This thesis is entitled Quantum Local Asymptotic Normality and other questions of Quantum Statistics ,. Quantum statistics are statistics on quantum objects. In classical statistics, we usually start from the data. Indeed, if we want to predict the weather, and can measure the wind or the temperature, we can measure both. On the other hand the laws of physics themselves forbid us to measure simultaneously the speed and the position of an electron. We therefore have to start with the observed object itself, and choose the best measurement for our purposes.\n\nMy main result is that, for all statistical purposes, numerous copies of the same spin (magnetic state of an electron) is equivalent to a Gaussian state of a quantum harmonic oscillator, typically laser light. We can extend this to higher dimensions. As an application, we get an (asymptotically) optimal estimation scheme for unknown spins. The idea is to transform the spins into laser light, and use the already known optimal estimation methods for laser light.\n\nThe thesis furthermore includes four smaller problems, notably how to estimate a unitary (natural) evolution very quickly, and how best to decide which is the state of a quantum object, among a finite number
This thesis is entitled Quantum Local Asymptotic Normality and other questions of Quantum Statistics ,. Quantum statistics are statistics on quantum objects. In classical statistics, we usually start from the data. Indeed, if we want to predict the weather, and can measure the wind or the temperature, we can measure both. On the other hand the laws of physics themselves forbid us to measure simultaneously the speed and the position of an electron. We therefore have to start with the observed object itself, and choose the best measurement for our purposes.\n\nMy main result is that, for all statistical purposes, numerous copies of the same spin (magnetic state of an electron) is equivalent to a Gaussian state of a quantum harmonic oscillator, typically laser light. We can extend this to higher dimensions. As an application, we get an (asymptotically) optimal estimation scheme for unknown spins. The idea is to transform the spins into laser light, and use the already known optimal estimation methods for laser light.\n\nThe thesis furthermore includes four smaller problems, notably how to estimate a unitary (natural) evolution very quickly, and how best to decide which is the state of a quantum object, among a finite number
Observations or measurements taken of a quantum system (a small number of fundamental particles) are inherently random.If the state of the system depends on unknown parameters, then the distribution of the outcome depends on these parameters too, and statistical inference problems result.Often one has a choice of what measurement to take, corresponding to different experimental set-ups or settings of measurement apparatus.This leads to a design problem-which measurement is best for a given statistical problem.This paper gives an introduction to this field in the most simple of settings, that of estimating the state of a spin-half particle given n independent copies of the particle.We show how in some cases asymptotically optimal measurements can be constructed.Other cases present interesting open problems, connected to the fact that for some models, quantum Fisher information is in some sense non-additive.In physical terms, we have non-locality without entanglement.
Observations or measurements taken of a quantum system (a small number of fundamental particles) are inherently random. If the state of the system depends on unknown parameters, then the distribution of the outcome depends on these parameters too, and statistical inference problems result. Often one has a choice of what measurement to take, corresponding to different experimental set-ups or settings of measurement apparatus. This leads to a design problem--which measurement is best for a given statistical problem. This paper gives an introduction to this field in the most simple of settings, that of estimating the state of a spin-half particle given n independent copies of the particle. We show how in some cases asymptotically optimal measurements can be constructed. Other cases present interesting open problems, connected to the fact that for some models, quantum Fisher information is in some sense non-additive. In physical terms, we have non-locality without entanglement.
The subject of this thesis, quantum estimation theory, is inspired by recent developments, both practical and theoretical, in the new field of quantum information and computation. The main task of quantum estimation theory, the subject of this thesis, is that given many identical copies of an unknown quantum system one wants to guess or estimate what the system is. This is done by performing an measurement and analyzing the data obtained from it.
A quantum state is represented by a density matrix. In some cases one does not need to know the whole density matrix, sometimes a partial characterization is enough. Chapter 2 of this thesis is about optimally estimating the spectrum, i.e. the eigenvalues, of a density matrix. This is useful, for example, for estimating the entanglement of a bipartite pure state. The results of chapter 2 tell us that one can optimally estimate bipartite entanglement with LOCC measurements on one of the two parties.
Unitary operations transform density matrices to density matrices. They are the gates of quantum information processing. Suppose one has one such unitary operation but one does not know what it does. In order to characterize it one needs to know how it transforms any input state. In order to know how a particular state is transformed, one can prepare that state, feed it to the unitary many times, and measure the output. One state is, of course, not enough. It turns out that the unitary can be completely characterized with a finite set of states, this is known as quantum process tomography. Unfortunately, in many cases it is difficult to have available in the lab all the input states needed to perform quantum process tomography. It is here where entanglement comes to the rescue. One pair of maximally entangled qubits, where one of them is used as input for the unitary gate and the other is left untouched, is sufficient. All the information about the unknown gate is contained in the joint output state. This is the problem treated in chapter 3. Now the output state needs to be measured. Again, one could measure the two qubits separately (LOCC) or jointly (collective measurement). Is there anything to gain by performing joint measurements? The answer is yes, optimal collective measurements are at least 3 times more accurate than any LOCC measurement.
Chapter 4 treats the problem of estimating commuting unitary gates. There it is shown that this extra information makes it unnecessary to use an entangled input state and to use collective measurements at the output.
In chapters 3 and 4 it is assumed that only one copy of the unitary gate is available. If, however, more than one copy (say N ) are available at the same time one could use an N-fold entangled state as input. If no entanglement is used, classical statistics tells us that the error of the estimation is proportional to 1/√N. In the literature, it has been shown that by using an entangled state in the input one can achieve an error proportional to 1/N. However, this has been shown only for qubit gates. In chapter 5 important steps are taken in the direction of showing the same thing for a general unitary gate, but this remains an open problem.
This thesis seeks to develop a general method for solving so-called quantum realizability problems, which are questions of the following form: under which conditions does there exist a quantum state exhibiting a given collection of properties? The approach adopted by this thesis is to utilize mathematical techniques previously developed for the related problem of property estimation which is concerned with learning or estimating the properties of an unknown quantum state. Our primary result is to recognize a correspondence between (i) property values which are realized by some quantum state, and (ii) property values which are occasionally produced as estimates of a generic quantum state. Chapter 3 reviews concepts of stability and norm minimization from geometric invariant theory and non-commutative optimization theory for the purposes of characterizing the flow of a quantum state under the action of a reductive group. Chapter 4 demonstrates how the gradient of this flow, also called the moment map, can be estimated by performing a covariant quantum measurement on a large number of identical copies of the quantum state. Chapter 5 outlines the correspondence between between the realizability of a moment map value on one hand and the asymptotic likelihood it is produced as an estimate on the other hand. By appropriately composing these moment map estimation schemes, we derive necessary and sufficient conditions for the existence of a quantum state jointly realizing any finite collection of moment maps. Chapter 6 applies these insights to the quantum marginal problem and is a duplication of arXiv:2211.00685.
This PHD thesis is concerned with uncertainty relations in quantum probability theory, state estimation in quantum stochastics, and natural bundles in differential geometry. After some comments on the nature and necessity of decoherence in open systems and its absence in closed ones, we prove sharp, state-independent inequalities reflecting the Heisenberg principle, the necessity of decoherence and the impossibility of perfect joint measurement. These bounds are used to judge how far a particular measurement is removed from the optimal one. We do this for a qubit interacting with the quantized EM field, continually probed using homodyne detection. We calculate to which extent this joint measurement is optimal. We then propose a two-step strategy to determine the (possibly mixed) state of n identically prepared qubits, and prove that it is asymptotically optimal in a local minimax sense, using `Quantum Local Asymptotic Normality' for qubits. We propose a physical implementation of QLAN, based on interaction with the quantized EM field. In differential geometry, a bundle is called `natural' if diffeomorphisms of the base lift to automorphisms of the bundle. We slightly extend this notion to `infinitesimal naturality', requiring only vector fields (infinitesimal diffeomorphisms) to lift. We classify these bundles. Physical fields that transform under (infinitesimal) space-time transformations must be described in terms of (infinitesimally) natural bundles. All spin structures are infinitesimally natural. Our framework thus encompasses Fermionic fields, for which the notion of a natural bundle is too restrictive. Interestingly, generalized spin structures (e.g. spin-c structures) are not always infinitesimally natural. We classify the ones that are. Depending on the gauge group at hand, this can significantly reduce the number of allowed space-time topologies.
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 nonparametric 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.