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

Abstract

We study the task of estimating a $d-$dimensional quantum state $\ddot{I}$ under the constraint that the measurement is $\hat{I}\pm -$gentle. Such measurements $M$ do not collapse the state; they issue both a random variable $R^M=\ddot{I}‰$ containing statistical information and a post-measurement state $\ddot{I}{}_{M\to \ddot{I}‰}$ such that $\Vert \ddot{I}{}_{M\to \ddot{I}‰}-\ddot{I}{\Vert }_{Tr}\le \hat{I}\pm$. 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 $d^3/(n\hat{I}{\pm }^2)$, instead of $d^2/n$ for general measurements. Moreover, for rank $r$ states with $r\le d$ we prove that the optimal minimax rate is $r{d}^2/(n\hat{I}{\pm }^2)$, instead of $rd/n$. Very surprisingly, the loss for gentleness $d/\hat{I}{\pm }^2$ 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.