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Ordered-Angle Coding for Exact Multiuser Unanimity Testing

Luis Adrián Lizama-Pérez

arXiv:2609.05645v1quant-phcs.CR

TL;DR

The paper asks whether scheduled common-angle rotations can finitely test unanimity among serial transformation-only users. It develops ordered-angle codes and proves exact discrimination within the deterministic-unanimity pulse family precisely for power-of-two user counts, while bounding calibration and readout errors.

  • Problem

    The paper asks whether scheduled common rotation magnitudes can form a finite relation code when users keep one private sign across a logical word.

    Source: §1 · p. 2

  • Method

    It derives a mixed-weight imitation law for nonadaptive words of deterministic-unanimity same-axis pulses and analyzes angular offsets and binary readout errors.

    Source: §6 · p. 8, §9.1 · p. 13

  • Results

    Within that pulse family, finite perfect words exist exactly for power-of-two n, with the dyadic word pulse-minimal at log2 n trials; odd primes admit balanced statistical words.

    Source: §1 · p. 2, §18 · p. 23

  • Takeaways & Limitations

    The exactness and minimality result is a code-design statement for the deterministic-unanimity Bell-readout family, not for arbitrary real-angle schedules or adaptive measurements.

    Source: §17 · p. 22

    The algebraic formulas are conditioned on valid binary classifications, while setting-dependent or adversarially controlled loss remains outside the security claim.

    Source: §16 · p. 22

Abstract

from arXiv · show

We introduce ordered-angle coding for binary-unanimity testing among $n$ transformation-only users in a serial quantum architecture inherited from Loop-Back communication. User $B_i$ fixes one private sign across a logical word and applies $R(s_i\alpha_j)$ in trial $j$. If $w$ users choose the negative sign, serial composition gives $R[(n-2w)\alpha_j]$. We show that every fixed same-axis pulse that is deterministic for both unanimous inputs under the binary Bell readout has $\alpha_j=q_j\pi/(2n)$ with integer $q_j$. For a nonadaptive word $\mathbf q=(q_1,\ldots,q_m)$, a mixed Hamming weight imitates the unanimous signature with probability \[ M_{\mathbf q}(w)=\prod_{j=1}^{m}\cos^2\!\left(\frac{\pi q_jw}{n}\right). \] Within this complete deterministic-unanimity pulse family, a finite perfect word exists if and only if $n$ is a power of two. For $n=2^r$, the dyadic word $(1,2,4,\ldots,2^{r-1})$ is exact and pulse-minimal with $m_{\min}=r=\log_2n$ trials. It replaces the $O(n^2\log(1/\varepsilon))$ worst-case burden of repeated smallest-angle tests by exact $O(\log n)$ verification. Odd primes instead admit balanced statistical words with uniform mixed-weight imitation $2^{-(p-1)}$. For actual operations $R(s_i\alpha_j+\delta_{ij})$, an ideal rejecting position is lifted to $\sin^2\Delta_j$, where $\Delta_j=\sum_i\delta_{ij}$. This yields explicit robustness bounds with angular and binary-readout errors. Bell entanglement is not required for the additive algebra, but it keeps the traveler locally maximally mixed in every honest trial. We therefore present the result as a coded multiuser relation primitive with optional raw conference-key-candidate use, not as a composably secure conference-key protocol.

1 Introduction

The paper asks whether scheduled common rotation magnitudes can encode a finite test of unanimity among serial transformation-only users. Within the complete deterministic-unanimity pulse family, it establishes exact power-of-two codes and distinguishes this relation primitive from a security claim.

2.1 Multiparty key agreement and restricted-user architectures

  • The serial architecture sends one Bell traveler through users who apply prescribed transformations, with Alice retaining the reference and making the final observation.Transformation-only operation is an architectural constraint, not a security theorem; related multiparty key-generation and restricted-user approaches are established [10] [11] [12] [13] [15] [16] [17] [18].Source: §1 · p. 1, §2.1 · p. 3

Main contributions

  • Ordered-angle coding schedules common rotation magnitudes across trials while each user keeps one private sign, so different positions probe mixed Hamming-weight classes.The work derives a word-imitation law and characterizes deterministic-unanimity pulses as α = qπ/(2n), with integer q.Source: §1 · p. 2, Main contributions · p. 2

1 Introduction

  • For n = 2^r, the dyadic word is exact and pulse-minimal at r = log2 n trials, replacing repeated smallest-angle tests that require O(n^2 log(1/ε)) trials.Finite exact discrimination within the deterministic-unanimity same-axis pulse family is possible precisely when n is a power of two; odd-prime counts admit balanced statistical words.Source: §1 · p. 2

Main contributions

  • The contribution is a coded multiuser relation primitive, not a composably secure conference-key protocol.Bell entanglement is separated from the codeword algebra, and active intermediate attacks remain outside the claim.Source: Main contributions · p. 2, §1 · p. 2

2.2 Rotation-based equality and quantum private comparison

  • The novelty is a position-labelled common-angle word whose scheduled magnitudes produce the product law and the exactness and minimality results, not rotation-based equality testing itself.Earlier Bell- and rotation-based comparison approaches include.Source: §2.2 · p. 3

2.3 Hamming-weight computation with one-qubit rotations

  • Summed rotations and Hamming-weight-dependent one-qubit processing have prior precedents [28], but this scheme uses a sequential traveler and fresh Bell-reference/complement tests.Its exactness and pulse-minimality claims concern this deterministic-unanimity readout family, not arbitrary one-qubit algorithms.Source: §2.3 · pp. 3–4

2.4 Pulse sequences and the Loop-Back lineage

  • The contribution is not using multiple pulses, but replacing repeated or filtered trials with complementary angular queries whose ordered signature is the logical observable.Pulse-sequence equality tests also exist in simultaneous-message interference protocols.Source: §2.4 · p. 4

3 Bell-reference model and rotation convention

  • The Bell reference is sampled privately for each trial, while all remote users share an axis and rotation magnitude within that trial.The magnitude may change between trials; invalid loss or occupancy events are discarded and probabilities are conditioned on valid binary classifications.Source: §3 · pp. 4–5, Figure 1 · p. 5

4 Why a Bell traveler? Algebraic equivalence and information geometry

Same-axis serial composition reduces the input pattern to its Hamming weight before measurement, independently of whether the reference is Bell-entangled. The Bell traveler instead changes the local information geometry by remaining locally maximally mixed.

4.1 Same-axis composition is carrier-independent

  • Same-axis rotations compose additively, so the accumulated rotation depends on the number of negative signs, a reduction that occurs before any Bell-specific measurement.Pure references with ⟨Y⟩ = 0 reproduce the same ideal overlap law and probability-level codeword algebra.Source: §4.1 · p. 6

4.2 What Bell changes

  • A traveler alone reveals no local Bloch direction in an honest Bell trial; collective rotations become observable when Alice reunites it with her retained subsystem and private reference.The Bell resource therefore contributes information geometry rather than the additive rotation algebra.Source: §4 · p. 5, §4.2 · p. 6
  • Randomized pure references remain rank one in each realized trial, so repeated route access can reveal relative changes absent from a single-point averaged state.Local maximal mixing is stronger than ensemble masking, but does not prove secrecy against a coherent probe that stores an ancilla across route segments.Source: §4.2 · p. 6

5 Single-pulse n-user family

A single deterministic-unanimity pulse gives both unanimous inputs the same binary signature, while mixed inputs can still imitate it. The complete pulse family is discretized by integer coefficients, and its smallest angle has a costly worst-case repetition burden.

5 Single-pulse n-user family

  • For a mixed weight w, the single-pulse reference probability is cos²[(n − 2w)α], with the complement probability equal to its complement.The effective rotation angle follows from the number of negative signs among users.Source: §5 · pp. 6–7
  • The smallest positive deterministic angle makes both unanimous branches deterministic and channel-identical, but a mixed Hamming weight can still yield nonzero soundness error.The worst mixed cases are w = 1 or n − 1.Source: §5 · p. 7
  • Repeated use of the smallest pulse requires O(n² log(1/ε)) trials to suppress the worst-case mixed imitation below a fixed target ε.For n = 2, the redesigned deterministic angle is π/4, distinct from the probabilistic α = π/8 operating point of the two-user predecessor [6].Source: §5 · p. 7
  • Every same-axis pulse deterministic for both unanimous signs has α = qπ/(2n) for integer q, and the unanimous effective operations agree up to global phase.Thus the integer coefficients parameterize the complete deterministic-unanimity pulse family rather than an arbitrary discretization.Source: §5 · p. 7

6 Two distinct uses of multiple Bell trials

The paper distinguishes repeating one angle to accumulate evidence from coding a sequence of angles that tests different mixed Hamming-weight classes. It then gives a word-imitation law and separates fixed-input discrimination from accepted-block mismatch under a prior.

6 Two distinct uses of multiple Bell trials

  • An ordered-angle codeword changes the common angle across trials to reject different mixed Hamming-weight classes, unlike repetition, which raises the same likelihood to a higher power.The same private input signs are retained throughout the word; the angle schedule changes between trials.Source: §6 · p. 8, Figure 2 · p. 8

6.1 Mode A: same-angle repetition

  • Repeated q = 1 trials add independent evidence without measuring anything new, while for n ≥ 4 angle diversity can ask different collective questions at the same pulse budget.For n = 3, q = 1 repetition is already the natural statistical amplifier.Source: §6.1 · p. 8

6.2 Mode B: angle-diverse codewords

  • For n = 4, two q = 1 trials leave worst mixed imitation at 1/4, whereas q = (1, 2) rejects every mixed weight in the ideal model.The two-position code uses signature (C, N).Source: §6.2 · p. 9

7 General ordered-angle codeword theory

  • For independent Bell trials, a mixed Hamming weight w imitates the full unanimity signature with probability given by the product of its per-trial match probabilities.The design problem selects an angle schedule subject to a pulse budget and implementation constraints.Source: §7 · p. 9

7.1 Aggregate acceptance and accepted-block mismatch

  • Fixed-weight imitation M_q(w) measures discrimination for one mixed configuration, while ε(q) also accounts for the larger number of mixed strings under the input prior.These quantities describe different questions and should not be conflated.Source: §7.1 · p. 10

8 Exact and pulse-minimal family for powers of two

Within the complete deterministic-unanimity same-axis pulse family and finite nonadaptive schedules, exact discrimination is possible precisely for power-of-two user counts. The dyadic word is pulse-minimal, with four- and eight-user examples illustrating its rejection pattern.

8 Exact and pulse-minimal family for powers of two

  • Finite exact words exist if and only if n = 2^r; for such n, the dyadic word uses r pulses and is pulse-minimal.Each pulse eliminates one 2-adic valuation class of mixed Hamming weights.Source: §8 · pp. 10–11
  • For n not a power of two, a mixed weight has nonzero imitation probability in every allowed trial; the impossibility result is limited to this pulse family and nonadaptive finite schedules.It does not cover arbitrary real-angle schedules with nondeterministic positions, adaptive measurements, or higher-dimensional resources.Source: §8 · pp. 10–11

8.1 Worked example: four users

  • For four users, two repeated π/8 pulses leave worst-case imitation at 1/4, while replacing the second with π/4 makes the two-pulse test deterministic.Under independent unbiased input bits, q = (1, 2) has zero intrinsic mixed fraction among accepted blocks.Source: §8.1 · p. 11

8.2 Worked example: eight users

  • For eight users, the word (1, 2, 4) rejects every mixed Hamming weight with certainty in three Bell trials.The pulses eliminate, respectively, the remaining valuation classes described in the section.Source: §8.2 · p. 12

9 Calibration and binary-readout robustness of exact words

The robustness analysis models additive per-user angular offsets and symmetric binary-readout flips for exact words. It derives conservative acceptance bounds and shows that the calibration tolerance tightens approximately as 1/n at a fixed discrimination target.

9.1 Additive per-user angular offsets

  • At an ideal rejecting position, additive offsets lift the exact zero match probability to sin^2 Δj, with Δj the sum of per-user offsets in that trial.The lift is quadratic for small accumulated angular error.Source: §9.1 · p. 13

9.2 Symmetric binary classification error and contrast

  • A symmetric binary-classification flip with probability ηj maps pre-readout match probability eμj to ηj + (1 − 2ηj)eμj.The model conditions on a valid two-photon event and does not assume a particular detector implementation.Source: §9.2 · p. 13

9 Calibration and binary-readout robustness of exact words

  • For an ideal exact word, explicit mixed-input and unanimous-input robustness bounds hold under independent valid trials, |δij| ≤ δmax, ηj ≤ ηmax < 1/2, and nδmax ≤ π/4.The bounds are conservative because they allow all user offsets in one trial to add coherently.Source: §9.2 · pp. 13–14

9.3 Numerical tolerance scale

  • With ideal binary readout, the conservative per-user calibration tolerance for sin^2(nδmax) < 1% tightens approximately as 1/n at a fixed discrimination target.The conventional Bloch-sphere angle tolerance is twice the amplitude-space value.Source: §9.3 · p. 14, Table 4 · p. 14, Figure 4 · p. 15

10 Balanced statistical words for odd prime user counts

Balanced words for odd prime user counts give every mixed Hamming weight the same imitation probability, and repeating the complete word amplifies the statistical test. The five-user example shows that angle diversity can reduce accepted-block contamination at a fixed trial budget.

10 Balanced statistical words for odd prime user counts

  • The balanced odd-prime word gives every mixed weight w = 1,...,p−1 the same probability of imitating the unanimous signature.The proof uses that multiplication by a nonzero residue permutes nonzero residues modulo p, with squared-cosine symmetry.Source: §10 · pp. 15–16

10.1 Repeated prime words

  • For p = 3, one prime word is a single α = π/6 Bell trial, and repetition gives ϵ3,r = 3/(4r + 3).For larger primes, the construction uses angle-diverse balanced words.Source: §10.1 · p. 16

10.2 Worked example: five users

  • At equal Bell-trial budgets, the five-user comparison contrasts q = 1 repetition with repetitions of the designed word (1, 2).The table caption identifies the compared schedules; the reported contamination values show the two-word improvement.Source: Table 5 · p. 16, §10.2 · p. 17
  • For five users, accepted-block mixed fraction falls from 15/31 (≈0.4839) after one (1, 2) word to 15/271 (≈0.05535) after two complete words.A third complete word, using six Bell trials, gives ϵ5,3 ≈0.00365.Source: §10.2 · p. 17
  • Figure 5 and Table 6 summarize prime-word scaling; larger primes have more mixed strings, leaving one word with aggregate error near one half, while repetition suppresses contamination.Table 6 reports aggregate mixed fraction among accepted blocks under independent unbiased input bits.Source: Figure 5 · p. 17, Table 6 · p. 17, §10.2 · p. 17

11 Composite user counts with odd factors

For n with an odd factor, a mixed weight creates an arithmetic obstruction to any finite perfect word in the deterministic-unanimity pulse family. Finite-budget schedules can still eliminate some classes exactly and suppress others statistically.

11 Composite user counts with odd factors

  • Writing n = 2^r d with odd d > 1, the mixed weight w = 2^r gives qw/n = q/d for every allowed coefficient, which cannot be half-integer.Thus no allowed pulse can annihilate that weight exactly, so no finite perfect word exists for such n.Source: §11 · p. 17
  • The obstruction applies to every n with an odd factor, rather than reflecting an insufficient search over word lengths.Its basis is the odd denominator in q/d for the specified mixed weight.Source: §11 · p. 17
  • For n = 6, q = (1, 2, 3) eliminates several Hamming classes exactly and suppresses unavoidable classes statistically.The design target is finite-budget minimax error or its prior-weighted analogue, not a nonexistent perfect word.Source: §11 · pp. 17–18

12 Resource accounting and the meaning of gain

The gain from ordered angles concerns relation verification, not the prior probability of unanimous inputs. Exact words can reduce trials at a fixed discrimination target, while early abort trades resource savings for additional transcript information.

12 Resource accounting and the meaning of gain

  • Angle diversity can improve relation discrimination, but it does not change the probability that independently chosen user bits are all equal.The accounting separates relation discrimination, accepted logical blocks per input, and true common bits per physical Bell trial; the utilization factor is not a secret-key rate.Source: §12 · p. 18
  • The factorization used for resource accounting assumes setting-independent validity events; adversarial or setting-dependent loss requires separate analysis.Source: §12 · p. 18

12.1 Equal-budget interpretation

  • At the same worst-case discrimination target, the exact dyadic word uses 3 trials rather than 30 for n = 8 and 4 rather than 119 for n = 16.This is a relation-verification gain, not an increase in the prior probability that independently chosen inputs are unanimous.Source: §12.1 · p. 18

12.2 Sequential early abort

  • Early rejection can reduce valid Bell trials spent on mixed inputs, but the stopping position can reveal Hamming-weight-class information to Alice.It is optional; full-word execution can disclose only the final pass/fail decision, though Alice still retains per-position records.Source: §12.2 · p. 18

13 Protocol specification

The protocol tests whether users' fixed private signs are unanimous by applying a scheduled word of serial rotations and comparing a binary record with the known unanimous signature. Its exact relation and privacy claims have explicit boundaries: statistical words retain residual error, and the work does not establish a composably secure conference-key protocol.

13 Protocol specification

  • Each user keeps one private input sign across the word; Alice and the users schedule angles, run fresh Bell trials with serial rotations, and record N, C, or erasure.A valid fixed-length word passes only when its complete binary record matches the unanimous signature; erasures are not binary outcomes.Source: §13 · p. 19

14 Operational interpretation and optional conference-key use

  • For an exact power-of-two word, ideal acceptance is equivalent to unanimity; statistical words retain residual mixed fraction ϵ(q).After an exact pass, users may retain their matching local bit as a raw common-key candidate, but this is not a conference-key security proof.Source: §13 · p. 19, §14 · p. 19

15.1 Common-sign final-channel identity

  • For modeled returned qubits and Alice's retained ancilla, unanimous 0^n and 1^n induce identical final channels when Alice cannot access travelers between user operations.The scope excludes multiple side-channel and out-of-mode probes and does not cover intermediate access.Source: §15.1 · p. 20

15.2 Transcript information for nonunanimous inputs

  • For nonunanimous inputs, the N/C record can reveal Hamming-weight-class information, and early abort additionally reveals the first failing position.Fixed-length execution removes abort-time leakage but not information in the complete outcome word.Source: §15.2 · p. 20

15.3 Local maximal mixing

  • Bell references keep the traveling subsystem locally maximally mixed, whereas pure qubits with ry = 0 can reproduce the same ideal rotational statistics.Local maximal mixing is a resource property of the implementation, not the mathematical origin of the ordered-angle theorem, and does not itself prove security against coherent multi-segment probes.Source: §15.3 · pp. 20–21

15.4 Explicit boundaries

  • The work does not establish composable secrecy, malicious-user security, protection against intermediate probing or listed side channels, nonunanimous-class privacy, finite-key bounds, or device independence.Practical deployment therefore remains outside these security claims.Source: §15.4 · p. 21

16 Photonic binary Bell readout

The binary Bell readout reduces the inherited comparator to two outcomes plus erasure. The discussion and conclusion present the code’s exactness and robustness alongside its resource rationale and security boundaries.

16 Photonic binary Bell readout

  • The comparator separates singlet and triplet outcomes into a 1+3 filter; separate outputs encode N, same-output records C, and loss or unverified occupancy is ∅.The binary readout avoids resolving all four Bell labels.Source: §16 · p. 21, Figure 6 · p. 22

17 Discussion

  • Within the deterministic-unanimity same-axis pulse family, log2 n positions are necessary and sufficient; this does not establish impossibility for arbitrary angles, adaptive measurements, or higher-dimensional resources.Source: §17 · p. 22
  • A bounded per-user angle offset lifts an ideal rejecting zero to sin2 ∆j, while the conservative calibration tolerance tightens approximately as 1/n at fixed false-acceptance target.Source: §17 · p. 22
  • Bell references keep the traveler locally maximally mixed while the relation is read through global correlation, but the work does not claim composable conference-key security.An accepted exact word can provide a raw conference-key candidate; malicious-user fairness, active probing, adversarial loss, and finite-key post-processing remain outside the claim.Source: §17 · p. 22, §18 · p. 23
  • Ordered angles improve verification, not the prior probability of spontaneous unanimity, which remains 2^(1-n) for independent unbiased user bits.Source: §17 · p. 22

18 Conclusion

  • The conclusion states that a finite perfect word exists exactly for power-of-two n, with a pulse-minimal dyadic word; odd primes instead admit balanced statistical words.Composite counts with odd factors motivate finite-budget optimization.Source: §18 · p. 23

Data and code availability

A Derivation of the pulse-match probability

The appendix derives the pulse-match rule, supplies a trigonometric identity used for the prime word, and describes how early aborts determine expected trial use.

A Derivation of the pulse-match probability

B Trigonometric identity for the prime word

C General early-abort enumeration

  • A block reaches trial j only if all earlier trials matched; expected valid trials weight the two unanimous strings and mixed classes by their binomial multiplicities.Source: C General early-abort enumeration · p. 24
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