Dynamical Coherence Measures

arXiv:2401.07127 · math-ph, math.MP, quant-ph · Submitted 2024-01-13 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Dynamical Coherence Measures".

Mira: Here's the gist of the paper. The gist: We present three measures of dynamical coherence of channels, which are generalizations based on generalized distance functions, and show that these new measures are monotone under free superchannels and convex.

Kai: Quantum coherence describes the existence of quantum interference, and it is often used in thermodynamics

1: .

Mira: Quantum coherence resource theory starts with free, i.e.

Kai: incoherent, states, which are diagonal states in a pre-fixed basis

1: .

Paper summary: Mira: To recap, this paper introduces three distinct measures based on generalized distance functions, such as relative entropy or trace distance. These are generalizations of several previous results that they've looked at before.

Kai: They focus on specific types of free operations—detection-incoherent, creation-incoherent, and detection-creation incoherent—and define the new measures based on these classes.

Lev: The paper shows that for a relative entropy of channels, the minimal distance to those free channels is a coherence measure of channels. This is defined by the functional C DI (N) equals min K in DI k delta N - delta Kk, which they call formula one point two >

Kai: They also show that this same functional becomes a coherence measure when you use a trace-one norm for the distance function, like the trace-one distance.

Mira: What's important is that they generalize both of these previous results and explicitly show that three other functionals are also coherence measures: C DI f (N) equals min K in DI f delta N, delta Kk, and C CI f (N) equals min K in CI k N delta, Delta K.

Lev: They prove that the trace-one distance satisfies the required conditions for being a coherence measure, and the quantum divergence of channels satisfies those same conditions for any divergence from definition two point one >

Kai: The proof shows that this quantum divergence, including relative entropy and diamond distance of channels, actually satisfies weak monotonicity, strong monotonicity, joint convexity, and stability >

Mira: And they define what those free channels are based on completely dephasing operators: detection incoherent means the effect is different when you apply a completely dephasing operation to system A versus system B > creation incoherent means the effect is different when you apply it to system B versus system A > and detection-creation incoherent means they are different in both directions.

Lev: The operational interpretation they give for these measures is that in a one-shot regime, they appear in the probability of guessing correctly if you obtained the target channel or the least distinguishable free operation, provided you can use only free measurements for the trace-one norm, and non-free measurements are allowed on the auxiliary system for the diamond norm >

Kai: So, it’s a formal way to connect these abstract distance functions to actual experimental possibilities when we're dealing with free operations.

Conclusion: Mira: So, looking at "Dynamical Coherence Measures," the main point is that they've introduced three distinct measures based on generalized distance functions, like relative entropy or trace distance.

Kai: They show these measures are monotone under free superchannels and convex, which is a big deal because it means they stay ordered when you apply operations that don't introduce new quantum interference.

Lev: That’s exactly right. It gives us a mathematical structure for coherence that respects the physical idea of what is "free" in terms of operations—things like detection or creation being incoherent >

Mira: So, why does this matter beyond the math? It connects these abstract measurements to real resource theory problems, like entanglement or correlations, which we already use to understand quantum states.

Kai: Because now we have quantifiable tools that are monotone and convex under those free operations. That means if you're trying to build a system with error correction or some kind of detection scheme, you know how the coherence measure you're using will behave when it gets processed by those kinds of channels >

Lev: It’s a formal way to handle these resource theory problems when you introduce free operations into the picture. That sets up the groundwork for how we might model these systems in real hardware experiments down the road >

Mira: The authors are Anna Vershynina and this work is out on arXiv, which means they're giving us a more robust way to study coherence in dynamic settings >

Kai: It’s about making sure our measurements don't just give us a number, but that the number respects the underlying physics of how we can manipulate those quantum systems >

Anna Vershynina

University of Houston

math-ph, math.MP, quant-ph

Submitted: 2024-01-13

Updated: 2026-10-18

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 89/100

The gist: The gist: We present three measures of dynamical coherence of channels, which are generalizations based on generalized distance functions, and show that these new measures are monotone under free

Key concepts

Dynamical Coherence: Quantum coherence refers to the existence of quantum interference. In this context, it relates to how channels preserve or change this interference property over time. It is often studied in thermodynamics and resource theory.
Free Operations: Free operations are quantum channels that do not create new coherence where none existed before. They map a set of incoherent states onto itself, meaning they are inherently non-coherent transformations in the context of this study.
Coherence Measures: These are mathematical tools based on generalized distance functions (like relative entropy or diamond distance) used to quantify the dynamical coherence of channels. They help determine how much coherence is present or preserved when a channel is transformed by other operations.
Free Superchannels: These are composite quantum operations constructed from free pre- and post-processing channels. The paper proves that the newly defined coherence measures behave predictably—they remain monotone (non-increasing) under these superchannel transformations.

Terminology

Coherence — 2× in this paper · explained in 2 episodes
Generalization — 1× in this paper · explained in 13 episodes
Trace — 1× in this paper · explained in 2 episodes

Summary

The gist: We present three measures of dynamical coherence of channels, which are generalizations based on generalized distance functions, and show that these new measures are monotone under free superchannels and convex.

Introduction to Dynamical Coherence

Quantum coherence describes the existence of quantum interference, and it is often used in thermodynamics [1]. Quantum coherence resource theory starts with free, i.e. incoherent, states, which are diagonal states in a pre-fixed basis [1]. Free operations are some quantum channels that do not create coherence where it was absent, in other words, map the set of incoherent states to itself [1]. Problems involving coherence included quantification of coherence [2], distribution [18], entanglement [6], operational resource theory [5], and correlations [13] >

Coherence Measures Based on Distance Functions

The new measures are based on the generalized distance function between channels, which for example could be the divergence or a trace-distance > In [36] it was shown that when the distance is measured by the relative entropy of channels (see Definition 2.3), the minimal distance to the free channels is a coherence measure of channels > Denote C as a set of free channels, taken to be either detection-incoherent (DI), creation-incoherent (CI), or detectioncreation incoherent (DCI) (see Definition 2.13) > Then it was shown that for a relative entropy of channels Dre, C(N) = min K∈C Dre(N, K). (1.1) is a coherence measure of channels >

Generalization of Existing Measures

The paper generalizes both the previous results and shows that the following functionals are coherence measures:

((

C DI f(N) = min K∈DI f(∆N, ∆K), (1.3) >

((

C CI f(N) = min K∈CI f(N ∆, K∆), (1.4) >

((

C DCI f(N) = min K∈DCI f(N, K). (1.5) >

Here f is a function acting as a distance between channels satisfying a particular list of properties, see Notation 3.1 > As an example, this function can be taken to be a relative entropy of channels, trace-one distance, or a diamond distance >

Properties and Proofs for Coherence Measures

The paper proves that the trace-one distance of channels satisfies the required conditions [32] > The quantum divergence of channels satisfies the above conditions for any divergence from Definition 2.1, in particular the relative entropy of channels and the diamond-distance of channels are such functions > The proof shows that D(N kM) satisfies weak monotonicity, strong monotonicity, joint convexity, and stability >

Coherence Measures in Dynamic Resource Theory

The paper defines three types of free channels based on completely dephasing operators:

Detection incoherent (DI) iff ∆EB NA→B = ∆EB NA→B∆EA >
Creation incoherent (CI) iff NA→B∆EA = ∆EB NA→B∆EA >
Detection-creation incoherent (DCI) or DIO iff ∆EB NA→B = NA→B∆EA >

The paper then defines coherence measures C of channels N satisfying:

(C1) Non-negativity: C(N) ≥ 0 and C(N) = 0 for free channels N >

C2. Monotonicity: for free superchannels Λ, the coherence measure is non-increasing, C(Λ(N)) ≤ C(N) >

Monotonicity under Free Superchannels

The paper demonstrates that all three coherence measures are monotone under free superchannels, specifically showing that for any free quantum superchannel Λ(NA→B) = KBE→D ◦ (NA→B ⊗ IE) ◦ MC→AE with free channels taken from a corresponding set of free operations (DI, CI, or DCI), all coherence measures are monotone, C C f(Λ(N)) ≤ C C f(N) >

Convexity and Linearity

The paper shows that the functionals defined in (3.11)-(3.13) are coherence measures of channels, i.e., they satisfy three conditions in Definition 2.

Improvements for AI systems

The AI system can quantify dynamical coherence of quantum channels using three measures:

C DI f (N) = min K∈DI f(∆N, ∆K), C CI f (N) = min K∈CI f(N ∆, K∆), and C DCI f (N) = min K∈DCI f(N, K). These measures are based on generalized distance functions such as the relative entropy of channels, trace-one distance, or the diamond distance.

The system can assess whether a channel is detection-incoherent (DI), creation-incoherent (CI), or detection-creation incoherent (DCI) by examining the relationship between its completely dephasing operator and the dephasing operators on the input and output systems, as defined in Definition 2.13.
The AI can evaluate whether a given quantum channel satisfies coherence measure properties such as Non-negativity, Weak monotonicity, Strong monotonicity, and Joint convexity by testing these conditions against the definitions provided in Section 3.1 and 2.4, which are essential for defining a valid coherence measure.