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.
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.
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.
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.
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 introduce a new cumulant-based quantum relative Rényi functional as a candidate quantum Rényi divergence, derived from the cumulant-generating function (CGF) of the quantum relative surprisal operator and extending the classical connection between Rényi divergence and statistical cumulants to the quantum setting. Unlike the Petz and sandwiched quantum Rényi divergences, the proposed construction is motivated by statistical structure rather than operator-algebraic or operational principles. The functional naturally admits a path-integral-like representation through the Lie-Trotter product expansion, providing a trajectory-based interpretation of quantum divergence in Hilbert space. On its natural non-regularized domain for α>1 under the support condition supp(ρ)⊆supp(σ), we establish several fundamental properties, including positivity, reduction to the classical case, additivity, unitary invariance, continuity, and monotonicity with respect to the Renyi parameter α. Whether the functional satisfies the quantum data-processing inequality (QDPI) under arbitrary CPTP maps remains open. To extend the analysis beyond the studied regime, we introduce a regularized version of the functional and study its behavior at α=0. We show that the resulting relative quantumness quantity vanishes if and only if the underlying states commute, yielding a necessary and sufficient characterization of non-commutativity. For commutativity-preserving (CoP) channels, we further conjecture a QDPI-type monotonicity relation for this quantity. Extensive numerical simulations provide strong evidence in support of this conjecture, with no violations observed for the CoP channels considered in this work.
This paper extends the Ontological Totality Functional (OTF) programme into the domain of quantum causal structure. Its central claim is that physical admissibility cannot be exhausted by state-level consistency or dynamical stability alone. It must also constrain the admissibility of causal organization itself. The paper introduces a causal defect functional for higher-order quantum processes and argues that definite causal order can emerge as a limiting case of admissible structure. Working in a finite-dimensional process-matrix setting, the paper defines the causal defect as the trace-norm distance to the set of valid causally separable processes with fixed normalization. It proves that this defect is non-negative, vanishes exactly on the causally separable set, and is convex. A free class of admissible supermaps is then identified: linear maps on processes that preserve validity, preserve causal separability, and are contractive in trace norm. Under these maps, the causal defect is monotone. An exact dual operational characterization is established: the causal defect equals the maximal bounded-test advantage over the causally separable set for Hermitian tests of bounded operator norm. In this sense, bounded causal witnesses arise as certified lower-bound functionals for admissibility. To connect the formalism to concrete physical scenarios, the paper studies a dephased-switch family W_eta = W_diag + eta X, where W_diag is causally separable and X encodes interference between orders. For this family, the defect is bounded above linearly in |eta|, while a witness with positive overlap on X yields a matching linear lower bound. As eta tends to zero, the process approaches definite-order admissibility. Within the OTF series, this paper functions as a frontier extension. Paper I introduced physical admissibility mathematically, Paper II studied admissibility in a tensor-cosmology channel, and Paper III interpreted admissibility ontologically. Paper IV shows that the same programme can be pushed into higher-order causal structure, where admissibility constrains not only which states and channels are physically stable but also which causal architectures can count as physically realized. Companion records in the OTF series:OTF I — DOI: 10.5281/zenodo.19068375OTF II — DOI: 10.5281/zenodo.19075284OTF Master Edition — DOI: 10.5281/zenodo.19077303