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arXiv:quant-ph/0012015 · Physical Review A

Optimal estimation of quantum dynamics

Antonio Acín, E. Jané, Guifré Vidal

Published Oct 18, 2001• 91 citations
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Abstract

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.

DOI
10.1103/physreva.64.050302

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•Jan 1, 2001•3 citations
We construct the optimal strategy for the estimation of an unknown unitary transformation U\ensuremath∈SU(d).U\ensuremath{\in}\mathrm{SU}(d).U\ensuremath∈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∈SU(2),U\ensuremath{\in}\mathrm{SU}(2),U\ensuremath∈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.
In a ubiquitous SU(2) dynamics, achieving the simultaneous optimal estimation of multiple parameters is significant but difficult. Using quantum control to optimize this SU(2) coding unitary evolution is one of solutions. We propose a method, characterized by the nested cross products of the coefficient vector X\mathbf{X}X of SU(2) generators and its partial derivative ${\ensuremath{\partial}}_{\ensuremath{\ell}}\mathbf{X}$, to investigate the control-enhanced quantum multiparameter estimation. Our work reveals that quantum control is not always functional in improving the estimation precision, which depends on the characterization of an SU(2) dynamics with respect to the objective parameter. This characterization is quantified by the angle ${\ensuremath{\alpha}}_{\ensuremath{\ell}}$ between X\mathbf{X}X and ${\ensuremath{\partial}}_{\ensuremath{\ell}}\mathbf{X}$. For an SU(2) dynamics featured by ${\ensuremath{\alpha}}_{\ensuremath{\ell}}=\ensuremath{\pi}/2$, the promotion of the estimation precision can get the most benefits from the controls. When ${\ensuremath{\alpha}}_{\ensuremath{\ell}}$ gradually closes to 0 or $\ensuremath{\pi}$, the precision promotion contributed to by quantum control correspondingly becomes inconspicuous. Until a dynamics with ${\ensuremath{\alpha}}_{\ensuremath{\ell}}=0$ or $\ensuremath{\pi}$, quantum control completely loses its advantage. In addition, we find a set of conditions restricting the simultaneous optimal estimation of all the parameters, but fortunately, which can be removed by using a maximally entangled two-qubit state as the probe state and adding an ancillary channel into the configuration. Lastly, a spin-1/21/21/2 system is taken as an example to verify the above-mentioned conclusions. Our proposal sufficiently exhibits the hallmark of control-enhancement in fulfilling the multiparameter estimation mission, and it is applicable to an arbitrary SU(2) parametrization process.
The problem of optimally estimating an unknown unitary quantum operation with the aid of entanglement is addressed. The idea is to prepare an entangled pair, apply the unknown unitary to one of the two parts, and then measure the joint output state. This measurement could be an entangled one or it could be separable (e.g., measurements which can be implemented with local operations and classical communication or LOCC). A comparison is made between these possibilities and it is shown that by using nonseparable measurements one can improve the accuracy of the estimation by a factor of 2(d+1)/d2(d+1)/d2(d+1)/d where d is the dimension of the Hilbert space on which U acts.
The theory of quantum state estimation is exploited here to investigate the most efficient strategies for this task, especially targeting a complete picture identifying optimal conditions in terms of Fisher information, quantum measurement, and associated estimator. The approach is specified to estimation of the phase of a qubit in a rotation around an arbitrary given axis, equivalent to estimating the phase of an arbitrary single-qubit quantum gate, both in noise-free and then in noisy conditions. In noise-free conditions, we establish the possibility of defining an optimal quantum probe, optimal quantum measurement, and optimal estimator together capable of achieving the ultimate best performance uniformly for any unknown phase. With arbitrary quantum noise, we show that in general the optimal solutions are phase dependent and require adaptive techniques for practical implementation. However, for the important case of the depolarizing noise, we again establish the possibility of a quantum probe, quantum measurement, and estimator uniformly optimal for any unknown phase. In this way, for qubit phase estimation, without and then with quantum noise, we characterize the phase-independent optimal solutions when they generally exist, and also identify the complementary conditions where the optimal solutions are phase dependent and only adaptively implementable.
We consider the problem of estimating an SU(d) quantum operation when n copies of it are available at the same time. It is well known that, if one uses a separable state as the input for the unitaries, the optimal mean square error will decrease as 1/n. However it is shown here that, if a proper entangled state is used, the optimal mean square error will decrease at a 1/n^2 rate. It is also shown that spherical 2-designs (e.g. complete sets of mutually unbiased bases and symmetric informationally complete positive operator valued measures) can be used to design optimal input states. Although 2-designs are believed to exist for every dimension, this has not yet been proven. Therefore, we give an alternative input state based on approximate 2-designs which can be made arbitrarily close to optimal. It is shown that measurement strategies which are based on local operations and classical communication between the ancilla and the rest of the system can be optimal.
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.