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arXiv:2607.20959

Large deviations in quantum dynamics and complexity

Xiangyu Cao, Jorge Kurchan

quant-phcond-mat.stat-mechhep-thSubmitted Jul 23, 2026• Announced Jul 24, 2026• 0 citations
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Abstract

We study three definitions of large deviation in many-body quantum dynamics: (i) via the full distribution of an extensive observable, (ii) via the distribution of measurement outcomes (from a continuous monitoring of the observable) over a time interval $t \le t_{\max}$, and (iii) via the distribution of expectation values over $t \le t_{\max}$. In generic systems without conservation laws, the large deviation function (i) reaches its longtime limit at $t \sim \mathcal{O}(1)$, independently of system size $N$. (ii) and (iii) reach their longtime limit at $t \sim e^{ N}$ and $t \sim \exp(e^{N})$, respectively. Before that, there is a {\it sharp} frontier between the explored and unexplored outcomes/expectation values; their distribution equals the longtime limit truncated at values that drift with $t_{\max}$. We propose that the evolution of these values with $t_{\max}$ provides a measure of quantum complexity.

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17 pages, 6 figures, comments welcome!

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We study three definitions of large deviation in many-body quantum dynamics: (i) via the full distribution of an extensive observable, (ii) via the distribution of measurement outcomes (from a continuous monitoring of the observable) over a time interval t≤tmax⁡t \le t_{\max}t≤tmax​, and (iii) via the distribution of expectation values over t≤tmax⁡t \le t_{\max}t≤tmax​. In generic systems without conservation laws, the large deviation function (i) reaches its longtime limit at t∼O(1)t \sim \mathcal{O}(1)t∼O(1), independently of system size NNN. (ii) and (iii) reach their longtime limit at t∼eNt \sim e^{ N}t∼eN and t∼exp⁡(eN)t \sim \exp(e^{N})t∼exp(eN), respectively. Before that, there is a {\it sharp} frontier between the explored and unexplored outcomes/expectation values; their distribution equals the longtime limit truncated at values that drift with tmax⁡t_{\max}tmax​. We propose that the evolution of these values with tmax⁡t_{\max}tmax​ provides a measure of quantum complexity.
The exact statistics of an arbitrary quantum observable is analytically obtained. Due to the probabilistic nature of a sequence of intermediate measurements and stochastic fluctuations induced by the interaction with the environment, the measurement outcomes at the end of the system's evolution are random variables. Here, we provide the exact large-deviation form of their probability distribution, which is given by an exponentially decaying profile in the number of measurements. The most probable distribution of the measurement outcomes in a single realization of the system transformation is then derived, thus achieving predictions beyond the expectation value. The theoretical results are confirmed by numerical simulations of an experimentally reproducible two-level system with stochastic Hamiltonian.
Quantum measurements are crucial for observing the properties of a quantum system, which, however, unavoidably perturb its state and dynamics in an irreversible way. Here we study the dynamics of a quantum system being subjected to a sequence of projective measurements applied at random times. In the case of independent and identically distributed intervals of time between consecutive measurements, we analytically demonstrate that the survival probability of the system to remain in the projected state assumes a large deviation (exponentially decaying) form in the limit of an infinite number of measurements. This allows us to estimate the typical value of the survival probability, which can therefore be tuned by controlling the probability distribution of the random time intervals. Our analytical results are numerically tested for Zeno-protected entangled states, which also demonstrate that the presence of disorder in the measurement sequence further enhances the survival probability when the Zeno limit is not reached (as it happens in experiments). Our studies provide a new tool for protecting and controlling the amount of quantum coherence in open complex quantum systems by means of tunable stochastic measurements.
Coexistence of different dynamical phases is a hallmark of glassy dynamics. This is well-studied in classical systems where the underlying theoretical framework is that of large deviation theory. The presence of a similar phase coexistence has been suggested in monitored quantum many-body systems, but the lack of suitable methods has yet prevented a systematic large deviation analysis. Here we present a tensor network framework that allows the application of large deviation theory to large quantum systems. Building on this, we locate a series of first-order dynamical phase transitions in a monitored discrete-time many-body quantum dynamics, at the level of the trajectory space. Crucially, our approach provides access not only to large-deviation statistics but also to conditioned quantum many-body states, enabling a microscopic characterization of the dynamical phases and their coexistence.
The exact statistics of an arbitrary measuring observable, describing specific features of an open quantum system, is analytically obtained. Due to the probabilistic nature of a sequence of intermediate measurements and stochastic fluctuations induced by the interaction with an external environment, the measurement outcomes at the end of the system evolution are random variables. Here, we provide the exact large deviation form of their probability distribution, given by an exponentially decaying profile in the number of measurements. The most probable distribution of the measurement outcomes in a single realization of the system transformation is then derived, thus achieving predictions beyond the expectation value.
We study the nonequilibrium dynamics of a homogeneous quantum Ising ring after a quench, in which the transverse field ggg suddenly changes from zero to a nonzero value. The long-timescale reduced dynamics of a single spin and of two nearest-neighbor spins, which involves the evaluation of expectation values of odd operators that break the fermion parity, is exactly obtained for finite-size but large rings through the use of a recently developed Pfaffian method [N. Wu, Phys. Rev. E 101, 042108 (2020)]. Time dependence of the transverse and longitudinal magnetizations (${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{z}\ensuremath{\rangle}}_{t}$ and ${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{x,y}\ensuremath{\rangle}}_{t}$), single-spin purity, expectation value of the string operator ${X}_{j}={\ensuremath{\prod}}_{l=1}^{j\ensuremath{-}1}{\ensuremath{\sigma}}_{l}^{z}{\ensuremath{\sigma}}_{j}^{x}$ (${\ensuremath{\langle}{X}_{j}\ensuremath{\rangle}}_{t}$), several equal-time two-site correlators (${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{x,z}{\ensuremath{\sigma}}_{j+1}^{x,z}\ensuremath{\rangle}}_{t}$, ${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{x}{\ensuremath{\sigma}}_{j+1}^{y}\ensuremath{\rangle}}_{t}$, and ${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{x}{\ensuremath{\sigma}}_{j+1}^{z}\ensuremath{\rangle}}_{t}$), and pairwise concurrence after quenches to different phases are numerically studied. Our main findings are that (i) The expectation value of a generic odd operator approaches zero in the long-time limit; (ii) ${\ensuremath{\langle}{X}_{j}\ensuremath{\rangle}}_{t}$ exhibits jjj-independent exponential decay for a quench to g=1g=1g=1 and the time at which ${\ensuremath{\langle}{X}_{j}\ensuremath{\rangle}}_{t}$ reaches its first maximum scales linearly with jjj; (iii) The single-spin purity dynamics is mainly controlled by ${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{x}\ensuremath{\rangle}}_{t}$ (${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{z}\ensuremath{\rangle}}_{t}$) for a quench to g<1g<1g<1 ($g\ensuremath{\ge}1$). For quenches to the disordered phase with $g\ensuremath{\gg}1$, the single-spin tends to be in the maximally mixed state and the transverse and longitudinal correlators ${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{z}{\ensuremath{\sigma}}_{j+1}^{z}\ensuremath{\rangle}}_{t}$ and ${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{x}{\ensuremath{\sigma}}_{j+1}^{x}\ensuremath{\rangle}}_{t}$, respectively, approaches $\ensuremath{-}0.25$ and 0.50.50.5 in the thermodynamic limit; (iv) The nearest-neighbor entanglement acquires a finite plateau value that increases with increasing ggg, and approaches a saturated value $\ensuremath{\sim}0.125$ for $g\ensuremath{\gg}1$.
We investigate waiting-time distributions (WTDs) of quantum jumps in continuously monitored quantum many-body systems, whose unconditional dynamics lead to the trivial infinite-temperature state. We demonstrate that the WTD of a half-chain subsystem exhibits an anomalous tail, markedly deviating from the Poissonian distribution in stark contrast to that of the whole system. By analyzing the spectral properties of the superoperator L0\mathscr L_0L0​, which is defined by removing the jump terms associated with the half-chain subsystem from the full Liouvillian, we find that the long-time behavior with the anomalous tail of the half-chain WTD is governed by the eigenvalue λ0 (<0)λ_0\:(<0)λ0​(<0) with the largest real part. We further reveal a qualitative change in the system-size dependence of λ0λ_0λ0​ as a function of the measurement strength: for sufficiently weak measurement, λ0λ_0λ0​ decreases proportionally to the system size, while for strong measurement, λ0λ_0λ0​ scales independently of the system size, signaling the persistence of the anomalous half-chain WTD in the thermodynamic limit. The WTD is extracted solely from the spacetime record of quantum jumps {ti,xi}\{t_i,x_i\}{ti​,xi​} and can be experimentally accessed without postselection. Our work establishes a spectral framework for understanding nontrivial WTDs in subsystems of monitored quantum dynamics and provides a novel diagnostics to assess many-body effects on WTDs.