...and 25 more