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We study quantum state certification using unentangled quantum measurements, namely measurements which operate only on one copy of the state at a time. When there is a common source of randomness available and the unentangled measurements are chosen based on this randomness, prior work has shown that copies are necessary and sufficient. We show a separation between algorithms with and without randomness. We develop a lower bound framework for both fixed and randomized measurements that relates the hardness of testing to the well-established Lüders rule. More precisely, we obtain lower bounds for randomized and fixed schemes as a function of the eigenvalues of the Lüders channel which characterizes one possible post-measurement state transformation. 

We consider distributed parameter estimation using interactive protocols subject to local information constraints such as bandwidth limitations, local differential privacy, and restricted measurements. We provide a unified framework enabling us to derive a variety of (tight) minimax lower bounds for different parametric families of distributions, both continuous and discrete, under any Lp loss. Our lower bound framework is versatile and yields “plug-and-play” bounds that are widely applicable to a large range of estimation problems, and, for the prototypical case of the Gaussian family, circumvents limitations of previous techniques. In particular, our approach recovers bounds obtained using data processing inequalities and Cramér–Rao bounds, two other alternative approaches for proving lower bounds in our setting of interest. Further, for the families considered, we complement our lower bounds with matching upper bounds. 

We study discrete distribution estimation under user-level local differential privacy (LDP). In user-level $$\varepsilon$$-LDP, each user has $$m\ge1$$ samples and the privacy of all $$m$$ samples must be preserved simultaneously. We resolve the following dilemma: While on the one hand having more samples per user should provide more information about the underlying distribution, on the other hand, guaranteeing the privacy of all $$m$$ samples should make the estimation task more difficult. We obtain tight bounds for this problem under almost all parameter regimes. Perhaps surprisingly, we show that in suitable parameter regimes, having $$m$$ samples per user is equivalent to having $$m$$ times more users, each with only one sample. Our results demonstrate interesting phase transitions for $$m$$ and the privacy parameter $$\varepsilon$$ in the estimation risk. Finally, connecting with recent results on shuffled DP, we show that combined with random shuffling, our algorithm leads to optimal error guarantees (up to logarithmic factors) under the central model of user-level DP in certain parameter regimes. We provide several simulations to verify our theoretical findings. 

We study the sample complexity of causal structure learning on a two-variable system with observational and experimental data. Specifically, for two variables X and Y, we consider the classical scenario where either X causes Y , Y causes X, or there is an unmeasured confounder between X and Y. We show that if X and Y are over a finite domain of size k and are significantly correlated, the minimum number of interventional samples needed is sublinear in k. We give a tight characterization of the tradeoff between observational and interventional data when the number of observational samples is sufficiently large. We build upon techniques for closeness testing and for non-parametric density estimation in different regimes of observational data. Our hardness results are based on carefully constructing causal models whose marginal and interventional distributions form hard instances of canonical results on property testing. 

We obtain tight minimax rates for the problem of distributed estimation of discrete distributions under communication constraints, where n users observing m samples each can broadcast only ℓ bits. Our main result is a tight characterization (up to logarithmic factors) of the error rate as a function of m, ℓ, the domain size, and the number of users under most regimes of interest. 

We study robust testing and estimation of discrete distributions in the strong contamination model. Our results cover both centralized setting and distributed setting with general local information constraints including communication and LDP constraints. Our technique relates the strength of manipulation attacks to the earth-mover distance using Hamming distance as the metric between messages (samples) from the users. In the centralized setting, we provide optimal error bounds for both learning and testing. Our lower bounds under local information constraints build on the recent lower bound methods in distributed inference. In the communication constrained setting, we develop novel algorithms based on random hashing and an L1-L1 isometry.