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arXiv:quant-ph/0510158 · Physical Review A

Optimal full estimation of qubit mixed states

E. Bagán, Manuel A. Ballester, Richard David Gill, Alex Monràs, R. Muñoz-Tapia

Published Mar 2, 2006• 89 citations
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Abstract

We obtain the optimal scheme for estimating unknown qubit mixed states when an arbitrary number $N$ 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 $N$, we give a prior-independent scheme which becomes optimal in the asymptotic limit of large $N$. 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.

DOI
10.1103/physreva.73.032301

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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.
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.
Given NNN identical copies of the state of a quantum two-level system, we analyze its optimal estimation. We consider two situations: general pure states and (pure) states restricted to lie on the equator of the Bloch sphere. We perform a complete and comprehensive analysis of the optimal schemes based on local measurements, and give results (optimal measurements, maximum fidelity, etc.) for arbitrary NNN, not necessarily large, within the Bayesian framework. We also make a comparative analysis of the asymptotic limit of these results with those derived from a (pointwise) Cram\'er-Rao type of approach. We give explicit schemes based on local measurements and no classical communication that saturate the fidelity bounds of the most general collective schemes.
The optimal and minimal measuring strategy is obtained for a two-state system prepared in a mixed state with a probability given by any isotropic a priori distribution. We explicitly construct the specific optimal and minimal generalized measurements, which turn out to be independent of the a priori probability distribution, obtaining the best guesses for the unknown state as well as a closed expression for the maximal mean-average fidelity. We do this for up to three copies of the unknown state in a way that leads to the generalization to any number of copies, which we then present and prove.
Abstract The central problem of quantum statistics is to devise measurement schemes for the estimation of an unknown state, given an ensemble of n independent identically prepared systems. For locally quadratic loss functions, the risk of standard procedures has the usual scaling of 1/ n . However, it has been noticed that for fidelity based metrics such as the Bures distance, the risk of conventional (non-adaptive) qubit tomography schemes scales as 1 / n for states close to the boundary of the Bloch sphere. Several proposed estimators appear to improve this scaling, and our goal is to analyse the problem from the perspective of the maximum risk over all states. We propose qubit estimation strategies based on separate adaptive measurements, and collective measurements, that achieve 1/ n scalings for the maximum Bures risk. The estimator involving local measurements uses a fixed fraction of the available resource n to estimate the Bloch vector direction; the length of the Bloch vector is then estimated from the remaining copies by measuring in the estimator eigenbasis. The estimator based on collective measurements uses local asymptotic normality techniques which allows us to derive upper and lower bounds to its maximum Bures risk. We also discuss how to construct a minimax optimal estimator in this setup. Finally, we consider quantum relative entropy and show that the risk of the estimator based on collective measurements achieves a rate O ( n − 1 log ⁡ n ) under this loss function. Furthermore, we show that no estimator can achieve faster rates, in particular the ‘standard’ rate n −1 .
By consuming multiple copies of an unknown qubit state, one can modify its purity while preserving the direction of its Bloch vector. We determine the maximum linear rate at which qubit states of different purities can be interconverted, allowing a nonzero error, quantified, for instance, by the trace distance, provided that it vanishes in the limit of infinitely many copies. Interestingly, the optimal conversion rate is determined by the two eigenvalues of the complex right-logarithmic-derivative (RLD) Fisher information matrix associated with SU(2)\mathrm{SU}(2)SU(2) rotations of the qubit state. When the output qubits have higher purity, corresponding to concentration, the optimal rate is given by the ratio of the maximum eigenvalues of the input and output RLD matrices. In contrast, when the output qubits have lower purity, corresponding to dilution, the optimal rate is given by the ratio of their minimum eigenvalues. Remarkably, both concentration and dilution can be implemented using SWAP tests as the only nontrivial two-qubit measurement primitive, together with ancillary qubits initially prepared in maximally mixed states, without requiring any additional two-qubit gates. Our work thus provides a novel operational interpretation of the full complex RLD Fisher information matrix. Crucially, its antisymmetric, purely imaginary part encodes geometric information beyond the statistical distance between density operators and plays an essential role in determining the optimal state-conversion rates.