Numerically exact open quantum system work statistics with process tensors

TL;DR

This work addresses the challenge of obtaining the full quantum work distribution for driven open systems beyond perturbative limits by introducing a numerically exact process-tensor framework that tracks work statistics via a work characteristic function along a generalized time axis. The method leverages PT-MPO representations to capture environmental influence efficiently, enabling non-perturbative, non-Markovian, and non-adiabatic regimes to be studied. Applying the framework to a Landauer erasure protocol reveals quantum features in the full work distribution that are invisible to low-order moments, and shows that a shortcut to adiabaticity can enhance erasure fidelity without necessarily altering mean or variance. The approach provides a versatile tool for optimizing quantum control and thermodynamics in near-term and future devices, with broad applicability to different bath models and driving schemes.

Abstract

Accurately quantifying the thermodynamic work costs of quantum operations is essential for the continued development and optimisation of emerging quantum technologies. This present a significant challenge in regimes of rapid control within complex, non-equilibrium environments - conditions under which many contemporary quantum devices operate and conventional approximations break down. Here, we introduce a process tensor framework that enables the computation of the full numerically exact quantum work statistics of driven open quantum systems. We demonstrate the utility of our approach by applying it to a Landauer erasure protocol operating beyond the weak-coupling, Markovian, and slow-driving limits. The resulting work probability distributions reveal distinct quantum signatures that are missed by low-order moments yet significantly impact the erasure fidelity of the protocol. Our framework delivers non-perturbative accuracy and detail in characterising energy-exchange fluctuations in driven open quantum systems, establishing a powerful and versatile tool for exploring thermodynamics and control in the operating regimes of both near-term and future quantum devices.

Figures (5)

• Figure 1: Schematic of the process-tensor framework for quantum work statistics. (i) Two-point measurement protocol (TPMP) used to define the work probability distribution (WPD). (ii) A driven open quantum system $S$ exchanges energy and information with its surrounding environment $E$, governed by a time-dependent Hamiltonian $\hat{H}(t)$ under external driving from a work source. (iii) Calculation of the work characteristic operator (WCO), $\hat{\rho}(\chi,t_f)$, via propagation along both the physical time axis $t$ and the counting-field axis $\chi$. (iv) Mapping the $t$ and $\chi$ axes onto a single generalised-time axis $\tau$ allows the WCO (now $\hat{\rho}(\tau)$) to be evolved under a non-completely-positive, trace-preserving (non-CPTP) map generated by appropriately defined forward $\hat{H}^{[f]}(\tau)$ and backward $\hat{H}^{[b]}(\tau)$ generalised-time-dependent Hamiltonians. (v) The forward and backward Hamiltonians are constructed from the time-dependent Hamiltonian $\hat{H}(t)$; their system components form the generalised-time-dependent Liouvillian $\mathcal{L}_S(\tau)$. (vi) For an open quantum system, propagation along the $\tau$ axis is performed using process-tensor matrix-product-operator (PT-MPO) techniques, which yield numerically exact dynamics. All work-counting complexity is contained within the system-local propagators $\mathcal{M}$, while the $\mathcal{Q}$ tensors, connected through inner-bonds that carry non-Markovian correlations, encode the environmental influence functional.
• Figure 2: Qubit erasure protocol. The TLS energy splitting begins far from the peak of the spectral density ($\epsilon_0\approx 0 \ll \Omega$), is brought close to resonance with the peak halfway through the protocol ($\epsilon_{\mathrm{max}} \approx \Omega$), and is then returned to its initial far-detuned value ($\epsilon_{f} = \epsilon_{0} \ll \Omega$). This cyclic design enables a thermodynamically consistent analysis of the work statistics. Throughout this work we use a maximum TLS energy splitting of $\epsilon_{\mathrm{max}} = 25\beta^{-1}$, an initial splitting of $\epsilon_{0} = 0.02\epsilon_{\mathrm{max}}$, and fixed environmental parameters $\Gamma=10\beta^{-1}$ and $\Omega=\epsilon_{\mathrm{max}}$. The coupling strength $\alpha$ and protocol duration $t_{f}$ are varied in subsequent figures.
• Figure 3: Erasure fidelity, coherence, and low-order work statistics as functions of protocol duration. (Top left) Erasure fidelity, defined as the overlap between the final reduced system state and the ground state of $\hat{H}_S(t_f)$. (Top right) TLS coherence in the energy eigenbasis of $\hat{H}_S(t_f)$, given by $\langle\hat{\sigma}_x\rangle$. (Bottom left) Mean work transfer. (Bottom right) Variance of work transfer. All quantities are evaluated at the end of the protocol and plotted against the protocol duration $t_f$ (logarithmic scale). Results are shown for different system-environment coupling strengths $\alpha$ (colours), for both the reference driving Hamiltonian in Eq. \ref{['eq: erasure hamiltonian']} (solid lines) and the STA-assisted protocol in Eq. \ref{['eq: H STA']} (circles, dotted lines). A generalised-time step of $\Delta\tau = 0.01$, SVD threshold $10^{-10}$, memory time $t_\text{mem}=5\beta$, and equilibration time $t_\mathrm{e} = 5\beta$ were used to ensure convergence. A tenth-order finite-difference method was employed to compute the work moments. Selected points (stars and crosses) mark the protocols for which the full work probability distributions are analysed in Fig. \ref{['fig: wpd']}.
• Figure 4: Full work probability distributions for representative Landauer erasure protocols. Work probability distributions for coupling strength $\alpha = 0.16$ and protocol durations $t_f = 4.5\beta$ (blue) and $t_f = 20\beta$ (red). Results are shown for both the driving Hamiltonian $\hat{H}_S(t)$ (solid curves) and the STA-assisted Hamiltonian $\hat{H}^{\mathrm{STA}}_S(t)$ (circles). The main panel highlights the broad positive-work distribution accompanied by heat dissipation during the erasure process. The inset displays the region around $W=0$, where closed-system transitions manifest as distinct peaks and we see a clear suppression of non-adiabatic work values when using the STA. Notably, the STA has little effect on the rest of the WPD, for either protocol duration. The WPDs shown here correspond to the highlighted points within Fig. \ref{['fig:dynamics-moments']} (blue and red stars and crosses) and were obtained by Fourier transforming converged WCFs, calculated using a generalised-time step $\Delta\tau = 0.01$, SVD threshold $10^{-10}$, and equilibration time $t_{\mathrm{e}} = 5\beta$. The largest counting field value used was $\chi_{\mathrm{max}}=200\beta$, after which the real and imaginary parts of the WCF had decayed to less than $10^{-3}$. The distribution is calculated with a fast Fourier transform (FFT), binned with $\Delta W = 0.002\beta^{-1}$. A slow decaying envelope of $\exp[-0.005\chi]$ was added to the WCF before performing the FFT, smoothing out some of the fast oscillations present in the WPD, without changing its shape qualitatively.
• Figure 5: Time integrated emission spectra of the TLS during the Landauer erasure protocols presented in Fig. \ref{['fig: wpd']} (left -- without STA, right -- with STA). We see similar oscillations in the emission spectra to those in the WPD for the faster protocol (blue), where the emission frequencies are, as expected, of the opposite sign to the work values in Fig. \ref{['fig: wpd']}.