Examining the validity of Schatten-$p$-norm-based functionals as coherence measures

Abstract

It has been asked by different authors whether the two classes of Schatten-$p$-norm-based functionals $C_p(\rho )={\min }_{\sigma \in \mathcal{I}}||\rho -\sigma |{|}_p$ and ${\tilde{C}}_p(\rho )=\Vert \rho -\Delta \rho {\Vert }_p$ with $p\ge 1$ are valid coherence measures under incoherent operations, strictly incoherent operations, and genuinely incoherent operations, respectively, where $\mathcal{I}$ is the set of incoherent states and $\Delta \rho$ is the diagonal part of density operator $\rho$. Of these questions, all we know is that $C_p(\rho )$ is not a valid coherence measure under incoherent operations and strictly incoherent operations, but all other aspects remain open. In this paper, we prove that (1) ${\tilde{C}}_1(\rho )$ is a valid coherence measure under both strictly incoherent operations and genuinely incoherent operations but not a valid coherence measure under incoherent operations, (2) $C_1(\rho )$ is not a valid coherence measure even under genuinely incoherent operations, and (3) neither $C_{p>1}(\rho )$ nor ${\tilde{C}}_{p>1}(\rho )$ is a valid coherence measure under any of the three sets of operations. This paper not only provides a thorough examination on the validity of taking $C_p(\rho )$ and ${\tilde{C}}_p(\rho )$ as coherence measures, but also finds an example that fulfills the monotonicity under strictly incoherent operations but violates it under incoherent operations.

8 pages

References in corpus (44)

• Quantum-enhanced measurements: beating the standard quantum limit
• Quantifying Coherence
• Quantum Coherence as a Resource
• Quantum Resource Theories
• Operational Resource Theory of Coherence
• Measuring Quantum Coherence with Entanglement
• Description of quantum coherence in thermodynamic processes requires constraints beyond free energy
• Robustness of coherence: An operational and observable measure of quantum coherence
• Vibrations, Quanta and Biology
• Converting Coherence to Quantum Correlations
• Intrinsic randomness as a measure of quantum coherence
• Frozen Quantum Coherence
• Quantum coherence in multipartite systems
• The general structure of quantum resource theories
• Trace-distance measure of coherence
• A mathematical theory of resources
• Quantum coherence and geometric quantum discord
• Robustness of asymmetry and coherence of quantum states
• Relating the Resource Theories of Entanglement and Quantum Coherence
• The Fidelity and Trace Norm Distances for Quantifying Coherence
• (Quantumness in the context of) Resource Theories
• Assisted distillation of quantum coherence
• A quantitative theory of coherent delocalization
• Distribution of quantum coherence in multipartite systems
• Alternative framework for quantifying coherence
• A Unified View of Quantum Correlations and Quantum Coherence
• Maximally coherent mixed states: Complementarity between maximal coherence and mixedness
• Resource Destroying Maps
• Genuine quantum coherence
• Quantum processes which do not use coherence
• Measure-Independent Freezing of Quantum Coherence
• One-shot coherence distillation
• Towards resource theory of coherence in distributed scenarios
• Deterministic Coherence Distillation
• Estimating coherence measures from limited experimental data available
• Non-asymptotic assisted distillation of quantum coherence
• Coherence non-generating channels
• Discordlike correlation of bipartite coherence
• Accessible Coherence and Coherence Distribution
• Coherence breaking channels and coherence sudden death
• Catalyst-Assisted Probabilistic Coherence Distillation for Mixed States
• Enhancing coherence of a state by stochastic strictly incoherent operations
• Flag Additivity in Quantum Resource Theories
• Superadditivity of convex roof coherence measures

Cited by in corpus (6)

• Universal Method to Estimate Quantum Coherence
• Skew information-based coherence generating power of quantum channels
• Witnessing quantum coherence with prior knowledge of observables
• Quantifying Coherence with Untrusted Devices
• The tightness of multipartite coherence from spectrum estimation
• Closed expressions for one-qubit states of convex roof coherence measures