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arXiv:quant-ph/0702211 · Physical Review A

Parameter estimation for mixed states from a single copy

Thomas Konrad, Otfried Gühne, Jürgen Audretsch, Hans J. Briegel

Published Jun 1, 2007• 2 citations
arXivHTMLPDF

Abstract

Given a single copy of a mixed state of the form $\ensuremath{\rho}=\ensuremath{\lambda}{\ensuremath{\rho}}_{1}+(1\ensuremath{-}\ensuremath{\lambda}){\ensuremath{\rho}}_{2}$, what is the optimal measurement to estimate the parameter $\ensuremath{\lambda}$ if ${\ensuremath{\rho}}_{1}$ and ${\ensuremath{\rho}}_{2}$ are known? We present a general strategy to obtain the optimal measurements employing a Bayesian estimator. The measurements are chosen to minimize the deviation between the estimated value and the true value of $\ensuremath{\lambda}$. We explicitly determine the optimal measurements for a general two-dimensional system and for important higher dimensional cases.

DOI
10.1103/physreva.75.062101

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Given a single copy of a mixed state of the form \ensuremathρ=\ensuremathλ\ensuremathρ1+(1\ensuremath−\ensuremathλ)\ensuremathρ2\ensuremath{\rho}=\ensuremath{\lambda}{\ensuremath{\rho}}_{1}+(1\ensuremath{-}\ensuremath{\lambda}){\ensuremath{\rho}}_{2}\ensuremathρ=\ensuremathλ\ensuremathρ1​+(1\ensuremath−\ensuremathλ)\ensuremathρ2​, what is the optimal measurement to estimate the parameter \ensuremathλ\ensuremath{\lambda}\ensuremathλ if \ensuremathρ1{\ensuremath{\rho}}_{1}\ensuremathρ1​ and \ensuremathρ2{\ensuremath{\rho}}_{2}\ensuremathρ2​ are known? We present a general strategy to obtain the optimal measurements employing a Bayesian estimator. The measurements are chosen to minimize the deviation between the estimated value and the true value of \ensuremathλ\ensuremath{\lambda}\ensuremathλ. We explicitly determine the optimal measurements for a general two-dimensional system and for important higher dimensional cases.
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.
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 study the estimation of the overlap between two unknown pure quantum states of a finite-dimensional system, given M and N copies of each type. This is a fundamental primitive in quantum information processing that is commonly accomplished from the outcomes of N swap tests, a joint measurement on one copy of each type whose outcome probability is a linear function of the squared overlap. We show that a more precise estimate can be obtained by allowing for general collective measurements on all copies. We derive the statistics of the optimal measurement and compute the optimal mean square error in the asymptotic pointwise and finite Bayesian estimation settings. Besides, we consider two strategies relying on the estimation of one or both states and show that, although they are suboptimal, they outperform the swap test. In particular, the swap test is extremely inefficient for small values of the overlap, which become exponentially more likely as the dimension increases. Finally, we show that the optimal measurement is less invasive than the swap test and study the robustness to depolarizing noise for qubit states.
In this work, we study Bayesian quantum parameter estimation given a finite number of uses of the process encoding one or more unknown physical quantities. We analyze the performance of different kinds of quantum metrological protocols, including the conventional parallel, sequential, or general, which may involve an indefinite causal order. We also analyze a class of hybrid adaptive greedy strategies, which are based on classical feedforward between quantum protocols to optimize the next round. Within each class, the central question is to determine the optimal strategy -- namely, the choice of optimal input state, control operations, measurement, and estimators. Using the formalism of higher-order operations, we develop an algorithm that searches for the optimal solution, and we provide a numerical implementation based on semidefinite programming. Our benchmark cases, specifically those against existing analytical solutions, demonstrate how powerful and precise our method is. We further demonstrate the strength of our algorithm in several examples, from single to multiparameter estimation, and with various prior distributions. Particularly, we find examples where the adaptive greedy strategy matches the performance of general strategies with indefinite causal order, and at the same time examples that showcase a strict hierarchy between all different classes.
We consider the problem of determining the mixed quantum state of a large but finite number of identically prepared quantum systems from data obtained in a sequence of ideal (von Neumann) measurements, each performed on an individual copy of the system. In contrast to previous approaches, we do not average over the possible unknown states but work out a ``typical'' probability distribution on the set of states, as implied by the experimental data. As a consequence, any measure of knowledge about the unknown state and thus any notion of ``best strategy'' (i.e. the choice of observables to be measured, and the number of times they are measured) depend on the unknown state. By learning from previously obtained data, the experimentalist re-adjusts the observable to be measured in the next step, eventually approaching an optimal strategy. We consider two measures of knowledge and exhibit all ``best'' strategies for the case of a two-dimensional Hilbert space. Finally, we discuss some features of the problem in higher dimensions and in the infinite dimensional case.