[Paper Review] Distribution of coherence in multipartite systems under entropic coherence measure

Kaifeng Bu, Lu Li+3 more|arXiv (Cornell University)|Oct 23, 2017

Quantum Information and Cryptography1 references3 citations

TL;DR

This paper introduces incoherent-quantum (IQ) coherence measures based on max- and min-relative entropies to quantify coherence in multipartite quantum systems. It establishes a lower bound on total coherence in terms of local coherence and genuine multipartite entanglement, and proves a monogamy relation that captures the non-sharable nature of coherence in multipartite settings.

ABSTRACT

The distribution of coherence in multipartite systems is one of the fundamental problems in the resource theory of coherence. To quantify the coherence in multipartite systems more precisely, we introduce new coherence measures, incoherent-quantum (IQ) coherence measures, on bipartite systems by the max- and min- relative entropies and provide the operational interpretation in certain subchannel discrimination problem. By introducing the smooth max- and min- relative entropies of incoherent-quantum (IQ) coherence on bipartite systems, we exhibit the distribution of coherence in multipartite systems: the total coherence is lower bounded by the sum of local coherence and genuine multipartite entanglement. Besides, we find the monogamy relationship for coherence on multipartite systems by incoherent-quantum (IQ) coherence measures. Thus, the IQ coherence measures introduced here truly capture the non-sharability of quantumness of coherence in multipartite context.

Motivation & Objective

• To address the lack of rigorous coherence measures that satisfy subadditivity and triangle inequality in multipartite systems.
• To characterize the distribution of coherence in multipartite systems beyond existing measures like l1-norm or relative entropy.
• To establish a monogamy relation for coherence that reflects the non-sharable nature of quantum coherence across multiple subsystems.
• To provide operational interpretations of the proposed coherence measures in subchannel discrimination tasks.
• To derive a lower bound on total coherence in terms of local coherence and genuine multipartite entanglement using smooth max- and min-relative entropies.

Proposed method

• Introduces incoherent-quantum (IQ) coherence measures on bipartite systems using max- and min-relative entropies.
• Defines smooth max- and min-relative entropies of IQ coherence to handle approximate state discrimination and operational tasks.
• Uses the duality between max- and min-relative entropies to derive chain rules and coherence inequalities.
• Applies the data processing inequality and trace norm bounds to relate coherence in subsystems and conditional states.
• Employs the conditional min- and max-entropy formalism to derive bounds on coherence in tripartite systems.
• Establishes equivalence between smooth entropy-based coherence measures and the relative entropy of coherence in the asymptotic limit.

Experimental results

Research questions

• RQ1 How can coherence in multipartite systems be quantified in a way that respects subadditivity and triangle inequality?
• RQ2 What is the operational significance of coherence measures based on max- and min-relative entropies in quantum information tasks?
• RQ3 Does a monogamy relation for coherence exist in multipartite systems when using entropic coherence measures?
• RQ4 Can the total coherence in a multipartite system be lower bounded by the sum of local coherence and genuine multipartite entanglement?
• RQ5 How do smooth max- and min-relative entropies of IQ coherence relate to the asymptotic relative entropy of coherence?

Key findings

• The total coherence in a multipartite system is lower bounded by the sum of local coherence and genuine multipartite entanglement, as quantified by the proposed IQ coherence measures.
• A monogamy relation for coherence is proven using IQ coherence measures, demonstrating that coherence cannot be freely shared among multiple subsystems.
• The smooth max- and min-relative entropies of IQ coherence provide a rigorous operational framework for subchannel discrimination problems.
• The chain rule for relative entropy of coherence is derived, showing that $ C_r(AB|C) \geq C_r(A|BC) + C_r(B|C) $, which implies a trade-off in coherence distribution.
• The proposed IQ coherence measures are shown to be equivalent to the relative entropy of coherence in the asymptotic limit, validating their operational consistency.
• The bound $ C^{ ext{max}}_{ ext{smooth}}(A|C) \leq C^{ ext{max}}_{ ext{smooth}}(AB|C) - C^{ ext{min}}_{ ext{smooth}}(B|C) $ is established, demonstrating the coherence trade-off in conditional systems.