We answer the historical distributed dense-coding question for the symmetric four-qubit W state in the local-unitary model of Bruss and collaborators. Two independent senders transmit one qubit each to different receivers. A complete Bell instrument at one receiver retains a classical-quantum output at the other; the established cq multiple-access coding theorem gives an achievable sum-rate supremum at least 3/2 + h2(1/4) = 2.311278124459... bits per resource copy. Both senders can achieve 9/8 bits per copy with vanishing uniform average error, establishing the original-model LOCC-DC classification. The Bell decomposition, Bell forwarding, cqMAC coding theorem and prior general MAC dense-coding framework are explicitly credited. The bounded primary-literature audit found no earlier explicit or established-equivalent achievement of the exact target in inspected sources; specific access and version limitations are documented. No categorical firstness or worldwide continued openness, exact optimal capacity, one-copy advantage, zero-error or maximal-error claim is made. AI tools were used extensively in solving, drafting, reproducing and adversarially reviewing the result. This is an unrefereed preprint without conventional human peer review. The package includes the manuscript PDF and standalone LaTeX source, a standard-library exact rational verifier, recorded results, a priority-audit supplement, license and file digests. Finite computations supplement the analytical proof and established asymptotic coding theorem.
We answer the historical distributed dense-coding question for the symmetric four-qubit W state in the local-unitary model of Bruss and collaborators. Two independent senders transmit one qubit each to different receivers. A complete Bell instrument at one receiver retains a classical-quantum output at the other; the established cq multiple-access coding theorem gives an achievable sum-rate supremum at least 3/2 + h2(1/4) = 2.311278124459... bits per resource copy. Both senders can achieve 9/8 bits per copy with vanishing uniform average error, establishing the original-model LOCC-DC classification. The Bell decomposition, Bell forwarding, cqMAC coding theorem and prior general MAC dense-coding framework are explicitly credited. The bounded primary-literature audit found no earlier explicit or established-equivalent achievement of the exact target in inspected sources; specific access and version limitations are documented. No categorical firstness or worldwide continued openness, exact optimal capacity, one-copy advantage, zero-error or maximal-error claim is made. AI tools were used extensively in solving, drafting, reproducing and adversarially reviewing the result. This is an unrefereed preprint without conventional human peer review. The package includes the manuscript PDF and standalone LaTeX source, a standard-library exact rational verifier, recorded results, a priority-audit supplement, license and file digests. Finite computations supplement the analytical proof and established asymptotic coding theorem.
We analyze implementations of bipartite unitaries in a distributed quantum computation setting using local operations and classical communication (LOCC) assisted by shared entanglement. We employ concepts and techniques developed in quantum Shannon theory to study an asymptotic scenario in which the two distant parties perform the same bipartite unitary on infinitely many pairs of input states generated by a completely random i.i.d. (independent and identically distributed) quantum information source. We analyze the minimum costs of resources of entanglement and classical communication per copy. For protocols consisting of two-round LOCC, we prove that an achievable rate tuple of costs of entanglement and classical communication is given by the “Markovianizing cost” of a tripartite state associated with the unitary, which is conjectured to be optimal as well. The Markovianizing cost can be computed by a finite-step algorithm.
We give a self-contained explicit counterexample for the existential unregularized bounded-region fermion question on printed page 371 of Oberwolfach Report 7/2025, with pointwise operator-norm topology distinguished from strong-operator topology on page 365. In dimension three, one fixed smooth nonnegative radial compactly supported pair potential and one fixed interaction ball give a fixed smooth normalized annihilation mode whose Heisenberg norm differences tend to the maximal value 2. Explicit packed finite-particle Slaters and times pi/(2N+1-kappa) witness the same vector-norm limit, a stronger vector asymptotic, and changing normal-state values tending to 4. Both diagonal-density and normal-ordered interactions are covered with ordered-pair coefficient 1 and no ultraviolet smearing. A fixed smooth-cutoff commutator, Slater second moment and two-sector Duhamel comparison give error O(N^(-1/6)+N^(-1/3)). The kinetic energy acts on all of R³. Continuity on every fixed vector and in every fixed normal state remains valid. Siebert's arXiv:2609.34402v1, submitted 28 September 2026 before the original 7 October candidate, supplies the earlier high-density cluster and localized phase mechanism. Its stated theorem concerns full-space difference potentials, but its proof directly adapts to this bounded symmetric cutoff kernel and yields the qualitative existential conclusion. Its Proposition 2.6 explicitly gives maximal norm 2 for the full-space constant-number reference. The present note gives an explicit free-Slater transfer to the fixed spatial cutoff. A separately checked audit-time energy bootstrap also recovers the cutoff maximal limit through the earlier localized phase mechanism. These cutoff implications are deductions made in the audit, not literal cutoff theorems attributed to the prior paper. No first solution, first quantitative refinement, historically certified novelty, or continued open status through October 2026 is claimed. The bounded primary-source priority audit records access and indexing limitations. This five-page unrefereed research note is accompanied by its identical authored PDF/LaTeX source, unchanged 26-file historical candidate and independently authored analytic, operator-algebra, computational, source/model, priority and priority-mapping audits. The fresh computational family has 2,668 exact finite predicates and 558 floating diagnostics; finite tests supplement the universal written proof, and numerical diagnostics are not rigorous interval enclosures or formal proof certificates. Historical optimized-assertion and inventory weaknesses are repaired by externally pinned closed regular-file validation and isolated nonoptimized checker execution, while all original bytes and the original 1/5 candidate count are preserved. Historical result outputs are not claimed byte-identical to current replays, and runtime source-byte checks are explicitly omitted. AI tools were used extensively in development, verification, adversarial review and writing. Independent AI-assisted review is not human or journal peer review; neither is claimed. No copied third-party source bodies are distributed. No all-potential, all-region, dimension-one/two, fixed-density/energy, thermodynamic, fixed-state-discontinuity or CAR-invariance result is asserted.
We analyze implementations of bipartite unitaries by means of local operations and classical communication (LOCC) assisted by shared entanglement. We employ concepts and techniques developed in the quantum Shannon theory to study an asymptotic scenario, in which two distant parties perform the same bipartite unitary on infinitely many pairs of inputs. We analyze minimum cost of entanglement and classical communication per copy. For two-round LOCC protocols, we derive a single-letter formula for the minimum cost of entanglement and classical communication, under an additional requirement that the error converges to zero faster than 1/n4, where n is the number of input pairs. The formula is given by the “Markovianizing cost” of a tripartite state associated with the unitary, which can be computed by a finite-step algorithm. We also derive a lower bound on the minimum cost of resources, which applies for protocols with arbitrary number of rounds.
This source-only preprint gives a four-qubit counterexample to the arbitrary-phase extension of the first-excited parity conjecture discussed in Wong, Bravyi, Gosset and Liu, Lee-Yang tensors and Hamiltonian complexity, arXiv:2602.03605v1, Section 5.6. For an unweighted four-cycle with phase pi on one edge and a common parameter 0 <= s < 1, it derives the complete 16-dimensional spectrum and an exact parity threshold. The rational value s = 2/3 is an explicit witness; a specified fixed-witness robustness interval is also proved. The original zero-phase conjecture and the general all-graph gap problem remain open. Ground-state uniqueness and the non-strict gap bound are known inputs, explicitly credited to Wei, arXiv:2609.26802v2, Corollary 5.8 and Section 5.4, and Bravyi, Gosset, Liu and Wong, arXiv:2607.14401v1, Theorem 1. The concrete phase counterexample, complete spectrum and threshold are distinguished from these known results. The primary-source comparison is dated and bounded; worldwide priority is not certified. Read the unchanged PHASE_EPR_FINAL_MANUSCRIPT_v0.1.3_R2_20261010.tex or .md first, together with the current public source/verification supplement and PUBLIC_RELEASE_NOTE. Older versions and administrative private-draft wording, if present as provenance, describe historical preparation stages and do not define this public version or its reviewed scope. Auxiliary six-qubit exploration is outside this paper's completed claim. Shunpei Shimizu initiated and directed this personal independent project, arranged verification, and coordinated the materials. No university or institutional affiliation is declared. AI systems performed research exploration, proof construction, computational checks, analytical review and manuscript preparation. The reviews are internal AI analyses; independent human verification of every proof, specialist human peer review and complete proof-assistant verification are not claimed. No PDF of editorial revision R2 has been generated or visually verified: the available compiler failed during environment setup. Editable TeX, a readable Markdown manuscript and verification materials are provided. This is a limited, unrefereed research preprint, released under Creative Commons Attribution 4.0.
This note records a self-contained exact reduction from every even-order symmetric matrix with binary nonnegative rational off-diagonal entries to a simple unweighted perfect-matching count, with polynomial encoding length, an explicit logarithmic integer-weight gadget, exact rational scaling, and weighted matching fibers. Citing the general-graph FPRAS of OpenAI, it derives a uniform rational hafnian FPRAS with exact zero detection and a rigorously specified bounded-bit total-variation approximate sampler. Logarithmic integer weight removal, binary rational weighted-to-unweighted equivalence, and counting self-reduction are established machinery and credited to earlier work, including Dell and collaborators, McQuillan, and Jerrum, Valiant, and Vazirani. The approximation breakthrough belongs to OpenAI. This is an attributed consequence and implementation note; no first-publication, new reduction, or independent solution of the base approximation problem is claimed. The supplement contains proofs, transparent code, exact finite enumeration and output-law checks, dependency and priority audits, and clean reproduction instructions. Finite checks do not certify the upstream FPRAS. The real upstream Lean statement was inspected, but no independent kernel rebuild was reproduced. Nonnegative rational inputs only; no signed or complex hafnian or unrestricted Gaussian-boson-sampling consequence. AI tools were used extensively in research, drafting and automated adversarial verification. The preprint has not undergone conventional human peer review or refereeing at publication. Automated reviews are not human peer review. Newly authored prose and data are CC BY 4.0; code is MIT as specified in the archive.
We investigate dense coding by imposing various locality restrictions to our decoder by employing the resource theory of asymmetry framework. In this task, the sender Alice and the receiver Bob share an entangled state. She encodes the classical information into it using a symmetric preserving encoder and sends the encoded state to Bob through a noiseless quantum channel. The decoder is limited to a measurement to satisfy a certain locality condition on the bipartite system composed of the receiving system and the preshared entanglement half. Our contributions are summarized as follows: First, we derive an achievable transmission rate for this task dependently of conditions of encoder and decoder. Surprisingly, we show that the obtained rate cannot be improved even when the decoder is relaxed to local measurements, two-way LOCCs, separable measurements, or partial transpose positive (PPT) measurements for the bipartite system. Moreover, depending on the class of allowed measurements with a locality condition, we relax the class of encoding operations to super-quantum encoders in the framework of general probability theory (GPT). That is, when our decoder is restricted to a separable measurement, theoretically, a positive operation is allowed as an encoding operation. Surprisingly, even under this type of super-quantum relaxation, the transmission rate cannot be improved. This fact highlights the universal validity of our analysis beyond quantum theory.