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Reply to Comment on "Scaling and universality at noisy quench dynamical quantum phase transitions"

This paper clarifies that the absence of dynamical quantum phase transitions in noisy mixed-state dynamics under the Uhlmann-Bures fidelity, as highlighted in a recent comment, does not contradict the authors' original findings because their study utilized a distinct, operationally defined protocol based on noise-averaged excitation probabilities rather than a mixed-state fidelity measure.

Original authors: S. Ansari, R. Jafari, A. Akbari, M. Abdi

Published 2026-06-30

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CC BY 4.0

Original authors: S. Ansari, R. Jafari, A. Akbari, M. Abdi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a chaotic dance floor where couples (quantum particles) are trying to move in perfect sync. Suddenly, a loud, unpredictable noise starts playing, messing up their steps. Physicists call this a "noisy quench." They want to know: at what point does the dance completely fall apart? This moment of total breakdown is called a Dynamical Quantum Phase Transition (DQPT).

Recently, a critic (J. Sirker) wrote a letter saying, "You can't find this breakdown moment if there is any noise at all." The authors of this paper (Ansari, Jafari, et al.) are writing back to say, "That's true for your way of measuring, but not for our way."

Here is the simple breakdown of their argument:

1. The Two Different Ways to Measure the Dance

The core of the disagreement is about how to measure the dancers after the noise stops.

The Critic's Method (The "Blurry Photo"):
The critic looks at the dancers after the noise and takes a "mixed-state" photo. Imagine taking a photo of a spinning fan; you see a blur. You can't tell exactly where any single blade is, only the average blur. The critic uses a very strict mathematical rule (called Uhlmann-Bures fidelity) to measure this blur.
- The Result: The critic proves that if you look at this "blur," the dance never actually breaks down completely. The noise just smoothes out the sharp edges. So, no DQPTs exist in this "blurry" view.
The Authors' Method (The "Rehearsal"):
The authors say, "Wait, we aren't looking at a blurry photo. We are looking at a specific rehearsal plan."
Their method works in two steps:
1. Step 1 (The Noise): They let the noise happen and calculate the average chance that a dancer ended up in a specific position (like, "There is a 50% chance the dancer is standing up").
2. Step 2 (The Clean Rehearsal): They take those average chances and say, "Okay, let's pretend we have a perfect dancer who matches those chances exactly." Then, they let this perfect dancer dance in silence (no noise) and watch what happens.
- The Result: In this "perfect dancer" scenario, the dance does break down sharply at specific moments. The authors see the DQPTs.

2. Why the Critic is Wrong About Their Method

The critic argued that the authors' method was flawed because it ignored "coherence" (the secret, invisible connections between dancers) and only looked at simple probabilities (like heads or tails).

The authors reply with a clever analogy involving a magic trick:

They explain that the "probability" of a dancer being in a certain spot isn't just a simple coin flip. In the quantum world, the probability is actually a mix of the dancer's position and their secret, invisible spin (coherence).
Even though they only measure the final position (the probability), that number secretly contains information about the invisible spin.
The Proof: They use a mathematical model (Landau-Zener) to show that the "probability" they measure is actually a recipe that includes both the visible position and the invisible spin. So, their method is sensitive to the quantum magic, just in a different way than the critic thought.

3. The Conclusion: Two Different Tools for Two Different Jobs

The authors conclude that there is no contradiction. It's like comparing a thermometer and a barometer.

The critic proved that a thermometer (the Uhlmann-Bures fidelity) can't detect the storm if it's raining too hard. That is a valid scientific fact.
However, the authors used a barometer (their two-step operational protocol). They showed that their tool can detect the storm, even in the rain.

The Takeaway:
The critic is right that if you look at the "blurry photo" of the noise, you won't see a sharp breakdown. But the authors' paper was never about the blurry photo. They were about a specific, two-step experiment where you measure the odds and then run a clean simulation. In that specific experiment, the sharp breakdown (DQPT) absolutely happens. The critic's "no-go" rule doesn't apply to the authors' specific experiment.

Technical Summary: Reply to Comment on "Scaling and universality at noisy quench dynamical quantum phase transitions"

Problem Statement
This paper addresses a critical comment by J. Sirker regarding the existence of dynamical quantum phase transitions (DQPTs) in noisy, mixed-state dynamics. Sirker argues that extending the Loschmidt echo from pure to mixed states is not unique; specifically, when defined via the Uhlmann-Bures fidelity of a noise-averaged density matrix, DQPTs cannot occur for any finite noise intensity. The authors of this reply clarify that their original work (Ref. [4]) did not utilize the Uhlmann-Bures mixed-state fidelity. Instead, it employed a distinct, operationally defined protocol. The central problem is to demonstrate that the "no-DQPT" theorem established for the Uhlmann-Bures return rate does not apply to the specific operational observable analyzed in Ref. [4], thereby resolving the apparent contradiction.

Methodology
The authors distinguish between two primary noise-averaging procedures for Loschmidt-type observables and define their own protocol as a third, distinct approach:

Standard Procedures: The paper reviews (i) averaging over single noise realizations (which washes out nonanalyticities) and (ii) averaging over the noisy ramp to produce a mixed state $\rho_k(t_f)$ , followed by a Uhlmann-Bures fidelity calculation (which yields no DQPTs for finite noise).
Operational Protocol (Ref. [4]): The authors reiterate their specific methodology:
- Stage 1 (Noisy Ramp): Noise acts only during the ramp from $t_i$ to $t_f$ . The system evolves according to a master equation, resulting in a mixed state $\rho_k(t_f)$ .
- Stage 2 (Coherent Post-Ramp): Rather than calculating the return rate directly from the mixed state, the authors construct a pure state $|\psi_k(0)\rangle$ at $t=0$ (post-ramp) whose populations match the noise-averaged excitation probabilities ( $p_k$ ) derived from the master equation.
- Evolution: This pure state is evolved coherently under the noiseless post-ramp Hamiltonian $H_f$ .
- Observable: The return rate is calculated as the logarithm of the Loschmidt echo of this coherent evolution.

To validate the physical content of this protocol, the authors perform a Landau-Zener analysis. They explicitly demonstrate that the excitation probability $p_k$ , used to construct the pure state, is not merely a classical population but depends on both diagonal and off-diagonal elements of the density matrix in the diabatic basis. This is shown via a unitary transformation to the adiabatic basis, proving that $p_k$ encodes coherence information.

Key Contributions and Results

Taxonomy Clarification: The paper establishes that the observable in Ref. [4] belongs to a different class of Loschmidt-type observables than the Uhlmann-Bures fidelity. The "no-DQPT" theorem proven for the latter is therefore inapplicable to the former.
Nonanalyticity Condition: Within the operational protocol, the dynamical free energy (return rate) is given by:
$g(t) = -\frac{1}{2\pi} \int_0^\pi dk \ln \left[ 1 - 4 p_k(1 - p_k) \sin^2(\epsilon_k^f t) \right]$
Nonanalyticities (DQPTs) arise when there exists a momentum $k^*$ such that the noise-averaged excitation probability $p_{k^*} = 1/2$ .
Coherence Retention: Through the Landau-Zener analysis, the authors refute the claim that their protocol is "blind" to coherence. They show that $p_k$ is a function of off-diagonal density matrix elements in the diabatic basis, ensuring the operational protocol retains quantum coherence information despite using a pure-state representation for the post-ramp evolution.
Experimental Motivation: The protocol is motivated by experimental feasibility in engineered two-level systems and quantum simulators, where mode-resolved excitation probabilities are directly measurable. The return rate corresponds to an interferometric signal generated by the second, coherent stage of the protocol.

Significance and Claims
The paper concludes that there is no direct contradiction between the Comment's theorem and the conclusions of Ref. [4]. The authors assert that:

The Comment correctly proves the absence of DQPTs for the Uhlmann-Bures mixed-state return rate.
However, this theorem does not invalidate the findings of Ref. [4], which investigated a different, experimentally motivated operational protocol based on population-preserving pure states.
Within this specific operational framework, DQPTs (nonanalyticities in the return rate) can and do occur.
The physical picture of dynamical phase structure in Ref. [4] is further supported by independent theoretical work (Ref. [19]) using reduced fidelity, reinforcing the validity of the operational approach.

The authors maintain that their work provides a valid operational definition for studying DQPTs in noisy environments, distinct from the fidelity-based definitions that preclude such transitions.

1. The Two Different Ways to Measure the Dance

2. Why the Critic is Wrong About Their Method

3. The Conclusion: Two Different Tools for Two Different Jobs

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