Large Deviation Functions for Open Quantum Systems with a Strong Symmetry

This paper proposes a method to derive genuine large-deviation rate functions for open quantum systems with strong symmetries by applying the Gärtner-Ellis theorem within operator space blocks and minimizing the resulting local rate functions, a scheme justified by dissipative freezing and demonstrated through analytical and three-spin models where nonanalyticities manifest as avoided level crossings.

The Gist

The Big Picture: Predicting the Unpredictable

Imagine you are watching a very complex, noisy quantum machine. You want to know how often it does something rare, like jumping to a specific state. In physics, we use a tool called a Large Deviation Function to predict the odds of these rare events. Think of this function as a "weather forecast" for the machine's behavior over a long time.

Usually, this forecast is smooth and easy to calculate. However, this paper deals with a special kind of machine that has a Strong Symmetry. Because of this symmetry, the machine gets "stuck" in different modes, making the forecast jagged and broken (mathematically, "nonanalytic"). The standard tools used to calculate these forecasts break down when the graph is jagged.

The authors of this paper propose a clever workaround: Don't look at the whole machine at once. Look at its separate rooms.

The Core Problem: The "Frozen" Machine

In a normal quantum system, if you start it in a mix of different states, it eventually settles into one unique, stable state. But in these special "symmetric" systems, something weird happens called Dissipative Freezing.

The Analogy:
Imagine a hotel with two separate wings (Wing A and Wing B) that are completely soundproof and have no doors connecting them.

• If you check in with a reservation that splits your time between both wings, the moment you wake up, you will find yourself in either Wing A or Wing B. You never move between them.
• Once you are in a wing, you stay there forever.
• The "Freezing" is the fact that the system randomly picks a wing and stays there, ignoring the other one.

Because the system gets "frozen" into one of these separate wings, the overall behavior of the machine is actually a mix of two different, distinct behaviors. If you try to draw a single smooth line to describe the whole hotel, the line will have a sharp, jagged break right in the middle where the two wings meet.

The Solution: The "Block-by-Block" Strategy

The paper argues that because the system gets frozen into these separate "blocks" (or wings), we shouldn't try to calculate the forecast for the whole hotel at once. Instead, we should:

1. Calculate the forecast for Wing A (ignoring Wing B).
2. Calculate the forecast for Wing B (ignoring Wing A).
3. Compare them. The final answer for the whole system is simply the "winner" (the one that is most likely to happen) at any given moment.

Mathematically, this means taking the minimum of the two separate forecasts. This works because, in the long run, the system will naturally follow the path of least resistance (the most probable path) within whichever wing it got frozen into.

The Proof: Two Test Cases

The authors tested this idea on two models:

1. A Simple Math Model: They created a theoretical system where they could solve the equations exactly. They showed that if you calculate the "local" forecasts for each block and then pick the lowest one, it perfectly matches the actual behavior of the system.
2. A Three-Spin Model: They looked at a system of three tiny magnets (spins) interacting with each other.
   • Without Symmetry Breaking: The system had the "frozen" behavior. The forecast graph had a sharp, jagged point (a "kink") right in the middle.
   • With Symmetry Breaking (Dephasing): They introduced a little bit of "noise" (dephasing) to the system, which is like opening a small door between the two hotel wings.
   • The Result: The sharp kink disappeared! The jagged line smoothed out into a curve. The authors used a mathematical technique called perturbation theory (like a gentle nudge) to show exactly how this "kink" vanishes. They found that the sharp point turns into a smooth "avoided crossing," similar to how two train tracks might look like they are about to crash but then curve away from each other.

The Takeaway

The paper solves a mathematical puzzle: How do we predict rare events in quantum systems that get "stuck" in different states?

The answer is: Break the problem down.
Instead of trying to force a smooth answer onto a broken system, calculate the smooth answers for the separate pieces and then combine them by picking the most likely outcome. This approach is justified by the physical reality that these systems "freeze" into one piece or the other, never mixing them.

The authors conclude that this method works perfectly for these specific symmetric systems and provides a clear way to understand how adding a little bit of noise (dephasing) smooths out the jagged behavior of the system.

Technical Summary

Technical Summary: Large Deviation Functions for Open Quantum Systems with a Strong Symmetry

Problem Statement
In open quantum systems possessing strong symmetries, the global scaled cumulant generating function (SCGF) is generally nonanalytic, particularly at the zero counting field. This nonanalyticity arises because strong symmetries decompose the system's operator space into independent blocks, leading to multiple steady states and "dissipative freezing," where quantum jump trajectories become confined to specific symmetry subspaces. Consequently, the standard Gärtner-Ellis theorem, which relies on the analyticity of the SCGF to derive the large-deviation rate function via a Legendre transform, cannot be directly applied to the global system. A straightforward Legendre transform of a nonanalytic SCGF yields a convex envelope, failing to capture the genuine nonconvex large-deviation rate function required to describe the system's fluctuation statistics.

Methodology
The authors propose a block-wise application of the Gärtner-Ellis theorem to resolve this issue. The methodology proceeds as follows:

1. Decomposition: The system's operator space is decomposed into independent diagonal blocks ($B_{\alpha\alpha}$) due to the strong symmetry.
2. Local Analysis: Within each diagonal block, the local SCGF ($\phi_\alpha(\lambda)$) is identified as the largest real part of the eigenvalues of the tilted generator restricted to that block. Since the dynamics within a single block lack the symmetry-induced degeneracy, these local SCGFs are assumed to be analytic.
3. Local Rate Functions: The local large-deviation rate functions ($I_\alpha(j)$) are obtained by applying the Legendre transform to the local analytic SCGFs, adhering to the standard Gärtner-Ellis procedure within each block.
4. Global Reconstruction: The global rate function $I(j)$ is reconstructed by taking the minimum of the local rate functions over all diagonal blocks where the initial state has non-zero coefficients:
   $$I(j) = \min_{\alpha} \{I_\alpha(j)\}$$
   This minimization reflects the "dissipative freezing" phenomenon, where the trajectory ensemble is dominated by the block with the most favorable (lowest) rate function in the long-time limit.

Key Contributions and Results
The paper validates this scheme through two specific models:

• Analytical Model: A system where the Hamiltonian and jump operator are proportional to a symmetry operator. The authors demonstrate that the global rate function is exactly the minimum of the local rate functions derived from the degenerate SCGFs, confirming the theoretical conjecture.
• Three-Spin Model: A system of three spins with XX interaction and dissipation on the first spin, possessing a permutation symmetry between the second and third spins.
  • Verification: Numerical simulations of quantum jump trajectories confirm that the global rate function obtained via the proposed minimization scheme matches the empirical data, while the standard global Legendre transform fails.
  • Dephasing Effects: The paper analyzes the effect of introducing local dephasing, which breaks the strong symmetry. It is shown that the nonanalytic point in the global SCGF vanishes, transforming the "level crossing" of the local SCGFs into an "avoided level crossing."
  • Perturbation Theory: Using degenerate perturbation theory, the authors quantify this transition. They derive that under weak dephasing, the system's relaxation dynamics and fluctuations are characterized by the perturbed superoperators within the diagonal blocks, with the Liouvillian gap emerging at the zero counting field.

Significance and Claims
The paper claims to provide a rigorous framework for calculating large-deviation rate functions in open quantum systems with strong symmetries, a regime where standard methods fail. The significance lies in:

1. Theoretical Resolution: It resolves the challenge of nonanalytic SCGFs by shifting the application of the Gärtner-Ellis theorem from the global space to the symmetry-invariant blocks.
2. Physical Justification: The approach is physically justified by the dissipative freezing phenomenon, linking the mathematical minimization of rate functions to the physical selection of symmetry subspaces by quantum trajectories.
3. Quantitative Insight: The work quantifies how symmetry-breaking perturbations (like dephasing) smooth out nonanalyticities, bridging the gap between systems with multiple steady states and those with a unique steady state.

The authors note that the paper focuses on presenting this basic idea and method, explicitly stating that they do not explore more complex scenarios such as multiple strong symmetries, non-Abelian groups, or open quantum many-body systems, which are left for future investigation.