Work Statistics and Quantum Trajectories: No-Click Limit and non-Hermitian Hamiltonians

TL;DR

This work develops a TPM-based framework for quantum work statistics in continuously monitored systems, where intermediate generalized measurements induce non-Hermitian evolution in no-click trajectories. By introducing the multi-measurement evolution operator and deriving the work-generating function, the authors show that Jarzynski-type relations are modified unless the dynamics are unital, and they quantify this with an efficacy parameter ${\gamma_t}$. They apply the formalism to a monitored 1D transverse-field Ising chain, revealing measurement-induced energy saturation and suppressed fluctuations due to a Zeno-like effect, with clear signatures at the no-click critical line $\gamma_c(h)$. The results connect non-Hermitian quantum dynamics, quantum trajectories, and many-body phase transitions, offering energy-based probes for monitoring-induced critical behavior with potential experimental realization. The work provides a concrete methodology to study thermodynamics of monitored quantum systems and paves the way for zeno-based control and sensing in quantum technologies.

Abstract

We investigate quantum work statistics within the standard two-point measurement (TPM) scheme in continuously monitored quantum systems, including the effects of generalized unitary evolution, possibly controlled by quantum circuit models, and multiple generalized measurements as well as post-selection of no-click trajectories. We derive an explicit expression for the work generating function that naturally incorporates non-Hermitian dynamics arising from quantum jump processes and reveals deviations from the standard Jarzynski equality due to measurement-induced asymmetries. We illustrate our theoretical framework by analyzing a one-dimensional transverse-field Ising model under local spin monitoring. In this model, increased measurement strength projects the system onto the no-click state, leading to a suppression of energy fluctuations and measurement-induced energy saturation, reminiscent of the quantum Zeno effect. Moreover, we find signatures of the measurement-induced transition observed in the no-click limit in the moments of the work distribution.

Figures (4)

• Figure 1: (a) Average work density $\langle w \rangle$ (normalized by system size $L$) and (b) its variance $\langle \Delta w^2 \rangle$ plotted as functions of the measurement strength $\gamma$ for the monitored one-dimensional transverse-field Ising model at different values of the transverse field $h$ at time $t=1.0$. The plots demonstrate that when $\gamma=0$, both quantities are zero, reflecting the coincidence of the initial and effective non-Hermitian Hamiltonians. As $\gamma$ increases, the non-Hermitian dynamics progressively project the system onto the no-click (spin-down) state, leading to a saturation of the work performed and a reduction in its fluctuations.
• Figure 2: Average work density $\langle w \rangle$ as a function of the measurement rate $\gamma$ for $h=\{0.3,0.6,0.9\}$ at long evolution time $t=5000$. A sharp rise appears at $\gamma \rightarrow 0^{+}$ when monitoring is first switched on. Each curve then shows a kink at a value of $\gamma$ that coincides with the analytic no-click criticality $\gamma_{c}(h)=4\sqrt{1-h^2}$ (blue dotted guides and labels). For larger $\gamma$, $\langle w \rangle$ saturates due to Zeno suppression.
• Figure 3: (a) Average work density $\langle w \rangle$ and (b) its variance $\langle \Delta w^2 \rangle$ as functions of the transverse field $h$ for various fixed measurement strengths $\gamma$ at time $t=1.0$. The plots illustrate that for a given $\gamma$, both quantities initially increase with $h$, reaching a maximum before decreasing as $h$ is further increased, reflecting the interplay between unitary dynamics and measurement backaction. At higher $\gamma$ values, a stronger transverse field is required to overcome the measurement-induced energy modifications.
• Figure 4: Efficacy $\gamma_t$ versus time $t$ for the monitored Ising chain at $J=1$ and $h=0.5$, comparing three measurement rates. The unitary case $\gamma=0$ yields $\gamma_t=1$ for all times. For moderate monitoring at $\gamma=2$ the curve shows a brief overshoot at initial times then decays as the non-Hermitian loss builds up. For stronger monitoring, at $\gamma=5$, the decay is faster and the initial bump is much smaller.