Trace-distance measure of coherence (1511.01854v2)

Abstract: We show that trace distance measure of coherence is a strong monotone for all qubit and, so called, $X$ states. An expression for the trace distance coherence for all pure states and a semi definite program for arbitrary states is provided. We also explore the relation between $l_1$-norm and relative entropy based measures of coherence, and give a sharp inequality connecting the two. In addition, it is shown that both $l_p$-norm- and Schatten-$p$-norm-based measures violate the (strong) monotonicity for all $p\in(1,\infty)$.

• The paper introduces an analytic expression and SDP method for computing trace-distance coherence in both pure and mixed quantum states.
• It compares trace distance with l1-norm and relative entropy measures, highlighting its competitive strong monotonicity in coherence quantification.
• The research demonstrates that l_p and Schatten-p norms violate monotonicity, reinforcing the reliability of trace distance as a coherence measure.

Trace-Distance Measure of Coherence

The paper "Trace-distance measure of coherence" by Swapan Rana et al. presents an in-depth exploration of the trace distance as a measure of quantum coherence. It addresses coherence quantification within the framework of quantum mechanics resource theory, emphasizing its importance in distinguishing quantum states from classical counterparts. This paper leverages the trace distance function ${\rho - \delta}_1$ to quantify coherence, where $\rho$ represents a quantum state and $\delta$ an incoherent state.

Key Contributions

The authors provide a rigorous treatment of the trace-distance measure of coherence, showing it to be a strong monotone for both qubit and $X$ states. Specifically, strong monotonicity is pivotal in resource theory as it ensures that coherence measures do not increase on average under incoherent operations. This includes the following significant advancements:

1. Analytic Expression for Trace Distance Coherence: The paper elaborates on analytical expressions for pure states and proposes a semi-definite programming (SDP) method for the computation of coherence in arbitrary states. This SDP formulation enables calculating the minimum trace distance between a quantum state and the set of incoherent states, thereby determining the trace distance coherence measure.
2. Comparison with Other Measures: The research investigates the trace distance measure in comparison to $l_1$-norm and relative entropy-based measures of coherence. The authors derive inequalities linking $l_1$-norm and relative entropy measures, revealing the trace distance as a competitive choice due to its inherent properties in strong monotonicity beyond qubit states.
3. Violation of Monotonicity for $l_p$-Norm and Schatten-$p$-Norm: The paper provides critical insight into the $l_p$-norm (for $p \in (1, \infty)$) and Schatten-$p$-norm based measures, demonstrating their failure to comply with the monotonicity condition. This finding narrows the focus on trace distance and $l_1$-norm measures as potentially more reliable coherence measures.

Numerical Evaluations and Conjectures

By demonstrating strong monotonicity in trace distance coherence through analytical and numerical methods, the authors conjecture that the trace distance measure maintains strong monotonicity across various states. However, it remains an open question whether this holds universally due to the complexity in extending these results beyond tested cases.

Implications and Future Directions

This paper has profound implications for developing quantum technologies where measurement and manipulation of coherence play crucial roles. Understanding and quantifying coherence provide foundational tools in quantum computation and communication protocols, such as quantum cryptography and quantum teleportation. The findings about the trace distance measure propose it as a robust metric for coherence in practical applications.

The future work could involve a deeper investigation into the trace distance across a broader set of quantum states, refinement of semi-definite programming techniques to handle larger and more complex quantum systems, and exploration of the relationship between coherence measures and technological applications in quantum networks.

In conclusion, the authors have notably contributed to coherence quantification by reinforcing the trace distance as a viable and reliable measure. This research inspires further theoretical developments and experimental verifications within quantum mechanics and information science.