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Register a new userOn the properties of the asymptotic incompatibility measure in multiparameter quantum estimation
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Added by
Alessandro Candeloro
Publication date
2021
fields
Physics
and research's language is
English
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We address the use of asymptotic incompatibility (AI) to assess the quantumness of a multiparameter quantum statistical model. AI is a recently introduced measure which quantifies the difference between the Holevo and the SLD scalar bounds, and can be evaluated using only the symmetric logarithmic derivative (SLD) operators of the model. At first, we evaluate analytically the AI of the most general quantum statistical models involving two-level (qubit) and single-mode Gaussian continuous-variable quantum systems, and prove that AI is a simple monotonous function of the state purity. Then, we numerically investigate the same problem for qudits ($d$-dimensional quantum systems, with $2 < d leq 4$), showing that, while in general AI is not in general a function of purity, we have enough numerical evidence to conclude that the maximum amount of AI is achievable only for quantum statistical models characterized by a purity larger than $mu_{sf min} = 1/(d-1)$. In addition, by parametrizing qudit states as thermal (Gibbs) states, numerical results suggest that, once the spectrum of the Hamiltonian is fixed, the AI measure is in one-to-one correspondence with the fictitious temperature parameter $beta$ characterizing the family of density operators. Finally, by studying in detail the definition and properties of the AI measure we find that: i) given a quantum statistical model, one can readily identify the maximum number of asymptotically compatibile parameters; ii) the AI of a quantum statistical model bounds from above the AI of any sub-model that can be defined by fixing one or more of the original unknown parameters (or functions thereof), leading to possibly useful bounds on the AI of models involving noisy quantum dynamics.
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Simultaneous quantum estimation of multiple parameters has recently become essential in quantum metrology. Although the ultimate sensitivity of a multiparameter quantum estimation in noiseless environments can beat the standard quantum limit that every classical sensor is bounded by, it is unclear whether the quantum sensor has an advantage over the classical one under realistic noise. In this work, we present a framework of the simultaneous estimation of multiple parameters with quantum sensors in a certain noisy environment. Our multiple parameters to be estimated are three components of an external magnetic field, and we consider the noise that causes only dephasing. We show that there is an optimal sensing time in the noisy environment and the sensitivity can beat the standard quantum limit when the noisy environment is non-Markovian.
When collective measurements on an infinite number of copies of identical quantum states can be performed, the precision limit of multi-parameter quantum estimation is quantified by the Holevo bound. In practice, however, the collective measurements are always restricted to a finite number of quantum states, under which the precision limit is still poorly understood. Here we provide an approach to study the multi-parameter quantum estimation with general $p$-local measurement where the collective measurements are restricted to at most $p$ copies of quantum states. We demonstrate the power of the approach by providing a hierarchy of nontrivial tradeoff relations for multi-parameter quantum estimation which quantify the incompatibilities of general $p$-local measurement. These tradeoff relations also provide a necessary condition for the saturation of the quantum Cramer-Rao bound under $p$-local measurement, which is shown reducing to the weak commutative condition when $prightarrow infty$. To further demonstrate the versatility of the approach, we also derive another set of tradeoff relations in terms of the right logarithmic operators(RLD).
The estimation of more than one parameter in quantum mechanics is a fundamental problem with relevant practical applications. In fact, the ultimate limits in the achievable estimation precision are ultimately linked with the non-commutativity of different observables, a peculiar property of quantum mechanics. We here consider several estimation problems for qubit systems and evaluate the corresponding quantumness R, a measure that has been recently introduced in order to quantify how much incompatible are the parameters to be estimated. In particular, R is an upper bound for the renormalized difference between the (asymptotically achievable) Holevo bound and the SLD Cramer-Rao bound (i.e. the matrix generalization of the single-parameter quantum Cramer-Rao bound). For all the estimation problems considered, we evaluate the quantumness R and, in order to better understand its usefulness in characterizing a multiparameter quantum statistical model, we compare it with the renormalized difference between the Holevo and the SLD-bound. Our results give evidence that R is a useful quantity to characterize multiparameter estimation problems, as for several quantum statistical model it is equal to the difference between the bounds and, in general, their behaviour qualitatively coincide. On the other hand, we also find evidence that for certain quantum statistical models the bound is not in tight, and thus R may overestimate the degree of quantum incompatibility between parameters.
We study the geometric measure of quantum coherence recently proposed in [Phys. Rev. Lett. 115, 020403 (2015)]. Both lower and upper bounds of this measure are provided. These bounds are shown to be tight for a class of important coherent states -- maximally coherent mixed states. The trade-off relation between quantum coherence and mixedness for this measure is also discussed.
The quantum multiparameter estimation is very different from the classical multiparameter estimation due to Heisenbergs uncertainty principle in quantum mechanics. When the optimal measurements for different parameters are incompatible, they cannot be jointly performed. We find a correspondence relationship between the inaccuracy of a measurement for estimating the unknown parameter with the measurement error in the context of measurement uncertainty relations. Taking this correspondence relationship as a bridge, we incorporate Heisenbergs uncertainty principle into quantum multiparameter estimation by giving a tradeoff relation between the measurement inaccuracies for estimating different parameters. For pure quantum states, this tradeoff relation is tight, so it can reveal the true quantum limits on individual estimation errors in such cases. We apply our approach to derive the tradeoff between attainable errors of estimating the real and imaginary parts of a complex signal encoded in coherent states and obtain the joint measurements attaining the tradeoff relation. We also show that our approach can be readily used to derive the tradeoff between the errors of jointly estimating the phase shift and phase diffusion without explicitly parameterizing quantum measurements.
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