This paper presents a series of general results about the optimal estimation of physical transformations in a given symmetry group. In particular, it is shown how the different symmetries of the problem determine different properties of the optimal estimation strategy. The paper also contains a discussion about the role of entanglement between the representation and multiplicity spaces and about the optimality of square-root measurements.
This paper presents a series of general results about the optimal estimation of physical transformations in a given symmetry group. In particular, it is shown how the different symmetries of the problem determine different properties of the optimal estimation strategy. The paper also contains a discussion about the role of entanglement between the representation and multiplicity spaces and about the optimality of square-root measurements.
The maximum likelihood strategy for the estimation of group parameters allows one to derive in a general fashion optimal measurements, optimal signal states, and their relations with other information theoretical quantities. These results provide deep insight into the general structure underlying optimal quantum estimation strategies. The entanglement between representation spaces and multiplicity spaces of the group action appears to be the unique kind of entanglement which is really useful for the optimal estimation of group parameters.
We derive the optimal input states and the optimal quantum measurements for estimating the unitary action of a given symmetry group, showing how the optimal performance is obtained with a suitable use of entanglement. Optimality is defined in a Bayesian sense, as minimization of the average value of a given cost function. We introduce a class of cost functions that generalizes the Holevo class for phase estimation, and show that for states of the optimal form all functions in such a class lead to the same optimal measurement. As a first application of the main result is the complete proof of the optimal efficiency in the transmission of a Cartesian reference frame. As a second application, we derive the optimal estimation of a completely unknown two-qubit maximally entangled state, provided that N copies of the state are available. In the limit of large N, the fidelity of the optimal estimation is shown to be $1\ensuremath{-}3∕(4N)$.
It is impossible to overstate the importance of symmetry in physics and mathematics. Symmetry arguments play a central role in a broad range of problems from simplifying a system of linear equations to a deep role in organizing the fundamental principles of physics. They are used, for instance, in Noether’s theorem to find the consequences of symmetry of a dynamics. For many systems of interest, the dynamics are sufficiently complicated that one cannot hope to characterize their evolution completely, whereas by appealing to the symmetries of the dynamical laws one can easily infer many useful results. \n \nIn part I of this thesis we study the problem of finding the consequences of symmetry of a (possibly open) dynamics from an information-theoretic perspective. The study of this problem naturally leads us to the notion of asymmetry of quantum states. The asymmetry of a state relative to some symmetry group specifies how and to what extent the given symmetry is broken by the state. Characterizing these is found to be surprisingly useful to constrain which final states of the system can be reached from a given initial state. Another motivation for the study of asymmetry comes from the field of quantum metrology and relatedly the field of quantum reference frames. It turns out that the degree of success one can achieve in many metrological tasks depends only on the asymmetry properties of the state used for metrology. We show that some ideas and tools developed in the field of quantum information theory are extremely useful to study the notion of asymmetry of states and therefore to find the consequences of symmetry of an open or closed system dynamics. \n \nIn part II of this thesis we present a novel application of symmetry arguments in the field of quantum estimation theory. We consider a family of multi-copy estimation problems wherein one is given n copies of an unknown quantum state according to some prior distribution and the goal is to estimate certain parameters of the given state. In particular, we are interested to know whether collective measurements are useful and if so to find an upper bound on the amount of entanglement which is required to achieve the optimal estimation. We introduce a new approach to this problem by considering the symmetries of the prior and the symmetries of the parameters to be estimated. We show that based on these symmetries one can find strong constraints on the amount of entanglement required to implement the optimal measurement. In order to infer properties of the optimal estimation procedure from the symmetries of the parameters and the prior we come up with a generalization of Schur-Weyl duality. Just as Schur-Weyl duality has many applications to quantum information theory and quantum algorithms so too does this generalization. In this thesis we explore some of these applications.
We study the problem of joint estimation of real squeezing and amplitude of the radiation field, deriving the measurement that maximizes the probability density of detecting the true value of the unknown parameters. More generally, we provide a solution for the problem of estimating the unknown unitary action of a nonunimodular group. Remarkably, in this case the optimal measurements do not coincide with the so called square-root measurements. We compare the efficiency of different quantum states for detecting unknown squeezing and displacement, providing strategies to improve the sensitivity, and showing how entanglement can be used to approximate perfect estimation.
We construct the optimal strategy for the estimation of an unknown unitary transformation $U\ensuremath{\in}\mathrm{SU}(d).$ This includes, in addition to a convenient measurement on a probe system, finding which is the best initial state on which U is to act. When $U\ensuremath{\in}\mathrm{SU}(2),$ such an optimal strategy can be applied to simultaneously estimate both the direction and the strength of a magnetic field, and shows how to use a spin 1/2 particle to transmit information about a whole coordinate system instead of only a direction in space.
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.