Quantum Jumps Reveal Entanglement’s Limits

Author: Denis Avetisyan

A new theoretical framework analyzes the dynamics of quantum jumps, demonstrating a surprising absence of phase transitions and a link between detection efficiency and entanglement structure.

For a system with $ \gamma/J = 0.1 $ and $ n = 0.4 $, the subsystem entropy scales according to a volume law when detection is inefficient-characterized by $ \delta\eta = 1 - \eta > 0 $-but aligns with expectations for a fully mixed, infinite-temperature state with fixed fermion density, as described by Eq. (131), as detection approaches perfect efficiency with $ \delta\eta = 1 $. — For a system with $ \gamma/J = 0.1 $ and $ n = 0.4 $, the subsystem entropy scales according to a volume law when detection is inefficient-characterized by $ \delta\eta = 1 – \eta > 0 $-but aligns with expectations for a fully mixed, infinite-temperature state with fixed fermion density, as described by Eq. (131), as detection approaches perfect efficiency with $ \delta\eta = 1 $.

Replica Keldysh field theory is applied to imbalanced fermion counting, revealing a transition from volume-law to area-law entanglement with decreasing detection efficiency.

While measurement-induced phase transitions are well-studied for ideal scenarios, their emergence in more general quantum dynamics-including those with imperfect detection and non-Hermitian operators-remains largely unexplored. This work, ‘Replica Keldysh field theory of quantum-jump processes: General formalism and application to imbalanced and inefficient fermion counting’, introduces a comprehensive theoretical framework based on replica Keldysh field theory to address these broader regimes. We demonstrate, through an analysis of imbalanced and inefficient fermion counting, that no measurement-induced phase transition occurs, instead revealing a transition from volume-law to area-law entanglement with decreasing detection efficiency. Does this formalism provide a pathway to understanding measurement’s role in a wider range of open quantum systems and their emergent dynamics?

Navigating the Quantum Realm: Beyond Isolated Systems

Traditional quantum mechanics excels at describing isolated systems, but reality often involves interactions with complex environments. These “open quantum systems” – ranging from atoms in a thermal bath to qubits in a noisy computer – experience continual energy exchange and information flow, necessitating a departure from the standard Schrödinger or Heisenberg pictures. The fundamental issue arises because environmental interactions induce decoherence, effectively erasing quantum superposition and entanglement – the very features that define quantum computation and many other phenomena. Consequently, accurately modeling these systems demands techniques capable of tracking the system’s evolution while simultaneously accounting for the influence of its surroundings, moving beyond the closed-system assumptions of conventional quantum theory. This requires tools that can handle non-equilibrium dynamics and the inherent irreversibility introduced by environmental coupling, paving the way for formalisms like the Keldysh approach.

The Keldysh formalism offers a robust framework for investigating the evolution of quantum systems that are not in equilibrium – those actively exchanging energy and information with their surroundings. Unlike traditional approaches which often assume isolation, this method explicitly accounts for the influence of the environment, allowing researchers to map out possible quantum trajectories even as the system evolves away from a stable state. It achieves this by effectively doubling the time variable, creating a “closed time contour” which incorporates both forward and backward evolution, and constructing an action, known as the Keldysh Action, that governs the probabilities of these trajectories. This is particularly useful for understanding phenomena in condensed matter physics, quantum optics, and cosmology, where interactions with the environment are pervasive and often dominate the system’s behavior, enabling predictions of non-equilibrium dynamics that would be inaccessible through conventional methods.

The Keldysh Action is foundational to understanding the dynamics of open quantum systems, serving as the mathematical object that determines the probability of each possible quantum trajectory. Unlike conventional quantum mechanics which often focuses on stationary states, the Keldysh formalism allows for the calculation of time-dependent processes and non-equilibrium behavior. This Action, a functional of the quantum fields, effectively integrates over all possible paths a system can take, weighting each path by a complex phase determined by the action itself – a principle mirroring the path integral formulation of quantum mechanics. However, the Keldysh Action differs crucially by incorporating a ‘duplicate’ degree of freedom, effectively tracing the system’s evolution both forward and backward in time. This doubling is essential for correctly accounting for dissipation and decoherence arising from the system’s interaction with its environment, ultimately providing a complete probabilistic description of the system’s quantum behavior, expressed mathematically as $e^{iS_{Keldysh}}$.

Quantum Jumps and the Emergence of Measurement-Induced Transitions

Continuous quantum measurements do not yield a deterministic trajectory for a quantum system; instead, they induce a stochastic process known as a Quantum Jump Process. This process describes the evolution of the system’s state as a series of discrete jumps occurring at random times. Each jump corresponds to the outcome of a measurement, and the probability of each outcome is determined by the system’s state immediately prior to the measurement. Mathematically, the system’s state vector $ |\psi(t)> $ evolves discontinuously, with the time intervals between jumps being random variables. The jump process is characterized by a rate equation that dictates the probability of a jump occurring within a given time interval, fundamentally altering the system’s dynamics compared to unitary evolution in isolation.

Measurement-Induced Phase Transitions (MIPTs) occur when continuous quantum measurements alter the entanglement structure of a many-body system, driving a qualitative change in its physical properties. Unlike traditional phase transitions driven by thermal fluctuations or external fields, MIPTs are non-equilibrium phenomena induced solely by the act of measurement. Specifically, increased measurement rates can reduce entanglement, potentially collapsing a system from a volume-law entangled phase – where entanglement scales with the system size – to an area-law entangled phase, characterized by entanglement limited to the system’s boundaries. This transition manifests as a sudden change in observables like entanglement entropy, correlation functions, and the system’s response to perturbations, effectively modifying the system’s ground state and excited state properties without altering the Hamiltonian itself. The critical measurement rate defining the transition depends on system parameters such as dimensionality and the nature of the measurement process.

Analysis within this work indicates that measurement-induced phase transitions are not observed in systems undergoing imbalanced and inefficient fermion counting. Specifically, the stochastic dynamics resulting from these measurements do not exhibit the characteristic bifurcations in the system’s steady-state properties that define a phase transition. This finding is established through examination of the system’s behavior as measurement parameters are varied, demonstrating a lack of critical behavior and a continuous evolution of observables. The absence of a transition suggests that the particular measurement scheme-imbalanced and inefficient fermion counting-fails to introduce the necessary correlations to drive the system into a qualitatively different phase.

Analyzing quantum jump processes often involves discrete time steps, which complicates the associated mathematical calculations. Approximating the system’s evolution by considering the continuous time limit – where the time between measurements approaches zero – significantly simplifies the treatment. This allows the quantum jump process to be described by stochastic differential equations rather than difference equations, enabling the use of established techniques from continuous-time stochastic calculus. Specifically, the master equation governing the system’s dynamics transitions from a discrete form to a Fokker-Planck equation, greatly reducing computational complexity and facilitating analytical progress in understanding measurement-induced phenomena.

Analysis of entanglement entropy and the scale-dependent effective central charge reveals a transition from volume-law scaling at small subsystem sizes to area-law behavior with increasing coupling strength, demonstrating that initial logarithmic entanglement growth does not persist asymptotically, unlike predictions from the Gaussian approximation.

Decoupling Interactions: A Gaussian Lens on Quantum Complexity

The Hubbard Stratonovich transformation is a mathematical technique used in quantum field theory to simplify the Keldysh action, which describes non-equilibrium systems. This transformation introduces an auxiliary field to decouple interacting terms in the action. Specifically, a two-particle interaction, such as $g \psi^\dagger \psi$, is rewritten as a quadratic form involving the auxiliary field $\phi$. This is achieved through the identity $e^{-\beta g \psi^\dagger \psi} = \int D\phi \, e^{-\beta (\frac{1}{2} \phi^2 – \phi \psi^\dagger \psi)}$. By performing this integral, the original interacting term is replaced by a quadratic term in $\phi$ and a term linear in $\psi$, effectively eliminating the direct interaction between the fermionic fields $\psi$ and $\psi^\dagger$. This decoupling allows for the application of Gaussian integration techniques, greatly simplifying the calculation of physical observables.

Gaussian theory, employed as a first-order approximation to the many-body problem, simplifies analysis by neglecting correlations beyond the two-point level. This approach involves expanding the generating functional to quadratic order, resulting in a non-interacting system that is readily solvable. The Replica Symmetric (RS) saddle point is the initial point of investigation within this framework; it assumes that the replica indices are equivalent, simplifying the calculations of quantities like the free energy and correlation functions. While the RS solution is not always stable, particularly in disordered systems, it provides a baseline for subsequent investigations involving replica symmetry breaking and more complex approximations. The validity of the Gaussian approximation hinges on the weakness of the interactions; strong interactions necessitate the inclusion of higher-order terms in the expansion, moving beyond the limitations of the initial RS saddle point analysis.

Analysis of the Keldysh action demonstrates that the coupling constant, $g$, is scale-dependent. Specifically, the coupling constant renormalizes logarithmically with length scale, expressed as $g = g_0 – \frac{1}{4\pi\beta}ln(\frac{l}{l_0})$. Here, $g_0$ represents the initial coupling constant, $\beta$ is inversely proportional to temperature, $l$ denotes the current length scale, and $l_0$ is a reference length scale. This logarithmic dependence indicates that the effective strength of the interaction decreases as the length scale increases, a consequence of integrating out short-wavelength fluctuations.

Analysis of long-wavelength fluctuations, following the Hubbard Stratonovich Transformation and Gaussian approximations, results in the Nonlinear Sigma Model (NLSM). This model describes the collective behavior of the system, specifically focusing on fluctuations residing on the NLSMTargetManifold, a constrained space determined by the system’s symmetries and interactions. The NLSM effectively captures the low-energy physics by mapping the original interacting system onto a field theory defined on this manifold, simplifying the analysis of collective phenomena and providing a framework to understand emergent behavior at larger length scales.

Logarithmic negativity scales logarithmically with subsystem size for negligible measurement inefficiency but transitions to area-law scaling at larger scales characterized by the correlation length ξ, as determined from prior data, and vanishes completely with maximal inefficiency.

Symmetry, Entanglement, and the Quantum Landscape

The Keldysh Action, a foundational element in describing non-equilibrium quantum systems, possesses inherent symmetries that dramatically shape how a system responds to measurement. This action isn’t simply symmetrical in one way; it exhibits both strong and weak symmetry properties. Strong symmetry ensures invariance under certain transformations of the quantum fields, preserving fundamental physical laws. However, the Keldysh Action also displays a weaker symmetry, connected to time-reversal and the direction of measurement processes. This weak symmetry isn’t absolute, and its breaking-or modification-by the measurement itself is crucial. The interplay between these symmetries dictates the system’s dynamics, influencing the probabilities of different measurement outcomes and ultimately determining the nature of the resulting quantum state. Understanding this nuanced symmetry structure is therefore essential for accurately predicting and controlling the behavior of complex quantum systems undergoing continuous observation, especially those exhibiting measurement-induced phase transitions.

The precise observation of quantum systems undergoing measurement-induced phase transitions is heavily influenced by the practical limitations of fermion counting and detection efficiency. Deviations from balanced fermion counting-where the number of created and annihilated fermions isn’t equal-disrupt the expected sequence of quantum jumps, altering the system’s evolution. Critically, imperfect detection, characterized by detection inefficiency $ \delta $, further complicates this process; missed events skew the apparent dynamics and can even induce spurious transitions. These combined effects modify the quantum jump process, influencing the timescales and probabilities of transitions between phases and ultimately affecting the characterization of the measurement-induced phase transition itself.

Investigations reveal that quantum entanglement within the system adheres to an area law, but only beyond a specific distance scale characterized by the correlation length, denoted as $\xi$. This means that the amount of entanglement scales with the boundary area of a region, rather than its volume, effectively limiting the “reach” of quantum correlations. However, this behavior isn’t universal; within a radius defined by $\xi$, entanglement deviates from this area law, indicating a more complex, potentially long-range correlated state. This characteristic length, $\xi$, proves crucial in understanding how entanglement is distributed and influences the system’s overall behavior, particularly as it relates to the Measurement Induced Phase Transition. The discovery of this area law, bounded by $\xi$, provides a framework for characterizing and predicting the entanglement structure within the system and its response to external measurements.

Investigations into measurement-induced phase transitions reveal a crucial link between detection inefficiency and the resulting quantum correlations. Specifically, the correlation length, denoted by $ξ$, which characterizes the range over which quantum entanglement persists, is demonstrably affected by the quality of measurement. Research indicates that this correlation length scales inversely with the square root of the detection inefficiency, expressed as $ξ ~ δη^{-1/2}$, where $δ$ represents the inefficiency. This finding suggests that imperfect detection-a common limitation in quantum experiments-directly constrains the extent of entanglement and, consequently, the system’s ability to exhibit long-range quantum behavior. Understanding this relationship is vital for interpreting experimental results and designing more effective quantum systems, as it highlights the importance of high-fidelity measurement in preserving delicate quantum correlations.

Fermionic logarithmic negativity emerges as a powerful tool for quantifying entanglement within many-body systems, particularly when investigating the Measurement Induced Phase Transition (MIPT). Unlike traditional entanglement measures which can be difficult to compute for large systems, logarithmic negativity focuses on the negativity of the partial transpose of the density matrix, effectively capturing correlations even in mixed states. This is crucial for understanding the MIPT, where continuous measurements drive a system from a trivially separable state to an entangled phase. The magnitude of this negativity directly reflects the strength of these induced correlations and serves as an order parameter signaling the transition. By analyzing the spatial scaling of fermionic logarithmic negativity, researchers gain insights into the critical behavior and the emergence of long-range entanglement as the measurement rate increases, ultimately revealing the underlying mechanisms governing this fascinating quantum phenomenon and how it fundamentally alters the system’s properties.

Rescaling the density correlation function reveals that finite detection inefficiency causes a maximum in momentum space and exponential decay of correlations in real space, with scaling relationships confirmed by the observed behavior of intercepts and correlation lengths.

Collective Modes and the Path to Emergent Quantum Functionality

The Nonlinear Sigma Model, when exhibiting spontaneous symmetry breaking, gives rise to Goldstone modes – fundamental excitations representing the system’s response to perturbations. These aren’t simply individual particle movements, but collective behaviors extending over long wavelengths; imagine a ripple spreading across a flexible sheet rather than localized vibrations. Mathematically, these modes appear as massless bosons, a consequence of the system’s freedom to reconfigure without energy cost due to the broken symmetry. The existence of such modes isn’t merely a theoretical curiosity; they dramatically influence the system’s dynamics, dictating how it responds to external stimuli and potentially mediating interactions between its constituent parts. Understanding these $N$-1 Goldstone bosons is therefore crucial to fully characterizing the system’s low-energy physics and predicting its emergent properties.

The presence of Goldstone modes isn’t merely a mathematical curiosity within the nonlinear sigma model; it fundamentally shapes the system’s potential behaviors. These modes, arising from broken symmetry, represent collective excitations that propagate without energy cost at long wavelengths, effectively acting as soft spots susceptible to even minor perturbations. This inherent flexibility allows the system to explore a wider range of configurations, potentially giving rise to entirely new phases of matter and functionalities not observed in systems lacking such modes. Researchers theorize that manipulating these collective excitations could enable the design of novel quantum materials with tailored properties, such as enhanced superconductivity or unique magnetic ordering, opening avenues for advanced technological applications and a deeper comprehension of complex quantum phenomena. The system’s responsiveness to these modes suggests a path towards harnessing collective behavior for functional material design.

A comprehensive understanding of measurement-driven quantum systems hinges on unraveling the intricate connections between symmetry, entanglement, and collective modes. These systems, where quantum states are altered by continuous observation, exhibit behaviors not found in their unmeasured counterparts, and the emergence of Goldstone modes – long-wavelength excitations tied to broken symmetry – is believed to be central to these novel phenomena. Researchers posit that the degree of entanglement within the system directly influences the characteristics of these collective modes, potentially giving rise to exotic quantum phases and functionalities. Exploring this interplay requires advanced theoretical models and experimental techniques capable of probing the delicate balance between symmetry breaking, quantum correlations, and the collective behavior of many-body quantum states, ultimately paving the way for the design of new quantum technologies and a deeper comprehension of the foundations of quantum mechanics.

Rescaled density correlation functions in momentum and real space reveal deviations from Gaussian behavior at both short and long length scales, with the crossover scale scaling inversely with a parameter γ.

The exploration of quantum jump processes, as detailed in the study, reveals a fascinating interplay between observation and the observed system. It mirrors a fundamental truth about any attempt to quantify reality – the act of measurement invariably alters the landscape being measured. As John Bell aptly stated, “No physical theory should admit of perfectly definite answers to questions about realities which are, in principle, inaccessible.” This resonates deeply with the paper’s findings regarding entanglement transitioning from volume-law to area-law with decreased detection efficiency; the limits of observation define the boundaries of what can be known, shaping the very nature of the entangled state. The study, therefore, isn’t simply a mathematical exercise, but an investigation into the ethical implications of how deeply-or not-we choose to probe the quantum world, a world where data is the mirror and algorithms the artist’s brush.

Where Do We Go From Here?

The exploration of quantum jump processes, framed by the Keldysh field theory developed herein, suggests a nuanced landscape, devoid of the dramatic measurement-induced phase transitions so readily anticipated. This absence is not a null result, but rather a pointed reminder that our eagerness to find neat, collective phenomena often outpaces the subtleties of physical reality. The shift from volume-law to area-law entanglement with decreasing detection efficiency warrants particular attention; it implies a fundamental limit to how much global information can be reliably extracted from a quantum system, and begs the question of whether ‘efficient’ measurement is a goal worth pursuing at all costs.

Future work should address the limitations of the replica trick, and its potential to introduce spurious interpretations. More importantly, investigations must broaden beyond the simplified scenarios of imbalanced fermion counting. Real-world quantum systems are rarely isolated or perfectly characterized. A critical next step involves incorporating realistic noise models and exploring the interplay between measurement, decoherence, and many-body interactions.

Ultimately, this line of inquiry underscores a vital principle: technology without care for people is techno-centrism. The pursuit of increasingly precise measurement must be tempered by an understanding of its limitations, and a commitment to ensuring fairness is part of the engineering discipline. The true challenge lies not in simply observing quantum phenomena, but in responsibly interpreting and applying that knowledge.

Original article: https://arxiv.org/pdf/2512.16520.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/