Quantum limits to parameter estimation Introduction Many experiments can be thought of as comprising two steps: (i) a preparation procedure in which the system to be measured is isolated and prepared, and the apparatus is initialized; and (ii) a measurement step in which the system is coupled to an apparatus and the measurement result recorded. The preparation procedure can be specified by a set of classical parameters, or settings of a physical device. The measurement results are random classical variables that will be correlated with the preparation procedure. In this chapter we are concerned with the case in which the classical parameters specifying the preparation of the state are imperfectly known. Then, assuming that the physical system is well understood, these correlations allow the unknown parameters to be estimated from the measurement results. As we saw in the last chapter, in quantum mechanics the results of measurements are generally statistical, even when one has complete knowledge of the preparation procedure. A single preparation step and measurement step might not be sufficient to estimate a parameter well. Thus it is common to repeat the two steps of preparation and measurement on a large number of systems, either all at one time or sequentially. Whether measuring one quantum system or many, one is faced with a number of questions. How should one prepare the system state? What sort of measurement should one make on the system?
Quantum limits to parameter estimation Introduction Many experiments can be thought of as comprising two steps: (i) a preparation procedure in which the system to be measured is isolated and prepared, and the apparatus is initialized; and (ii) a measurement step in which the system is coupled to an apparatus and the measurement result recorded. The preparation procedure can be specified by a set of classical parameters, or settings of a physical device. The measurement results are random classical variables that will be correlated with the preparation procedure. In this chapter we are concerned with the case in which the classical parameters specifying the preparation of the state are imperfectly known. Then, assuming that the physical system is well understood, these correlations allow the unknown parameters to be estimated from the measurement results. As we saw in the last chapter, in quantum mechanics the results of measurements are generally statistical, even when one has complete knowledge of the preparation procedure. A single preparation step and measurement step might not be sufficient to estimate a parameter well. Thus it is common to repeat the two steps of preparation and measurement on a large number of systems, either all at one time or sequentially. Whether measuring one quantum system or many, one is faced with a number of questions. How should one prepare the system state? What sort of measurement should one make on the system?
The concept of a quantum state represents one of the most fundamental pillars of the paradigm of quantum theory.'3 Contrary to its mathematical elegance and convenience in calculations, the physical interpretation of a quantum state is not so transparent. The problem is that the quantum state (described either by a state vector, or density operator or a phase-space probability density distribution) does not have a well defined objective status, i.e. a state vector is not an objective property of a particle. According to Peres (see,1 p. 374): "There is no physical evidence whatsoever that every physical system has at every instant a well defined state. .. In strict interpretation of quantum theory these mathematical symbols [i.e., state vectorsi represent statistical information enabling us to compute the probabilities of occurrence of specific events." Once this point of view is adopted then it becomes clear that any "measurement" or a reconstruction of a density operator (or its mathematical equivalent) can be understood exclusively as an expression of our knowledge about the quantum mechanical state based on a certain set of measured data. To be more specific, any quantum-mechanical reconstruction scheme is nothing more than an a posteriori estimation of the density operator of a quantum-mechanical (microscopic) system based on data obtained with the help of a macroscopic measurement apparatus.3 The quality of the reconstruction depends on the "quality" of the measured data and the efficiency of the reconstruction procedure with the help of which the data analysis is performed. In particular, we can specify three different situations. Firstly, when all system observables are precisely measured. In this case the complete reconstruction of an initially unknown state can be performed (we will call this the reconstruction on the complete observation level) . Secondly, when just part of the system observables is precisely measured then one cannot perform a complete reconstruction of the measured state. Nevertheless, the reconstructed density operator still uniquely determines mean values of the measured observables (we will denote this scheme as reconstruction on incomplete observation levels) . Finally, when measurement does not provide us with sufficient information to specify the exact mean values (or probability distributions) but only the frequencies of appearances of eigenstates of the measured observables, then one can perform an estimation (e.g. reconstruction based on quantum Bayesian inference) which is the "best" with respect to the given measured data and the a priori knowledge about the state of the measured system.
We examine the conditions under which quantum parameter estimation is compatible with the concept of a ``true'' quantity value: a unique value of the estimated quantity that exists prior to and independently of measurement. While quantum parameter estimation typically assumes that the quantity to be estimated is a parameter, we argue that this assumption can be lifted, opening the way to a more general formulation of estimation that may apply even in situations where the quantity to be estimated is no longer a parameter, but instead a quantum variable susceptible to quantum fluctuations. We identify three assumptions that characterize the way estimation is often presented in the quantum metrology literature, namely, as a procedure for inferring an unknown ``true value", which is assumed to be unique and to exist prior to and independently of measurement. Using our more general framework, we prove a no-go theorem showing that the preceding assumptions are jointly incompatible with two further assumptions: the completeness of the quantum state (no `hidden variables'), and the presence of entanglement between the probe systems and the system under measurement. We note that the presence of such entanglement is a generic prediction of standard quantum theory. Consequently, if one is not willing to accept hidden variables, nor willing to modify standard quantum theory by introducing a new physical mechanism that destroys entanglement at some scale or regime, then a realist account of estimation is ultimately impossible.
MEASUREMENT INTERACTIONS In the present interpretation terms such as ‘measurement,’ ‘observation,’ ‘find,’ ‘classical,’ and ‘apparatus’ do not appear in the formulation of quantum mechanics. Yet it must of course be true that quantum mechanics can be used to predict features of the results of measurements – in particular their statistics. What is required, then, is an account in quantum mechanical terms of those physical interactions capable of being used to perform measurements of quantum dynamical variables, from which it follows that the predicted probabilities for specified behavior of certain quantum systems in these interactions coincide with what, on a more conventional interpretation, are simply postulated as the quantum mechanical probabilities for various measurement results on a system with a given quantum state. It is necessary to see how this consequence may be derived, not only in order to explicate the Born rules, but also to explain the role of quantum states within the present interpretation. The first step toward the required account is to analyze the nature of an interaction capable of being used to perform a measurement of the value of a quantum dynamical variable in a quantum system. Any such interaction will be called an M -type interaction, whether or not it is in fact used to perform a measurement. The basic characteristic of an M -type interaction is that it establishes a correlation between the initial dynamical state of one quantum system σ (the object system) and the final dynamical state of another quantum system α (the apparatus system).
Introduction I place a tape measure with one end on the floor next to a table, read the height of the table from the tape, and record the result in a notebook. What are the essential features of this measurement process ? The key point is the establishment of a correlation between a physical property (the height) of a measured system (the table) and a suitable record (in the notebook), which is itself a physical property of some other system. It will be convenient in what follows to think of this record as part of the measuring apparatus, which consists of everything essential to the measuring process apart from the measured system. Human beings are not essential to the measuring process. The height of a table could be measured by a robot. In the modern laboratory, measurements are often carried out by automated equipment, and the results stored in a computer memory or on magnetic tape, etc. While scientific progress requires that human beings pay attention to the resulting data, this may occur a long time after the measurements are completed. In this and the next chapter we consider measurements as physical processes in which a property of some quantum system, which we shall usually think of as some sort of “particle”, becomes correlated with the outcome of the measurement, itself a property of another quantum system, the “apparatus”. Both the measured system and the apparatus which carries out the measurement are to be thought of as parts of a single closed quantum mechanical system. This makes it possible to apply the principles of quantum theory developed in earlier chapters.
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.
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.