Abstract
Measurements of an object's temperature are important in many disciplines, from astronomy to engineering, as are estimates of an object's spatial configuration. We present the quantum optimal estimator for the temperature of a distant body based on the black body radiation received in the far-field. We also show how to perform separable quantum optimal estimates of the spatial configuration of a distant object, i.e. imaging. In doing so we necessarily deal with multi-parameter quantum estimation of incompatible observables, a problem that is poorly understood. We compare our optimal observables to the two mode analogue of lensed imaging and find that the latter is far from optimal, even when compared to measurements which are separable. To prove the optimality of the estimators we show that they minimise the cost function weighted by the quantum Fisher information---this is equivalent to maximising the average fidelity between the actual state and the estimated one.
Highlights
The quantum Fisher information (QFI) is a prevalent figure of merit in the field of quantum parameter estimation [5, 16, 18, 19, 21]
Since the QFI is a property of the state alone and does not depend on a particular measurement scheme, the precision in parameter estimates is determined by the uncertainty in the state only [4], and is fundamental in nature and cannot be reduced by improving measurement apparatus
We show that the spatial configuration of a thermally radiating source is determined by spatial correlations in the far-field
Summary
The quantum Fisher information (QFI) is a prevalent figure of merit in the field of quantum parameter estimation [5, 16, 18, 19, 21]. It is quite straightforward to determine the statistics of these independent modes and calculate the quantum Fisher information to determine the optimal measurements for simultaneous temperature and solid angle estimates. It has been stated before that the optimal measurement for temperature estimates is photon number counting, [11, 27] These results rely upon the assumption that we can measure a single thermal mode in the far-field, an assumption that we claim cannot be satisfied without knowledge of the exact spatial properties of the source. We find it important to consider the coherence between the spatial modes, and this becomes the essential parameter to estimate if we wish to glean spatial information about the source. We find a POVM which is independent of the parameter values and yet is close to optimal, which is of great practical importance
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