Comment on: "Scaling and Universality at Noisy Quench Dynamical Quantum Phase Transitions"

Dynamical quantum phase transitions (DQPTs) are investigated in two-band models subjected to noise, and the authors rigorously demonstrate that standard protocols used to study these transitions, such as replacing a mixed state with an excitation probability pure state, are fundamentally flawed and insensitive to decoherence. This comment proves that DQPTs do not survive noise averaging under the protocol considered in the referenced work, showing that sharp transitions observed in prior studies are artifacts of an unsuitable approximation scheme.

The gist

For any non-zero noise amplitude, a properly chosen Loschmidt return rate, which is sensitive to noise, will never show DQPTs in two-band models.

Critique of the Pure State Approximation

The authors of the referenced work approximate the noise-averaged mixed state obtained from a master equation by replacing it with a pure state characterized by its excitation probability. The comment rigorously shows that this approximation is exponentially poor in the thermodynamic limit. Theorem 1 proves that except for trivial cases, the solution of the Lindblad Master equation (4) is a mixed state because the purity is a non-increasing function, meaning any starting pure state will generally evolve into a mixed state.

Rigorous Proof Against DQPTs

Theorem 2 establishes that in two-dimensional Hilbert spaces, the Loschmidt echo can have zeroes if and only if both the initial state and final state are pure. Since Theorem 1 proves that the noise-averaged density matrix is impure, this implies that for any non-zero noise amplitude, there are no DQPTs under the noise-averaging procedure considered in Ref. [1]. This result is derived from analyzing the Uhlmann-Bures metric, which is sensitive to coherences.

Insensitivity of Interferometric Protocols

The paper demonstrates that even if results are reinterpreted as an exact interferometric protocol (like the Pancharatnam Loschmidt echo), they do not resolve the issue of noise. The interferometric quantity depends only on a single projection, such as r · nˆ, and is completely independent of the length∥r∥ of the Bloch vector. Since noise causes a reduction in purity (shrinking of∥r∥), this diagnostic carries no information about decoherence caused by noise.

Effect of Noise Averaging on DQPTs

The authors investigate three inequivalent ways to obtain proper noise averages:

1. Case 1: Using the Master equation to obtain the noise-averaged initial mixed state, which leads to a return rate that shows no DQPTs for any non-zero noise.

2. Case 2: Averaging over pure state realizations, where the probability that a finite fraction of realizations has the exact same critical time is zero, ensuring the averaged echo remains smooth.

3. Case 3: Averaging individual return rates at different noise realizations, which also results in a smooth function because non-analyticities occur at different random times.

Conclusion on Noise and DQPTs

Overall, the paper concludes that averaging over noise then always does smooth out dynamical quantum phase transitions showing that the sharp transitions observed in Ref. [1] are artefacts of an unsuitable protocol. The chosen protocol cannot answer how noise affects DQPTs because it is insensitive to decoherence. However, the second theorem suggests that DQPTs are possible in n-band models with n > 2 even for mixed states, though this is expected only in fine-tuned models.

Alternative Protocols

The comment discusses alternative natural ways to average over noise and shows that in all of them DQPTs are smoothed out. This confirms that the sharp transitions observed in the original study are artifacts of an unsuitable protocol. The paper also notes that while individual noise realizations can show DQPTs, a proper noise averaging will always smooth out these features. The core finding is that the chosen protocol fails to reflect the actual physics because it neglects coherences and decoherence. This fundamental issue remains even if results are interpreted as an exact interferometric quantity. In summary, for any non-zero noise amplitude, no DQPTs exist under the noise-averaging procedure considered in Ref. [1] for two-band models. The second theorem shows that in n-band models with n > 2 DQPTs are possible even for mixed states although this is expected to happen only in some fine-tuned models and not in generic noisy or finite-temperature systems.

How it works

The analysis hinges on the distinction between the Loschmidt echo based on a pure state approximation and one based on the proper metric, such as the Uhlmann-Bures metric. The latter is sensitive to coherences, which are destroyed by noise (as shown by Theorem 1), whereas the former ignores them. The paper proves that for two-band models, this sensitivity to coherences is necessary to detect DQPTs under noise.

Based on the provided scientific comment, here are the specific improvements that can be made to AI systems, categorized by the underlying physical/mathematical concepts they would need to incorporate:

) 1. Develop a Decoherence-Aware State Representation Module: Instead of approximating noise-averaged mixed states with pure states (as done in Ref. [1]), the AI system must utilize metrics that account for quantum coherences, such as the Uhlmann-Bures metric or full density matrix evolution.

2. Integrate Rigorous Purity Monitoring: The system should explicitly track the purity of the evolving state (e.g., using the purity derivative equation derived in Theorem 1). This allows the AI to distinguish between dynamics driven by coherent Hamiltonian evolution and those dominated by decoherence/dissipation, which is crucial for accurately predicting non-equilibrium behavior.

3. Implement Noise-Averaging Protocols: The system should adopt alternative, physically rigorous methods for averaging over noise realizations (Cases 2 and 3 in Section V) rather than the simplistic noise-averaged pure state approach. This involves either averaging over pure states generated by individual noise realizations or averaging return rates at the end, ensuring that effects like decoherence are not artificially smoothed out.

4. Dynamic Phase Diagram Prediction: The AI system should be trained to predict phase transitions based on the correct mathematical framework (i.e., incorporating coherence effects). Specifically, it must learn that DQPTs only exist in noise-induced phases when the density matrices are genuinely mixed and coherent information is preserved, rather than relying on artifacts from pure-state approximations.

5. Model Higher-Dimensional Complexity: The AI should be capable of handling systems beyond the two-band model (e.g., n-band models) where DQPTs might be possible for mixed states (as hinted in Conclusion VI). This requires extending the underlying quantum simulation or phase diagram mapping to higher Hilbert spaces where the support orthogonality condition for zero echo can be met even if states are impure.

The improved AI system could perform the following specific tasks:

1. Identify and classify non-equilibrium dynamics (quenches/ramps) in complex quantum systems by determining whether observed critical points (DQPTs) are genuine physical phenomena or artifacts of an inadequate measurement or approximation protocol.

2. Design resilient experimental setups by predicting which noise regimes will preserve the sharpness of dynamical phase transitions, as opposed to smoothing them out, based on the system's coherence structure.

3. Accurately simulate and predict the critical velocities and noise strengths required for DQPTs in noisy environments, accounting for the fact that increased noise actually decreases this velocity (as shown in Ref. [1]).

4. Diagnose the presence of decoherence effects during quantum evolution by monitoring changes in state purity, providing a direct measure of how noise is eroding quantum information.

5. Generate physically meaningful dynamical phase diagrams that correctly distinguish between phases with and without DQPTs, accurately representing the role of noise as a source of novel phases (like the multiple critical momentum modes phase).

In Ref. [1], dynamical quantum phase transitions (DQPTs) -- non-analyticities in the Loschmidt return rate at critical times -- are investigated in the presence of noise for a two-band model. The authors report that DQPTs persist even after averaging over the noise and they use their results to derive dynamical phase diagrams. The protocol used approximates the noise-averaged mixed state, obtained using a master equation, by a pure state, characterized by its excitation probability. In this comment we rigorously show that: (1) This approximation is exponentially poor in the thermodynamic limit. (2) When using the correct metric, the Loschmidt echo of two density matrices in any two-dimensional Hilbert space can become zero if and only if both density matrices are pure, ruling out DQPTs for non-zero noise. (3) An a posteriori reinterpretation of the results as an interferometric protocol is possible but such a protocol is unsuitable to investigate the effects of noise on DQPTs because it is inherently blind to decoherence. We also investigate alternative natural ways to average over noise realizations and show that in all of them DQPTs are smoothed out.