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

The Gist

Imagine you are watching a complex dance performance. In the world of quantum physics, this dance is called a "Dynamical Quantum Phase Transition" (DQPT). It happens when a system is suddenly jolted (a "quench") and then evolves over time. Scientists look for specific moments in this dance where the system completely "forgets" its starting position. In a perfect, silent room, these moments of total forgetting happen sharply and clearly, like a dancer freezing mid-air.

A recent study (referred to as Ref. [1]) tried to see what happens to this dance when the room is noisy—filled with static, interference, and chaos. They claimed that even with the noise, the dancers still freeze at specific moments, and they mapped out a new "phase diagram" showing different types of noisy dances.

Jesko Sirker, the author of this comment, is here to say: "Wait a minute. The math they used to draw that map is fundamentally broken."

Here is a breakdown of Sirker's argument using simple analogies:

1. The "Pure State" Mistake: Ignoring the Blur

In the noisy study, the researchers calculated the average effect of the noise. This average creates a "mixed state"—think of it as a blurry photograph where the dancer is slightly out of focus because of the shaking camera.

However, the noisy study made a huge shortcut. They took that blurry photo and said, "Let's pretend this is actually a sharp, clear photo (a 'pure state') again, just with a different level of brightness." They threw away the "blur" (the loss of quantum coherence) and only kept the "brightness" (the probability of being in a certain spot).

Sirker's Analogy: Imagine trying to understand a foggy day by looking at a photo of the sun and saying, "Okay, the sun is 50% bright, so let's pretend it's a perfectly clear day with a sun that is 50% bright." You have lost the most important part of the fog: the fact that you can't see clearly. By pretending the blurry state is sharp, the researchers ignored the very thing noise does: it destroys the delicate connections (coherences) that make quantum mechanics work.

2. The "Two-Door" Theorem: Why Noise Kills the Freeze

Sirker proves two mathematical rules (Theorems) that act like a bouncer at a club:

• Rule 1: If you have noise, your state is always blurry (mixed). It can never be perfectly sharp (pure) unless the noise is magically turned off.
• Rule 2: In the specific type of quantum system being studied (which acts like a two-door room), the "Loschmidt Echo" (the measure of how much the system forgets its start) can only hit zero (total forgetting) if both the starting state and the ending state are perfectly sharp (pure).

The Conclusion: Since noise makes the state blurry (Rule 1), and you need two sharp states to get a "zero" result (Rule 2), it is mathematically impossible to get a "zero" result in a noisy system. The "freezing" moments (DQPTs) that the noisy study claimed to find cannot exist when you use the correct math.

3. The "Interferometer" Trap

The author suggests that the noisy study might have accidentally been measuring something else entirely: an "interferometric protocol."

The Analogy: Imagine you are trying to measure how much a car's engine is vibrating.

• The Correct Way: You measure the vibration of the whole car, including the rattling parts.
• The Flawed Way (What the noisy study did): You take the car apart, measure how much one specific bolt is shaking, and then pretend that bolt represents the whole car.

Sirker argues that the method used in the noisy study is like measuring only the bolt. It is "blind" to the most important effect of noise: decoherence (the loss of the quantum connection). Because their method ignores the loss of connection, it falsely predicts that the "freezing" moments still happen.

4. The Real Result: Smoothing Out the Dance

When Sirker applies the correct mathematical tools (the Uhlmann-Bures metric, which properly handles blurry, noisy states) and averages over the noise correctly, the sharp "freezing" moments disappear.

Instead of a sharp cliff where the system freezes, the graph becomes a smooth hill. The noise doesn't just shift the timing of the transition; it washes the transition out completely. The "three phases" (including the new noise-created phase) claimed in the original study are an illusion created by using the wrong math.

Summary

The original paper claimed that quantum phase transitions survive noise and create new, strange phases. Sirker's comment argues that this is impossible because:

1. Noise makes quantum states "blurry."
2. You cannot get a "total zero" result (the signature of the transition) if the state is blurry.
3. The original authors tried to fix the blur by pretending it was sharp, which is a mathematically invalid shortcut.
4. When you do the math correctly, the sharp transitions simply smooth out and vanish.

The "new phase" created by noise is a mirage; in reality, noise just makes the quantum dance less distinct, not more complex.

Technical Summary

Technical Summary: "Scaling and Universality at Noisy Quench Dynamical Quantum Phase Transitions"

Problem Statement
The paper addresses a critical inconsistency in the study of Dynamical Quantum Phase Transitions (DQPTs) in the presence of noise. DQPTs are defined as non-analyticities (cusps) in the Loschmidt return rate, occurring at critical times when the Loschmidt echo $L(t)$ vanishes. Recent work (Ref. [1]) investigated DQPTs in a two-band model subjected to a linear ramp with Gaussian white noise. The authors of Ref. [1] reported that DQPTs persist even after averaging over noise realizations, deriving a dynamical phase diagram with three distinct phases, including a novel phase created by noise.

The central methodological flaw identified in Ref. [1] is the approximation of the noise-averaged mixed state (obtained via a master equation) as a pure state characterized solely by its excitation probability. This comment argues that this approximation neglects the decoherence effects inherent to noise, which are crucial for understanding non-equilibrium dynamics in open systems.

Methodology
The author employs rigorous quantum information theory and statistical mechanics to critique the approximation used in Ref. [1]. The methodology involves:

1. Master Equation Analysis: The noise-averaged state is treated as the solution to a Lindblad master equation of the dephasing type. The author analyzes the time evolution of the purity $P(t) = \text{tr}[\rho^2(t)]$ to demonstrate that, for non-trivial couplings, the system evolves from a pure state into a mixed state.
2. Metric Selection: The paper argues that for mixed states, the standard Loschmidt amplitude definition ($L(t) = \langle \Psi(0) | \Psi(t) \rangle$) is invalid. Instead, the proper metric is the Uhlmann-Bures metric, defined as $L(t) = \text{tr}\sqrt{\sqrt{\rho(0)}\rho(t)\sqrt{\rho(0)}}$.
3. Theoretical Proofs: Two rigorous theorems are derived to establish the conditions under which the Loschmidt echo can vanish in the presence of noise.
4. Bloch Sphere Representation: For the specific case of a two-band model (where the Hilbert space factorizes into 2D subspaces for each momentum mode), the density matrices are represented using Bloch vectors to explicitly calculate the fidelity and demonstrate the impossibility of zero overlap for mixed states.
5. Approximation Analysis: The paper analyzes the "pure-state approximation" used in Ref. [1], showing that replacing a mixed state with a pure state based only on excitation probability is exponentially poor in the thermodynamic limit because it discards off-diagonal coherences.

Key Contributions and Results

• Theorem 1 (Purity Decay): It is proven that for a system starting in a pure state and evolving under a dephasing Lindblad master equation (where the noise operator does not commute with the Hamiltonian), the purity is a monotonically decreasing function. Consequently, the state becomes mixed for any non-zero noise amplitude $\xi \neq 0$.
• Theorem 2 (Vanishing Echo Condition): In a two-dimensional Hilbert space, the Loschmidt echo $L(t)$ (calculated via the Uhlmann-Bures metric) can be zero if and only if both the initial state $\rho(0)$ and the final state $\rho(t)$ are pure. If either state is mixed, the support of the density matrices cannot be orthogonal, and $L(t)$ cannot vanish.
• Refutation of Ref. [1] Findings: Since the noise-averaged state in Ref. [1] is mixed (by Theorem 1) and the system is a two-band model (factorizing into 2D subspaces), Theorem 2 dictates that the Loschmidt echo cannot be zero. Therefore, the non-analyticities (DQPTs) reported in Ref. [1] are artifacts of the incorrect pure-state approximation. The correct dynamical phase diagram contains only a "no-DQPT" phase for any non-zero noise.
• Critique of Interferometric Protocols: The paper notes that while the results of Ref. [1] could be reinterpreted as an interferometric protocol, such protocols are inherently blind to decoherence. Consequently, they are unsuitable for investigating the true effects of noise on DQPTs.
• Smoothing of DQPTs: By considering alternative natural ways to average over noise realizations (using the correct mixed-state metrics), the author demonstrates that in all valid protocols, the sharp non-analyticities characteristic of DQPTs are smoothed out by noise.

Significance and Claims
The paper claims to rigorously invalidate the main findings of Ref. [1] regarding the existence of noise-induced DQPT phases. It asserts that:

1. The approximation of a noise-averaged mixed state by a pure state is exponentially poor in the thermodynamic limit.
2. Protocols that neglect coherences (off-diagonal elements) in the density matrix fail to capture the essential physics of decoherence.
3. When the correct Uhlmann-Bures metric is applied to the noise-averaged mixed state, DQPTs are strictly ruled out for any non-zero noise amplitude in the considered two-band models.

The work serves as a corrective to the literature, emphasizing that the study of noise effects on DQPTs requires a formalism that accounts for mixed states and decoherence, rather than relying on pure-state approximations that artificially preserve non-analyticities.