Quantum Corridor IV: Zero Relative-Entropy Dissipation, Fixed Points, and Reversible Structure

Kenneth Searcy

Published Mar 10, 2026• 0 citations

Abstract

When dissipation vanishes, the next question is what structure remains. This paper is the fourth quantum installment in the Forced Structure Series. It follows Quantum Corridor III, which identified the generator-level relative-entropy dissipation rate sigma(rho|pi) for strictly positive states evolving under an admissible Lindblad semigroup relative to a faithful stationary state and proved its nonnegativity. The present paper addresses the next forced question: the equality condition sigma(rho|pi) = 0. It proves unconditionally that zero dissipation is first-order stationarity of the relative-entropy decay curve at t = 0. It then shows that, under an explicit entropy-decay convexity hypothesis, zero dissipation propagates to constancy of the relative entropy along the full semigroup orbit. From that constancy, equality in the quantum data processing inequality follows for every semigroup map on the pair (rho, pi), and the state is therefore placed inside the entropy-reversible structure determined by the semigroup relative to pi. Under a further named hypothesis identifying the reversible sector with the fixed-point sector, the paper obtains a fixed-point characterization of the zero set. In the commuting limit, the result reduces to the classical vanishing entropy-production and zero-current condition established in the classical generator corridor. The paper is explicit about scope: the fixed-point biconditional is proved only under the stated added hypotheses, not claimed in full generality.

DOI: 10.5281/zenodo.18944221

Related papers

Quantum Corridor III: Generator-Level Relative-Entropy Dissipation and Unraveling-Invariant Irreversibility

Kenneth Searcy

Irreversibility can be fixed before any path-space choice is made. This paper is the third quantum installment in the Forced Structure Series. It begins from the results of Quantum Corridor I and Quantum Corridor II, where admissible noncommutative maps were fixed as completely positive trace-preserving maps, quantum relative entropy was fixed as the corridor divergence, and admissible continuous-time evolution was fixed in the GKSL/Lindblad semigroup setting. The central result of the present paper is that these earlier corridor commitments already determine an intrinsic generator-level quantity: the instantaneous decay rate of the relative entropy D(Phi_t(rho)||pi) along an admissible semigroup Phi_t = e^(tL) relative to a faithful stationary state pi. In finite dimension, for strictly positive initial states, this rate is shown to equal -Tr(L(rho)(log rho - log pi)), and it is therefore nonnegative. The quantity is generator-determined and independent of any unraveling or path-space realization. In the commuting limit, it reduces to the classical generator-level entropy production rate. The paper is explicit about scope: it proves a canonical generator-level relative-entropy dissipation rate, not a full thermodynamic characterization, not a complete equality condition, and not yet a path-space bridge theorem.

Mar 10, 20260 citations

Quantum Path-Space Corridor II: Record-Level Irreversibility and Generator-Level Relative-Entropy Dissipation

Kenneth Searcy

Proves the jump-branch bridge theorem in the quantum corridor series: under the Chetrite–Mallick reversed-jump entropy-production hypotheses, the expected record-level irreversibility of a fixed jump unraveling equals the time integral of the intrinsic generator-level relative-entropy dissipation along the unconditioned Lindblad orbit. Establishes the instantaneous mean-rate identity, verifies the zero stationary case, and shows reduction to the classical path-space/generator correspondence in the commuting sector, while explicitly preserving genuine unraveling dependence beyond the mean.

Mar 11, 20260 citations

Quantum Path-Space Corridor I: Lindblad Unravelings, Record Reversal, and Pathwise Irreversibility

Kenneth Searcy

Quantum path space begins only after non-uniqueness is faced honestly. This paper is the first path-space installment of the quantum branch in the Forced Structure Series. It begins from the results of Quantum Corridor I and Quantum Corridor II, where admissible noncommutative maps were fixed as completely positive trace-preserving maps and continuous-time admissible evolution was fixed in the Lindblad semigroup setting. The central structural point of the present paper is that a Lindblad generator does not by itself determine a unique quantum trajectory law or path-space object. Rather than treating that as a side remark, the paper makes it the first explicit result. It then identifies the additional structure required for a history-level lift: a chosen jump-type unraveling together with a monitoring interpretation and record space. Within that setting, the paper defines forward and reversed record laws relative to a faithful stationary state and identifies the pathwise irreversibility functional as the log-likelihood ratio between those laws. Its expectation is the relative entropy between the forward and reversed record measures and is therefore nonnegative. In the commuting adapted limit, the construction reduces to the previously established classical path-space corridor. The paper is explicit about scope: it does not claim a unique quantum path space forced by the generator alone, and it treats only the jump-unraveling branch, leaving unraveling comparison and coarse-graining questions for the next step in the series.

Mar 9, 20260 citations

Quantum Corridor I: CPTP Admissibility, Quantum Relative-Entropy Monotonicity, and Recovery

Kenneth Searcy

The quantum corridor begins at admissibility, not at trajectories. This paper gives the first quantum step in the Forced Structure corridor series. Working within a finite-dimensional operator-algebraic extension of the classical framework, it identifies the admissible noncommutative replacement for the classical map/divergence/equality lane. The central direct result is that if admissible transformations preserve normalization, preserve positivity, and remain consistent under extension by an untouched ancilla, then the admissible maps are exactly completely positive trace-preserving maps. Once that map class is fixed, Umegaki quantum relative entropy supplies a canonical monotone divergence, and equality in that monotonicity law is characterized by Petz recoverability. In the commuting limit, the noncommutative corridor reduces to the classical one. The paper is explicit about scope: it does not claim to derive all of quantum theory from prior ontology, and it does not yet derive continuous-time quantum dynamics or quantum path space. Its role in the series is narrower and structural: to fix the map-level quantum corridor that the next continuous-time paper will extend.

Mar 7, 20260 citations

Quantum Corridor II: Continuous-Time Admissibility, Quantum Semigroups, and Lindblad Structure

Kenneth Searcy

Continuous-time quantum admissibility forces the generator lane. This paper is the second quantum installment in the Forced Structure corridor series. It begins from the map-level result of Quantum Corridor I, where admissible noncommutative state transformations were fixed as completely positive trace-preserving maps, and asks what additional structure is required to pass to continuous-time evolution. The paper makes those assumptions explicit: time-indexed admissibility, identity at zero, semigroup composition, time-homogeneity, and norm continuity at the origin. Under those assumptions, the admissible evolution lies in the standard uniformly continuous quantum dynamical semigroup setting. In finite dimension, this places the generator in the setting of the Gorini–Kossakowski–Sudarshan–Lindblad theorem, so the admissible generator class is characterized by GKSL/Lindblad structure. In the commuting limit, the restricted generator reduces to the classical rate-matrix generator, showing that the quantum generator corridor is a continuation of the classical branch rather than a detached formalism. The paper is explicit about scope: it does not rederive the GKSL theorem from scratch, and it does not yet address quantum path space, trajectories, or history-level irreversibility.

Mar 9, 20260 citations

Information-geometric bounds on nonequilibrium relaxation in quantum Markov dynamics

Xiao-Kan Guo, Zhiqiang Huang

In this paper, we study the quantum information-geometric structure underlying short-time nonequilibrium relaxation in quantum Markov dynamics, and derive bounds on the nonequilibrium correction to the short-time relaxation curvature. These bounds generalize the information-geometric structure underlying Auconi's classical nonequilibrium relaxation inequality to quantum Markov dynamics. We consider the von Neumann relative entropy as quantum divergence, and parametrize perturbations through the Kubo-Mori map, which converts the second-order expansion of the relative entropy into the Bogoliubov-Kubo-Mori (BKM) inner product at all temperatures. For a quantum Markov semigroup with full-rank stationary state, the induced tangent-space generator admits a canonical decomposition into a BKM-symmetric and a BKM-antisymmetric part, and the leading nonequilibrium curvature correction takes the exact form of the expectation of the commutator between the dissipative and the transport part of the generator. This correction is bounded by the product of a BKM-symmetric dissipative activity and a transport-sector activity. The bound is verified on a noncommuting qubit model and a driven-dissipative qutrit, and the full chain is confirmed numerically on a two-dimensional Fokker-Planck steady state. In the high-temperature and overdamped limit the quantum formula reduces to the position-space Fokker-Planck expression and reproduces Auconi's entropy-production bound with the correct temperature factor.

quant-pharXiv:2609.09599Sep 10, 20260 citations