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arXiv:1603.04346 · Physical Review A

Universal optimal estimation of the polarization of light with arbitrary photon statistics

Lu Zhang, Kam Wai Clifford Chan, Pramode K. Verma

Published Mar 30, 2016• 4 citations
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Abstract

A universal and optimal method for the polarimetry of light with arbitrary photon statistics is presented. The method is based on the continuous maximum-likelihood positive operator-valued measure (ML-POVM) for pure polarization states over the surface of the Bloch sphere. The success probability and the mean fidelity are used as the figures of merit to show its performance. The POVM is found to attain the collective bound of polarization estimation with respect to the mean fidelity. As demonstrations, explicit results for the $N$ photon Fock state, the phase-randomized coherent state (Poisson distribution), and the thermal light are obtained. It is found that the estimation performances for the Fock state and the Poisson distribution are almost identical, while that for the thermal light is much worse. This suggests that thermal light leaks less information to an eavesdropper and hence could potentially provide more security in polarization-encoded quantum communication protocols than a single-mode laser beam as customarily considered. Finally, comparisons against an optimal adaptive measurement with classical communications are made to show the better and more stable performance of the continuous ML-POVM.

DOI
10.1103/physreva.93.032137

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A universal and optimal method for the polarimetry of light with arbitrary photon statistics is presented. The method is based on the continuous maximum-likelihood positive operator-valued measure (ML-POVM) for pure polarization states over the surface of the Bloch sphere. The success probability and the mean fidelity are used as the figures of merit to show its performance. The POVM is found to attain the collective bound of polarization estimation with respect to the mean fidelity. As demonstrations, explicit results for the NNN photon Fock state, the phase-randomized coherent state (Poisson distribution), and the thermal light are obtained. It is found that the estimation performances for the Fock state and the Poisson distribution are almost identical, while that for the thermal light is much worse. This suggests that thermal light leaks less information to an eavesdropper and hence could potentially provide more security in polarization-encoded quantum communication protocols than a single-mode laser beam as customarily considered. Finally, comparisons against an optimal adaptive measurement with classical communications are made to show the better and more stable performance of the continuous ML-POVM.
A continuous positive operator-valued measurement (POVM) is described for the polarization state of a single photon propagating in a known spatiotemporal mode. This POVM is shown to be the maximum-likelihood quantum measurement for estimating that polarization state. Moreover, the maximum-likelihood polarization estimate derived from this POVM enables uniformly optimal measure-and-prepare state transmission via classical communication---at the F‾=2/3\overline{F}=2/3F=2/3 classical limit on average teleportation fidelity---for any pure state of polarization. When two continuous-POVM polarimeters are used to detect the singlet state of a pair photons that are in distinct spatial modes, the conditional probability density for one polarimeter's output given the other's encompasses the quantum interference patterns seen in standard singlet-state polarization analysis. For single-photon illumination, a 1:M1:M1:M optical splitter followed by a collection of MMM projective polarization measurements, whose bases are uniformly distributed over the Poincar\'e sphere, yields observation statistics that converge in distribution to those of the continuous-POVM polarimeter as $M\ensuremath{\rightarrow}\ensuremath{\infty}$.
We present the experimental results for a method used to perform polarimetry on ensembles of single photons. Our setup is based on a measurement method known to be optimal for estimating the state of two-level systems. The setup has no moving parts and is sensitive to weak sources (emitting single photons) of light as it relies on photon counting, and has potential applications in both classical polarization measurements and quantum communication scenarios. In our implementation, we are able to reconstruct the Stokes parameters of pure polarization states with an average fidelity of 99.9%.
A direct derivation is given for the optimal mean fidelity of quantum state estimation of a ddd-dimensional unknown pure state with its NNN copies given as input, which was first obtained by Hayashi in terms of an infinite set of covariant positive operator valued measures (POVM's) and by Bru\ss{} and Macchiavello establishing a connection to optimal quantum cloning. An explicit condition for POVM measurement operators for optimal estimators is obtained, by which we construct optimal estimators with finite POVMs using exact quadratures on a hypersphere. These finite optimal estimators are not generally universal, where universality means the fidelity is independent of input states. However, any optimal estimator with finite POVM for M(>N)M(>N)M(>N) copies is universal if it is used for NNN copies as input.
We consider how the theory of optimal quantum measurements determines the maximum information available to the receiving party of a quantum key distribution (QKD) system employing linearly independent but nonorthogonal quantum states. Such a setting is characteristic of several practical QKD protocols. Due to nonorthogonality, the receiver is not able to discriminate unambiguously between the signals. To understand the fundamental limits that this imposes, the quantity of interest is the maximum mutual information between the transmitter (Alice) and the receiver, whether legitimate (Bob) or an eavesdropper (Eve). To find the optimal measurement—taken individually or collectively—we use a framework based on operator algebra and general results derived from singular-value decomposition, achieving optimal solutions for von Neumann measurements and positive operator-valued measures (POVMs). The formal proof and quantitative analysis elaborated for two signals allow us to conclude that optimal von Neumann measurements are uniquely defined and provide a higher information gain compared to POVMs. Interestingly, collective measurements not only do not provide additional information gain with respect to individual ones but also suffer from a gain reduction in the case of POVMs. Published by the American Physical Society 2024
A direct derivation is given for the optimal mean fidelity of quantum state estimation of a d-dimensional unknown pure state with its N copies given as input, which was first obtained by Bruss and Macchiavello by establishing a connection to optimal quantum cloning. An explicit condition for POVM measurement operators for optimal estimators is obtained, by which we show optimal estimators with finite POVM exist but not generally universal, where universality means the fidelity is independent of input states. However any optimal estimator for M(>N) copies is universal if it is used for N copies as input.