Preprint by Seth Douglas (seth.douglas@gmail.com). A finite closed mixed-apparatus construction approximates any Schur channel of Gram rank k in half-diamond distance using a physical apparatus of dimension of order epsilon^-2 (sqrt(k) + ln d), with a maximally mixed initializer of rank of order sqrt(k). A universal entropy converse 2Q - P >= S(J) and an exact extreme-channel cost P, Q >= log2 k give, for a known extreme rank-two ququart Schur channel, a matched leading-order exact-versus-vanishing-error separation: exact service needs P = Q = T, while error T^-4 admits Q = T/2 + O(log T) and P = O(log T). A separately scoped scheduled-delivery theorem compares sublogarithmic and superlogarithmic delay under mandatory arrivals, a common release schedule and observation-only stopping. Includes the complete manuscript, TeX source, reproducibility scripts and partial Lean coverage. Scoped licenses: the manuscript, PDF and TeX are licensed under Creative Commons Attribution 4.0 International (CC-BY-4.0). Original code and accompanying software documentation are MIT-licensed. These are separate file-specific licenses, not a choice of license for every file. See LICENSE.md and LICENSES/ in the archive. The aggregate Other (Open) category represents this mixed-license package. Lean coverage is partial: fourteen scoped dependency closures, not the headline analytic theorems. The construction is a probabilistic existence proof; no efficient compiler, arbitrary-channel upper bound, caller-paced delivery guarantee or sharp logarithmic-delay frontier is claimed. AI assistance is disclosed in the manuscript. This is a separate sequel to the closed repeated-use memory paper, DOI 10.5281/zenodo.22779673.
Preprint by Seth Douglas (seth.douglas@gmail.com). A finite closed mixed-apparatus construction approximates any Schur channel of Gram rank k in half-diamond distance using a physical apparatus of dimension of order epsilon^-2 (sqrt(k) + ln d), with a maximally mixed initializer of rank of order sqrt(k). A universal entropy converse 2Q - P >= S(J) and an exact extreme-channel cost P, Q >= log2 k give, for a known extreme rank-two ququart Schur channel, a matched leading-order exact-versus-vanishing-error separation: exact service needs P = Q = T, while error T^-4 admits Q = T/2 + O(log T) and P = O(log T). A separately scoped scheduled-delivery theorem compares sublogarithmic and superlogarithmic delay under mandatory arrivals, a common release schedule and observation-only stopping. Includes the complete manuscript, TeX source, reproducibility scripts and partial Lean coverage. Scoped licenses: the manuscript, PDF and TeX are licensed under Creative Commons Attribution 4.0 International (CC-BY-4.0). Original code and accompanying software documentation are MIT-licensed. These are separate file-specific licenses, not a choice of license for every file. See LICENSE.md and LICENSES/ in the archive. The aggregate Other (Open) category represents this mixed-license package. Lean coverage is partial: fourteen scoped dependency closures, not the headline analytic theorems. The construction is a probabilistic existence proof; no efficient compiler, arbitrary-channel upper bound, caller-paced delivery guarantee or sharp logarithmic-delay frontier is claimed. AI assistance is disclosed in the manuscript. This is a separate sequel to the closed repeated-use memory paper, DOI 10.5281/zenodo.22779673.
Preprint by Seth Douglas (seth.douglas@gmail.com) on the dimension and actual initial-entropy rate region of a closed device serving repeated uses of one fixed quantum channel against adaptive inputs. Includes the complete manuscript, appendices and reproducibility materials. Scoped licenses: the manuscript, appendices, PDF and TeX are licensed under Creative Commons Attribution 4.0 International (CC-BY-4.0). Original code and accompanying software documentation are MIT-licensed. These are separate file-specific licenses, not a choice of license for every file. See LICENSE.md and LICENSES/ in the archive. The aggregate Other (Open) category represents this mixed-license package. Lean coverage is partial: a finite quantum closing identity and six arithmetic lemmas, not the full entropy or asymptotic theorem. AI assistance is disclosed in the manuscript. No efficient circuit synthesis or AK resolution is claimed.
Preprint by Seth Douglas (seth.douglas@gmail.com) on quantitative recovery from finitely many calibrated sharp binary observables, with complete Appendix A and reproducibility materials. Scoped licenses: the manuscript, PDF and TeX are licensed under Creative Commons Attribution 4.0 International (CC-BY-4.0). Original code and accompanying software documentation are MIT-licensed. These are separate file-specific licenses, not a choice of license for every file. See LICENSE.md and LICENSES/ in the archive. The aggregate Other (Open) category represents this mixed-license package. Lean coverage is partial: calibrated projections imply finite-word operator-norm stability; no numbered recovery theorem is fully formalized. Language models assisted exploration and checking; the author retains responsibility for the claims.
How much memory does it take to apply the same quantum channel many times, when each output must be released before the next input arrives and every retained qubit counts? One-use invariants do not decide it. Two explicit channels on dimension 1664 have identical normalized Choi spectra (32 eigenvalues 1/32), the same maximal complementary entropy 5, and zero asymptotic purity cost. At the same optimal exchange rate, one is implemented exactly with five memory qubits for any number of uses, while the other needs Omega(n^(1/(2 log_2(107)+2))) memory qubits for n uses, at every fixed error at most 1/16 and whatever purity is available; memory O(n^(1/3) log^(4/3) n) suffices for it. The hard channel is built from Houghton's group H_3. Behind the example is a general theory: the achievable exchange and purity rates, the purity cost kappa, which vanishes exactly on the closure of finite tracial factorizations, and memory bounds from how well a channel's environment is approximated by finite ones, and, for channels built from finite presentations, from the approximation profile of the group. Version 1.1 adds where the expensive channels lie: on the relative boundary of the closure; any fixed depolarizing admixture brings them down to polylogarithmic memory at the rank rate; on a face containing the Houghton channel a single moment of a trace on the group decides expense; and one free fermion per site already forces polynomial memory. Version 1.1.1 adds a map of what the main separation depends on (published inputs, companion results and new arguments), sets finite-budget and asymptotic-rate statements side by side, cites the companion papers at their published versions, pins the tested software versions, and makes every check refuse to run under optimized Python. The mathematical results are unchanged. Version 1.1.2 corrects attributions and makes two headline statements precise. The separation of Theorem A is stated at the exchange budget 5n/2, and the channels that need polynomial memory at every exchange rate are those with purity limited to O(log n). Credits added or corrected: Bisio, D'Ariano, Perinotti and Chiribella, and Bisio, D'Ariano, Perinotti and Sedlak; Boes et al. for the square-root dephasing construction; Lie, Son, Boes, Ng and Wilming for the exact case of the tracialization lemma; Faist, Berta and Brandao for the relation to the thermodynamic capacity; Haagerup, Musat and Rordam for the moment criteria; Thoma, Johnson, Watrous, Horodecki-Oppenheim-Winter and Lie-Jeong; Cornulier for the Baumslag-Solitar bound. Known results are moved out of numbered statements. The mathematical results and the numbering are unchanged. All operational bounds use complete-experiment error against adaptive observers with quantum references. The general rate-region achievability result uses a cited closed-device theorem; the explicit separation does not. This preprint archive includes the PDF, TeX source, finite verification scripts, and scoped Lean arithmetic proofs. The artifacts do not formally verify the entire manuscript; their scope and external dependencies are documented in verification/README.md.
How much memory does a device need to apply one quantum channel n times, if each output must leave before the next input arrives? The paper counts the qubits the device keeps, the qubits it exchanges for fresh ones, and the purity it is given, with error measured on the whole experiment against an adaptive observer. Immediate release has a price. For the qutrit Werner-Holevo channel, vanishing error requires log2 3 bits of purity per use, a known rate; the paper adds a bound at finite error. If all outputs may leave together, O(log n) bits suffice in total at polynomially small error, refining a known zero rate. At error n^-2 the change comes when outputs wait in batches of about log n, and the zero supplied-purity-rate side holds for every unital channel. Purity separates cheap from expensive repeated use along one set of channels, the closure of those that a finite, maximally mixed bath implements exactly: outside it, linear purity is known to be needed at small error, hence linear memory when fresh qubits are maximally mixed. The paper proves a converse for memory, keeping maximally mixed fresh qubits and the optimal exchange rate: a channel in the closure with a flat probe (an input that leaves the minimal environment maximally mixed) can be served with sublinear memory at any error that is not exponentially small. For channels built from finitely many group relations, memory at fixed error is at most logarithmic, divergent but sublinear, or linear, and some are linear exactly when some finitely presented group is not hyperlinear. Along the way, Rybar and Ziman's question on repeating a channel without resets is answered. At high precision, the memory of n uses of such a channel equals, up to squared logarithms, the cost of the best single use that never leaves the channel's support. That cost is the bath entropy plus n times the entropy one use adds to the bath. For Houghton's group, devices acting through the group need and attain memory n^(1/3) up to logarithmic factors. Whether the single-use law holds at fixed error is open. This archive includes the paper, its LaTeX source and the finite numerical checks; they support but do not replace the proofs.
A device that applies a quantum channel over and over must release each output before the next input arrives. How much memory does it need? Suppose every fresh qubit the device receives is maximally mixed. In companion work we showed that the answer then turns on one set: the channels that a unitary with a finite, maximally mixed bath implements, exactly or in the limit. A channel in this set with a flat probe, an input that leaves its environment maximally mixed, can be served with sublinear memory. A channel outside the set needs linear memory at small error, however many qubits the device exchanges. No algorithm can decide on which side of this line a channel lies. A computable map sends each program to an explicit Schur channel with Gaussian-rational entries. If the program halts, logarithmic memory suffices; if it runs forever, linear memory is needed at an error computed from the program. The same line recasts Connes' embedding problem. It has a positive answer exactly when every factorizable channel with a flat probe can be served with sublinear memory; Schur channels already suffice. Just past the line, the cost switches on at the statistical scale. Take the qutrit Werner-Holevo family at distance h/sqrt(n) beyond its boundary point. Independent uses of the boundary channel imitate it with error tending to 2 Phi(h/(2 sigma)) - 1, where Phi is the standard normal distribution function and sigma = 5 sqrt(2)/27. We prove that no device drawing on purity that grows more slowly than sqrt(n) does better, whatever its memory. This archive includes the paper, its LaTeX source, an exact certificate and the finite numerical checks; they support but do not replace the proofs.
Exact Binary Projections for Array-RQMC: Joint Laws and Pathwise-Preserving Execution Aoi Kawasaki. Preprint; not peer reviewed. Contribution For a specified two-dimensional Sobol net with linear matrix scrambling and a digital shift, the complete rank-ordered binary innovation vector is uniform over exactly N outcomes at N = 2^m paths. An odd mask and a fair bit describe the law, requiring m independent fair bits per step under the stated ideal randomization model. Independent adjacent antithetic pairs have the same covariance but need not have the same higher-order law; exact nonlinear witnesses give variances 0 versus 2 and 4 versus 2. This characterization enables direct generation without generating unused coordinates or sorting the point set. A second rewrite maintains exactly the reference state order for stopped scalar binary chains with monotone branches, including absorption and floating-point ties. Measured results and limits Across eight fixed benchmark cases in two input encodings, state-order maintenance reduced warm execution time by 37.2% at N = 512 and 67.5% at N = 4,096 relative to direct binary generation with full state sorting. The encodings do not create sixteen independent model laws. The measurements are from one CPU/software stack, not a held-out hardware or dynamics study. The joint-law equivalence of the library and direct generators is an ideal-law statement, not equality under the same finite integer seed. Direct and order-maintained execution do preserve the same finite-seed trajectories and outputs under the stated arithmetic conditions. No new convergence rate or universal variance advantage is claimed. Comparisons with the measured CRN and antithetic implementations remain workload- and reuse-dependent. Avoidable mask initialization was included in the measured baselines; it is not an intrinsic CRN cost. Startup and calibration can outweigh warm-time savings. The publication retains those recorded costs. Included materials and reproducibility scope The record contains the manuscript PDF and a self-contained reproducibility capsule with manuscript/build sources, three array kernels, 19 tests, saved table/figure inputs, two figures, dependency versions, and SHA-256 inventories. The tests include exact finite enumeration, within-project separate reference implementations, absorption and floating-point tie checks, and saved-data arithmetic. They are not third-party peer review. The capsule does not include every raw calibration/validation observation, bootstrap sample, timing round, or baseline/calibration driver. Regenerating saved tables and figures is not a rerun of the full original experiment. No GPU, pretrained model, private workspace, or external dataset is required. Rights and disclosure Publication materials and saved data: CC BY 4.0. Code: MPL-2.0, with per-file scope in LICENSES.txt. Generative-AI assistance is disclosed in the manuscript. The version-specific archival identifier is doi:10.5281/zenodo.22728405.