The uncontrolled endpoint channel can conceal how well a quantum process can protect a logical task under time-resolved control. We formulate a fixed-task resource theory of continuous process tensors with physical access interfaces, entrance-erasing free objects, and budgeted free conversions. Complete-record total-variation error measures performance; the calibrated capability is monotone under conversions that preserve a common free baseline. For unknown-state transmission the baseline is exactly $1-d^{-2}$. Two qubit processes with the same endpoint channel have optimal errors $0$ and $p=\sin^2(2gh)$ at the same finite pulse budget in an ideal-transparent system-only window architecture, forcing any endpoint-only predictor to err by at least $p/2$ on one of them. An exactly solvable family shows how the allocation of a fixed interaction phase around a control window sets the attainable error, and a nontrivial perfect echo requires and attains total control action $π$ in that architecture. The separation extends directly to every qubit unitary target at a budget that includes its boundary gate, while unitary tasks in any dimension have baseline $1-d^{-2}$ whenever a budget-feasible free realization gives the depolarizing output. Complete-record implementation bounds quantify coherent and dissipative control imperfections. Continuous-process randomized compilation supplies physical instrument pullbacks; compatible completion conditions extend their action, and certified dynamical bounds yield achievable task-error and capability guarantees. The resulting theory identifies temporal structure and control investment that endpoint characterization alone cannot resolve.
The uncontrolled endpoint channel can conceal how well a quantum process can protect a logical task under time-resolved control. We formulate a fixed-task resource theory of continuous process tensors with physical access interfaces, entrance-erasing free objects, and budgeted free conversions. Complete-record total-variation error measures performance; the calibrated capability is monotone under conversions that preserve a common free baseline. For unknown-state transmission the baseline is exactly 1−d−2. Two qubit processes with the same endpoint channel have optimal errors 0 and p=sin2(2gh) at the same finite pulse budget in an ideal-transparent system-only window architecture, forcing any endpoint-only predictor to err by at least p/2 on one of them. An exactly solvable family shows how the allocation of a fixed interaction phase around a control window sets the attainable error, and a nontrivial perfect echo requires and attains total control action π in that architecture. The separation extends directly to every qubit unitary target at a budget that includes its boundary gate, while unitary tasks in any dimension have baseline 1−d−2 whenever a budget-feasible free realization gives the depolarizing output. Complete-record implementation bounds quantify coherent and dissipative control imperfections. Continuous-process randomized compilation supplies physical instrument pullbacks; compatible completion conditions extend their action, and certified dynamical bounds yield achievable task-error and capability guarantees. The resulting theory identifies temporal structure and control investment that endpoint characterization alone cannot resolve.
Traditional discrete randomized compiling (RC) tailors errors with random control frames at logical-gate boundaries, but fails to capture intra-instrument control evolution or persistent system-environment coupling. In memory-bearing environments, errors correlate across times, not just single gates. We build continuous-process randomized compilation (CPRC) in the continuous process tensor (cPT) framework, unifying random unitary control trajectories, pointwise-covariant finite-duration instruments, and environmental dynamics on one time axis. We study four protocols: fixed Pauli refresh, bounded Cayley paths, unitary Brownian motion, Ornstein-Uhlenbeck driving. We obtain three key results: (1) Pointwise covariance ensures each trajectory exactly reproduces the target logic without system-environment coupling; a closed-frame counterexample proves endpoint-only compensation causes first-order instrument errors. (2) Independent Pauli conjugations over full intervals diagonalize error histories at boundaries, but classical labels retain environment-induced correlations, confirmed by an exact static-bath solution. (3) We derive a total-variation error bound for full adaptive output records under bounded centered coupling and mixing conditions. We prove the Fock norm convergence of the entire trajectory-averaged cPT for finite non-Gaussian environments and Gaussian baths with time-integrable covariance, including the uncut Drude spectrum. We numerically validate the theory in a qubit model via output probabilities, operator conditional mutual information, and all two-particle coefficients for both qubit and thermal Drude baths. This work extends RC from discrete circuits to continuous-time memory-bearing quantum processes, offering a unified framework, quantitative tools, implementable schemes, and benchmarks for non-Markovian error tailoring.
Paper 6 of 11 — A General Theory of Answerability The eleven-paper TUS OS mathematical research programme Abstract This paper develops a general theory of how the exact resource cost of representing an independently fixed native system depends on the operational access contract imposed on that system. Rather than assigning a single intrinsic “dimension” to a process or operational object, the paper treats realisation as a fully typed problem determined jointly by the native system, the access contract, the admitted realisation category and the tolerated error. The framework defines contract-indexed achievable resource regions, declared-memory closure, lawful contract restriction and resource monotonicity. Strengthening a contract can require a realisation to preserve or recover distinctions that a weaker behavioural contract is permitted to compress, while the independently fixed native system itself remains unchanged. Lawful contract restriction induces reverse inclusion of achievable regions, but strict separation requires additional lower and upper certificates. For stochastic processes, the paper introduces causal-state transparency as a recovery obligation imposed on an exact autonomous predictor at a declared inspection boundary. All persistent quantum, classical, controller, history and source-side information available at that boundary is included in the resource account. A direct-sum classical–quantum identity-capacity theorem proves that uniform worst-case recovery probability τ across N intrinsic causal classes requires identity capacity at least ⌈τN⌉. The theorem is then applied to two external stochastic-process families. Periodically modulated decay processes exhibit an unbounded finite separation: exact autonomous prediction has minimum identity capacity 2, exact causal-state transparency has minimum capacity N, and every fixed positive transparency threshold forces a lower bound growing as ⌈τN⌉. For the dual-Poisson process, an exact qubit predictor exists under the weak prediction contract, while every positive uniform worst-case transparency threshold makes the finite-capacity achievable region empty. This is a contract-induced loss of finite feasibility on the same native process, not a claim that the process itself ceases to exist or that no larger infinite-capacity realisation could exist. The paper then returns to the TUS OS discovery instance from which the general problem arose. The same independently fixed 64-state canonical target admits a complete static quantum-operational Atlas of exact minimum Hilbert dimension 4, while recovery-complete LANDING selected-line instrument access, GUIDANCE selected-line readout and GUIDANCE selected-line-and-endpoint joint-instrument access each have exact minimum Hilbert dimension 64 within the same declared representation category. The native target does not change; the operational access obligation does. Together, the TUS OS, periodically modulated decay and dual-Poisson results establish three distinct exact contract-relative regimes: strict finite separation: 4 → 64; unbounded finite separation: 2 → N; finite-capacity breakdown: 2 → ∅finite for every τ > 0. These are three instances of one general structural law, not three identical recovery mechanisms. TUS OS uses admitted experiment families that collectively separate its canonical carriers, whereas the stochastic-process results use direct causal-state decoding at a declared inspection boundary. The results establish that finite-capacity behavioural sufficiency does not imply positive uniform worst-case recoverability of native predictive identity, and that exact representation cost is not a context-free property of the native system alone. It depends on the complete typed obligation imposed on an admitted realisation. The paper does not claim physical quantum execution, quantum advantage, a universal scalar resource, equivalence of TUS OS recovery with causal-state transparency, or finite-capacity breakdown for prior-weighted dual-Poisson transparency. Supporting General Theory Programme This paper is Paper 6 in the TUS OS quantum-operational and Answerability research programme. Building on Papers 1–5, it develops the general theory of contract-relative quantum realisations of classical processes, showing that quantum resource requirements are not intrinsic to a process alone but depend on the governing observational, recovery, and transparency contract. Paper 7 extends this into operational observability and the classification of contract-induced resource transitions; Papers 8–9 develop generic Answerability theory and calculus; Paper 10 proves the protected and executable realisation of TUS OS within that calculus; and Paper 11 closes the programme with A General Theory of Answerability, establishing Structural Identity, Recognition, the canonical mandatory protected core, and lawful Realisation Freedom. Paper 1 — Canonical Tetrahedral Qubit-State Geometry in the TUS OS TriadXOR Completion, Receipt Verification and the Qubit-Geometric Semantic WitnessEstablishes the canonical tetrahedral geometry, unique XOR completion, exact qubit-SIC representation and centred-face trine geometry. Paper 2 — The TUS OS Operational CalculusClassical Construction of Exact Qubit Prepare–Measure Behaviour from a Receipt-Custodied Engine ArchitectureEstablishes exact full-carrier qubit prepare–measure behaviour from the deterministic classical engine under the declared Coordinate-Incidence Operational Harness. For each lawful foundation, the 64 six-bit carriers map to exactly 27 anchor-relative qubit preparations, while the separately declared face Harness gives the exact trine table. Paper 3 — From Finite Relational Structure to Qubit Operational FormAmbient Foundations, Canonical Subatlas, Exact Global Quantum Dimension, GUIDANCE Witness, Sequential Closure and Global Representation in TUS OSEstablishes the repeated-common-effect static Atlas, its exact minimum complex Hilbert dimension 4, exact carrier-reconstruction structure and deterministic predictive-state minimum 64 for the frozen complete principal future language. Paper 4 — Technical Volume I — Formal Foundations and the Complete-Future CorpusNative Contract, Statistical Discrimination and the Invariant Probability ModuleEstablishes the frozen complete principal terminal future-experiment language, complete statistical separation of distinct carrier pairs, exact parity-resolved discrimination results, complement reachability and the 33-dimensional rational invariant probability module. Paper 4 — Technical Volume II — Orthogonality, Process Closure and Native Triadic SynthesisEstablishes universal complement-support orthogonality and its propagation under exact common-CPTP dynamics, the resulting common-CPTP process bounds, recovery-complete LANDING and GUIDANCE results, exact 64-dimensional selected-line instrument/readout minima at their declared contracts, an exact classical–quantum hybrid construction, and the native structural/predictive quotient correspondence. The state-only common-CPTP minimum remains bounded within 32–64 rather than being fixed at 64. Paper 5 — TUS OS as an Anchored Analog Quantum SystemNative Witnesses, Predictive Synthesis, and Access-Contract Resource SeparationDefines Analog Quantum Views and Anchored Analog Quantum Systems, proves that TUS OS is a proper Anchored Analog Quantum System on its 64-input abstract canonical corpus, elevates the inherited structural/predictive correspondence into the Predictive Synthesis Identity Theorem, and composes the recognition, predictive, resource and downstream-conservation results into the TUS OS Anchored Analog Quantum System Theorem. Paper 6 — Contract-Relative Quantum Realisations of Classical ProcessesPredictive Compression, Vanishing Worst-Case Causal-State Transparency, and Finite-Dimensional BreakdownGeneralises the access-dependent resource phenomenon beyond the TUS-specific discovery corpus by formalising the contract-relative achievable-resource object (S,C,𝔐,ε) ⟼ Ach(S,C;𝔐,ε), and establishes three exact contract-relative regimes: strict finite separation, unbounded finite separation and finite-capacity breakdown. These are instantiated respectively by the TUS OS 4→64 separation, the periodically modulated decay 2→N exact prediction-to-transparency separation, and dual-Poisson finite-capacity breakdown for every positive uniform worst-case transparency threshold. Paper 7 — Operational Observability and Contract-Induced Resource Transitions in Quantum RealisabilityStructural Classification, Family Scaling, and Finite-Resource BreakdownDevelops a structural classification theory for contract-induced representation-resource transitions, introducing the Contract Distinction System (CDS), CRC cofinal/refinement equivalence, and Certified Observability Transport Theory (COTT). It separates internal divergence from family divergence, classifies baseline-feasible strengthenings as resource-neutral, strict finite, finite-resource breakdown, or UNRESOLVED, and proves exact and asymptotic witnesses including an interface-induced 2→3 separation, the TUS OS 4→64 transition, incrementally unbounded finite separation for periodically modulated decay processes, and finite-resource breakdown for positive-threshold dual-Poisson transparency. Paper 8 — The Resource Geometry of Answerable InferenceIrreducible Answerability Structure, Minimal Sufficient Cores, and Certified Inference-Resource BoundariesDevelops the resource geometry of answerable inference. It formalises lawful-discharge semantics, source-level inferential obligations, answerability-sufficient reductions and irreducible sufficient cores, while keeping semantic structure, certified lower bounds and operational feasibility independently constituted. Gate E establishes a local answerability-to-resource threshold result; globally, exact closure identifies the Pareto-minimal inference-resource
The same quantum channel can hide different resource costs Cloc=ε(μM−μ0)+kBT0[j=1∑Mh(q−μj+μj−1)−Mh(q)]. Two finite supply devices can implement exactly the same complete quantum channel while depleting different amounts of locally recoverable free energy. Even perfect endpoint process tomography need not determine that supply cost. This paper proves an explicit counterexample, then identifies the extra records that reconstruct the resource without fitting a relaxation rate: the intermediate mean excitation numbers, together with independently known preparation and energy scales. For a qubit contact with fixed population contraction Λ, prepared supply population q, initial system population p0, and M contacts, write Δ=(q−p0)(1−Λ). The full dynamically attainable resource interval is Cmin=εΔ+kBT0[h(q−Δ)−h(q)],Cmax=εΔ+MkBT0[h(q−Δ/M)−h(q)]. Both endpoints are constructively attainable when idle contacts are allowed. Every intermediate value occurs with the same complete reduced channel. The paper supplies the realizing contact angles, the refinement limit, an explicit correction for outgoing-unit coherence, and finite-error bounds that include uncertain energy and temperature scales. The collective extension gives the exact Dicke collision kernel, a controlled weak-collision limit, finite-contact parameter identification, and a stationary Otto-cycle resource balance. The reconstruction archive independently builds the joint unitary, partial traces, full reduced superoperators and a second Markov propagator. It reproduces the published collective-engine benchmark and includes all numerical cases, code, plot data and checksums. Scientific status. These are conditional mathematical theorems with reproducible synthetic verification. They extend the QTT physical-contact program by separating endpoint selection from resource selection. The chosen exchange contact has a matched conventional quantum realization. The work neither reports a laboratory detection nor claims an engine-specific axiom-only value of its coupling. Its operational advance is an explicit resource-selection criterion and a reconstruction that requires no coupling fit. Dependencies: QTT Physical Contact Selection and Predictive Transfer and Completed-Event Transport and Quantum Coherence. Independent benchmark: Garcez and colleagues, collective Dicke Otto engine. Corpus Tree. The PDF is the main paper and default preview. The ZIP contains the complete editable and executable reconstruction package.
A fixed complete ideal unitary gate can be implemented by infinitely many time-dependent control paths, which are invisible to the endpoint but may affect the system's response to environmental noise. In a two-atom Rydberg system with six-segment piecewise-constant drives, we construct distinct controls that share the same full unitary endpoint up to a global phase. We compute the interaction-picture dissipative response operator K_z = ∫_0^T U_z^{-1}(t) D U_z(t) dt, where D is the local dephasing generator. For two main lifted controls (denoted CW and CCW) we find ||K_z-K_{z0}||_F/16 ≈ 1.6×10^{-4}, and the exact noisy-channel distance D_E(γ) = (1/16)||E_z(T;γ)-E_{z0}(T;γ)||_F scales as γ^{0.9965}. The first-order prediction \hat D_E^{(1)}(γ) = γ ||K_z-K_{z0}||_F/16, with no parameters fitted to the channel differences, matches the exact distance to within 1.1% relative error for γ ≤ 0.03 μs^{-1}, while the residual scales as γ^{1.9979}. These results establish that the ideal executed path quantitatively predicts weak-noise channel separation for controls that share the same complete unitary. An exploratory non-collinear control (ALT) is also constructed and shows a smaller but similarly predictable response (e_max = 1.079%). Whether the path-response predictor can prospectively select a lower-loss control for a fixed task remains an open direction. The claims are limited to the declared two-atom model and control family; production compilation, hardware validation, and universal geometric principles remain outside the present scope.
Quantum records may be well encoded upstream yet fail to support downstream inference when routing, decoder resources, memory, or comparison channels are restricted. We analyze finite routed quantum-record networks using task-specific Bayes risk. Three exact results provide the constructive core. First, a single Bell-coded pair supplied as the same complete input to GLOBAL and finite-round LOCC-late decoders yields the risk gap qμ/2, where q is pair-delivery probability and μ retained coherence; memory policy determines whether this access is preserved. Second, fixing local binary experiments with error rates 0.20 and 0.30, we determine the entire compatible joint-risk interval [0.15, 0.20] and an attaining decoder with minimax risk 0.20 that does not know the coupling. Third, a binary sufficient-statistic delivery law is attained by a three-record majority compression with nonzero raw risk, yielding a tight physical reconstruction defect. A generated-outcome model gives an exact receiver-availability boundary, while four reduced-description witnesses and continuity bounds delineate when source, state/resource, local-coupling, or clock/memory information is insufficient. The results distinguish record formation, receiver use, memory retention, and task-adequate comparison within standard finite quantum processes. Author manuscript, preprint version 1.0. This manuscript is not a peer-reviewed journal publication. A separate technical Supplement provides complete model definitions, proofs, channel specifications and numerical details. Reproducibility materials are available separately as Software release 1.3 on OSF: https://doi.org/10.17605/OSF.IO/5TDRB. The preprint and reproducibility record are separate, linked scholarly outputs.
Quantum combs are powerful conceptual tools for capturing multi-time processes in quantum information theory, constituting the most general quantum mechanical process. But, despite their causal nature, they lack a meaningful physical connection to time -- and are, by and large, arguably incompatible with it without extra structure. The subclass of quantum combs which assumes an underlying process is described by the so-called process tensor framework, which has been successfully used to study and characterise non-Markovian open quantum systems. But, although process tensors are motivated by an underlying dynamics, it is not a priori clear how to connect them to a continuous process tensor object mathematically -- leaving an uncomfortable conceptual gap. In this work, we take a decisive step toward remedying this situation. We introduce a fully continuous process tensor framework by showing how the discrete multi-partite Choi matrix becomes a vector in bosonic Fock space, which is intrinsically and rigorously defined in the continuum. With this equipped, we lay out the core structural elements of this framework and its properties. This translation allows for an information-theoretic treatment of multi-time correlations in the continuum via the analysis of their continuous matrix product state representatives. Our work closes a gap in the quantum information literature, and opens up the opportunity for the application of many-body physics insights to our understanding of quantum stochastic processes in the continuum.