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Physical Review A

Continuous positive-operator-valued measurement of photon polarization

Jeffrey H. Shapiro

Published May 22, 2008• 2 citations
Publisher

Abstract

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 $\overline{F}=2/3$ 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 $1:M$ optical splitter followed by a collection of $M$ 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 $M\ensuremath{\rightarrow}\ensuremath{\infty}$.

DOI
10.1103/physreva.77.052330

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•Nov 2, 2009•2 citations
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→\ensuremath∞M\ensuremath{\rightarrow}\ensuremath{\infty}M\ensuremath→\ensuremath∞.
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.
In the quantum theory of measurement, the positive operator-valued measure (POVM) is an important concept, and its implementation can be useful. A POVM consists of a set of non-negative quantum-mechanical Hermitian operators that add up to the identity. The probability that a quantum system is in a particular state is given by the expectation value of the POVM operator corresponding to that state. Following a brief review of the mathematics and mention of the history of POVMs in quantum theory, a particular implementation of a POVM for use in the measurement of nonorthogonal photon polarization states is reviewed. The implementation consists simply of a Wollaston prism, a mirror, two beam splitters, a polarization rotator and three phototubes arranged in an interferometric configuration, and it is shown analytically that the device faithfully represents the POVM. Based on Neumark's extension theorem, the two-dimensional Hilbert space of the POVM implementation can be embedded in the three-dimensional Hilbert space of an ordinary projective-valued measure. Also, analytical expressions are given for the maximum Renyi information loss from the device to a disturbing probe, and for the error and inconclusive rates induced by the probe. Various aspects of the problem of probe optimization are elaborated.
Positive-operator-value measures (POVM's) are the most general class of quantum measurements. We propose a setup in which all possible POVM's of a single-photon polarization state (corresponding to all possible sets of two-dimensional Kraus operators) can be implemented easily using linear optics elements. This method makes it possible to experimentally realize any projective orthogonal, projective nonorthogonal or nonprojective sets of any number of POVM operators. Furthermore, our implementation only requires vacuum ancillas and is deterministic rather than probabilistic. Thus it realizes every POVM with the correct set of output states. We give the settings required to implement two different well-known nonorthogonal projective POVM's.
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%.
The statistical ensemble formalism of Kim et al [J. Opt. Soc. Am. A4, 433 (1987)] offers a realistic model for characterizing the effect of stochastic nonimage-forming optical media on the state of polarization of transmitted light. With suitable choice of the Jones ensemble, various Mueller transformations-some of which are hitherto unknown-are deduced. It is observed that the ensemble approach is formally identical to the positive-operator-valued measures (POVMs) on the quantum density matrix. This observation, in combination with the recent suggestion by Ahnert and Payne [Phys. Rev. A71, 012330-1 (2005)]-in the context of generalized quantum measurement on single photon polarization states-that linear optics elements can be employed in setting up all possible POVMs enables us to propose a way of realizing different types of Mueller devices.
This thesis describes a linear-optical device for performing generalized quantum measurements \non quantum bits (qubits) encoded in photon polarization, the implementation \nof said device, and its use in two diff erent but related experiments. The device works by \ncoupling the polarization degree of freedom of a single photon to a `mode' or `path' degree \nof freedom, and performing a projective measurement in this enlarged state space in order \nto implement a tunable four-outcome positive operator-valued measure (POVM) on the \ninitial quantum bit. In both experiments, this POVM is performed on one photon from a \ntwo-photon entangled state created through spontaneous parametric down-conversion. \nIn the fi rst experiment, this entangled state is viewed as a two-qubit photonic cluster \nstate, and the POVM as a means of increasing the computational power of a given resource \nstate in the cluster-state model of quantum computing. This model traditionally \nachieves deterministic outputs to quantum computations via successive projective measurements, \nalong with classical feedforward to choose measurement bases, on qubits in a highly entangled \nresource called a cluster state; we show that `virtual qubits' can be appended to a \ngiven cluster by replacing some projective measurements with POVMs. Our experimental \ndemonstration fully realizes an arbitrary three-qubit cluster computation by implementing \nthe POVM, as well as fast active feed-forward, on our two-qubit photonic cluster state. \nOver 206 diff erent computations, the average output delity is 0.9832 +/- 0.0002; furthermore \nthe error contribution from our POVM device and feedforward is only of order 10^-3, less \nthan some recent thresholds for fault-tolerant cluster computing. \nIn the second experiment, the POVM device is used to implement a deterministic \nprotocol for remote state preparation (RSP) of arbitrary photon polarization qubits. RSP \nis the act of preparing a quantum state at a remote location without actually transmitting \nthe state itself. We are able to remotely prepare 178 diff erent pure and mixed qubit \nstates with an average delity of 0.995. Furthermore, we study the the fidelity achievable \nby RSP protocols permitting only classical communication, without shared entanglement, \nand compare the resulting benchmarks for average fidelity against our experimental results. \nOur experimentally-achieved average fi delities surpass the classical thresholds whenever \nclassical communication alone does not trivially allow for perfect RSP.