arXiv subjects

Arick Grootveld

Publications and source records attributed to Arick Grootveld.

6 recordsLinked to original sources

arXiv2026

Degree Sequence Reconstruction from Subgraph Traces

The goal of degree sequence reconstruction is to recover the ordered vector of degrees of an unknown graph from vertex deleted traces, where each vertex is deleted independently with probability $p$. We provide two algorithms for reconstruction; the first uses rejection sampling to reduce the problem to an estimation problem for a mixture distribution. Combined with prior trace reconstruction results, this gives a reconstruction algorithm using $\Exp{\tilde O (n^{1/3})}$ traces, although no sub-exponential time decoder is known. Our other approach involves recovering certain graph invariants, degree moments, that can identify a graphs degree sequence. Extremal polynomial bounds show that $\tilde \Theta(n^{1/2})$ degree moments are necessary and sufficient to reconstruct the degree sequence, which leads to an algorithm with $\Exp{\tilde O(n^{1/2})}$ trace complexity. The same polynomial machinery yields a sub-exponential time decoder for the degree sequence from the moments. Additionally, we give an $O(n^{3})$ upper bound and a $\Omega(n^2)$ lower bound for the trace complexity of recovering the number of edges.

cs.IT↗

arXiv2026

A Quantum Method of Types

The method of types is a fundamental tool in classical information theory, with applications ranging from composite hypothesis testing and universal source coding to the capacity of arbitrarily varying channels. In this work we introduce an empirical operator acting as a quantum analog of the empirical distribution. We show that this empirical operator satisfies combinatorial and large-deviation bounds, which in combination describe a quantum method of types. As an application, we use our method to prove a universal achievability result for composite quantum hypothesis testing.

cs.IT↗

arXiv2026

The Optimal Rate Function in Covariant Quantum State Tomography

The problem of quantum tomography is to estimate an unknown quantum state $\rho$ from a measurement of $n$ copies of $\rho$. One can ask which tomography protocol, i.e.\ which choice of multi-copy measurement, gives the best possible estimate of $\rho$. To do so, we characterize tomography protocols by their \emph{rate function}, which governs the exponential rate at which a protocol assigns probability to a particular estimate $\sigma$ of the true state $\rho$. This rate function is a quantum mechanical generalization of the classical relative entropy between the true state and its estimate, and depends on the choice of protocol. It is bounded by the quantum relative entropy, and we show that this bound is sharp: for any $\rho$ and $\sigma$ we construct a family of protocols whose rate functions converge to the quantum relative entropy $D(\sigma\|\rho)$. We consider the family of covariant tomography protocols; these are the basis independent state estimation schemes that assume no prior information about $\rho$ and $\sigma$. Keyl described a specific tomography protocol based on Schur sampling, and conjectured that among all covariant tomography protocols it has the largest possible rate function for all $\sigma$ and $\rho$. We prove this conjecture. The resulting rate function is an annealed version of quantum relative entropy, due to the cost of learning the eigenbasis in covariant quantum state tomography.

quant-ph↗

arXiv2026

Asymptotically Optimal Quantum Universal Quickest Change Detection

This paper investigates the quickest change detection of quantum states in a universal setting: specifically, where the post-change quantum state is not known a priori. We establish the asymptotic optimality of a two-stage approach in terms of worst average delay to detection. The first stage employs block POVMs with classical outputs that preserve quantum relative entropy to arbitrary precision. The second stage leverages a recently proposed windowed-CUSUM algorithm that is known to be asymptotically optimal for quickest change detection with an unknown post-change distribution in the classical setting.

quant-ph↗

arXiv2026

Asymptotically Optimal Tests for One- and Two-Sample Problems

In this work, we revisit the one- and two-sample testing problems: binary hypothesis testing in which one or both distributions are unknown. For the one-sample test, we provide a more streamlined proof of the asymptotic optimality of Hoeffding's likelihood ratio test, which is equivalent to the threshold test of the relative entropy between the empirical distribution and the nominal distribution. The new proof offers an intuitive interpretation and naturally extends to the two-sample test where we show that a similar form of Hoeffding's test, namely a threshold test of the relative entropy between the two empirical distributions is also asymptotically optimal. A strong converse for the two-sample test is also obtained.

cs.IT↗

arXiv2025

Towards Quantum Universal Hypothesis Testing

Hoeffding's formulation and solution to the universal hypothesis testing (UHT) problem had a profound impact on many subsequent works dealing with asymmetric hypotheses. In this work, we introduce a quantum universal hypothesis testing framework that serves as a quantum analog to Hoeffding's UHT. Motivated by Hoeffding's approach, which estimates the empirical distribution and uses it to construct the test statistic, we employ quantum state tomography to reconstruct the unknown state prior to forming the test statistic. Leveraging the concentration properties of quantum state tomography, we establish the exponential consistency of the proposed test: the type II error probability decays exponentially quickly, with the exponent determined by the trace distance between the true state and the nominal state.

cs.IT↗