This preprint proves a sharp purity bound for arbitrary complex pure states in tensor products of antisymmetric pair spaces, across the physical cut consisting of the first system from each pair versus the second system from each pair. For every finite number m of pairs and local dimensions at least two, purity is at most 2^(-m) and entanglement entropy is at least m bits, with equality characterized by products of normalized decomposable two-forms. It derives entanglement of formation m for m copies of the normalized antisymmetric state and, using the published regularization theorem, entanglement cost one ebit for every local dimension at least two. The archive provides original formal sources and a precise account of their scope; it does not claim a complete Lean translation of every ordinary LOCC execution. Research directed and curated by Matthew Liston. The mathematics, proofs, formalization and manuscript preparation substantially involve AI systems; paper-specific attribution and checking limits are disclosed in the paper and archive. No independent human proof verification, external endorsement or peer review is claimed. Licensing: original manuscript and documentation CC BY 4.0; original code MIT. The inherited QICLean SchattenNorm helper retains Apache 2.0 and its original attribution. The archive provides the exact file-level license mapping. Manuscript revision: 22 October 2026. Research director's note — Matthew Liston: This project is motivated by philosophical exploration and curiosity about AI-assisted research. My aim in sharing these results is to make them available for scrutiny, correction and further development, so that others can build on whatever proves useful to mathematics and science. Correspondence: atemporalwalk@gmail.com.
This preprint proves a sharp purity bound for arbitrary complex pure states in tensor products of antisymmetric pair spaces, across the physical cut consisting of the first system from each pair versus the second system from each pair. For every finite number m of pairs and local dimensions at least two, purity is at most 2^(-m) and entanglement entropy is at least m bits, with equality characterized by products of normalized decomposable two-forms. It derives entanglement of formation m for m copies of the normalized antisymmetric state and, using the published regularization theorem, entanglement cost one ebit for every local dimension at least two. The archive provides original formal sources and a precise account of their scope; it does not claim a complete Lean translation of every ordinary LOCC execution. Research directed and curated by Matthew Liston. The mathematics, proofs, formalization and manuscript preparation substantially involve AI systems; paper-specific attribution and checking limits are disclosed in the paper and archive. No independent human proof verification, external endorsement or peer review is claimed. Licensing: original manuscript and documentation CC BY 4.0; original code MIT. The inherited QICLean SchattenNorm helper retains Apache 2.0 and its original attribution. The archive provides the exact file-level license mapping. Manuscript revision: 22 October 2026. Research director's note — Matthew Liston: This project is motivated by philosophical exploration and curiosity about AI-assisted research. My aim in sharing these results is to make them available for scrutiny, correction and further development, so that others can build on whatever proves useful to mathematics and science. Correspondence: atemporalwalk@gmail.com.
Entanglement cost is the asymptotic rate of Bell pairs required to prepare a quantum state. Its regularized definition has made exact evaluation difficult, even for isotropic states (i.e., depolarized maximally entangled states). In this work, we determine the entanglement cost of every qubit isotropic state and, more generally, every two-qubit Bell-diagonal state. Our proof uses a family of suitably tuned qubit semigroups to transform a general log-Sobolev entropy bound into supporting lines of Wootters' function. A recent exact tensorization theorem of Dong et al. [arXiv:2606.17729] extends these bounds to arbitrarily many copies, yielding a general lower bound on entanglement cost. Matching this bound with the entanglement of formation also determines the exact cost for a broader class of two-qubit states. For qudit isotropic states, we derive another general lower bound from the full depolarizing 2→3 norm. This bound substantially improves on the PPT-relative entropy of entanglement and nearly matches the entanglement of formation, leaving a gap of at most 2% of logd that vanishes as d grows. Finally, the exact qubit results and qudit bounds extend to the entanglement cost of preparing quantum depolarizing channels under both parallel and sequential strategies.
We determine the preshared entanglement required for spatially separated parties, restricted to local operations and classical communication, to attain globally optimal probabilistic two-copy purification of arbitrary bipartite pure states under depolarizing noise. In every local dimension d≥2, one shared maximally entangled qubit pair suffices: local controlled-SWAP operations exactly reproduce the globally optimal successful transformation. Conversely, any finite-dimensional preshared resource state that attains the same benchmark, even under a positive-partial-transpose relaxation, must have entanglement of formation at least one ebit. For pure resources with one ebit, or for two-qubit resources including mixed states, attaining the benchmark is possible only for states with exactly two nonzero Schmidt weights, both equal to 1/2, namely those locally unitarily equivalent to a maximally entangled qubit pair. These findings provide a benchmark for evaluating the entanglement demands of noise management in quantum networks and modular quantum computers.
This preprint studies learning an unknown bounded-degree n-qubit Hamiltonian in a known Pauli basis from independent copies of a single-temperature Gibbs state. For fixed locality, interaction degree and fixed positive rational inverse temperature and coefficient bound, it gives a polynomial-resource algorithm using product single-qubit Pauli measurements, without geometry, a gap, correlation decay or a minimum nonzero coefficient. The output has rational coefficients, support inside the true support and total coefficient error at most the requested tolerance with the stated success probability. The temperature-dependent exponents are extremely large; the result is asymptotic. The refinement certificate is formalized in Lean conditionally on explicit analytic inputs. Initialization, iteration, solver, sampling and full resource bounds remain paper proofs. A companion source note preserves the separate hidden-Majorana discussion and its open calibration problem; it is not a proof of that broader learner. Research directed and curated by Matthew Liston. The mathematics, proofs, formalization and manuscript preparation substantially involve AI systems; paper-specific attribution and checking limits are disclosed in the paper and archive. No independent human proof verification, external endorsement or peer review is claimed. Licensing: original manuscript and documentation CC BY 4.0; original code MIT. The archive provides the exact file-level license mapping. Manuscript revision: 22 October 2026. Research director's note — Matthew Liston: This project is motivated by philosophical exploration and curiosity about AI-assisted research. My aim in sharing these results is to make them available for scrutiny, correction and further development, so that others can build on whatever proves useful to mathematics and science. Correspondence: atemporalwalk@gmail.com.
Multipartite entanglement is the premier resource for quantum technologies. Yet, its exact quantification in the laboratory is notoriously challenging, typically requiring the full knowledge of high dimensional quantum states. Here, we construct observable bounds to multipartite entanglement for systems of arbitrary size, which are functions of the local and global state purities, and correlation functions. First, we derive experimentally accessible upper and lower limits to both the bipartite entanglement of formation and the squashed entanglement of bipartite systems, by leveraging cornerstone results of quantum information theory: the entropy strong subadditivity inequality and the Koashi-Winter monogamy relation. Then, we convert them into bounds to the entanglement up to degree k for arbitrary states, and to the genuine k-partite entanglement, by employing a recently proposed method. Finally, we analytically and numerically test these results, by bounding the multipartite entanglement of several relevant states and mixtures, including the important classes of GHZ, Dicke, W states, and random pure states.
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.
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.