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

• URL: http://arxiv.org/abs/2607.24491v1
• Date: Mon, 27 Jul 2026 14:28:07 GMT
• Title: Optimal estimation of high-dimensional quantum states using locally gentle measurements
• Abstract summary: We study the task of estimating a $d-$dimensional quantum state $$ under the constraint that the measurement is $-$gentle.<n>We show that the optimal minimax estimation rate in Frobenius norm is of order $d3/(n 2)$, instead of $d2/n$ for general measurements.
• Score: 1.1470070927586014
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study the task of estimating a $d-$dimensional quantum state $ρ$ under the constraint that the measurement is $α-$gentle. Such measurements $M$ do not collapse the state; they issue both a random variable $R^M = ω$ containing statistical information and a post-measurement state $ρ_{M \to ω}$ such that $\|ρ_{M\to ω} - ρ\|_{Tr} \leq α$. 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 $d^3/(n α^2)$, 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 $rd^2/(n α^2)$, instead of $rd/n$. Very surprisingly, the loss for gentleness $d/α^2$ 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.

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