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IEEE International Symposium on Information Theory (ISIT)

A Heisenberg limit for quantum region estimation

Michael Walter, Joseph M. Renes

Published Jun 1, 2014• 1 citation
Publisher

Abstract

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.

DOI
10.1109/isit.2014.6875008

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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 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.
The Heisenberg limit traditionally provides a lower bound on the phase uncertainty scaling as $1/\ensuremath{\langle}N\ensuremath{\rangle}$, where $\ensuremath{\langle}N\ensuremath{\rangle}$ is the mean number of photons in the probe. However, this limit has a number of loopholes which potentially might be exploited to achieve measurements with even greater accuracy. Here we close these loopholes by proving a completely rigorous form of the Heisenberg limit for the average error over all phase shifts, applicable to any estimate of a completely unknown phase shift. Our result gives a completely general, constraint-free, and nonasymptotic statement of the Heisenberg limit. It holds for all phase estimation schemes, including multiple passes, nonlinear phase shifts, multimode probes, and arbitrary measurements.
Abstract Heisenberg scaling characterizes the ultimate precision of parameter estimation enabled by quantum mechanics, which represents an important quantum advantage of both theoretical and technological interest. Here, we present a comprehensive and rigorous study of the attainability of strong, global notions of Heisenberg scaling (in contrast to the commonly studied local estimation based on e.g. quantum Fisher information) in the fundamental problem of quantum metrology, in noisy environments. As our first contribution, we formally define two useful notions of Heisenberg scaling in global estimation respectively based on the average estimation error and the limiting distribution of estimation error (which we highlight as a practically important figure of merit). A main result of this work is that for the standard phase damping noise, an O ( n −1 ) noise rate is a necessary and sufficient condition for attaining global Heisenberg scaling. We first prove that O ( n −1 ) is an upper bound on the noise rate for Heisenberg scaling to be possible, and then show by constructing a ‘robust’ estimation procedure that global Heisenberg scaling in both senses can indeed be achieved under Θ( n −1 ) noise. In addition, we provide a practically more friendly adaptive protocol using only an one-qubit memory, which achieves global Heisenberg scaling in terms of limiting distribution as well under O ( n −1 ) noise.
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 Quantum multiparameter estimation involves estimating multiple parameters simultaneously and can be more precise than estimating them individually. Our interest here is to determine fundamental quantum limits to the achievable multiparameter estimation precision in the presence of noise. We first present a lower bound to the estimation error covariance for a noisy initial probe state evolving through a noiseless quantum channel. We then present a lower bound to the estimation error covariance in the most general form for a noisy initial probe state evolving through a noisy quantum channel. We show conditions and accordingly measurements to attain these estimation precision limits for noisy systems. We see that the Heisenberg precision scaling of 1/N can be achieved with a probe comprising N particles even in the presence of noise. In fact, some noise in the initial probe state or the quantum channel can serve as a feature rather than a bug, since the estimation precision scaling achievable in the presence of noise in the initial state or the channel in some situations is impossible in the absence of noise in the initial state or the channel. However, a lot of noise harms the quantum advantage achievable with N parallel resources, and allows for a best precision scaling of 1/N{\bf{1}}/\sqrt{{\boldsymbol{N}}}1/N​ 1 / N . Moreover, the Heisenberg precision limit can be beaten with noise in the channel, and we present a super-Heisenberg precision limit with scaling of 1/N2 for optimal amount of noise in the channel, characterized by one-particle evolution operators. Furthermore, using γ-particle evolution operators for the noisy channel, where γ > 1, the best precision scaling attainable is 1/N2γ, which is otherwise known to be only possible using 2γ-particle evolution operators for a noiseless channel.
Quantum precision enhancement is of fundamental importance for the development of advanced metrological optical experiments, such as gravitational wave detection and frequency calibration with atomic clocks. Precision in these experiments is strongly limited by the 1/√N shot noise factor with N being the number of probes (photons, atoms) employed in the experiment. Quantum theory provides tools to overcome the bound by using entangled probes. In an idealized scenario this gives rise to the Heisenberg scaling of precision 1/N. Here we show that when decoherence is taken into account, the maximal possible quantum enhancement in the asymptotic limit of infinite N amounts generically to a constant factor rather than quadratic improvement. We provide efficient and intuitive tools for deriving the bounds based on the geometry of quantum channels and semi-definite programming. We apply these tools to derive bounds for models of decoherence relevant for metrological applications including: depolarization, dephasing, spontaneous emission and photon loss. Quantum metrology employs the properties of quantum states to further enhance the accuracy of some of the most precise measurement schemes to date. Here, a method for estimating the upper bounds to achievable precision in quantum-enhanced metrology protocols in the presence of decoherence is presented.