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AIP conference proceedings

Quantum limits on optical phase estimation accuracy from classical rate-distortion theory

Ranjith Nair

Published Jan 1, 2014• 0 citations
Publisher

Abstract

The classical information-theoretic lower bound on the distortion of a random variable upon transmission through a noisy channel is applied to quantum-optical phase estimation. An approach for obtaining Bayesian lower bounds on the phase estimation accuracy is described that employs estimates of the classical capacity of the relevant quantum-optical channels. The Heisenberg limit for lossless phase estimation is derived for arbitrary probe state and prior distributions of the phase, and shot-noise scaling of the phase accuracy is established in the presence of nonzero loss for a parallel entanglement-assisted strategy with a single probe mode.

DOI
10.1063/1.4903098

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The classical information-theoretic lower bound on the distortion of a random variable upon transmission through a noisy channel is applied to quantum-optical phase estimation. An approach for obtaining Bayesian lower bounds on the phase estimation accuracy is described that employs estimates of the classical capacity of the relevant quantum-optical channels. The Heisenberg limit for lossless phase estimation is derived for arbitrary probe state and prior distributions of the phase, and shot-noise scaling of the phase accuracy is established in the presence of nonzero loss for a parallel entanglement-assisted strategy with a single probe mode.
Abstract We derive fundamental lower bounds on the performance of optical metrology and communication systems in a Bayesian framework. The derivation uses classical rate-distortion theory in conjunction with bounds on the capacity to transmit classical information of various optical channels specified by the system design. The bounds are expressed in terms of the system parameters, the prior probability distribution of the parameter, and the average energy, i.e. number of photons E in the probe state. For phase estimation, the bounds pertain to a cyclic mean squared error criterion incorporating the cyclic nature of the phase. In the absence of optical loss, our bounds are applicable to multimode linear phase modulation schemes (including ancilla-assisted ones), and to nonlinear modulations on a single mode. The bounds display inverse-quadratic Heisenberg-limit scaling of the cyclic mean square error with respect to E . In the presence of any finite amount of loss, a lower bound on ancilla-assisted phase estimation with standard quantum limit scaling is derived, which is shown to be little different from a similar bound for coherent-state probes. For systems involving a single optical mode, we also obtain lower bounds on the mean squared error of estimating any classical parameter, and on the average error probability of any M -ary communication system under an average energy constraint. The bounds are valid for arbitrary quantum measurements, for any prior probability distribution, and do not rely on unbiasedness assumptions.
The laws of quantum mechanics place fundamental limits on the accuracy of measurements and, therefore, on the estimation of unknown parameters of a quantum system. In this paper, we prove lower bounds on the size of confidence regions reported by any region estimator for a given ensemble of probe states and probability of success. Our bounds are derived from a previously unnoticed connection between the size of confidence regions and the error probabilities of a corresponding binary hypothesis test. In group-covariant scenarios, we find that there is an ultimate bound for any estimation scheme, which depends only on the representation-theoretic data of the probe system, and we evaluate its asymptotics in the limit of many systems, establishing a general Heisenberg limit for region estimation. We apply our results to several examples, in particular, to phase estimation, where our bounds allow us to recover the well-known Heisenberg and shot-noise scaling.
We propose a framework to variationally obtain detectable capacity bounds for quantum channels. The proposal of this framework is motivated by the difficulty of estimating the von Neumann entropy of an unknown quantum state. Instead of estimating the von Neumann entropy at the channel output, we propose to estimate the state purity---which can be measured by a single measurement setting---followed by bounding the von Neumann entropy from above and below. This procedure leads to new upper and lower bounds on various communication rates of quantum channels for some fixed input states. Then, by utilizing the variational method to find optimal input states we obtain lower bounds on (i) the quantum capacity of arbitrary channels, (ii) the entanglement-assisted classical capacity of arbitrary channels, and (iii) the classical capacity of covariant channels. Corresponding to these lower bounds, we also obtain upper bounds on (i) NNN-shot coherent information, (ii) the entanglement-assisted classical capacity, and (iii) the NNN-shot Holevo capacity of arbitrary quantum channels. All these bounds can be estimated by a single measurement setting without needing a full process tomography or any further a priori knowledge, e.g., preferred basis of the channel.
We study the role of quantum entanglement (particle entanglement and mode entanglement) in optical phase estimation by employing the mode description and the particle description of bosonic probe states. To determine the physical resources responsible for Heisenberg scaling (attainable only in the noiseless case), we restrict our analysis to noiseless quantum-limited optical phase estimation. The quantum Fisher information (QFI) is expressed as a function of the first- and second-order optical coherence functions. The resulting form of the QFI elucidates the deriving metrological resources for quantum phase estimation: field intensity and photon detection correlations. Our analysis confirms that mode entanglement is not required for quantum-enhanced interferometry, whereas particle entanglement is a necessary requirement. Furthermore, the derived forms of the QFI equations are summations of two terms: the classical (shot-noise-limit scaling) term and the quantum (Heisenberg scaling) term. This allows us to clearly identify the physical resources responsible for quantum enhancement in optical phase estimation.
The laws of quantum mechanics place fundamental limits on the accuracy of measurements and therefore on the estimation of physical parameters by a quantum system. In this work, we prove lower bounds on the size of confidence regions reported by any region estimator for a given ensemble of probe states and probability of success. Our bounds are derived from a previously unnoticed connection between the size of confidence regions and the error probabilities of a corresponding binary hypothesis test. In group-covariant scenarios, we find that there is an ultimate bound for any estimation scheme which depends only on the representation-theoretic data of the probe system, and we evaluate its asymptotics in the limit of many systems, establishing a general “Heisenberg limit” for region estimation. We apply our results to several scenarios, in particular to phase estimation, where our bounds strengthen the well-known Heisenberg and shot-noise scaling.
The quantum Fisher information and quantum correlation parameters are employed to study the application of nonclassical light to the problem of parameter estimation. It is shown that the optimal measurement sensitivity of a quantum state is determined by its intermode correlations (which depends on path entanglement) and intramode correlations (which depends on the photon statistics). In light of these results, we consider the performance of quantum-enhanced optical interferometers. Furthermore, we propose a Heisenberg-limited metrology protocol involving standard elements from passive and active linear optics, for which the quantum Cram\'er-Rao bound is saturated with an intensity measurement. Interestingly, the quantum advantage for this scheme is derived solely from the nonclassical photon statistics of the probe state and does not depend on entanglement. We study the performance of this scheme in the presence of realistic losses and consequently predict a substantial enhancement over the shot-noise limit with current technological capabilities.