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Register a new userOptimal estimation of joint parameters in phase space
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Added by
Marco Giovanni Genoni
Publication date
2012
fields
Physics
and research's language is
English
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We address the joint estimation of the two defining parameters of a displacement operation in phase space. In a measurement scheme based on a Gaussian probe field and two homodyne detectors, it is shown that both conjugated parameters can be measured below the standard quantum limit when the probe field is entangled. We derive the most informative Cramer-Rao bound, providing the theoretical benchmark on the estimation and observe that our scheme is nearly optimal for a wide parameter range characterizing the probe field. We discuss the role of the entanglement as well as the relation between our measurement strategy and the generalized uncertainty relations.
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In the context of multiparameter quantum estimation theory, we investigate the construction of linear schemes in order to infer two classical parameters that are encoded in the quadratures of two quantum coherent states. The optimality of the scheme built on two phase-conjugate coherent states is proven with the saturation of the quantum Cramer--Rao bound under some global energy constraint. In a more general setting, we consider and analyze a variety of $n$-mode schemes that can be used to encode $n$ classical parameters into $n$ quantum coherent states and then estimate all parameters optimally and simultaneously.
By using a systematic optimization approach we determine quantum states of light with definite photon number leading to the best possible precision in optical two mode interferometry. Our treatment takes into account the experimentally relevant situation of photon losses. Our results thus reveal the benchmark for precision in optical interferometry. Although this boundary is generally worse than the Heisenberg limit, we show that the obtained precision beats the standard quantum limit thus leading to a significant improvement compared to classical interferometers. We furthermore discuss alternative states and strategies to the optimized states which are easier to generate at the cost of only slightly lower precision.
Measurement incompatibility is a distinguishing property of quantum physics and an essential resource for many quantum information processing tasks. We introduce an approach to verify the joint measurability of measurements based on phase-space quasiprobability distributions. Our results therefore establish a connection between two notions of non-classicality, namely the negativity of quasiprobability distributions and measurement incompatibility. We show how our approach can be applied to the study of incompatibility-breaking channels and derive incompatibility-breaking sufficient conditions for bosonic systems and Gaussian channels. In particular, these conditions provide useful tools for investigating the effects of errors and imperfections on the incompatibility of measurements in practice. To illustrate our method, we consider all classes of single-mode Gaussian channels. We show that pure lossy channels with 50% or more losses break the incompatibility of all measurements that can be represented by non-negative Wigner functions, which includes the set of Gaussian measurements.
Entanglement does not correspond to any observable and its evaluation always corresponds to an estimation procedure where the amount of entanglement is inferred from the measurements of one or more proper observables. Here we address optimal estimation of entanglement in the framework of local quantum estimation theory and derive the optimal observable in terms of the symmetric logarithmic derivative. We evaluate the quantum Fisher information and, in turn, the ultimate bound to precision for several families of bipartite states, either for qubits or continuous variable systems, and for different measures of entanglement. We found that for discrete variables, entanglement may be efficiently estimated when it is large, whereas the estimation of weakly entangled states is an inherently inefficient procedure. For continuous variable Gaussian systems the effectiveness of entanglement estimation strongly depends on the chosen entanglement measure. Our analysis makes an important point of principle and may be relevant in the design of quantum information protocols based on the entanglement content of quantum states.
Paired estimation of change in parameters of interest over a population plays a central role in several application domains including those in the social sciences, epidemiology, medicine and biology. In these domains, the size of the population under study is often very large, however, the number of observations available per individual in the population is very small (emph{sparse observations}) which makes the problem challenging. Consider the setting with $N$ independent individuals, each with unknown parameters $(p_i, q_i)$ drawn from some unknown distribution on $[0, 1]^2$. We observe $X_i sim text{Bin}(t, p_i)$ before an event and $Y_i sim text{Bin}(t, q_i)$ after the event. Provided these paired observations, ${(X_i, Y_i) }_{i=1}^N$, our goal is to accurately estimate the emph{distribution of the change in parameters}, $delta_i := q_i - p_i$, over the population and properties of interest like the emph{$ell_1$-magnitude of the change} with sparse observations ($tll N$). We provide emph{information theoretic lower bounds} on the error in estimating the distribution of change and the $ell_1$-magnitude of change. Furthermore, we show that the following two step procedure achieves the optimal error bounds: first, estimate the full joint distribution of the paired parameters using the maximum likelihood estimator (MLE) and then estimate the distribution of change and the $ell_1$-magnitude of change using the joint MLE. Notably, and perhaps surprisingly, these error bounds are of the same order as the minimax optimal error bounds for learning the emph{full} joint distribution itself (in Wasserstein-1 distance); in other words, estimating the magnitude of the change of parameters over the population is, in a minimax sense, as difficult as estimating the full joint distribution itself.
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