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

Abstract

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 $20/(3n{N}^2)$ for even $n$ and $6/(n{N}^2)$ 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.