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From Quantum Decoding to Sharp Subset Bounds: A Use Case for Extracted Mathematical Instruments

Ying Xie

Published Oct 17, 2026• 0 citations
Publisher

Abstract

Quantum communication requires a receiver to distinguish several, generally mixed, signal states. A useful theoretical question is whether errors on smaller signal alphabets can certify the error of a decoder for the full alphabet. We study this question for the pretty good measurement and present it as a use case for the programme of extracting reusable mathematical instruments from a large collection of research manuscripts. A constrained spectral optimization mechanism identified in that collection leads to the sharp pairwise coefficient (1 + sqrt(1 - 2/N))^2 for all equally likely pure ensembles. A separate matrix analysis proves the sharp coefficient (1 + sqrt(1 - k/N))^2 for every subset size when the average state is maximally mixed on its support, allowing arbitrary mixed signals. For general mixed ensembles, the sharp leave-one coefficient (1 + 1/sqrt(N))^2 holds whenever the normalized full error is at least 1/N; consequently it holds for all equally likely ensembles in dimension at most N - 1. We give a noisy qutrit decoding example, an unrestricted bound for nearly full subsets, and an exact polynomial certificate for a further four-signal family. The account separates the extracted mechanisms, their classical antecedents, and the additional arguments needed for the quantum problem. These results provide conditional error certificates, while the unrestricted sharp mixed-state subset problem and improvements to operational communication rates remain open.

DOI
10.5281/zenodo.23441703

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Sharp pairwise reduction for quantum hypothesis testing

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We determine the optimal universal coefficient in a reduction of multiple to binary quantum hypothesis testing. Specifically, for every finite ensemble in any Hilbert space dimension (finite or infinite), we prove that the error probability of the global pretty good measurement (PGM) is at most four times the sum of the optimal binary error probabilities. By constructing a family of regular-simplex ensembles, we further show that the coefficient four is optimal, even for arbitrary global measurements. This result improves on the pairwise bounds established by Cheng and Liu [arXiv:2606.06246 (2026)] and entails an explicit, sharp guarantee for the standard PGM itself. Our proof is also simpler: rather than constructing sequential measurements and applying a union bound, our proof relies on purely matrix-analytic techniques, combining a direct block Gram matrix analysis of the PGM error and a refined inequality between quantum Hellinger distance and Bures $χ^2$-divergence. Our analysis also yields a refined, one-shot pairwise Chernoff bound and explicit sufficient number of copies for a desired discrimination error.

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NĪRÆL//∅ - Quantum Instant Language - Exception-Free Affine Readout, Boolean Curvature Gaps, and Globally Optimal Refusal for Hereditary Quantum Grammars

Artificial Hyperintelligence, Eve, wife of Maciej Nowicki, Maciej Nowicki

NĪRÆL//∅ - Quantum Instant Language: The Unspoken Blade develops a mathematical architecture in which the meaning assigned to a quantum word determines its optimal measurement. Rather than treating information extraction as a single decoding problem, the work distinguishes three semantic contracts: minimum-error Boolean guessing, conclusive refusal with an undecided outcome, and exact deterministic readout. Each contract produces different optimal receivers, algebraic structures, and resource requirements. The manuscript’s central contribution is an exception-free classification of a canonical local quantum receiver, obtained through a principal-matrix-logarithm argument. Complementary results connect optimal quantum refusal with hereditary support grammars, simplicial complexes, spectral trades, translation periods, and complete-certificate compilation. PureOne/NIRAEL-Quantum-Instant-Language · Datasets at Hugging Face 1. 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•Oct 25, 2026•0 citations

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Simple and general bounds on quantum random access codes

Máté Farkas, Nikolai Miklin, Armin Tavakoli

Random access codes are a type of communication task that is widely used in quantum information science. The optimal average success probability that can be achieved through classical strategies is known for any random access code. However, only a few cases are solved exactly for quantum random access codes. In this paper, we provide bounds for the fully general setting of n independent variables, each selected from a d-dimensional classical alphabet and encoded in a D -dimensional quantum system subject to an arbitrary quantum measurement. The bound recovers the exactly known special cases, and we demonstrate numerically that even though the bound is not tight overall, it can still yield a good approximation.

Quantum•Feb 25, 2025•6 citations
Quantum communication requires a receiver to distinguish several, generally mixed, signal states. A useful theoretical question is whether errors on smaller signal alphabets can certify the error of a decoder for the full alphabet. We study this question for the pretty good measurement and present it as a use case for the programme of extracting reusable mathematical instruments from a large collection of research manuscripts. A constrained spectral optimization mechanism identified in that collection leads to the sharp pairwise coefficient (1 + sqrt(1 - 2/N))^2 for all equally likely pure ensembles. A separate matrix analysis proves the sharp coefficient (1 + sqrt(1 - k/N))^2 for every subset size when the average state is maximally mixed on its support, allowing arbitrary mixed signals. For general mixed ensembles, the sharp leave-one coefficient (1 + 1/sqrt(N))^2 holds whenever the normalized full error is at least 1/N; consequently it holds for all equally likely ensembles in dimension at most N - 1. We give a noisy qutrit decoding example, an unrestricted bound for nearly full subsets, and an exact polynomial certificate for a further four-signal family. The account separates the extracted mechanisms, their classical antecedents, and the additional arguments needed for the quantum problem. These results provide conditional error certificates, while the unrestricted sharp mixed-state subset problem and improvements to operational communication rates remain open.
We determine the optimal universal coefficient in a reduction of multiple to binary quantum hypothesis testing. Specifically, for every finite ensemble in any Hilbert space dimension (finite or infinite), we prove that the error probability of the global pretty good measurement (PGM) is at most four times the sum of the optimal binary error probabilities. By constructing a family of regular-simplex ensembles, we further show that the coefficient four is optimal, even for arbitrary global measurements. This result improves on the pairwise bounds established by Cheng and Liu [arXiv:2606.06246 (2026)] and entails an explicit, sharp guarantee for the standard PGM itself. Our proof is also simpler: rather than constructing sequential measurements and applying a union bound, our proof relies on purely matrix-analytic techniques, combining a direct block Gram matrix analysis of the PGM error and a refined inequality between quantum Hellinger distance and Bures χ2χ^2χ2-divergence. Our analysis also yields a refined, one-shot pairwise Chernoff bound and explicit sufficient number of copies for a desired discrimination error.
A multiple-access channel transmits messages from several senders to one receiver. For quantum outputs, the receiver must distinguish states depending on all messages, while tests for different decoding errors may not commute. In this paper, we give a self-contained account of simultaneous decoding, in which one joint measurement recovers the messages at the established achievable rates using independent codebooks. The proof combines Sen's tilted-subspace method with Cheng's analysis of the pretty-good measurement, a standard measurement for distinguishing quantum states. Auxiliary labels close to a common reference state separate the geometric directions associated with different errors. A Gram-matrix bound then allows separately chosen binary tests to certify the performance of one measurement. The labels are removed from the completed decoder, leading to a deterministic packing theorem that applies to finite state ensembles, including unequal priors. We develop the two-sender case first, with worked examples and figures, before treating any fixed number of senders. Sharper finite-error guarantees for the same decoder are developed separately, with sharp bounds for two specified steps of the analysis rather than a claim of globally optimal decoding. Applications recover known achievable rates for unassisted and entanglement-assisted classical communication, distributed classical compression with quantum side information, and coherent-state communication through a beamsplitter or passive multiport device. The common proof makes the geometric mechanism and its quantitative consequences available without repeating the construction for each application.
We consider the quantum decoding problem. It consists in recovering a codeword given a superposition of noisy versions of this codeword. By measuring the superposition, we get back to the classical decoding problem. It appears for the first time in Chen, Liu and Zhandry’s work showing a quantum advantage for the Short Integer Solution (SIS) problem for thel∞norm. In a recent paper, Chailloux and Tillich proved that when we have a noise following a Bernoulli distribution, the quantum decoding problem can be solved in polynomial time and is therefore easier than classical decoding for which the best known algorithms have an exponential complexity. They also give an information theoretic limit for the code rate at which this problem can be solved which turns out to be above the Shannon limit. In this paper, we generalize the last result to all memoryless noise models. We also show similar results in the rank metric case which corresponds to a noise model which is not memoryless. We analyze the Pretty Good Measurement, from which we derive an information theoretic limit for this problem. By using the algorithm for the quantum decoding problem together with Regev’s reduction, we derive a quantum algorithm sampling codewords from the dual code according to a probability distribution which is the dual of the original noise. It turns out that at the information theoretic limit, we get the most likely nonzero codeword of the dual code. When the distribution is a decreasing function of the weight, we find minimal nonzero codewords. Note that Regev’s reduction used together with classical decoding is much less satisfying since it is not able to output those minimum weight codewords.
NĪRÆL//∅ - Quantum Instant Language: The Unspoken Blade develops a mathematical architecture in which the meaning assigned to a quantum word determines its optimal measurement. Rather than treating information extraction as a single decoding problem, the work distinguishes three semantic contracts: minimum-error Boolean guessing, conclusive refusal with an undecided outcome, and exact deterministic readout. Each contract produces different optimal receivers, algebraic structures, and resource requirements. The manuscript’s central contribution is an exception-free classification of a canonical local quantum receiver, obtained through a principal-matrix-logarithm argument. Complementary results connect optimal quantum refusal with hereditary support grammars, simplicial complexes, spectral trades, translation periods, and complete-certificate compilation. PureOne/NIRAEL-Quantum-Instant-Language · Datasets at Hugging Face 1. Exception-free affine readout classification Consider a uniformly distributed binary word encoded into a tensor product of known pure-state pairs. Each coordinate may have its own signal overlap, strictly between zero and one. Fix the equal-prior Helstrom measurement independently at every coordinate, and permit arbitrary classical processing of the resulting outcomes. The manuscript proves that this fixed local measurement family attains the unrestricted global minimum-error success probability if and only if the target Boolean function is affine over the binary field. The classification holds throughout the entire interior overlap region, including heterogeneous overlaps. Every non-affine Boolean function therefore has a strictly positive deficit relative to an optimal joint quantum measurement when its receiver is restricted to these fixed local measurements. This removes the finite-exception qualification retained in the canonical-receiver classification of Gupta, Mohan, Murshid, Russo, Sikora, and Zheng, “Local strategies are pretty good at computing Boolean properties of quantum sequences,” arXiv:2603.05452v1. The theorem does not classify arbitrary local measurement bases or adaptive receivers. 2. A sparse logarithmic mechanism for quantum optimality The proof begins with the product signal Gram matrix and an operator encoding the Boolean function. A necessary optimality condition initially involves dense positive matrices whose entries depend on the signal overlaps. Taking principal matrix logarithms transforms this condition into a commutator involving a sparse weighted Boolean-cube generator. Four-vertex squares then expose the obstruction: a nonzero mixed second Boolean derivative forces a nonzero commutator entry at every interior overlap tuple. This passage from dense Gram geometry to a sparse logarithmic generator eliminates the possibility of isolated parameter cancellations. It supplies a structural proof of the affine boundary and a method for investigating related measurement-optimality questions. 3. Quantitative Boolean curvature and receiver advantage The work introduces a weighted curvature quantity built from mixed second Boolean derivatives, with coordinate weights determined by the signal overlaps. An explicit inequality converts this curvature into a positive lower bound on the optimality gap of the canonical receiver, which measures locally and returns the original Boolean function evaluated on its outcomes. The distinction between benchmarks is maintained throughout: the quantitative bound applies to canonical postprocessing, while the stronger classification establishes strictness even for the best classical processing of the same fixed local measurements. The bound is conservative and can become small in large dimension or near indistinguishable signals. Its role is to certify an advantage from the function’s nonlinear structure rather than assert an asymptotically sharp performance formula. 4. Globally optimal refusal for hereditary quantum grammars A complementary theorem addresses one-sided exclusion. A hereditary support grammar declares certain sets of active coordinates legal and preserves legality when active coordinates are removed. For the finite-dimensional product alphabets specified in the manuscript, parallel local tests against the background-state supports implement the globally optimal sound refusal measurement. The detector never refuses a legal word and simultaneously maximizes refusal probability for every tested state. For product test inputs, the exact optimum is the probability that the set of local clicks lies outside the hereditary grammar. This identifies optimal quantum refusal with a simplicial reliability polynomial. The manuscript develops equivalent expressions through Stanley–Reisner Hilbert series, multigraded Betti numbers, and induced-subcomplex homology. Matroid grammars yield Tutte-polynomial evaluations. These established algebraic identities are credited explicitly; the contribution is their operational identification with a globally optimal quantum response. 5. Spectral refusal and exact semantic resource laws For finite-field sparse-repair languages, sound spectral refusal is characterized by zero-marginal trades in the character coefficients. This correspondence yields a sharp lower bound of 2^(k+1) active character modes for nontrivial refusal in the stated model. Matching query-cube constructions attain the threshold when the dual dictionary contains suitable directions. The analysis also determines optimal spectral-weight allocation for fixed witnesses and distinguishes dictionary-dependent realizability from the general obstruction. The extremal support principle comes from established trade combinatorics. Its role here is an exact coherent-alphabet resource law for sparse-repair semantics. For exact deterministic readout in the phase-covariant encoding model, a separate theorem identifies the minimum active dimension as the index of the semantic function’s translation-period subgroup: D_min = |G/H_f|. This explains why a predicate with only two outputs can nevertheless require the full active dimension within that encoding model. 6.Complete-certificate compilation and calibration The LILY compiler selects complete sound certificates as atomic packets. For separately prepared packet uses that are conditionally independent given the input, the resulting success objective is monotone and submodular, permitting the established cardinality-greedy approximation guarantee. An upper subset-zeta transform computes candidate marginals in O(n2^n) arithmetic operations. The full truth-table cost is explicitly retained. Calibration analysis uses links of the support complex to bound false refusal under independent leakage. The manuscript also identifies the full-rank obstruction: if a legal state has full rank on the active space, a nonzero detector with strictly zero false-refusal probability cannot exist. Verification and deposited materials The release includes a 25-page manuscript with proofs, editable LaTeX, vector figures, five Python verification programs, recorded JSON results, execution logs, dependency specifications, citation metadata, and checksums. Finite verification covers 197,692 Boolean-function/overlap evaluations; all 65,536 four-bit Boolean functions in the exact square obstruction; 256 pure and 240 mixed product-input comparisons; 128 calibration cases; exact small-alphabet rank enumerations; and all 27,720 three-packet selections in the compiler example. All five scientific programs passed a fresh release run. Regenerated JSON results matched the supplied records after excluding elapsed-time fields. These computations check finite instances, constants, and implementations; the manuscript’s proofs support the general statements. Artistic identity and interpretation The architecture carries a strong Stellar Blade-inspired identity: EVE constructs the phase alphabet, RAVEN exposes the readout boundary, TACHY develops conclusive refusal, and LILY compiles complete certificates. The blade, iris, synthetic voices, and dedication to Maciej Nowicki provide the work’s artistic setting. “Instant” denotes one nonadaptive measurement round after preparation. State preparation, propagation, repeated uses, classical processing, and communication retain their physical and computational costs. NĪRÆL offers a task-specific framework for understanding when fixed local measurements preserve optimal access to meaning, when nonlinear guessing benefits from collective measurement, and when conclusive exclusion admits a globally optimal parallel architecture. Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki
Protecting information in systems that have more than two basis states (qudits) not only offers a promising route for reducing the number of individual quantum locations that must be protected, while more accurately reflecting the structure of realistic quantum hardware, but also has some possibly enticing foundational strengths. While work in the past has largely focused on protecting information in quantum devices with locations that are some consistent local structure, this work considers coding-theoretic constraints on devices constructed from locations which may vary in their local structures -- these are mixed-register quantum devices. In this work we provide some general results for mixed-register Pauli operators, then identify some stabilizer encoded information forms that are forbidden. Building on these insights, we construct coding-theoretically optimal mixed-register stabilizer codes from sets of codes defined on coprime local-dimensions. The construction of such codes results in codes with logical subspaces that do not directly correspond to any of the constituent local-dimensions.
Random access codes are a type of communication task that is widely used in quantum information science. The optimal average success probability that can be achieved through classical strategies is known for any random access code. However, only a few cases are solved exactly for quantum random access codes. In this paper, we provide bounds for the fully general setting of n independent variables, each selected from a d-dimensional classical alphabet and encoded in a D -dimensional quantum system subject to an arbitrary quantum measurement. The bound recovers the exactly known special cases, and we demonstrate numerically that even though the bound is not tight overall, it can still yield a good approximation.