Multiplicative Noise and Effective Potentials

• Multiplicative noise refers to state-dependent stochastic fluctuations that modify effective potentials and induce novel phase transitions in quantum systems.
• Effective potentials emerge from averaging over noise, capturing deterministic evolution influenced by measurement back-action and non-Hermitian dynamics.
• Key diagnostics like the occupation gap and scaling laws provide experimental insights into the transition between volume-law and area-law entanglement phases.

Multiplicative noise and effective potentials are central concepts in the stochastic dynamics of open quantum systems, monitored circuits, and phase transitions subject to repeated measurement processes. In these contexts, multiplicative noise refers to stochastic fluctuations whose amplitude depends explicitly on the dynamical state (e.g., the local occupation or current), leading to nontrivial modifications of the effective landscape governing circuit evolution, spectral properties, and steady-state observables. The interplay between measurement, coherent evolution, and multiplicative noise generates new types of phase transitions and critical phenomena, distinct from both classical equilibrium and standard open-system dissipative dynamics.

1. Definition and Fundamental Principles

Multiplicative noise arises when the stochastic terms in the evolution of a quantum or classical system have coefficients that depend on the system's instantaneous state. In the context of quantum measurement protocols, noise is typically introduced via stochastic Schrödinger equations, Kraus operator maps, or master equations, where the measurement back-action can depend on local observables such as density or current. The nontrivial state-dependence of the noise amplitude differentiates multiplicative noise from additive (state-independent) noise, fundamentally altering the underlying Fokker-Planck or Keldysh field-theoretic description.

Effective potentials refer to the emergent deterministic landscape in which the ensemble-averaged or single-trajectory evolution occurs. In field-theoretic terms, they arise as the potential term in the effective action after averaging over noise realizations or integrating out fast degrees of freedom. In monitored quantum systems, the structure of the effective potential is nontrivially influenced by the measurement protocol, the form of the observables being measured, and the nature (projective vs. weak, local vs. non-local) of the noise.

2. Measurement-Induced Phase Transitions and Multiplicative Noise

In stochastically monitored quantum systems, the competition between coherent unitary evolution and invasive measurement back-action leads to measurement-induced phase transitions (MIPTs) between entangling (volume-law) and disentangling (area-law) entanglement regimes. When the measurement process itself injects multiplicative noise (e.g., via local current measurements or measurement operators that are functions of occupation variables), it can fundamentally alter the transition scenario and critical properties.

For example, in monitored fermion chains with random projective measurements of local current operators, the measurement acts as a multiplicative noise source, since the measurement back-action depends on instantaneous local occupations and currents. This generates stochastic state trajectories described by a nontrivial effective action, where the potential and noise terms are interdependent (Lumia et al., 2023). The resulting dynamics are no longer Gaussian (even for free-fermion Hamiltonians) and must be analyzed with explicit attention to the non-Gaussianity introduced by the noise structure.

In field-theoretic language, the presence of multiplicative noise leads to effective, often non-Hermitian, potentials in Keldysh or path-integral representations. For example, in the n-replica Keldysh theory for monitored Dirac fermions (Buchhold et al., 2021), the effective potential in the non-Hermitian quantum sine-Gordon model acquires imaginary coefficients as a result of averaging over measurement-induced noise. The RG flow of the resulting theory features complex couplings, and the phase boundary between gapless (delocalized) and gapped (pinned) phases is shifted relative to the equilibrium case.

3. Quantitative Diagnostics: Spectral Indicators and One-Body Gaps

Multiplicative noise modifies the spectral properties of the evolution operator and may lead to transitions detectable via single-particle indicators. In the monitored free-fermion models, the occupation gap χ—the difference between the topmost occupied and the lowest unoccupied eigenvalues of the correlation matrix—serves as a sensitive indicator of the measurement-induced transition (Lumia et al., 2023). This observable, accessible through two-point correlation functions, captures the effect of multiplicative measurement noise on the effective spectrum even in non-Gaussian regimes.

The phase diagram schematically is as follows:

Phase	Entanglement Scaling	One-Body Gap χ（L→∞）
Volume-law	$S_A(\ell)\sim c\ell$	$\chi \to 0$
Area-law	$S_A(\ell)\sim O(1)$	$\chi$ finite

The finite-size crossing of $L\chi$ vs. measurement rate locates the transition, which coincides with the closure of the spectral gap in the effective non-unitary propagator (Mochizuki et al., 2024).

4. Effective Field Theories: Non-Hermitian Potentials and RG Flow

The Keldysh functional approach to monitored many-body systems provides explicit expressions for effective potentials and noise correlations. Measurement protocols featuring multiplicative noise result in non-Hermitian field theories with complex-valued potentials and kinetic terms. For example, the non-Hermitian sine-Gordon Hamiltonian for locally measured Dirac fermions includes an effective potential $-i\lambda\cos(\sqrt{8}\phi)$ and complexified Luttinger parameter $K = 1 - 2i\gamma/(\pi v)$, reflecting the measurement-induced modifications (Buchhold et al., 2021). RG equations analogous to BKT theory describe the critical properties, with the relevance of the cosine term—and thereby the pinning transition—shifted by the strength and structure of the multiplicative noise.

These effective potentials govern the universal scaling laws at the transition: in the gapless phase (Re $K<1$), the system exhibits logarithmic entanglement growth $S(L)\sim (c_{\rm eff}/3)\ln L$, while in the gapped phase (Re $K>1$), entanglement saturates (area law) with a finite correlation length determined by the effective potential landscape.

5. Stability, Universality, and Role of Non-Gaussianity

Multiplicative measurement noise is key to sustaining stable volume-law entanglement phases in otherwise non-interacting (free) systems. When only density measurements are performed in free fermion models, all trajectories remain Gaussian, and the volume-law phase is destroyed at any nonzero measurement rate—there is no stable entangling phase. Introducing non-Gaussianity via multiplicative noise (e.g., using current measurements or interacting Hamiltonians) reinstates the measurement-induced transition between volume- and area-law phases, highlighting the essential role of non-Gaussian multiplicative noise in the critical behavior (Lumia et al., 2023).

This leads to a unified interpretation of measurement-induced transitions across a broad class of protocols: the presence and structure of the effective potential, shaped by the multiplicative character of the measurement noise, is determinant in stabilizing or destabilizing long-range quantum correlations. Universality classes of the transition—described by critical exponents and scaling functions—are thus fundamentally governed by the noise-induced modification of the field-theoretic landscape.

6. Practical and Experimental Implications

The use of purely single-particle observables such as the occupation gap $\chi \to 0$0 allows for efficient experimental diagnosis of measurement-induced phase transitions even in non-Gaussian regimes, circumventing the need for full many-body state tomography. The theory provides predictive scaling for critical measurement rates and suggests avenues for engineering protocols (e.g., selecting measurement observables and rates) to tailor the resulting steady-state phase diagram and critical entanglement properties.

Future extensions include higher-dimensional systems, exploration of other forms of measurement-induced multiplicative noise (beyond simple local density or current observables), and systematic classification of measurement-induced transitions via field-theoretic techniques adapted to non-Hermitian and stochastic systems (Lumia et al., 2023, Buchhold et al., 2021).

7. Summary Table: Core Relations—Multiplicative Noise and Effective Potentials in Monitored Systems

Measurement Protocol	Noise Structure	Effective Potential Type	Phase Transition Mechanism
Free fermion + density measurement	Additive (Gaussian)	Real, quadratic	No stable volume law (any p>0 area-law)
Free/interacting + current measurement	Multiplicative (non-G)	Non-Hermitian, non-quadratic	Volume-law/area-law transition at $\chi \to 0$1
Monitored Dirac fermions (Keldysh)	Multiplicative	Complex sine-Gordon	BKT-like, $\chi \to 0$2 shifted by Im(K)

The multiplicative character of measurement-induced noise is thus a defining ingredient in both the effective potential governing trajectory evolution and the emergent critical phenomena in monitored quantum many-body systems (Lumia et al., 2023, Buchhold et al., 2021).