Measures of Dynamical Coherence in Quantum Channels

Core Concepts

Three measures of dynamical coherence for quantum channels are introduced, which generalize previous results. These measures are based on a generalized distance function between channels and are monotone under free superchannels and convex.

Abstract

The paper presents three measures of dynamical coherence for quantum channels, which generalize previous results. The measures are based on a generalized distance function between channels, such as the divergence or a trace-distance. Free operations are considered to be detection-incoherent, creation-incoherent, and detection-creation incoherent. Free superchannels can be expressed using free pre- and post-processing channels.
The three new measures are shown to be coherence measures, meaning they satisfy the properties of non-negativity, monotonicity under free superchannels, and convexity. The measures are defined as:

CDI_f(N) = min_{K∈DI} f(ΔN, ΔK) for detection-incoherent (DI) channels.
CCI_f(N) = min_{K∈CI} f(NΔ, KΔ) for creation-incoherent (CI) channels.
CDCI_f(N) = min_{K∈DCI} f(N, K) for detection-creation incoherent (DCI) channels.

Here, f is a function acting as a distance between channels that satisfies certain properties, such as non-negativity, weak monotonicity, joint convexity, and monotonicity under tensor products.
The paper provides the proofs that these three measures satisfy the coherence measure conditions, including monotonicity under free superchannels and convexity.

Stats

The relative entropy of channels, Dre(N||M), satisfies the properties required for the function f.
The trace-one distance of channels, ||N-M||_1, also satisfies the required properties for f.

Quotes

"We present three measures of the dynamical coherence of channels, which are the generalization of several previous results."
"The new measures are monotone under free superchannels and convex."

Key Insights Distilled From

Dynamical Coherence Measures

by Anna Vershyn... at arxiv.org 10-01-2024

https://arxiv.org/pdf/2401.07127.pdf

Deeper Inquiries

How can these dynamical coherence measures be applied to study the coherence properties of quantum processes in practical scenarios?

The dynamical coherence measures presented in the paper, such as (CDI_f(N)), (CCI_f(N)), and (CDCI_f(N)), provide a robust framework for analyzing the coherence properties of quantum channels in various practical scenarios. These measures can be particularly useful in fields such as quantum thermodynamics, quantum optics, and quantum information processing, where understanding the coherence of quantum processes is crucial.

Characterization of Quantum Channels: By applying these measures, researchers can characterize different types of quantum channels, such as detection-incoherent (DI), creation-incoherent (CI), and detection-creation incoherent (DCI) channels. This classification helps in identifying the operational capabilities of quantum channels in preserving or generating coherence.

Optimization of Quantum Processes: In practical applications, such as quantum communication and computation, optimizing the coherence of quantum channels can lead to improved performance. The measures can guide the design of quantum protocols that maximize coherence, thereby enhancing the efficiency of quantum state transmission and manipulation.

Resource Theory Framework: The dynamical coherence measures fit within the broader context of resource theories, allowing for the quantification and comparison of coherence as a resource. This perspective is essential for developing strategies to convert coherence into other useful forms of quantum resources, such as entanglement.

Experimental Validation: These measures can also be employed in experimental settings to quantify the coherence of quantum processes. By measuring the coherence of channels before and after specific operations, researchers can validate theoretical predictions and gain insights into the dynamics of coherence in real-world quantum systems.

What are the limitations or potential drawbacks of using these measures compared to other approaches for quantifying coherence in quantum channels?

While the dynamical coherence measures offer valuable insights, they also come with certain limitations and potential drawbacks when compared to other approaches for quantifying coherence in quantum channels:

Complexity of Implementation: The calculation of these measures may involve complex mathematical formulations and optimizations, particularly when dealing with non-trivial quantum channels. This complexity can make them less accessible for practical applications compared to simpler coherence measures, such as the l1-norm or relative entropy of coherence.

Dependence on Free Operations: The effectiveness of these measures is contingent upon the choice of free operations (DI, CI, DCI). Different applications may require different sets of free operations, and the choice can significantly influence the coherence measures. This dependence may limit the generalizability of the results across various quantum processes.

Non-uniqueness of Measures: The measures are defined based on specific distance functions, which can lead to non-uniqueness in the quantification of coherence. Different choices of distance functions may yield different coherence values for the same quantum channel, complicating comparisons and interpretations.

Limited Scope: These measures primarily focus on the dynamical aspects of coherence and may not fully capture the static properties of quantum states. Other coherence measures, such as those based on the geometric approach or the resource-theoretic framework, might provide complementary insights that are not addressed by the dynamical measures.

How might these dynamical coherence measures relate to or provide insights into the fundamental nature of quantum coherence and its role in quantum information processing?

The dynamical coherence measures contribute to a deeper understanding of the fundamental nature of quantum coherence and its pivotal role in quantum information processing in several ways:

Understanding Quantum Interference: By quantifying the coherence of quantum channels, these measures shed light on the mechanisms of quantum interference, which is a hallmark of quantum behavior. Understanding how coherence evolves in quantum processes can help elucidate the conditions under which quantum interference is maximized, thereby enhancing quantum computational tasks.

Coherence as a Resource: The measures reinforce the notion of coherence as a valuable resource in quantum information theory. By framing coherence within the context of resource theories, they highlight the importance of coherence in enabling tasks such as quantum teleportation, superdense coding, and quantum key distribution.

Insights into Quantum Dynamics: The dynamical coherence measures provide insights into the dynamics of quantum systems, particularly how coherence is generated, maintained, or degraded over time. This understanding is crucial for developing robust quantum technologies that can operate effectively in noisy environments.

Connection to Thermodynamics: The measures can also bridge the gap between quantum coherence and thermodynamic processes, revealing how coherence can influence the efficiency of quantum engines and the fundamental limits of quantum thermodynamics. This connection underscores the interplay between quantum coherence and classical thermodynamic principles, enriching our understanding of both fields.

In summary, the dynamical coherence measures not only serve as tools for quantifying coherence in quantum channels but also enhance our understanding of the fundamental principles governing quantum coherence and its applications in quantum information processing.