Informationally complete measurements allow the estimation of expectation values of any operator on a quantum system, by changing only the data processing of the measurement outcomes. In particular, an informationally complete measurement can be used to perform quantum tomography, namely to estimate the density matrix of the quantum state. The data processing is generally nonunique, and can be optimized according to a given criterion. In this paper we provide the solution of the optimization problem which minimizes the variance in the estimation. We then consider informationally complete measurements performed over bipartite quantum systems focusing attention on universally covariant measurements, and compare their statistical efficiency when performed either locally or globally on the two systems. Among global measurements we consider the special case of Bell measurements, which allow us to estimate the expectation of a restricted class of operators. We compare the variance in the three cases: local, Bell, and unrestricted global---and derive conditions for the operators to be estimated such that one type of measurement is more efficient than the other. In particular, we find that for factorized operators and Bell projectors the Bell measurement always performs better than the unrestricted global measurement, which in turn outperforms the local one. For estimation of the matrix elements of the density operator, the relative performances depend on the basis on which the state is represented, and on the matrix element being diagonal or off-diagonal, however, with the global unrestricted measurement generally performing better than the local one.
Informationally complete measurements allow the estimation of expectation values of any operator on a quantum system, by changing only the data processing of the measurement outcomes. In particular, an informationally complete measurement can be used to perform quantum tomography, namely to estimate the density matrix of the quantum state. The data processing is generally nonunique, and can be optimized according to a given criterion. In this paper we provide the solution of the optimization problem which minimizes the variance in the estimation. We then consider informationally complete measurements performed over bipartite quantum systems focusing attention on universally covariant measurements, and compare their statistical efficiency when performed either locally or globally on the two systems. Among global measurements we consider the special case of Bell measurements, which allow us to estimate the expectation of a restricted class of operators. We compare the variance in the three cases: local, Bell, and unrestricted global---and derive conditions for the operators to be estimated such that one type of measurement is more efficient than the other. In particular, we find that for factorized operators and Bell projectors the Bell measurement always performs better than the unrestricted global measurement, which in turn outperforms the local one. For estimation of the matrix elements of the density operator, the relative performances depend on the basis on which the state is represented, and on the matrix element being diagonal or off-diagonal, however, with the global unrestricted measurement generally performing better than the local one.
The expectation value of an arbitrary operator O can be obtained via a universal measuring apparatus that is independent of O, by changing only the data-processing of the outcomes. Such a ``universal detector'' performs a joint measurement on the system and on a suitable ancilla prepared in a fixed state, and is equivalent to a positive operator valued measure (POVM) for the system that is ``informationally complete''. The data processing functions generally are not unique, and we pose the problem of their optimization, providing some examples for covariant POVM's, in particular for SU(d) covariance group.
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.
In standard optical tomographic methods, the off-diagonal elements of a density matrix ρ are measured indirectly. Thus, the reconstruction of ρ, even if it is based on linear inversion, typically magnifies small errors in the experimental data. Recently, an optimal tomography solution measuring all the elements of ρ one-by-one without error magnification has been theoretically proposed. We implemented this method for two-qubit polarization states. For comparison, we also experimentally implemented other well-known tomographic protocols, either based solely on local measurements (of, e.g., the Pauli operators and James-Kwiat-Munro-White projectors) or with mutually unbiased bases requiring both local and global measurements. We reconstructed seventeen separable, partially and maximally entangled two-qubit polarization states. Our experiments show that our method has the highest stability against errors in comparison to other quantum tomographies. In particular, we demonstrate that each optimally-reconstructed state is embedded in an uncertainty circle of the smallest radius, both in terms of trace distance and disturbance. We explain how to experimentally estimate uncertainty radii for all the implemented tomographies and show that, for each reconstructed state, the relevant uncertainty circles intersect indicating the approximate location of the corresponding physical density matrix.
This thesis will focus on deriving informationally-optimal quantum state tomographic measurements on single-qubit systems, with the definition of informationally optimal to be defined as those measurements which maximize the average information gain. The informationally-optimal measurements that will be covered include projective measurements (formally building off of the work of [1] and putting it on firm, information-theoretic foundations), and more general types of quantum measurements called directional (also known as rank-one) positive operator-valued measure (POVM) measurements, and, finally, adaptive directional POVM measurements. For projective measurements, we build on the work of Wootters and Fields ([1]) and show, via analytical methods and maximizing the average information gained from the measurements, that orthogonal projective measurements are optimal. We then go on to define what we call directional POVMs, which are POVMs whose elements are given by weighted projection operators. We consider these measurements in the context of static quantum state tomography (QST), wherein we use the same choice of measurement throughout our analysis/experiment. By examining these directional POVMs acting on single qubit systems, we show that the most informationally optimal of these types of measurements are informationally complete directional POVMs that are both minimal, informationally complete (MIC) and symmetric, informationally complete (SIC) POVMs, simultaneously. Finally, we deal with the case of adaptive directional POVM measurements.
Quantum state tomography (QST) remains the gold standard for benchmarking and verification of near-term quantum devices. While QST for a generic quantum many-body state requires an exponentially large amount of resources, most physical quantum states are structured and can often be represented by a much smaller number of parameters, making efficient QST potentially possible. A prominent example is a matrix product state (MPS) or a matrix product density operator (MPDO), which is believed to represent most physical states generated by one-dimensional (1D) quantum devices. We study whether a general MPS/MPDO state can be recovered with bounded errors using only a number of state copies polynomial in the number of qubits, which is necessary for efficient QST. To make this question practically interesting, we assume only local measurements of qubits directly on the target state. By using a local symmetric informationally complete positive operator-valued measurement (SIC-POVM), we provide a positive answer to the above question for a variety of common many-body quantum states, including typical short-range entangled states, random MPS/MPDO states, and thermal states of one-dimensional Hamiltonians. In addition, we also provide an affirmative no answer for certain long-range entangled states such as a family of generalized GHZ states, but with the exception of target states that are known to have real-valued wavefunctions. Our answers are supported by a near-perfect agreement between an efficient calculation of the Cramer-Rao bound that rigorously bounds the sample complexity and numerical optimization results using a machine learning assisted maximal likelihood estimation (MLE) algorithm. This agreement also leads to an optimal QST protocol using local SIC-POVM that can be practically implemented on current quantum hardware and is highly efficient for most 1D physical states.
Despite multipartite entanglement being a global property of a quantum state, a number of recent works have made it clear that it can be quantified using only local measurements. This is appealing because local measurements are the easiest to implement on current quantum hardware. However, it remains an open question what protocol one should use in order to minimize the resources required to estimate multipartite entanglement from local measurements alone. In this work, we construct and compare several estimators of multipartite entanglement based solely on the data from local measurements. We first construct statistical estimators for a broad family of entanglement measures using local randomized measurement (LRM) data before providing a general criterion for the construction of such estimators in terms of projective 2-designs. Importantly, this allows us to derandomize the multipartite estimation protocol based on LRMs. In particular, we show how local symmetric, informationally complete positive operator-valued measures enable multipartite entanglement quantification with only a single measurement setting. For all estimators, we provide both the classical postprocessing cost and rigorous performance guarantees in the form of analytical upper bounds on the number of measurements needed to estimate the measures to any desired precision.