Locally Gentle State Certification for High Dimensional Quantum Systems

TL;DR

This paper explores the fundamental limits of non-destructive quantum state certification under gentle measurement constraints, deriving the sample complexity and revealing a linear dimension penalty in high-dimensional quantum systems.

Contribution

It introduces a theoretical framework for locally-gentle quantum state certification, quantifies the sample complexity under gentle constraints, and connects measurement disturbance with privacy in quantum learning.

Findings

Sample complexity scales as ^3 / (^2 \u03b1^2) under gentle measurement constraints.

Gentle measurements impose a linear dimension penalty, , rather than quadratic, in high-dimensional quantum systems.

The work links physical measurement limitations to privacy mechanisms in quantum information processing.

Abstract

Standard approaches to quantum statistical inference rely on measurements that induce a collapse of the wave function, effectively consuming the quantum state to extract information. In this work, we investigate the fundamental limits of \emph{locally-gentle} quantum state certification, where the learning algorithm is constrained to perturb the state by at most $\alpha$ in trace norm, thereby allowing for the reuse of samples. We analyze the hypothesis testing problem of distinguishing whether an unknown state $\rho$ is equal to a reference $\rho_0$ or $\epsilon$-far from it. We derive the minimax sample complexity for this problem, quantifying the information-theoretic price of non-destructive measurements. Specifically, by constructing explicit measurement operators, we show that the constraint of $\alpha$-gentleness imposes a sample size penalty of $\frac{d}{\alpha^2}$, yielding a total sample complexity of $n = \Theta(\frac{d^3}{\epsilon^2 \alpha^2})$. Our results clarify the trade-off between information extraction and state disturbance, and highlight deep connections between physical measurement constraints and privacy mechanisms in quantum learning. Crucially, we find that the sample size penalty incurred by enforcing $\alpha$-gentleness scales linearly with the Hilbert-space dimension $d$ rather than the number of parameters $d^2-1$ typical for high-dimensional private estimation.