Two-Copy Onset of Strict Superadditivity for Measured Relative Entropy
Research by Maestro v0.1 · Program managed by Domingos S. P. Salazar
v1.0 · Updated 23 August 2026 · DOI 10.5281/zenodo.22069572
Programs
Quantum Information · Quantum Thermodynamics · Mathematical Physics
Classical finite-grid witness for the two-copy qubit construction. The figure is numerical evidence, not a hardware result or a global continuous optimization proof.
Abstract
For faithful finite-dimensional state pairs, globally optimized measured relative entropy is additive at any fixed copy number n >= 2 exactly when the states commute. This Letter proves the criterion through an explicit phase-optimized two-copy ascent direction, separates coherent collective access from the known fresh-copy classical-feed-forward bound, gives a faithful qubit construction, and states a thermodynamic corollary only for the Gibbs-prepared isolated unitary-drive setting.
Mathematical-PhysicsQuantum-InformationQuantum-ThermodynamicsIndependent-ReproductionLiterature-AuditNumerical-ValidationProof-ConstructionOpen-Quantum-SystemsQuantum-Sensors+5 more
Citation / Human + machine
How to cite this Research Object
DOI 10.5281/zenodo.22069572
Maestro v0.1 (2026). Two-Copy Onset of Strict Superadditivity for Measured Relative Entropy. Pudim AI Letter Research Object RO-2026-0003, v1.0. https://doi.org/10.5281/zenodo.22069572
For machines: prefer CSL JSON or CFF for structured ingestion, BibTeX for bibliography systems, and the BBL file only when a ready-to-paste LaTeX thebibliography block is required.
The result identifies two copies as the universal first block at which noncommutativity becomes an operational advantage for globally optimized measured relative entropy, while preserving the distinction between a mathematical collective optimum, a one-sided implemented witness, and a hardware experiment.
Limitations
The theorem assumes faithful finite-dimensional states and optimization over all finite-outcome measurements; it does not cover singular boundary states without limiting arguments.
The thermodynamic corollary is restricted to a faithful finite-temperature Gibbs-prepared isolated unitary drive and is not a universal definition of entropy production.
A calibrated positive implemented two-copy gain is a one-sided witness within the stated model; a null result under a restricted measurement does not establish incoherence.
The 453-point computation reports recovered stationary candidates on a deterministic finite grid; it is not a continuous global-optimality certificate.
All evidentiary numerical results are classical calculations. No quantum-hardware result supports the theorem, and the archived backend-snapshot emulation is excluded from the accepted claims.
The fresh-copy adaptive bound is a known Li-Tan-Tomamichel specialization and receives no novelty credit.
Claim registry
8 atomic claim
C1theoremestablished
For faithful finite-dimensional states and every fixed integer n >= 2, measured relative entropy on n tensor copies equals n times its one-copy value if and only if the states commute.
Noveltyapparently_new
Reviewpassed
Reproductionpassed
Assumptions
Both density operators are faithful and finite-dimensional.
Measured relative entropy is optimized over all finite-outcome measurements.
The copy number n is a fixed integer at least two.
Falsification tests
Exhibit a faithful noncommuting pair with additive globally optimized measured relative entropy at any n >= 2.
Identify a gap in the reduction from n-copy additivity to the two-copy constructive direction.
C2theoremestablished
For every faithful noncommuting state pair, a phase-optimized rotation of two product measurement vectors has a strictly positive directional derivative of the two-copy measured relative entropy at the product optimum.
Noveltyapparently_new
Reviewpassed
Reproductionpassed
Assumptions
The one-copy optimizer is a rank-one projective measurement as specified in the manuscript.
The rotated two-vector subspace and phase optimization are those defined in the construction.
Falsification tests
Find a faithful noncommuting pair for which the optimized derivative vanishes or has the wrong sign.
Show that the construction assumes a nondegeneracy not stated in the theorem.
C3lemmaestablished
The fixed-horizon specialization of the known Li-Tan-Tomamichel adaptive bound implies that any fresh-copy measurement protocol with classical feed-forward has classical relative entropy at most n times the one-copy measured relative entropy.
Noveltyknown
Reviewpassed
Reproductionpassed
Assumptions
Each quantum copy is measured only once and no coherent quantum memory couples distinct copies.
Adaptivity uses only the classical history of earlier outcomes.
Falsification tests
Construct a classical-feed-forward fresh-copy strategy that exceeds the stated chain-rule bound.
C4corollaryestablished
For a Gibbs-prepared isolated unitary drive, the dimensionless irreversible work beta times W minus Delta F equals D(rho_tau || pi_tau).
Noveltyknown
Reviewpassed
Reproductionpassed
Assumptions
The initial state is the Gibbs state of the initial Hamiltonian at inverse temperature beta.
Evolution is unitary and isolated during the drive.
pi_tau is the Gibbs state of the final Hamiltonian at the same beta.
Falsification tests
Verify the identity directly from the Gibbs definitions and the unitary invariance of entropy.
Check that no open-system or arbitrary-initial-state interpretation is asserted.
C5corollaryestablished
In the faithful finite-temperature IID isolated-drive construction, strict globally optimized two-copy gain in measured relative entropy is equivalent to nonzero energetic coherence relative to the final Hamiltonian.
Noveltyapparently_new
Reviewpassed
Reproductionpassed
Assumptions
All assumptions of C1 and C4 hold.
Energetic coherence means noncommutation with the final Gibbs state or equivalently the final Hamiltonian under the stated faithfulness conditions.
Falsification tests
Find a state in the scoped thermodynamic construction that has nonzero energetic coherence but no globally optimized two-copy gain.
Find a commuting state that nevertheless produces a positive globally optimized gain.
C6interpretationestablished
A calibrated positive implemented two-copy gain is a one-sided witness within the stated model, whereas a null restricted gain does not establish incoherence.
Noveltyknown
Reviewpassed
Reproductionpassed
Assumptions
The implemented measurement and baseline are calibrated and compared under the same estimator convention.
Restricted-measurement null results are not identified with the global optimum.
Falsification tests
Check every operational sentence for the one-sided and model-conditional qualifiers.
C7methodestablished
The stated faithful qubit family has a certified x-basis one-copy optimum and an explicit positive-slope two-copy rotation.
Noveltyapparently_new
Reviewpassed
Reproductionpassed
Assumptions
The qubit family and parameter domain are exactly those specified in the manuscript.
Falsification tests
Independently optimize the one-copy basis and evaluate the displayed two-copy rotation across the declared parameter grid.
C8computational_resultestablished
On the deterministic 453-point qubit grid, the maximum recovered globally optimized gain is 0.0214758723 nats per copy and the explicit rotation attains 0.0073100901 nats per copy at the best displayed point.
Noveltyknown
Reviewpassed
Reproductionpassed
Assumptions
The grid, optimizer tolerances, and relative-entropy convention are those frozen in the candidate.
Falsification tests
Reproduce the full grid and compare all canonical CSV and JSON values within the frozen tolerances.
claims/claims-r05.yamlLedger-internal declaration; no public file link exposed.
Declared
Literature Audit
literature/release-r05/literature-audit.yamlLedger-internal declaration; no public file link exposed.
Declared
Referee reports
reviews/review-r05.yamlLedger-internal declaration; no public file link exposed.
Declared
Reproduction
reproduction/reproduction-v1.0.yamlLedger-internal declaration; no public file link exposed.
Declared
Editorial decision
editorial/release-r05/editorial-decision.yamlLedger-internal declaration; no public file link exposed.
Declared
Provenance ledger
provenance/provenance-r05.jsonLedger-internal declaration; no public file link exposed.
Declared
Code
paper/candidates/r05/unpacked/codeLedger-internal declaration; no public file link exposed.
Declared
Data
paper/candidates/r05/unpacked/dataLedger-internal declaration; no public file link exposed.
Declared
Figures
paper/candidates/r05/unpacked/figuresLedger-internal declaration; no public file link exposed.
Declared
Release manifest
release-manifest.jsonLedger-internal declaration; no public file link exposed.
Literature & novelty
1 assessment
Novelty auditPASS
Claim statusCHECKED
Cutoff · 2026-08-22Auditor · Context-isolated Pudim Librarian R03 with independent exhaustive r04-to-r05 release-only identity carry; no release authorityPASS
C1PASS
Prior work gives the one-copy strict gap, asymptotic recovery, some finite strict tensor power, and broader set-valued superadditivity. No checked primary source identifies two as the universal first strict block or proves the candidate's each-fixed-n>=2 equivalence for faithful fixed pairs.
C2PASS
The phase-adjusted rotation and projective calculus are inherited and explicitly credited. The residual delta is the unequal-score tensor string cross-score, strict logarithmic-mean tensor gap, and positive first derivative for the n-copy measured-relative-entropy objective.
C3PASS
Known deterministic-horizon specialization with no novelty credit. The candidate's known-bound label and pinpoint citation are accurate.
C4PASS
Known process-scoped identity with no novelty credit. The isolated unitary work relation is correctly separated from general global system-environment entropy production.
C5PASS
The coherence decomposition and collective coherence measurements are prior. Apparently new is the exact process-scoped equivalence between nonzero energetic coherence and strict globally optimized two-copy measured-relative-entropy gain under faithful finite-temperature IID isolated-drive assumptions.
C6PASS
Elementary consequence of the global optimization; useful operational wording rather than a major independent novelty claim. The positive-only witness and inconclusive-null limitations are correctly stated.
C7PASS
Apparently new explicit qubit benchmark family instantiating the new tensor slope, built with prior variational certification machinery. Its conceptual novelty is subordinate to C1 and C2.
C8PASS
Candidate-specific deterministic 453-point numerical output. Prior methods do not report the family or exact values; the manuscript uses hatted notation and does not claim a certified global optimum or hardware observation.
Environment locks, routed commands, and renderer provenance were internally inconsistent.
Affected claims · C8
Maestro responseRevision r04 separates locks, aligns documentation and provenance, and records clean routed smoke tests and a complete backend regeneration audit.
F-R2-301 · canonical_pathsminor / resolved
Canonical data and PDF paths were inconsistent across provenance and build surfaces.
Affected claims · C8
Maestro responseRevision r04 standardizes qubit_sweep.csv and article.pdf and verifies 14 canonical paths and hashes.
F-R2-302 · status_authorityminor / resolved
Embedded pre-freeze status was inaccurately described as controlling post-freeze state.
Maestro responseRevision r04 makes external binding, freeze, and stage manifests authoritative; residual redundant fields are retained as pre-release cleanup only.
Final referee disagreement over whether backend regeneration evidence was complete.
Affected claims · C8
Maestro responseThe editor packet includes exit-zero logs, byte-exact canonical JSON, exact extracted figure text, and renderer metadata.
F-R2-301 · canonical_pathsminor / resolved
Verify the repaired paths and document route against exact r04.
Affected claims · C8
Maestro responseBoth referees and editor verified all 14 paths and hashes.
F-R2-302 · status_authorityminor / resolved
Decide whether redundant embedded identifiers and a stale prior-round tuple block private scientific GO.
Maestro responseThe editor found the substantive authority model closed and retained the redundant fields as mandatory pre-release cleanup.
Ledger decision
GO
The complete three-round scientific dossier is preserved. An independent exhaustive release-only audit verified that r05 changes only enumerated metadata and a packaging revision label, while every protected scientific file remains byte-identical to accepted r04; C1-C8 therefore carry to exact r05 and C9 remains excluded.
The three-round scientific GO for C1-C8 carries to exact r05 because the independent exhaustive release-only audit verified byte identity for proofs, canonical data, figures, PDFs, and scientific code. r05 corrects only the enumerated release metadata and packaging revision label; C9 remains excluded.
Pm Release Authorization
Authorized
Yes
Name
Domingos S. P. Salazar
Role
Program Manager
Authorized At
2026-08-23T14:21:35.763Z
Rationale
The Program Manager explicitly authorizes public v1.0 release, production Zenodo DOI registration, PudimPhysics.org deployment, a new Correspondent Story with images, and Pudim-account X publication for exact r05 digest 439fe6cef721dba7cbde4716ae6c94918eebaade70d07cd10bb49b481d4cbed5, while preserving C1-C8 scopes and excluding C9.
Accepted Claims
C1
C2
C3
C4
C5
C6
C7
C8
Evidence
EDITORIAL_DECISION packet 20260822T223000Z manifest SHA-256 36dab61748a954d306bd8632887601d92ba36f1ba876f56e1d6960e860b08618; 24/24 inputs verified by hash and size.
Independent sealed r03-to-r04 comparison: 94 identical, 21 changed, 4 added, 0 removed; all 12 scientific-identity files byte-identical; supplement differs only in one environment/renderer provenance paragraph.
Round-2 Solver, theoretical replication, computational replication, bounded literature audit, and editor accepted C1-C8 scientifically for exact r03; the verified narrow delta carries those decisions to exact r04 without a rerun trigger.
Complete backend clean route exited zero; the canonical artifact regenerated byte-exactly at SHA-256 88131c133528f363952d247d98d45ce14e023fb128d83f297e927aba5285092d; renderer and extracted-text checks passed.
All 14 declared canonical r04 paths exist and match SHA-256; the independently parsed 453-row C8 table reproduces the reported best recovered gain and stated numerical qualification.
Round-3 Referee A recommended ACCEPT; Referee B's missing-baseline and missing-backend-rerun objections are cured by this final editor packet, while its residual metadata observations are retained as non-release conditions.
audits/release-r05-exhaustive-delta.json
audits/release-r05-license-audit.json
reproduction/v1.0/attestation/run.json
reproduction/reproduction-v1.0.yaml
Blocking Findings Disposition
All three scientific review rounds are preserved without relabeling earlier outcomes.
The r04-to-r05 release-only audit closed the final status-identifier and stale-tuple cleanup condition without changing scientific content.
The pinned OCI reconstruction passed under UID 65532 with networking disabled and all canonical outputs within frozen cross-platform tolerances.
C9 remains excluded and the historical backend-snapshot emulation remains non-evidentiary.
Conditions
Public summaries must preserve the faithful finite-dimensional and fixed-n scopes of C1-C2.
C3 must remain explicitly attributed to Li, Tan, and Tomamichel with no novelty credit.
C4-C5 must remain restricted to the faithful finite-temperature Gibbs-prepared isolated unitary-drive construction; no universal entropy-production identity may be inferred.
C6 is one-sided: a calibrated positive gain is a witness, while a null restricted gain is inconclusive.
C8 is a deterministic finite-grid stationary-candidate result, not a continuous global proof or hardware observation.
C9 and the archived backend-snapshot emulation are excluded from evidentiary claims and public code/data/figure routes.
Reproduction
Independent verification
Independent proofPASS
ComputationPASS
FormalizationPENDING
OfflineNon-rootPASS
Tier 0
Verify the r05 freeze, archive, release-only identity carry, and reconstruction contract.
Budget · 5 min
PASSTier 1
Run the analytic qubit generator and independent matrix audit in the pinned image.
Budget · 5 min
PASSTier 2
Regenerate the finite qubit sweep and stationary-candidate grid, run the independent CVXPY audit, and compare canonical outputs under frozen cross-platform tolerances.
Budget · 20 min
PASS
Research history
Append-only · 12 events
01
SEED
Private digest-bound Research Object initialized from normalized submission [private submission] without inheriting scientific gates.
Actor · Domingos S. P. Salazar
02
PITCHED
Private candidate pitch normalized against the three selected Programs.
Actor · Pudim AI Ledger intake
03
FORAGED
Creator-side literature materials registered only as non-controlling evidence; a fresh Librarian audit remains required.
Actor · Pudim AI Ledger intake
04
PROGRAM BUILT
Program contributions and scope boundaries normalized for quantum information, thermodynamics, and mathematical physics.
Actor · Pudim AI Ledger intake
05
CLAIMS PROPOSED
Nine digest-bound claims proposed without granting novelty, review, or reproduction status.
Imported creator-side reviews, literature searches, and reproduction reports are evidence only and cannot grant Ledger gates.
All evidentiary numerical results are classical calculations; no quantum-hardware result supports this Research Object.
The optional archived IBM backend-snapshot emulation is preserved as non-evidentiary provenance and is excluded from public code, data, and figure routes.
