arXiv subjects

Henning Stein

Publications and source records attributed to Henning Stein.

3 recordsLinked to original sources

arXiv2026

Optimal estimation of high-dimensional quantum states using locally gentle measurements

We study the task of estimating a $d-$dimensional quantum state $\rho$ under the constraint that the measurement is $\alpha-$gentle. Such measurements $M$ do not collapse the state; they issue both a random variable $R^M = \omega$ containing statistical information and a post-measurement state $\rho_{M \to \omega}$ such that $\|\rho_{M\to \omega} - \rho\|_{Tr} \leq \alpha$. We describe gentle measurements and their connection to quantum differential privacy. Our results show that the optimal minimax estimation rate in Frobenius norm is of order $d^3/(n \alpha^2)$, instead of $d^2/n$ for general measurements. Moreover, for rank $r$ states with $r\leq d$ we prove that the optimal minimax rate is $rd^2/(n \alpha^2)$, instead of $rd/n$. Very surprisingly, the loss for gentleness $d/\alpha^2$ scales with the ambient dimension of the Hilbert space, rather than the number of parameters $rd$, typically seen in classical differential privacy. We propose optimal gentle measurements and indicate how they can be physically implemented using an ancillary state and a CNOT gate to entangle it with the initial state. We notice that the resulting random variable has a likelihood that satisfies local differential privacy. Lower bounds are proven through a new quantum information-theoretic inequality applied to well chosen families of states in the manifold of (small-rank) quantum states.

math.ST↗

arXiv2026

Locally Gentle State Certification for High Dimensional Quantum Systems

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.

quant-ph↗

arXiv2025

Sample-optimal learning of quantum states using gentle measurements

Gentle measurements of quantum states do not entirely collapse the initial state. Instead, they provide a post-measurement state at a prescribed trace distance $\alpha$ from the initial state together with a random variable used for quantum learning of the initial state. We introduce here the class of $\alpha-$locally-gentle measurements ($\alpha-$LGM) on a finite dimensional quantum system which are product measurements on product states and prove a strong quantum Data-Processing Inequality (qDPI) on this class using an improved relation between gentleness and quantum differential privacy. We further show a gentle quantum Neyman-Pearson lemma which implies that our qDPI is asymptotically optimal (for small $\alpha$). This inequality is employed to show that the necessary number of quantum states for prescribed accuracy $\epsilon$ is of order $1/(\epsilon^2 \alpha^2)$ for both quantum tomography and quantum state certification. Finally, we propose an $\alpha-$LGM called quantum Label Switch that attains these bounds. It is a general implementable method to turn any two-outcome measurement into an $\alpha-$LGM.

quant-ph↗