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arXiv:quant-ph/0411133 · Physical Review A

Optimal phase estimation for qubits in mixed states

Giacomo Mauro D’Ariano, Chiara Macchiavello, Paolo Perinotti

Published Oct 26, 2005• 15 citations
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Abstract

We address the problem of optimal estimation of the relative phase for two-dimensional quantum systems in mixed states. In particular, we derive the optimal measurement procedures for an arbitrary number of qubits prepared in the same mixed state.
DOI
10.1103/physreva.72.042327

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We obtain the optimal scheme for estimating unknown qubit mixed states when an arbitrary number NNN of identically prepared copies is available. We discuss the case of states in the whole Bloch sphere as well as the restricted situation where these states are known to lie on the equatorial plane. For the former case we obtain that the optimal measurement does not depend on the prior probability distribution provided it is isotropic. Although the equatorial-plane case does not have this property for arbitrary NNN, we give a prior-independent scheme which becomes optimal in the asymptotic limit of large NNN. We compute the maximum mean fidelity in this asymptotic regime for the two cases. We show that within the pointwise estimation approach these limits can be obtained in a rather easy and rapid way. This derivation is based on heuristic arguments that are made rigorous by using van Trees inequalities. The interrelation between the estimation of the purity and the direction of the state is also discussed. In the general case we show that they correspond to independent estimations whereas for the equatorial-plane states this is only true asymptotically.
Phase estimation is used in many quantum algorithms, particularly in order to estimate energy eigenvalues for quantum systems. When using a single qubit as the probe (used to control the unitary we wish to estimate the eigenvalue of), it is not possible to measure the phase with a minimum mean-square error. In standard methods, there would be a logarithmic (in error) number of control qubits needed in order to achieve this minimum error. Here, we show how to perform this measurement using only two control qubits, thereby reducing the qubit requirements of the quantum algorithm. To achieve this task, we prepare the optimal control state one qubit at a time, at the same time as applying the controlled unitaries and inverse quantum Fourier transform. As each control qubit is measured, it is reset to |0⟩ then entangled with the other control qubit, so only two control qubits are needed.
We address the problem of estimating the phase $\ensuremath{\phi}$ given NNN copies of the phase-rotation gate ${u}_{\ensuremath{\phi}}$. We consider, for the first time, the optimization of the general case where the circuit consists of an arbitrary input state, followed by any arrangement of the NNN phase rotations interspersed with arbitrary quantum operations, and ending with a general measurement. Using the polynomial method, we show that, in all cases where the measure of quality of the estimate $\stackrel{\texttildelow{}}{\ensuremath{\phi}}$ for $\ensuremath{\phi}$ depends only on the difference $\stackrel{\texttildelow{}}{\ensuremath{\phi}}\ensuremath{-}\ensuremath{\phi}$, the optimal scheme has a very simple fixed form. This implies that an optimal general phase estimation procedure can be found by just optimizing the amplitudes of the initial state.
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 discuss the problem of estimating a qubit mixed state. We give the optimal estimation that can be inferred from any given set of measurements. For collective measurements and for a large number N of copies, we show that the error in the estimation varies as 1/N.1/N.1/N. For local measurements, we focus on the simpler case of states lying on the equatorial plane of the Bloch sphere. We show that the error using plain tomography varies as 1/N1/4,{1/N}^{1/4},1/N1/4, while our approach leads to an error proportional to 1/N3/4.{1/N}^{3/4}.1/N3/4.