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.
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≤k≤⌊n/2⌋. In the balanced qubit case, k=⌊n/2⌋ and N=2n, the variance is asymptotic to 20/(3nN2) for even n and 6/(nN2) 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.
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.
We investigate multipartite entanglement via the statistical properties of pure quantum states of n-qubits. By analyzing the distribution of purity among balanced bipartitions, we compare Haar-typical states, uniformly distributed on the unit sphere of states, with Hadamard states, being characterized by equal weights in the computational basis. We analyze different ensembles of Hadamard states characterized by their phase distributions. Through analytical and numerical calculations, we show that Hadamard states exhibit, on average, a higher degree of entanglement than Haar-typical states. In addition, we show that a particular class of Hadamard states, characterized by real coefficients with alternating signs, known as hypergraph states, appears especially relevant in the search for maximally multipartite entangled states, both for their structural simplicity and the increased likelihood of sampling highly entangled states. These results identify Hadamard states as a tractable yet promising class for exploring multipartite entanglement structures and advancing the characterization of maximally multipartite entangled quantum states.
We analyze the properties of entangled random pure states of a quantum system partitioned into two smaller subsystems of dimensions N and M. Framing the problem in terms of random matrices with a fixed-trace constraint, we establish, for arbitrary N≤M, a general relation between the n-point densities and the cross moments of the eigenvalues of the reduced density matrix, i.e., the so-called Schmidt eigenvalues, and the analogous functionals of the eigenvalues of the Wishart-Laguerre ensemble of the random matrix theory. This allows us to derive explicit expressions for two-level densities, and also an exact expression for the variance of von Neumann entropy at finite N,M. Then, we focus on the moments E{K^{a}} of the Schmidt number K, the reciprocal of the purity. This is a random variable supported on [1,N], which quantifies the number of degrees of freedom effectively contributing to the entanglement. We derive a wealth of analytical results for E{K^{a}} for N=2 and 3 and arbitrary M, and also for square N=M systems by spotting for the latter a connection with the probability P(x_{min}^{GUE}≥sqrt[2N]ξ) that the smallest eigenvalue x_{min}^{GUE} of an N×N matrix belonging to the Gaussian unitary ensemble is larger than sqrt[2N]ξ. As a by-product, we present an exact asymptotic expansion for P(x_{min}^{GUE}≥sqrt[2N]ξ) for finite N as ξ→∞. Our results are corroborated by numerical simulations whenever possible, with excellent agreement.
We demonstrate that absolutely maximally entangled (AME) states consisting of N=4n qudits with n∈{1,2,3,…}, each of even local dimension d, cannot be realized as canonical graph states over the ring Zd. This result imposes strong constraints on AME states in composite local dimensions and characterizes the limitations of graph-state constructions for highly entangled multipartite quantum systems. Furthermore, for d≡2(mod4) this obstruction, combined with the prime-power decomposition of stabilizer states, excludes all pure stabilizer AME states (including four quhexes), while clarifying the distinction between canonical Zd graph states and stabilizer constructions over prime-power factors. We also discuss a composite construction of mixed k-uniform states. For (N,d)=(4,6), the construction yields a rank-two mixed 2-uniform stabilizer state of purity 1/2, which is optimal among normalized stabilizer projectors.
Abstract We characterize the achievable second-moment concentration exponent for block-product fidelity quantum kernels whose per-block ensembles have a maximally mixed first moment. For n = mb qubits partitioned into m independent b-qubit blocks, with no entangling gate crossing a block boundary, the exponent c, defined by E[κ²] = 2⁻ᶜⁿ, satisfies c ∈ [1, c(b)] for every finite b, where c(b) = [(b−1) + log₂(2ᵇ+1)]/b increases monotonically from log₂ 3 ≈ 1.585 at b=1 to 2 as b→∞. The lower endpoint is attained exactly by digitizing (basis-encoding) block ensembles, proved here; the upper endpoint follows from the optimal second frame potential of state 2-designs (Nakata et al.) [8], which we apply rather than re-derive. We validate the resulting scaling against idealized Haar-random blocks and finite-depth parameterized circuits, with propagated standard errors, obtaining relative error from −2.9% to +0.14% for block sizes b=2 through 6, tightening as b grows. We identify a structural limitation of single-qubit (b=1) blocks: a continuous data parameter spans only a one-dimensional subset of the Bloch sphere, which has zero measure with respect to the Haar measure, so no purely data-driven single-qubit encoding can realize an exact 2-design; a depth study shows the corresponding purity converging non-monotonically, consistent with this limitation. We further distinguish two notions of practical resolvability — the shot-noise threshold implied by the variance exponent, and the finite-shot consequence for pairwise similarity ranking — report each together with its associated agreement criterion, and calibrate the relationship between them on finite-depth circuits across a shot-budget-by-qubit-count grid (180 points, median heuristic error 0.017). Finally, we relate this result to adjacent constructions that also avoid concentration through different mechanisms — covariant kernels, permutation-equivariant barren-plateau-free kernels, and a Rydberg-blockade many-body kernel — each of which lies outside the hypothesis class considered here.
Absolutely maximally entangled states require every reduction of at most half of the parties to be maximally mixed, a condition that is both rigid and often impossible for qubit systems. Previous work introduced the quantum extremal number, which maximizes the number of exactly maximally mixed half-body marginals, and determined the exact value Qex(8,4)=56. The present work develops a robust extension of this extremal problem. For a subsystem A, the marginal maximal-mixing defect is defined by DA=2∣A∣Tr(ρA2)−1=2∣A∣ρA−2∣A∣IA22, and Qex,εD(n,k) is defined as the maximum number of k-body marginals satisfying DA≤ε in an n-qubit pure state. This counting problem differs from approximate k-uniformity, which requires all k-body marginals to obey a common error bound.
For pure states on 4m qubits, the following local stability inequality is established: ∑i∈TDT∖{i}≥1(∣T∣=2m+1). It follows that, whenever ε<1/(2m+1), the hypergraph of ε-good 2m-subsets is K2m+1(2m)-free. Combined with the known exact eight-qubit construction, this yields the stability plateau Qex,εD(8,4)=56,0≤ε<51. For odd systems of 2k+1 qubits, the exact forbidden hypergraph Hk is used to derive explicit finite-error stability radii. In particular, Qex,εD(9,4)≤120 for 0≤ε<1/17. These results turn exact quantum Turán obstructions into quantitative robustness statements and identify intervals on which quantum extremal numbers are stable under imperfect marginal mixedness.