About

Michael Nussbaum is a professor in the Department of Mathematics at Cornell University. He holds a Ph.D. obtained in 1979 and a Dr. Sc. earned in 1990 from the Academy of Sciences Berlin, Germany. His research focuses on applying ideas and concepts of mathematical statistics to the emerging field of quantum statistics, which develops in the context of technological breakthroughs in quantum engineering and communication. His work emphasizes the probabilistic nature of quantum measurements and the importance of statistical inference in analyzing and validating quantum experiments. Nussbaum's research involves the analysis of quantum statistical models and their classical counterparts, exploring non-commutative statistical decision theory with connections to operator algebra, quantum information, and quantum probability. A key aspect of his work is understanding how complex statistical models can be approximated by simpler ones in the asymptotic limit, facilitating the transfer of optimal procedures and enhancing the conceptual understanding of asymptotic inference. His contributions include investigating hypothesis testing, discrimination between quantum states, and Gaussian approximation of nonparametric quantum statistical models.

Research topics

• Physics
• Quantum mechanics
• Mathematics
• Statistical physics
• Applied mathematics
• Mathematical optimization
• Statistics
• Mathematical analysis
• Combinatorics
• Algorithm

Selected publications

• Asymptotic inference in a stationary quantum time series

  arXiv (Cornell University) · 2025-11-30

  preprintOpen access1st authorCorresponding

  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.

  PublisherOA PDFDOI

• Asymptotic Equivalence for Nonparametric Regression

  arXiv (Cornell University) · 2024-12-19 · 39 citations

  preprintOpen accessSenior author

  We consider a nonparametric model $\mathcal{E}^{n},$ generated by independent observations $X_{i},$ $i=1,...,n,$ with densities $p(x,θ_{i}),$ $i=1,...,n,$ the parameters of which $θ_{i}=f(i/n)\in Θ$ are driven by the values of an unknown function $f:[0,1]\rightarrow Θ$ 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 $Y_{i}=Γ(f(i/n))+\varepsilon _{i},$ where $\varepsilon_{1},...,\varepsilon _{n}$ are i.i.d. standard normal r.v.'s, the function $Γ(θ):Θ\rightarrow \mathrm{R}$ satisfies $Γ^{\prime}(θ)=\sqrt{I(θ)}$ and $I(θ)$ is the Fisher information corresponding to the density $p(x,θ).$

  PublisherOA PDFDOI

• On the optimal error exponents for classical and quantum antidistinguishability

  Letters in Mathematical Physics · 2024 · 11 citations

  • Mathematics
  • Statistical physics
  • Quantum mechanics
  PublisherDOI

• Asymptotic Equivalence for Nonparametric Generalized Linear Models

  arXiv (Cornell University) · 2024-12-19

  preprintOpen accessSenior author

  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 $Δ$; 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 Hölder ball with exponent $β>\frac 12 ,$ we establish global asymptotic equivalence to observations of a signal $Γ(f(t))$ in Gaussian white noise, where $Γ$ 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.

  PublisherOA PDFDOI

• A functional Hungarian construction for sums of independent random variables

  arXiv (Cornell University) · 2024-12-19 · 1 citations

  preprintOpen accessSenior author

  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 Komlós-Major-Tusnády 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 Hölder ball of smoothness $1/2$. This specific form of a strong approximation is useful for proving asymptotic equivalence of statistical experiments.

  PublisherOA PDFDOI

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

  Bernoulli · 2023 · 2 citations

  Senior authorCorresponding
  • Mathematics
  • Applied mathematics
  • Statistical physics

  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.

  PublisherDOI

• On the optimal error exponents for classical and quantum antidistinguishability

  arXiv (Cornell University) · 2023-09-07

  preprintOpen access

  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 $ψ$-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.

  PublisherOA PDFDOI

• Minimax nonparametric estimation of pure quantum states

  The Annals of Statistics · 2022-02-01

  articleSenior author

  In classical statistics, Pinsker’s theorem provides an exact asymptotic minimax bound in nonparametric estimation, improving upon optimal rates of convergence results. We obtain a quantum version of the theorem by establishing asymptotic minimax results for estimation of the displacement vector in a quantum Gaussian white noise model, given by a sequence of shifted vacuum states. Analogous results are then obtained for estimation of a general pure state from an ensemble of identically prepared, independent quantum systems, using the recently established local asymptotic equivalence to the quantum Gaussian white noise model. Optimality holds with respect to the most fundamental distance measure for quantum states, that is, trace norm distance, and in a true quantum sense, allowing for all possible measurements. Adaptive estimators are also obtained for the above cases. As an application, we obtain asymptotic minimax adaptive estimators for Wigner functions of pure states.

  PublisherDOI

• Local Asymptotic Normality and Optimal Estimation of low-rank Quantum Systems

  arXiv (Cornell University) · 2021 · 1 citations

  Senior authorCorresponding
  • Mathematics
  • Applied mathematics
  • Statistical physics

  In classical statistics, a statistical experiment consisting of $n$ i.i.d observations from d-dimensional multinomial distributions can be well approximated by a $d-1$ dimensional Gaussian distribution. In a quantum version of the result it has been shown that a collection of $n$ qudits 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. We also outline a two-stage procedure for the estimation of the low-rank qudit, where we obtain an estimator which is sharp minimax optimal. For the estimation of a linear functional of the quantum state, we construct an estimator, analyze the risk and use quantum LAN to show that our estimator is also optimal in the minimax sense.

  Publisher

• Local Asymptotic Equivalence of Pure States Ensembles and Quantum Gaussian White Noise

  RePEc: Research Papers in Economics · 2017-01-01

  preprintSenior author

  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. ;Classification-JEL: 62B15; 62G05; 62G10; 81P50

  Publisher

Recent grants

• Asymptotic Equivalence of Statistical Experiments

  NSF · $374k · 2003–2009

• New Horizons in Statistical Decision Theory

  NSF · $370k · 2014–2020

• Asymptotic Methods in Quantum Statistics

  NSF · $240k · 2008–2011

• Asymptotic Inference for Locally Stationary Processes

  NSF · $369k · 2011–2015

Frequent coauthors

• Mădălin Guţǎ

  27 shared

• Cristina Butucea

  École Nationale de la Statistique et de l'Administration Économique

  27 shared

• Arleta Szkoła

  Max Planck Institute for Mathematics in the Sciences

  11 shared

• Harrison H. Zhou

  Yale University

  10 shared

• Georgi K. Golubev

  Cornell University

  10 shared

• Ion Grama

  6 shared

• Wolfgang Karl Härdle

  National Yang Ming Chiao Tung University

  6 shared

• Pengsheng Ji

  5 shared