Entanglement diagnostics of stochastic-resetting steady states: theory and measurement on superconducting qubits

Abstract

Stochastic resetting drives quantum systems into nonequilibrium stationary states unreachable under uninterrupted unitary evolution, and their entanglement depends on whether the reset history is resolved or averaged over. We study this interplay in the transverse-field Ising chain, deriving closed-form results for two qubits and evaluating chains of up to 12 qubits exactly. In the two-qubit system, these closed-form results show that phase matching between energy eigenstates produces resonances, and their visibility in the stationary state is determined by the symmetry of the reset state. Measurements on an IBM superconducting processor show the entropies evaluated before and after the reset average closely following noiseless predictions with no fitted parameter, while concurrence lies below prediction across most of the hardware measurements. The two diagnostics respond differently because they read different components of the same reconstructed state, and a depolarizing model captures the concurrence suppression with one fitted parameter. The difference between the two entropies, i. e., the Jensen gap of entanglement entropy, equals the Holevo information that a subsystem retains about the time since the last reset. With a Bell-state reset, this information grows with chain length across a half-chain cut over the sizes studied, without visible saturation at the smaller reset probabilities. Since reset times are sampled and recorded classically, the trajectory-averaged entropy, and with it the Jensen gap, is obtained without postselection, in contrast to monitored circuits, where trajectory-resolved entanglement requires sampling exponentially rare measurement records.

The paper asks how stochastic resetting distinguishes entanglement in individual realizations from entanglement and mixedness in the stationary ensemble. It studies the transverse-field Ising model with global resets to a product or Bell-pair state, using analytic two-qubit calculations, exact state-vector evolution up to $N=12$, and two-qubit tomography on an IBM superconducting processor. For reset probability $p$, the stationary ensemble has geometric age weights $p(1-p)^m$. Energy-basis coherences survive averaging unchanged when $(E_i-E_j)T\in2\pi\mathbb Z$, provided the reset state overlaps both eigenstates. Two-qubit parity and overlap selection rules determine which resonances appear. Trajectory-averaged entropy can additionally resolve periodic trajectory patterns absent from averaged-state diagnostics. The difference $\Delta S_A=S(\bar\rho_A)-\sum_m p(1-p)^mS[\rho_A(m)]$ equals the Holevo information about time since the last reset retained by subsystem $A$, with $\Delta S_A\leq\min\{|A|,h(p)/p\}$. Half-chain gaps grow over the studied sizes for tiled-Bell resets, reaching $2.55$ bits; the size-independent age-entropy bound precludes indefinite extensive growth. Classically recorded reset ages permit age-resolved tomography without postselection, but require additional statistics. Hardware entropies closely follow noiseless predictions, whereas concurrence is generally suppressed. A single-parameter local depolarizing model fits the $p=0.05$ concurrence data with strength $1.56\times10^{-3}$ per step and RMS deviation $0.010$. Limitations include unresolved asymptotic size dependence, hardware measurements restricted to two qubits, and a phenomenological noise model without a fitted-parameter uncertainty. Reduced-state entropy ceases to quantify trajectory entanglement under noise; no measurement-induced transition is established.