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Replica Keldysh field theory of quantum-jump processes: General formalism and application to imbalanced and inefficient fermion counting

This paper develops a comprehensive replica Keldysh field theory to unify the description of measurement-induced phase transitions in both efficient and inefficient quantum-jump processes, demonstrating through analytical and numerical studies of imbalanced fermion counting that inefficient detection introduces a finite correlation length leading to distinct area-law and volume-law scaling behaviors in entanglement and subsystem entropy.

Original authors: Felix Kloiber-Tollinger, Lukas M. Sieberer

Published 2026-07-01

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CC BY 4.0

Original authors: Felix Kloiber-Tollinger, Lukas M. Sieberer

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 complex dance performance by a troupe of invisible dancers (quantum particles). In the world of quantum physics, these dancers don't just move on their own; they are constantly being watched. The way they move depends entirely on how the audience (the measurement device) watches them.

This paper builds a new "rulebook" (a mathematical framework) to understand what happens to these dancers when the audience is imperfect, distracted, or watching them in a very specific, chaotic way.

Here is the breakdown of their findings using everyday analogies:

1. The Old Rulebook vs. The New One

Previously, scientists had a great rulebook for two specific scenarios:

Perfect Watching: The audience sees every single move perfectly.
Fixed Rhythm: The audience checks in at a steady, unchanging beat.

However, real life is messier. Sometimes the audience misses steps (inefficient detection), and sometimes the dancers move at speeds that depend on where they are currently standing (state-dependent rates). The old rulebook couldn't handle this.

The Paper's Contribution: The authors created a new, universal "Replica Keldysh Field Theory." Think of this as a master translator that can take any chaotic, imperfect observation of quantum particles and turn it into a clear mathematical story. It works for both bosons (particles that like to clump together, like photons) and fermions (particles that avoid each other, like electrons).

2. The Experiment: The "Fermion Counting" Game

To test their new rulebook, the authors set up a simulation of a one-dimensional line of fermions (like a row of people passing a ball).

The Action: People are randomly gaining balls (creation) or losing balls (annihilation).
The Twist 1 (Imbalance): They gain balls at a different rate than they lose them. It's not a fair game; the flow is lopsided.
The Twist 2 (Inefficiency): The "scorekeeper" (the detector) is sleepy. They only catch a fraction of the ball gains and losses. The rest happen in the dark, unseen.

3. What They Found: The "Entanglement" Dance

In quantum physics, "entanglement" is like a deep, invisible bond between dancers. If they are highly entangled, they move as a single unit, no matter how far apart they are. Scientists measure this "bond" using Entropy.

Scenario A: The Perfect Scorekeeper (Efficient Detection)

Even when the scorekeeper sees everything, but the game is lopsided (imbalanced rates):

The Result: The dancers eventually stop being deeply connected over long distances. The "bond" (entanglement) follows an "Area Law."
The Analogy: Imagine a long line of people holding hands. If the line gets too long, the people at the very end stop feeling the tension of the people at the very beginning. They only feel the people right next to them.
The Surprise: Before they stop being connected, there is a "Goldilocks zone" (a middle distance) where the dancers act like they are in a perfect, critical state (like a fluid or a critical phase). They show signs of "conformal invariance," meaning they look the same whether you zoom in or out, but this only happens for a specific range of distances.

Scenario B: The Sleepy Scorekeeper (Inefficient Detection)

When the scorekeeper misses some jumps (inefficient detection):

The Result: The "bond" (entanglement) is cut off much sooner.
The Analogy: Imagine the dancers are in a foggy room. If the scorekeeper misses steps, it's like the fog gets thicker. Beyond a certain distance (the "correlation length"), the dancers are completely disconnected.
The Catch: While the true quantum bond (measured by "logarithmic negativity") disappears and follows the "Area Law," the total confusion or "entropy" of the system actually goes up. It follows a "Volume Law."
Why? This is because the system becomes "mixed." It's no longer a pure, coordinated dance; it's a chaotic mess of possibilities. The "Volume Law" here doesn't mean they are deeply connected; it means the system is just messy and unpredictable because the observer is missing information.

4. The Big Picture Connection

The most significant claim of the paper is that it bridges two worlds that scientists usually keep separate:

Measurement-Induced Dynamics: Systems that change because we are watching them.
Driven Open Systems: Systems that are constantly pushed and pulled by their environment (like a car engine running).

The authors show that if you turn down the "efficiency" of the watcher to zero, the "measurement" problem smoothly turns into a standard "open system" problem. This means the math used to study how watching a system changes it is now directly connected to the math used to study how a system behaves when it's just interacting with a noisy environment.

Summary

The Tool: A new mathematical framework to study quantum particles under imperfect, chaotic observation.
The Finding: Even if the observation is lopsided, the particles eventually stop being deeply connected over long distances (Area Law).
The Nuance: If the observer is lazy (inefficient), the system looks messy and chaotic (Volume Law entropy), but the actual quantum connection is still cut off (Area Law entanglement).
The Bridge: This work proves that "watching" a quantum system and "pushing" it with an environment are two sides of the same coin, describable by the same underlying physics.

Technical Summary: Replica Keldysh Field Theory of Quantum-Jump Processes

Problem Statement
Measurement-induced phase transitions (MIPTs) have been extensively studied in systems undergoing projective or continuous measurements of Hermitian observables, typically assuming perfect detection. However, a comprehensive theoretical framework has been lacking for more general scenarios, specifically:

Quantum-jump processes involving non-Hermitian jump operators, where jump rates are state-dependent and fluctuate in time.
Inefficient detection, where only a fraction of quantum jumps are registered, leading to mixed-state dynamics rather than pure-state trajectories.
Existing analytical methods, such as replica Keldysh field theories, have been limited to quantum state diffusion or random projective measurements with perfect detection. There was no general formalism capable of averaging over quantum trajectories with state-dependent jump rates and imperfect detection efficiency within a field-theoretic approach.

Methodology
The authors develop a replica Keldysh field theory for general quantum-jump processes in bosonic and fermionic many-body systems. The methodology proceeds through several key steps:

Kraus Operator Formulation: The dynamics are described in discrete time using Kraus operators. The evolution is split into "no-jump" continuous evolution (governed by an effective non-Hermitian Hamiltonian) and discontinuous quantum jumps.
Rescaling and Continuous-Time Limit: To handle the continuous-time limit ( $\Delta t \to 0$ ) within the replica framework, the authors introduce rescaled Kraus operators. This allows the trajectory average to be expressed as an average over jump times and types, leading to a Poissonian distribution of jumps in the continuum limit.
Replica Trick: To compute nonlinear observables (e.g., entanglement entropy), the authors employ the replica trick. They construct a $R$ -replica Keldysh partition function, where the trajectory average is performed before taking the replica limit $R \to 1$ . This requires careful handling of the rescaling factors to ensure a nontrivial continuous-time limit.
Field Theory Construction: The partition function is mapped to a functional integral over coherent states (Grassmann variables for fermions). The action consists of:
- A Liouvillian term (unitary evolution and dissipative contributions).
- A decay term (from the anti-Hermitian part of the effective Hamiltonian).
- A jump term encoding the stochastic quantum jumps.
Inefficient Detection: Inefficiency is modeled by splitting time steps into a monitored substep (efficient detection) and an unmonitored substep (deterministic Lindbladian evolution). This introduces a detection efficiency parameter $\eta \in [0, 1]$ . The resulting field theory interpolates between pure-state trajectories ( $\eta=1$ ) and deterministic mixed-state dynamics ( $\eta=0$ ).
Effective Field Theory: For the specific application, the authors perform a Hubbard-Stratonovich transformation to decouple fermionic bilinears, deriving a Nonlinear Sigma Model (NLSM) describing the Goldstone modes associated with replica symmetry breaking. They analyze the symmetries of the action (unitary, particle-hole) to determine the target manifold of the NLSM.

Application: Imbalanced and Inefficient Fermion Counting
The formalism is applied to a 1D lattice of free fermions subject to monitored gain and loss (fermion counting) with:

Imbalanced rates: Gain ( $\gamma_+$ ) and loss ( $\gamma_-$ ) occur at different rates.
Inefficient detection: Only a fraction $\eta$ of jumps is detected.

Key Results

Imbalanced but Efficient Counting ( $\eta=1, \gamma_+ \neq \gamma_-$ ):
- The system exhibits no measurement-induced phase transition.
- Entanglement obeys an area law for any nonzero jump rate on sufficiently large scales.
- However, an extended quantum-critical regime exists between two parametrically separated length scales: $l_0 \sim \gamma^{-1}$ and $l_c \sim \gamma^{-2}$ .
- Within this regime, the system displays signatures of emergent conformal invariance, including algebraic decay of density correlations ( $C_l \sim l^{-2}$ ) and approximately logarithmic entanglement growth ( $S_\ell \sim \ln \ell$ ).
- This behavior is captured by an NLSM with a target manifold $SU(R)$ (or $SU(2R)/Sp(R)$ with particle-hole symmetry). Renormalization group (RG) flow of the NLSM coupling drives the system to strong coupling at scale $l^* \sim l_0 e^{4\pi \beta g_0}$ , leading to the eventual crossover to area-law scaling.
Inefficient Detection ( $\eta < 1$ ):
- Inefficiency introduces a mass term in the NLSM proportional to the detection inefficiency $\delta\eta = 1 - \eta$ .
- This mass generates a finite correlation length $\xi \sim \delta\eta^{-1/2}$ .
- Subsystem Entropy: The von Neumann entropy exhibits volume-law scaling at large scales due to the mixed character of the trajectories (statistical entropy).
- Genuine Entanglement: The fermionic logarithmic negativity, a measure of genuine entanglement, obeys an area law for subsystem sizes larger than $\xi$ .
- In the limit $\eta \to 0$ , the system reduces to a deterministic Lindbladian evolution with a trivial, separable steady state.
Numerical Validation:
- Direct numerical simulations of quantum-jump trajectories (up to $L=1000$ ) confirm the analytical predictions.
- Simulations verify the scaling of the crossover momentum $q_c \sim \gamma^2$ and length $l_c \sim \gamma^{-2}$ .
- They confirm the scaling of the correlation length $\xi \sim \delta\eta^{-1/2}$ and the transition from logarithmic to area-law scaling in the logarithmic negativity.

Significance and Claims
The paper claims to provide a general and versatile theoretical foundation for studying measurement-induced phenomena across a wide class of monitored and open quantum systems. Its primary contributions are:

Unification: It establishes a direct connection between measurement-induced dynamics (pure-state trajectories) and the nonequilibrium steady states of driven open quantum matter (mixed-state dynamics) by incorporating inefficient detection.
Generality: It extends replica Keldysh field theory to general quantum-jump processes with non-Hermitian operators and state-dependent rates, a scenario previously inaccessible to analytical treatment.
Physical Insight: It demonstrates that measurement-induced criticality (the extended critical regime) is robust against imbalanced rates but fragile against detection inefficiency, which introduces a mass that destroys long-range entanglement.
Bridge to Open Systems: The framework links MIPTs to the study of driven open quadratic fermionic systems, offering a tool to analyze trajectory unravelings of models with gain-loss dissipation, topological phases, and non-Hermitian skin effects.

The authors maintain a modest stance, noting that while their formalism allows for the study of symmetry breaking in trajectories versus the mixed steady state, the specific application presented focuses on free fermions. They suggest that the framework opens the door to investigating more complex interacting systems and broader questions regarding topology and symmetry classifications in continuously monitored systems.