arXiv subjects

Michael Nussbaum

Publications and source records attributed to Michael Nussbaum.

13 recordsLinked to original sources

arXiv2025

Asymptotic inference in a stationary quantum time series

We consider a statistical model of a n-mode quantum Gaussian state which is shift invariant and also gauge invariant. Such models can be considered analogs of classical Gaussian stationary time series, parametrized by their spectral density. Defining an appropriate quantum spectral density as the parameter, we establish that the quantum Gaussian time series model is asymptotically equivalent to a classical nonlinear regression model given as a collection of independent geometric random variables. The asymptotic equivalence is established in the sense of the quantum Le Cam distance between statistical models (experiments). The geometric regression model has a further classical approximation as a certain Gaussian white noise model with a transformed quantum spectral density as signal. In this sense, the result is a quantum analog of the asymptotic equivalence of classical spectral density estimation and Gaussian white noise, which is known for Gaussian stationary time series. In a forthcoming version of this preprint, we will also identify a quantum analog of the periodogram and provide optimal parametric and nonparametric estimates of the quantum spectral density.

math.ST↗

arXiv2024

Asymptotic Equivalence for Nonparametric Regression

We consider a nonparametric model $\mathcal{E}^{n},$ generated by independent observations $X_{i},$ $i=1,...,n,$ with densities $p(x,\theta_{i}),$ $i=1,...,n,$ the parameters of which $\theta _{i}=f(i/n)\in \Theta $ are driven by the values of an unknown function $f:[0,1]\rightarrow \Theta $ in a smoothness class. The main result of the paper is that, under regularity assumptions, this model can be approximated, in the sense of the Le Cam deficiency pseudodistance, by a nonparametric Gaussian shift model $Y_{i}=\Gamma (f(i/n))+\varepsilon _{i},$ where $\varepsilon_{1},...,\varepsilon _{n}$ are i.i.d. standard normal r.v.'s, the function $\Gamma (\theta ):\Theta \rightarrow \mathrm{R}$ satisfies $\Gamma ^{\prime}(\theta )=\sqrt{I(\theta )}$ and $I(\theta )$ is the Fisher information corresponding to the density $p(x,\theta ).$

math.ST↗

arXiv2024

A functional Hungarian construction for sums of independent random variables

We develop a Hungarian construction for the partial sum process of independent non-identically distributed random variables. The process is indexed by functions $f$ from a class $\mathcal{H}$, but the supremum over $f\in $ $\mathcal{H}$ is taken outside the probability. This form is a prerequisite for the Koml\'{o}s-Major-Tusn\'{a}dy inequality in the space of bounded functionals $l^{\infty }(\mathcal{H})$, but contrary to the latter it essentially preserves the classical $n^{-1/2}\log n$ approximation rate over large functional classes $\mathcal{H}$ such as the H\"{o}lder ball of smoothness $1/2$. This specific form of a strong approximation is useful for proving asymptotic equivalence of statistical experiments.

math.PR↗

arXiv2024

Asymptotic Equivalence for Nonparametric Generalized Linear Models

We establish that a non-Gaussian nonparametric regression model is asymptotically equivalent to a regression model with Gaussian noise. The approximation is in the sense of Le Cam's deficiency distance $\Delta $; the models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. Our result concerns a sequence of independent but not identically distributed observations with each distribution in the same real-indexed exponential family. The canonical parameter is a value $f(t_i)$ of a regression function $f$ at a grid point $t_i$ (nonparametric GLM). When $f$ is in a H\"{o}lder ball with exponent $\beta >\frac 12 ,$ we establish global asymptotic equivalence to observations of a signal $\Gamma (f(t))$ in Gaussian white noise, where $\Gamma $ is related to a variance stabilizing transformation in the exponential family. The result is a regression analog of the recently established Gaussian approximation for the i.i.d. model. The proof is based on a functional version of the Hungarian construction for the partial sum process.

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arXiv2023

On the optimal error exponents for classical and quantum antidistinguishability

The concept of antidistinguishability of quantum states has been studied to investigate foundational questions in quantum mechanics. It is also called quantum state elimination, because the goal of such a protocol is to guess which state, among finitely many chosen at random, the system is not prepared in (that is, it can be thought of as the first step in a process of elimination). Antidistinguishability has been used to investigate the reality of quantum states, ruling out $\psi$-epistemic ontological models of quantum mechanics [Pusey et al., Nat. Phys., 8(6):475-478, 2012]. Thus, due to the established importance of antidistinguishability in quantum mechanics, exploring it further is warranted. In this paper, we provide a comprehensive study of the optimal error exponent -- the rate at which the optimal error probability vanishes to zero asymptotically -- for classical and quantum antidistinguishability. We derive an exact expression for the optimal error exponent in the classical case and show that it is given by the multivariate classical Chernoff divergence. Our work thus provides this divergence with a meaningful operational interpretation as the optimal error exponent for antidistinguishing a set of probability measures. For the quantum case, we provide several bounds on the optimal error exponent: a lower bound given by the best pairwise Chernoff divergence of the states, a single-letter semi-definite programming upper bound, and lower and upper bounds in terms of minimal and maximal multivariate quantum Chernoff divergences. It remains an open problem to obtain an explicit expression for the optimal error exponent for quantum antidistinguishability.

quant-ph↗

arXiv2021

Minimax estimation of low-rank quantum states and their linear functionals

In classical statistics, a well known paradigm consists in establishing asymptotic equivalence between an experiment of i.i.d. observations and a Gaussian shift experiment, with the aim of obtaining optimal estimators in the former complicated model from the latter simpler model. In particular, a statistical experiment consisting of $n$ i.i.d observations from d-dimensional multinomial distributions can be well approximated by an experiment consisting of $d-1$ dimensional Gaussian distributions. In a quantum version of the result, it has been shown that a collection of $n$ qudits (d-dimensional quantum states) of full rank can be well approximated by a quantum system containing a classical part, which is a $d-1$ dimensional Gaussian distribution, and a quantum part containing an ensemble of $d(d-1)/2$ shifted thermal states. In this paper, we obtain a generalization of this result when the qudits are not of full rank. We show that when the rank of the qudits is $r$, then the limiting experiment consists of an $r-1$ dimensional Gaussian distribution and an ensemble of both shifted pure and shifted thermal states. For estimation purposes, we establish an asymptotic minimax result in the limiting Gaussian model. Analogous results are then obtained for estimation of a low-rank qudit from an ensemble of identically prepared, independent quantum systems, using the local asymptotic equivalence result. We also consider the problem of estimation of a linear functional of the quantum state. We construct an estimator for the functional, analyze the risk and use quantum local asymptotic equivalence to show that our estimator is also optimal in the minimax sense.

math.ST↗

arXiv2017

Local asymptotic equivalence of pure quantum states ensembles and quantum Gaussian white noise

Quantum technology is increasingly relying on specialised statistical inference methods for analysing quantum measurement data. This motivates the development of "quantum statistics", a field that is shaping up at the overlap of quantum physics and "classical" statistics. One of the less investigated topics to date is that of statistical inference for infinite dimensional quantum systems, which can be seen as quantum counterpart of non-parametric statistics. In this paper we analyse the asymptotic theory of quantum statistical models consisting of ensembles of quantum systems which are identically prepared in a pure state. In the limit of large ensembles we establish the local asymptotic equivalence (LAE) of this i.i.d. model to a quantum Gaussian white noise model. We use the LAE result in order to establish minimax rates for the estimation of pure states belonging to Hermite-Sobolev classes of wave functions. Moreover, for quadratic functional estimation of the same states we note an elbow effect in the rates, whereas for testing a pure state a sharp parametric rate is attained over the nonparametric Hermite-Sobolev class.

math.ST↗

arXiv2013

Attainment of the multiple quantum Chernoff bound for certain ensembles of mixed states

We consider the problem of detecting the true quantum state among r possible ones, based on measurements performed on n of copies of a finite dimensional quantum system. It is known that the exponent for the rate of decrease of the averaged error probability cannot exceed the multiple quantum Chernoff bound (MQCB) defined as the worst case (smallest) quantum Chernoff distance between any possible pair of the r states. This error exponent is attainable for r pure states, but for the general case of mixed states only attainability up to a factor 1/3 is known. Here we show that the MQCB is attainable for mixed states if there is a pair which is closer in quantum Chernoff distance than 1/6 times the distance between all other pairs.

quant-ph↗

arXiv2012

Sharp adaptive nonparametric testing for Sobolev ellipsoids

We consider testing for presence of a signal in Gaussian white noise with intensity 1/sqrt(n), when the alternatives are given by smoothness ellipsoids with an L2-ball of (squared) radius rho removed. It is known that, for a fixed Sobolev type ellipsoid of smoothness beta and size M, a rho which is of order n to the power -4 beta/(4 beta+1)} is the critical separation rate, in the sense that the minimax error of second kind over alpha-tests stays asymptotically between 0 and 1 strictly (Ingster, 1982). In addition, Ermakov (1990) found the sharp asymptotics of the minimax error of second kind at the separation rate. For adaptation over both beta and M in that context, it is known that a loglog-penalty over the separation rate for rho is necessary for a nonzero asymptotic power. Here, following an example in nonparametric estimation related to the Pinsker constant, we investigate the adaptation problem over the ellipsoid size M only, for fixed smoothness degree beta. It is established that the sharp risk asymptotics can be replicated in that adaptive setting, if rho tends to zero slower than the separation rate. The penalty for adaptation here turns out to be a sequence tending to infinity arbitrarily slowly.

math.ST↗

arXiv2012

An asymptotic error bound for testing multiple quantum hypotheses

We consider the problem of detecting the true quantum state among $r$ possible ones, based of measurements performed on $n$ copies of a finite-dimensional quantum system. A special case is the problem of discriminating between $r$ probability measures on a finite sample space, using $n$ i.i.d. observations. In this classical setting, it is known that the averaged error probability decreases exponentially with exponent given by the worst case binary Chernoff bound between any possible pair of the $r$ probability measures. Define analogously the multiple quantum Chernoff bound, considering all possible pairs of states. Recently, it has been shown that this asymptotic error bound is attainable in the case of $r$ pure states, and that it is unimprovable in general. Here we extend the attainability result to a larger class of $r$-tuples of states which are possibly mixed, but pairwise linearly independent. We also construct a quantum detector which universally attains the multiple quantum Chernoff bound up to a factor 1/3.

quant-ph↗

arXiv2010

Exponential error rates in multiple state discrimination on a quantum spin chain

We consider decision problems on finite sets of hypotheses represented by pairwise different shift-invariant states on a quantum spin chain. The decision in favor of one of the hypotheses is based on outputs of generalized measurements performed on local states on blocks of finite size. We assume existence of the mean quantum Chernoff distances of any pair of states from the given set and refer to the minimum of them as the mean generalized quantum Chernoff distance. We establish that this minimum specifies an asymptotic bound on the exponential decay of the averaged probability of rejecting the true state in increasing block size, if the mean quantum Chernoff distance of any pair of the hypothetic states is achievable as an asymptotic error exponent in the corresponding binary problem. This assumption is in particular fulfiled by shift-invariant product states (i.i.d. states). Further, we provide a constructive proof for the existence of a sequence of quantum tests in increasing block size, which achieves an asymptotic error exponent which is equal to the mean generalized quantum Chernoff distance of the given set of states up to a factor, which depends on the set itself. It can be arbitrary close to 1 and is not less than $1/m$ for $m$ being the number of different pairs of states from the set considered.

quant-ph↗

arXiv2009

The Chernoff lower bound for symmetric quantum hypothesis testing

We consider symmetric hypothesis testing in quantum statistics, where the hypotheses are density operators on a finite-dimensional complex Hilbert space, representing states of a finite quantum system. We prove a lower bound on the asymptotic rate exponents of Bayesian error probabilities. The bound represents a quantum extension of the Chernoff bound, which gives the best asymptotically achievable error exponent in classical discrimination between two probability measures on a finite set. In our framework, the classical result is reproduced if the two hypothetic density operators commute. Recently, it has been shown elsewhere [Phys. Rev. Lett. 98 (2007) 160504] that the lower bound is achievable also in the generic quantum (noncommutative) case. This implies that our result is one part of the definitive quantum Chernoff bound.

quant-ph↗

arXiv2009

Asymptotic equivalence of spectral density estimation and gaussian white noise

We consider the statistical experiment given by a sample of a stationary Gaussian process with an unknown smooth spectral density f. Asymptotic equivalence, in the sense of Le Cam's deficiency Delta-distance, to two Gaussian experiments with simpler structure is established. The first one is given by independent zero mean Gaussians with variance approximately the value of f in points of a uniform grid (nonparametric Gaussian scale regression). This approximation is closely related to well-known asymptotic independence results for the periodogram and corresponding inference methods. The second asymptotic equivalence is to a Gaussian white noise model where the drift function is the log-spectral density. This represents the step from a Gaussian scale model to a location model, and also has a counterpart in established inference methods, i.e. log-periodogram regression. The problem of simple explicit equivalence maps (Markov kernels), allowing to directly carry over inference, appears in this context but is not solved here.

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