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.
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.
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.
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.
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.
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.
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.
We develop a claim-graded theory of quantum records and history measures from completely positive instruments. Class operators and Hilbert-valued event measures generate strongly positive decoherence functionals and a canonical history Hilbert space. In the declared linear-instrument class, event probabilities are affine in the state and additive in effects, hence have Born trace form; an independent physical exponent coordinate is incompatible with this structure. Outcome probabilities factor through effects, whereas conditional post-record states factor through complete CP operations. A single Stinespring dilation yields both the unconditional channel and the conditional states, so no second collapse map is required. In a gap-protected record parent, the fixed algebra retains outcome and CP-operation data while erasing implementation gauge, and its center supplies objective event propositions; nevertheless, deterministic unconditional CPTP evolution cannot select one pure central branch from a mixed central state while fixing all pure branches. Explicit jump–diffusion record dilations realize the same reduced dephasing semigroup with inequivalent path laws, proving that the reduced channel does not select a unique physical unravelling. Competing admissible information and work utilities likewise select inequivalent readouts. Bounded marked-Poisson parents, compensated interaction-picture measures, modularly damped Wick classes, and Boolean–Möbius/Borel-resummed local nets provide projective strongly positive histories in named finite and quasi-local classes. By contrast, global extensive event counts, positive-Poisson continuum uniformization, raw all-Fock absolute semivariation, and fixed final energy damping obey class-level No-Go theorems. The paper establishes no physical -flow, no unique unravelling, no universal utility, no deterministic collapse theorem, and no completed interacting relativistic QFT history measure.