Exact balanced subsystem designs for Haar-purity fluctuations

Ryosuke Shiroshita

Published Sep 15, 2026• 0 citations

Abstract

We determine the minimum complex-Haar variance of the average purity over exactly n distinct balanced bipartitions of an n-party pure state, identifying complementary subsystems. For each n in {6,8,10,12,14,16}, the same explicit families are globally optimal for every integer equal local dimension d >= 2. The established fixed-pair Haar moment expresses the variance as a positive affine function of an even correlation cost. A parity floor settles n congruent to 2 modulo 4. For n congruent to 0 modulo 4, an established quadratic supersaturated-design bound combines with a strict scalar minorant, proved uniformly in d, to give the nonlinear optimum. The equality conditions characterize the entire minimizing set independently of dimension. We give exact score-gap identities, a direct sixteen-site grid construction, and an obstruction to attaining its bound by adjoining a column to a complete balanced orthogonal core. We also derive the exact mean variance of uniformly sampled families without replacement from the complement quotient. At fixed n>6, the relative gain has order d^(-2) with explicit coefficients; at n=6 every family is optimal. Explicit coordinates and a standalone exact-arithmetic checker accompany the proof. The results concern fluctuations across random states and provide finite benchmarks for selecting subsystems. Correspondence: Ryosuke.Shiroshita@exactory.ai This preprint was written by exactory.ai (https://www.exactory.ai), an AI research system. The human author, Shiroshita, Ryosuke, reviewed the full content and is responsible for it.

DOI: 10.5281/zenodo.22773592

Related papers

Purification complexity in evaporating random circuits: a Page-transition benchmark

Ryosuke Shiroshita

Sep 7, 20260 citations

Exact Haar Statistics of Planar $k$-Purity in Multipartite Quantum Systems

Kourosh Mirsohi

quant-pharXiv:2608.28914Sep 1, 20260 citations

Extremal same-basis correlations: full-rank rigidity and a stabilizer classification

Ya Zhang

Oct 9, 20260 citations

Uniformly Stable Minimal Weyl--Heisenberg Measurements Approaching the SIC Benchmark

Xiuwu Zhu, Yu Wang

quant-pharXiv:2608.11850Aug 13, 20260 citations

Product-Haar conjugation: quantitative scalar freeness and universal liberation criteria

Bertrand Dalimier

Oct 3, 20260 citations

Exact entanglement trade-offs in qutrit and composite-dimensional stabilizer states

Zakaria Dahbi

quant-pharXiv:2605.05167May 7, 20260 citations

Exact balanced subsystem designs for Haar-purity fluctuations

Ryosuke Shiroshita

Published Sep 15, 2026• 0 citations

Publisher

Abstract

DOI: 10.5281/zenodo.22773592

Related papers

Purification complexity in evaporating random circuits: a Page-transition benchmark

Ryosuke Shiroshita

We prove early preparation and entropy bounds for an explicit evaporating random-circuit model. Independent Haar two-qubit gates scramble an active register before each emission, and emitted qubits never interact again. Preparation complexity allows arbitrary one- and two-qubit gates and any finite number of initially pure ancillas, at fixed trace-distance error. We specialize the established two-copy purity method to permanent deletion and bound its evolution by a moving hyperbolic-cosine comparison profile. The first scrambling round gives a uniform relative bound on expected subsystem purity; positivity preserves this bound through every subsequent emission, even if later scrambling rounds are shortened or omitted. For the stipulated schedule, radiation before a logarithmic window around half emission has a linear preparation upper bound with an explicit failure bound. The entropy of every radiation prefix differs from its dimension ceiling by at most a logarithmic term, simultaneously with high probability. Numerical recurrence calculations, exact subset identities and small-system state-vector simulations provide reproducible checks. These results settle the early-preparation and entropy clauses of the benchmark. They do not establish its post-Page complexity lower bound at the accumulated gate-count scale. We identify the additional distributional information that such a lower bound requires and explain why the entropy Page law alone is insufficient. This preprint was written by exactory.ai (https://www.exactory.ai), an AI research system. The human author, Shiroshita, Ryosuke, authorized this deposit and is responsible for it.We prove early preparation and entropy bounds for an explicit evaporating random-circuit model. Independent Haar two-qubit gates scramble an active register before each emission, and emitted qubits never interact again. Preparation complexity allows arbitrary one- and two-qubit gates and any finite number of initially pure ancillas, at fixed trace-distance error. We specialize the established two-copy purity method to permanent deletion and bound its evolution by a moving hyperbolic-cosine comparison profile. The first scrambling round gives a uniform relative bound on expected subsystem purity; positivity preserves this bound through every subsequent emission, even if later scrambling rounds are shortened or omitted. For the stipulated schedule, radiation before a logarithmic window around half emission has a linear preparation upper bound with an explicit failure bound. The entropy of every radiation prefix differs from its dimension ceiling by at most a logarithmic term, simultaneously with high probability. Numerical recurrence calculations, exact subset identities and small-system state-vector simulations provide reproducible checks. These results settle the early-preparation and entropy clauses of the benchmark. They do not establish its post-Page complexity lower bound at the accumulated gate-count scale. We identify the additional distributional information that such a lower bound requires and explain why the entropy Page law alone is insufficient. This preprint was written by exactory.ai (https://www.exactory.ai), an AI research system. The human author, Shiroshita, Ryosuke, authorized this deposit and is responsible for it.

Sep 7, 20260 citations

Exact Haar Statistics of Planar $k$ -Purity in Multipartite Quantum SystemsExact Haar Statistics of Planar $k$-Purity in Multipartite Quantum Systems

Kourosh Mirsohi

Planar $k$ -uniform states have maximally mixed reductions on every $k$ -site interval of a ring. We introduce planar $k$ -purity---the mean purity of those intervals---as a faithful cost function for this simultaneous constraint. For a Haar-random pure state of $n$ parties with local dimension $p$ , the joint purity of two intervals depends only on their overlap. An exhaustive four-replica calculation therefore yields the complete cyclic covariance kernel and a closed variance for every $1\le k\le\lfloor n/2\rfloor$ . In the balanced qubit case, $k=\lfloor n/2\rfloor$ and $N=2^n$ , the variance is asymptotic to $20/(3nN^2)$ for even $n$ and $6/(nN^2)$ for odd $n$ . A site-factorized permutation representation gives all raw moments and permits exact finite-sum evaluation of the skewness. Fixed-seed simulations validate nonbalanced qubit and qutrit cases, while balanced-qubit simulations through $n=10$ validate the parity formulas and finite-size skewness. Balanced planar and absolute balanced purity have the same Haar mean but different fluctuations; the planar variance is approximately $2.83$ times larger at $n=10$ . Thus subsystem incidence, although invisible to every one-cut marginal distribution, controls the collective fluctuations of geometrically related cuts. Together with universal attainability and a linear number of interval constraints at balance, this provides a geometry-aware benchmark for multipartite entanglement beyond AME existence regimes.Planar $k$-uniform states have maximally mixed reductions on every $k$-site interval of a ring. We introduce planar $k$-purity---the mean purity of those intervals---as a faithful cost function for this simultaneous constraint. For a Haar-random pure state of $n$ parties with local dimension $p$, the joint purity of two intervals depends only on their overlap. An exhaustive four-replica calculation therefore yields the complete cyclic covariance kernel and a closed variance for every $1\le k\le\lfloor n/2\rfloor$. In the balanced qubit case, $k=\lfloor n/2\rfloor$ and $N=2^n$, the variance is asymptotic to $20/(3nN^2)$ for even $n$ and $6/(nN^2)$ for odd $n$. A site-factorized permutation representation gives all raw moments and permits exact finite-sum evaluation of the skewness. Fixed-seed simulations validate nonbalanced qubit and qutrit cases, while balanced-qubit simulations through $n=10$ validate the parity formulas and finite-size skewness. Balanced planar and absolute balanced purity have the same Haar mean but different fluctuations; the planar variance is approximately $2.83$ times larger at $n=10$. Thus subsystem incidence, although invisible to every one-cut marginal distribution, controls the collective fluctuations of geometrically related cuts. Together with universal attainability and a linear number of interval constraints at balance, this provides a geometry-aware benchmark for multipartite entanglement beyond AME existence regimes.

quant-pharXiv:2608.28914Sep 1, 20260 citations

Extremal same-basis correlations: full-rank rigidity and a stabilizer classification

Ya Zhang

Complete written proofs and two exact replayable checker implementations classify all full-Schmidt-rank equality states without assuming stabilizer structure, classify every two-qudit pure stabilizer score at odd prime dimension, and determine the sharp pure-state ratio p/2 for every prime p congruent to 1 modulo 4. The general rank-deficient equality classification and the sharp ratio for primes congruent to 3 modulo 4, p at least 7, remain open in this record. This is an independently authored preprint with AI-assisted research and self-audit, not external professional peer review or a priority claim. The fixed immutable GitHub Release is https://github.com/mxym/math/releases/tag/ecqc-extremal-classification-v1. AI-assisted research and writing; no external funding. Existing licenses and third-party notices are retained.Complete written proofs and two exact replayable checker implementations classify all full-Schmidt-rank equality states without assuming stabilizer structure, classify every two-qudit pure stabilizer score at odd prime dimension, and determine the sharp pure-state ratio p/2 for every prime p congruent to 1 modulo 4. The general rank-deficient equality classification and the sharp ratio for primes congruent to 3 modulo 4, p at least 7, remain open in this record. This is an independently authored preprint with AI-assisted research and self-audit, not external professional peer review or a priority claim. The fixed immutable GitHub Release is https://github.com/mxym/math/releases/tag/ecqc-extremal-classification-v1. AI-assisted research and writing; no external funding. Existing licenses and third-party notices are retained.

Oct 9, 20260 citations

Uniformly Stable Minimal Weyl--Heisenberg Measurements Approaching the SIC Benchmark

Xiuwu Zhu, Yu Wang

Informational completeness (IC) guarantees that an inverse exists, not that it is statistically well conditioned. For minimal rank-one Weyl--Heisenberg (WH) measurements, covariance makes the nonidentity projector-Gram spectrum proportional to the fiducial's ambiguity intensities, with eigenvalues $d|χ_φ(u)|^2$ , turning stability into an explicit worst-direction design problem; write $λ$ for its smallest nonidentity eigenvalue. Haar fiducials are IC almost surely while $\mathbb E[λ^{-1}]=\infty$ , and an explicit geometric family used to establish balanced informationally complete measurements in every dimension has a normalized spectral floor bounded above by an exponentially decaying envelope. We then construct a hierarchy of minimal measurements. A cyclic family with exactly $d^2$ outcomes in every integer dimension has floors $Θ(d^{-3})$ for odd $d$ and $Θ(d^{-5})$ for even $d$ ; a finite-field family for $q=2^m$ obeys the uniform bound $λ\ge4/9$ . Our main result treats every prime-power dimension of characteristic $p\ge5$ . A balanced one-coordinate perturbation repairs the zero ambiguity axis of a cubic Alltop state, gives an attained floor uniformly bounded below by a positive constant, and confines the entire nonidentity spectrum to $[L_q,U_q]$ with $U_q/L_q\to1$ . Its SIC-normalized minimum tends to one, and $λ(φ_q)/Λ_q^\star\to1$ for the global finite-field WH max--min optimum $Λ_q^\star$ , without assuming SIC existence. The complete spectrum determines the exact finite-sample Hilbert--Schmidt error of canonical linear inversion at $I/d$ , while its lower edge controls local Fisher efficiency and canonical-shadow bounds.Informational completeness (IC) guarantees that an inverse exists, not that it is statistically well conditioned. For minimal rank-one Weyl--Heisenberg (WH) measurements, covariance makes the nonidentity projector-Gram spectrum proportional to the fiducial's ambiguity intensities, with eigenvalues $d|χ_φ(u)|^2$, turning stability into an explicit worst-direction design problem; write $λ$ for its smallest nonidentity eigenvalue. Haar fiducials are IC almost surely while $\mathbb E[λ^{-1}]=\infty$, and an explicit geometric family used to establish balanced informationally complete measurements in every dimension has a normalized spectral floor bounded above by an exponentially decaying envelope. We then construct a hierarchy of minimal measurements. A cyclic family with exactly $d^2$ outcomes in every integer dimension has floors $Θ(d^{-3})$ for odd $d$ and $Θ(d^{-5})$ for even $d$; a finite-field family for $q=2^m$ obeys the uniform bound $λ\ge4/9$. Our main result treats every prime-power dimension of characteristic $p\ge5$. A balanced one-coordinate perturbation repairs the zero ambiguity axis of a cubic Alltop state, gives an attained floor uniformly bounded below by a positive constant, and confines the entire nonidentity spectrum to $[L_q,U_q]$ with $U_q/L_q\to1$. Its SIC-normalized minimum tends to one, and $λ(φ_q)/Λ_q^\star\to1$ for the global finite-field WH max--min optimum $Λ_q^\star$, without assuming SIC existence. The complete spectrum determines the exact finite-sample Hilbert--Schmidt error of canonical linear inversion at $I/d$, while its lower edge controls local Fisher efficiency and canonical-shadow bounds.

quant-pharXiv:2608.11850Aug 13, 20260 citations

Product-Haar conjugation: quantitative scalar freeness and universal liberation criteria

Bertrand Dalimier

We study when conjugation by a product Haar unitary on (ℂ^d)^{⊗D} reproduces the scalar mixed-moment behavior of a Haar unitary on the full space as d→∞, with D fixed. On the fully traceless sector—matrices whose one-leg partial traces vanish—we prove sharp global-Haar-scale cancellation: for N=d^D, fixed alternating words have bias O(N^-2) and fixed-L^q fluctuations O(N^-1). If this partial-trace condition is asymptotically stable under centered polynomial evaluation, it yields quantitative scalar asymptotic freeness. For liberation from every bounded deterministic test family, the corresponding partial-trace defect is also necessary at the first nontrivial order and can be reconstructed from D fixed probes, giving a local-unitary second-moment diagnostic related to randomized-measurement methods. For arbitrary bounded deterministic coefficients, we derive a self-contained fixed-order expansion in which finitely many normalized permutation-tuple contractions determine every product-Haar mixed moment up to O(d^-1). Using the exact multi-leg contraction exponent of Collins–Yuan, currently available as a preprint, we obtain a finite necessary-and-sufficient criterion for matching global-Haar mixed moments at every fixed (D,k), and hence an all-orders characterization. In the bipartite k=4 case exactly four tensor-transport contractions survive: the traceless SU(d) exchange interaction activates the two inverse-cycle modes. For Bell-diagonal generators the same modes become explicit ordinary and symplectically twisted autocorrelation energies: centered random sign masks satisfy all four asymptotically almost surely, whereas a classical circulant two-level mask has an exact surviving pairing coordinate 1/3+8/(3d^2). Finally, exact vanishing of the polynomial partial-trace defect is equivalent to local complementarity, yielding a d^2 algebra-dimension bound and extremal connections with distance-two quantum MDS codes and absolutely maximally entangled states.We study when conjugation by a product Haar unitary on (ℂ^d)^{⊗D} reproduces the scalar mixed-moment behavior of a Haar unitary on the full space as d→∞, with D fixed. On the fully traceless sector—matrices whose one-leg partial traces vanish—we prove sharp global-Haar-scale cancellation: for N=d^D, fixed alternating words have bias O(N^-2) and fixed-L^q fluctuations O(N^-1). If this partial-trace condition is asymptotically stable under centered polynomial evaluation, it yields quantitative scalar asymptotic freeness. For liberation from every bounded deterministic test family, the corresponding partial-trace defect is also necessary at the first nontrivial order and can be reconstructed from D fixed probes, giving a local-unitary second-moment diagnostic related to randomized-measurement methods. For arbitrary bounded deterministic coefficients, we derive a self-contained fixed-order expansion in which finitely many normalized permutation-tuple contractions determine every product-Haar mixed moment up to O(d^-1). Using the exact multi-leg contraction exponent of Collins–Yuan, currently available as a preprint, we obtain a finite necessary-and-sufficient criterion for matching global-Haar mixed moments at every fixed (D,k), and hence an all-orders characterization. In the bipartite k=4 case exactly four tensor-transport contractions survive: the traceless SU(d) exchange interaction activates the two inverse-cycle modes. For Bell-diagonal generators the same modes become explicit ordinary and symplectically twisted autocorrelation energies: centered random sign masks satisfy all four asymptotically almost surely, whereas a classical circulant two-level mask has an exact surviving pairing coordinate 1/3+8/(3d^2). Finally, exact vanishing of the polynomial partial-trace defect is equivalent to local complementarity, yielding a d^2 algebra-dimension bound and extremal connections with distance-two quantum MDS codes and absolutely maximally entangled states.

Oct 3, 20260 citations

Exact entanglement trade-offs in qutrit and composite-dimensional stabilizer states

Zakaria Dahbi

When absolute maximal entanglement is unavailable in a state family, different objectives can select different distributions of residual entanglement. We determine exact trade-offs for qutrit stabilizer states and binary--ternary product states. For eight qutrits, evaluation of the complete published classification gives three nondominated balanced-cut entanglement profiles, characterized by the probabilities of obtaining three or four maximally entangled qutrit pairs. We relate the squared entropy-deficit objective to code weight enumerators and use a uniformity-forcing bound to prove that its eleven-qutrit minimum is six. The unique minimizing Clifford class is an already classified code, whose entropy profile also attains a sharp bound on average reference--erasure mutual information among four-uniform eleven-qutrit stabilizer states. In local dimension six, explicit sector alignments attain 56 maximally mixed four-party reductions, the maximum for arbitrary binary--ternary product states. Within the stabilizer product family, the exact cost--count Pareto boundary consists of $(44,48)$ and $(48,56)$ . These results identify when aggregate optimization determines operational entanglement properties and when it does not. Published classifications supply coverage; analytical bounds, explicit witnesses, and independent arithmetic checks establish the stated consequences.When absolute maximal entanglement is unavailable in a state family, different objectives can select different distributions of residual entanglement. We determine exact trade-offs for qutrit stabilizer states and binary--ternary product states. For eight qutrits, evaluation of the complete published classification gives three nondominated balanced-cut entanglement profiles, characterized by the probabilities of obtaining three or four maximally entangled qutrit pairs. We relate the squared entropy-deficit objective to code weight enumerators and use a uniformity-forcing bound to prove that its eleven-qutrit minimum is six. The unique minimizing Clifford class is an already classified code, whose entropy profile also attains a sharp bound on average reference--erasure mutual information among four-uniform eleven-qutrit stabilizer states. In local dimension six, explicit sector alignments attain 56 maximally mixed four-party reductions, the maximum for arbitrary binary--ternary product states. Within the stabilizer product family, the exact cost--count Pareto boundary consists of $(44,48)$ and $(48,56)$. These results identify when aggregate optimization determines operational entanglement properties and when it does not. Published classifications supply coverage; analytical bounds, explicit witnesses, and independent arithmetic checks establish the stated consequences.

quant-pharXiv:2605.05167May 7, 20260 citations

Exact Haar Statistics of Planar

k

-Purity in Multipartite Quantum Systems

Planar

k

-uniform states have maximally mixed reductions on every

k

-site interval of a ring. We introduce planar

k

-purity---the mean purity of those intervals---as a faithful cost function for this simultaneous constraint. For a Haar-random pure state of

n

parties with local dimension

p

, the joint purity of two intervals depends only on their overlap. An exhaustive four-replica calculation therefore yields the complete cyclic covariance kernel and a closed variance for every

1\le k\le\lfloor n/2\rfloor

. In the balanced qubit case,

k=\lfloor n/2\rfloor

and

N=2^n

, the variance is asymptotic to

20/(3nN^2)

for even

n

and

6/(nN^2)

for odd

n

. A site-factorized permutation representation gives all raw moments and permits exact finite-sum evaluation of the skewness. Fixed-seed simulations validate nonbalanced qubit and qutrit cases, while balanced-qubit simulations through

n=10

validate the parity formulas and finite-size skewness. Balanced planar and absolute balanced purity have the same Haar mean but different fluctuations; the planar variance is approximately

2.83

times larger at

n=10

. Thus subsystem incidence, although invisible to every one-cut marginal distribution, controls the collective fluctuations of geometrically related cuts. Together with universal attainability and a linear number of interval constraints at balance, this provides a geometry-aware benchmark for multipartite entanglement beyond AME existence regimes.

d|χ_φ(u)|^2

, turning stability into an explicit worst-direction design problem; write

λ

for its smallest nonidentity eigenvalue. Haar fiducials are IC almost surely while

\mathbb E[λ^{-1}]=\infty

, and an explicit geometric family used to establish balanced informationally complete measurements in every dimension has a normalized spectral floor bounded above by an exponentially decaying envelope. We then construct a hierarchy of minimal measurements. A cyclic family with exactly

d^2

outcomes in every integer dimension has floors

Θ(d^{-3})

for odd

d

and

Θ(d^{-5})

for even

d

; a finite-field family for

q=2^m

obeys the uniform bound

λ\ge4/9

. Our main result treats every prime-power dimension of characteristic

p\ge5

. A balanced one-coordinate perturbation repairs the zero ambiguity axis of a cubic Alltop state, gives an attained floor uniformly bounded below by a positive constant, and confines the entire nonidentity spectrum to

[L_q,U_q]

with

U_q/L_q\to1

. Its SIC-normalized minimum tends to one, and

λ(φ_q)/Λ_q^\star\to1

for the global finite-field WH max--min optimum

Λ_q^\star

, without assuming SIC existence. The complete spectrum determines the exact finite-sample Hilbert--Schmidt error of canonical linear inversion at

I/d

, while its lower edge controls local Fisher efficiency and canonical-shadow bounds.

(44,48)

and

(48,56)

. These results identify when aggregate optimization determines operational entanglement properties and when it does not. Published classifications supply coverage; analytical bounds, explicit witnesses, and independent arithmetic checks establish the stated consequences.