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

arXiv:2607.24491v1 Announce Type: cross Abstract: 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.

Abstract: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.

Statistics Theory (math.ST); Quantum Physics (quant-ph)

arXiv:2607.24491 [math.ST]

(or arXiv:2607.24491v1 [math.ST] for this version)

https://doi.org/10.48550/arXiv.2607.24491 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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

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