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arXiv:2607.24491

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

Cristina Butucea, Jan Johannes, Henning Stein

math.STquant-phSubmitted Jul 27, 2026• Announced Jul 28, 2026• 0 citations
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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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In differential privacy (DP), we want to query a database about n users, in a way that “leaks at most ε about any individual user,” even conditioned on any outcome of the query. Meanwhile, in gentle measurement, we want to measure n quantum states, in a way that “damages the states by at most α,” even conditioned on any outcome of the measurement. In both cases, we can achieve the goal by techniques like deliberately adding noise to the outcome before returning it. This paper proves a new and general connection between the two subjects. Specifically, we show that on products of n quantum states, any measurement that is α-gentle for small α is also O( α) -DP, and any product measurement that is ε-DP is also O( ε√n) -gentle.

ACM Symposium on Theory of Computing (STOC)•Jun 20, 2019•98 citations
We study the task of estimating a d−d-d−dimensional quantum state ρρρ under the constraint that the measurement is α−α-α−gentle. Such measurements MMM do not collapse the state; they issue both a random variable RM=ωR^M = ωRM=ω containing statistical information and a post-measurement state ρM→ωρ_{M \to ω}ρM→ω​ such that ∥ρM→ω−ρ∥Tr≤α\|ρ_{M\to ω} - ρ\|_{Tr} \leq α∥ρM→ω​−ρ∥Tr​≤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 d3/(nα2)d^3/(n α^2)d3/(nα2), instead of d2/nd^2/nd2/n for general measurements. Moreover, for rank rrr states with r≤dr\leq dr≤d we prove that the optimal minimax rate is rd2/(nα2)rd^2/(n α^2)rd2/(nα2), instead of rd/nrd/nrd/n. Very surprisingly, the loss for gentleness d/α2d/α^2d/α2 scales with the ambient dimension of the Hilbert space, rather than the number of parameters rdrdrd, 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.
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 ααα in trace norm, thereby allowing for the reuse of samples. We analyze the hypothesis testing problem of distinguishing whether an unknown state ρρρ is equal to a reference ρ0ρ_0ρ0​ or εεε-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 ααα-gentleness imposes a sample size penalty of dα2\frac{d}{α^2}α2d​, yielding a total sample complexity of n=Θ(d3ε2α2)n = Θ(\frac{d^3}{ε^2 α^2})n=Θ(ε2α2d3​). 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 ααα-gentleness scales linearly with the Hilbert-space dimension ddd rather than the number of parameters d2−1d^2-1d2−1 typical for high-dimensional private estimation.
Gentle measurements of quantum states do not entirely collapse the initial state. Instead, they provide a post-measurement state at a prescribed trace distance α from the initial state together with a random variable used for quantum learning of the initial state. We introduce here the class of α−locally-gentle measurements (α−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 α). This inequality is employed to show that the necessary number of quantum states for prescribed accuracy ϵ is of order 1∕(ϵ2α2) for both quantum tomography and quantum state certification. Finally, we propose an α−LGM called quantum Label Switch that attains these bounds. It is a general implementable method to turn any two-outcome measurement into an α−LGM.
Differential privacy provides a rigorous framework for quantifying privacy leakage in data analysis, while its quantum extensions have become increasingly relevant with the development of quantum computing and quantum machine learning. In this work, we introduce and study quantum probabilistic local differential privacy, a relaxation of quantum local differential privacy in which the privacy constraint is allowed to fail on a spectral violation event with low probability. This quantity can be interpreted as the probability under the quantum superoperation of a quantum privacy-loss violation, and is closely related to the acceptance probability of the quantum Neyman-Pearson test at a small threshold. We investigate the basic structural properties of this privacy notion and clarify its relationship with existing forms of quantum differential privacy. We show the properties of quantum probabilistic local differential privacy under tensor-product composition and unitary post-processing, while it is in general neither convex nor closed under post-processing by arbitrary quantum channels. We further characterize when depolarizing noise satisfies quantum probabilistic local differential privacy under several representative scenarios. Finally, we connect quantum probabilistic privacy constraints with statistical inference by deriving a lower bound on probabilistically privatized contraction coefficients in terms of the hockey-stick divergence. As an application, we obtain sample complexity bounds of probabilistically privated asymmetric and symmetric quantum hypothesis testing. These results provide a systematic foundation for studying probabilistic privacy guarantees in quantum information processing and their operational consequences for private quantum statistical inference.
In differential privacy (DP), we want to query a database about n users, in a way that "leaks at most eps about any individual user," even conditioned on any outcome of the query. Meanwhile, in gentle measurement, we want to measure n quantum states, in a way that "damages the states by at most alpha," even conditioned on any outcome of the measurement. In both cases, we can achieve the goal by techniques like deliberately adding noise to the outcome before returning it. This paper proves a new and general connection between the two subjects. Specifically, we show that on products of n quantum states, any measurement that is alpha-gentle for small alpha is also O(alpha)-DP, and any product measurement that is eps-DP is also O(eps*sqrt(n))-gentle. Illustrating the power of this connection, we apply it to the recently studied problem of shadow tomography. Given an unknown d-dimensional quantum state rho, as well as known two-outcome measurements E_1,...,E_m, shadow tomography asks us to estimate Pr[E_i accepts rho], for every i in [m], by measuring few copies of rho. Using our connection theorem, together with a quantum analog of the so-called private multiplicative weights algorithm of Hardt and Rothblum, we give a protocol to solve this problem using O((log m)^2 (log d)^2) copies of rho, compared to Aaronson's previous bound of ~O((log m)^4 (log d)). Our protocol has the advantages of being online (that is, the E_i's are processed one at a time), gentle, and conceptually simple. Other applications of our connection include new lower bounds for shadow tomography from lower bounds on DP, and a result on the safe use of estimation algorithms as subroutines inside larger quantum algorithms.
When sensitive information is encoded in data, it is important to ensure the privacy of information when attempting to learn useful information from the data. There is a natural tradeoff whereby increasing privacy requirements may decrease the utility of a learning protocol. In the quantum setting of differential privacy, such tradeoffs between privacy and utility have so far remained largely unexplored. In this work, we study optimal privacy-utility tradeoffs for both generic and application-specific utility metrics when privacy is quantified by (ε,δ)(\varepsilon,δ)(ε,δ)-quantum local differential privacy. In the generic setting, we focus on optimizing fidelity and trace distance between the original state and the privatized state. We show that the depolarizing mechanism achieves the optimal utility for given privacy requirements. We then study the specific application of learning the expectation of an observable with respect to an input state when only given access to privatized states. We derive a lower bound on the number of samples of privatized data required to achieve a fixed accuracy guarantee with high probability. To prove this result, we employ existing lower bounds on private quantum hypothesis testing, thus showcasing the first operational use of them. We also devise private mechanisms that achieve optimal sample complexity with respect to the privacy parameters and accuracy parameters, demonstrating that utility can be significantly improved for specific tasks in contrast to the generic setting. In addition, we show that the number of samples required to privately learn observable expectation values scales as Θ((εβ)−2)Θ((\varepsilon β)^{-2})Θ((εβ)−2), where ε∈(0,1)\varepsilon \in (0,1)ε∈(0,1) is the privacy parameter and βββ is the accuracy tolerance. We conclude by initiating the study of private classical shadows, which promise useful applications for private learning tasks.
In differential privacy (DP), we want to query a database about n users, in a way that “leaks at most ε about any individual user,” even conditioned on any outcome of the query. Meanwhile, in gentle measurement, we want to measure n quantum states, in a way that “damages the states by at most α,” even conditioned on any outcome of the measurement. In both cases, we can achieve the goal by techniques like deliberately adding noise to the outcome before returning it. This paper proves a new and general connection between the two subjects. Specifically, we show that on products of n quantum states, any measurement that is α-gentle for small α is also O( α) -DP, and any product measurement that is ε-DP is also O( ε√n) -gentle.