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Quantum indirect estimation theory and $AB$-infocompleteness

Giacomo Mauro D’Ariano, Paolo Perinotti, Massimiliano Federico Sacchi

Published Aug 29, 2007• 0 citations
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Abstract

We introduce the quantum indirect estimation theory, which provides a general framework to address the problem of which ensemble averages can be estimated by means of an available set of measuring apparatuses, e. g. estimate the ensemble average of an observable by measuring other observable. A main ingredient in this approach is that of informationally complete (infocomplete in short) measurements, which allow to estimate the ensemble average of any arbitrary system operator, as for quantum tomography. This naturally leads to the more stringent concept of AB-informationally complete measurements, by which one can estimate jointly all the moments of two incompatible observables A and B. After analyzing all general properties of such measurements, we address the problem of their optimality, and we completely solve the case of qubits, showing that a sigma_x sigma_y-infocomplete measurement is less noisy than any infocomplete one. We will also discuss the relation between the concept of AB-infocompleteness and the notion of joint measurement of observables A and B.

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We introduce the quantum indirect estimation theory, which provides a general framework to address the problem of which ensemble averages can be estimated by means of an available set of measuring apparatuses, e.g., estimating the ensemble average of an observable by measuring other observables. A main ingredient in this approach is that of informationally complete (infocomplete) measurements, which allow us to estimate the ensemble average of any arbitrary system operator, as for quantum tomography. This naturally leads to the more stringent concept of $AB$-informationally complete measurements, by which one can estimate jointly all the moments of two incompatible observables $A$ and $B$. After analyzing all general properties of such measurements, we address the problem of their optimality, and we completely solve the case of qubits, showing that a ${\ensuremath{\sigma}}_{x}{\ensuremath{\sigma}}_{y}$-infocomplete measurement is less noisy than any infocomplete one. We will also discuss the relation between the concept of $AB$ infocompleteness and the notion of joint measurement of observables $A$ and $B$.

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This thesis is concerned with several topics related to concept of incompatibility of quan- tum observables. The operational description of quantum theory is given, in which incom- patibility is expressed in terms of joint measurability. A connection between symmetric informationally complete positive operator-valued measures and mutually unbiased bases is given, and examples of this connection holding based on investigations in Mathematica are presented. An extension of the Arthurs-Kelly measurement model is then given, where the measured observable is calculated, thereby generalising the results given previously in the literature. It is shown that in the case of prior correlations between measurement probes there exists the possibility that a measurement of both probes leads to marginal observables with smaller statistical spread than if measurements are performed on the individual probes. This concept is then highlighted by considering two probe states that allow for this reduction in spread, and the required conditions for success are given. Fi- nally, error-error relations for incompatible dichotomic qubit observables are considered in the case of state-dependent and independent error measures. Quantities that arise in the state-independent measures case, which were previously presented geometrically, have been given operational meaning, and optimal approximating schemes in both cases are compared. Limitations regarding the state-dependent optimal approximations, and experimental work built upon this construction are also discussed.

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Quantum indirect estimation theory and ABABAB-infocompleteness
We introduce the quantum indirect estimation theory, which provides a general framework to address the problem of which ensemble averages can be estimated by means of an available set of measuring apparatuses, e. g. estimate the ensemble average of an observable by measuring other observable. A main ingredient in this approach is that of informationally complete (infocomplete in short) measurements, which allow to estimate the ensemble average of any arbitrary system operator, as for quantum tomography. This naturally leads to the more stringent concept of AB-informationally complete measurements, by which one can estimate jointly all the moments of two incompatible observables A and B. After analyzing all general properties of such measurements, we address the problem of their optimality, and we completely solve the case of qubits, showing that a sigma_x sigma_y-infocomplete measurement is less noisy than any infocomplete one. We will also discuss the relation between the concept of AB-infocompleteness and the notion of joint measurement of observables A and B.
We introduce the quantum indirect estimation theory, which provides a general framework to address the problem of which ensemble averages can be estimated by means of an available set of measuring apparatuses, e.g., estimating the ensemble average of an observable by measuring other observables. A main ingredient in this approach is that of informationally complete (infocomplete) measurements, which allow us to estimate the ensemble average of any arbitrary system operator, as for quantum tomography. This naturally leads to the more stringent concept of ABABAB-informationally complete measurements, by which one can estimate jointly all the moments of two incompatible observables AAA and BBB. After analyzing all general properties of such measurements, we address the problem of their optimality, and we completely solve the case of qubits, showing that a ${\ensuremath{\sigma}}_{x}{\ensuremath{\sigma}}_{y}$-infocomplete measurement is less noisy than any infocomplete one. We will also discuss the relation between the concept of ABABAB infocompleteness and the notion of joint measurement of observables AAA and BBB.
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.
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.
This thesis is concerned with several topics related to concept of incompatibility of quan- tum observables. The operational description of quantum theory is given, in which incom- patibility is expressed in terms of joint measurability. A connection between symmetric informationally complete positive operator-valued measures and mutually unbiased bases is given, and examples of this connection holding based on investigations in Mathematica are presented. An extension of the Arthurs-Kelly measurement model is then given, where the measured observable is calculated, thereby generalising the results given previously in the literature. It is shown that in the case of prior correlations between measurement probes there exists the possibility that a measurement of both probes leads to marginal observables with smaller statistical spread than if measurements are performed on the individual probes. This concept is then highlighted by considering two probe states that allow for this reduction in spread, and the required conditions for success are given. Fi- nally, error-error relations for incompatible dichotomic qubit observables are considered in the case of state-dependent and independent error measures. Quantities that arise in the state-independent measures case, which were previously presented geometrically, have been given operational meaning, and optimal approximating schemes in both cases are compared. Limitations regarding the state-dependent optimal approximations, and experimental work built upon this construction are also discussed.