Exact Law of Quantum Reversibility under Gaussian Pure Loss

Abstract

Classical reverse diffusion is generated by changing the drift at fixed noise. We show that the quantum version of this principle obeys an exact law with a sharp phase boundary. For Gaussian pure-loss dynamics -- the canonical model of continuous-variable decoherence in optical attenuation channels, squeezed-light interferometric sensing, and superconducting bosonic architectures -- complete positivity, the requirement that the dynamics remain physical even for systems entangled with an ancilla, creates an exact phase boundary at which the minimum reverse cost vanishes, fixes the reverse-noise budget on both sides, and makes pure nonclassical targets dynamically singular. The minimum reverse cost vanishes exactly at a critical squeezing-to-thermal ratio and is strictly positive away from it, with a sharp asymmetry: below the boundary, standard reverse prescriptions such as the fixed-diffusion Bayes reverse remain feasible at mild cost; above it, these prescriptions become infeasible, the covariance-aligned generator remains CP-feasible and uniquely optimal, and the cost can be severe. The optimal reverse noise is locked to the state's own fluctuation geometry and simultaneously minimizes the geometric, metrological, and thermodynamic price of reversal. For multimode trajectories, the exact cost is additive in a canonical set of mode-resolved data, and a globally continuous protocol attains this optimum on every mixed-state interval. If a pure nonclassical endpoint is included, the same pointwise law holds for every $t>0$, but the optimum diverges as $2/t$: exact reversal of a pure quantum state is dynamically unattainable. These results establish an exact law of quantum reversibility in the canonical pure-loss setting and provide a sharp benchmark for broader theories of quantum reverse diffusion.

Figures (3)

• Figure 1: Sharp phase boundary for Gaussian quantum reversibility under pure loss. (a) Minimum eigenvalue of the fixed-diffusion Bayes reverse CP matrix $M^{\mathrm{Bayes}}$ versus squeezing $r$ and thermal variance $\nu$ (vacuum $=1$). The dashed curve is the exact threshold $\nu=\cosh(2r)$. Published $1550\,\mathrm{nm}$ squeezing demonstrations Mehmet2011Meylahn2022 are overlaid by mapping directly reported squeezed and anti-squeezed quadrature variances to $(r,\nu)$; all displayed operating points lie in the sector $\nu<\cosh(2r)$, locating current hardware in the reverse-noise-forced region of the phase diagram. (b) Exact optimal reverse cost $Z_{\min}$ for the unrestricted instantaneous one-mode Gaussian reverse problem. The cost vanishes exactly on the dashed curve and is strictly positive on both sides, with qualitatively different regimes above and below.
• Figure 2: Pure-endpoint singularity of the exact Gaussian reversibility law. For pure squeezed targets, the exact pointwise Gaussian reverse optimum diverges as $Z_{\min}(t)\sim 2/t$ as $t\to 0^+$. (a) The optimal cost approaches the universal asymptote $2/t$. (b) The rescaled quantity $t Z_{\min}(t)$ collapses to the universal coefficient $2$, independent of the squeezing parameter $r$.
• Figure 3: Reverse-cost comparison of three Gaussian reverse protocols for a squeezed-thermal target under pure loss ($\nu=3$). The exact optimum $Z_{\min}$ (solid), the best isotropic protocol with $D=cI_2$ (dashed), and the fixed-diffusion Bayes reverse (dotted). The vertical dashed line marks the exact phase boundary $\cosh(2r)=\nu$, where $Z_{\min}$ vanishes. Below threshold, all three protocols are feasible but naive prescriptions pay a strict penalty relative to the exact optimum; above threshold, both Bayes and isotropic protocols become infeasible while the covariance-aligned generator remains CP-feasible and uniquely optimal---the difference is not a formal refinement but the difference between feasible and impossible reversal for standard prescriptions in experimentally relevant regimes. Above threshold, isotropic repair does not eventually restore complete positivity because exact covariance matching simultaneously drives the reverse drift, so the symplectic term $i(K\sigma+\sigma K^T)$ grows with the injected isotropic noise and overwhelms the diagonal repair in the most squeezed direction.