Marcin Jarzyna

Quantum Limits in Optical Interferometry

Abstract Nonclassical states of light find applications in enhancing the performance of optical interferometric experiments, with notable example of gravitational-wave detectors. Still, the presence of decoherence hinders significantly the performance of quantum-enhanced protocols. In this review, we summarize the developments of quantum metrology with particular focus on optical interferometry and derive fundamental bounds on achievable quantum-enhanced precision in optical interferometry taking into account the most relevant decoherence processes including: phase diffusion, losses, and imperfect interferometric visibility. We introduce all the necessary tools of quantum optics as well as quantum estimation theory required to derive the bounds. We also discuss the practical attainability of the bounds derived and stress, in particular, that the techniques of quantum-enhanced interferometry which are being implemented in modern gravitational-wave detectors are close to the optimal ones.

Quantum interferometry with and without an external phase reference

We discuss the role of an external phase reference in quantum interferometry. We point out inconsistencies in the literature with regard to the use of the quantum Fisher information (QFI) in phase estimation interferometric schemes. We discuss the interferometric schemes with and without an external phase reference and show a proper way to use QFI in both situations.

Compatibility in multiparameter quantum metrology

Simultaneous estimation of multiple parameters in quantum metrological models is complicated by factors relating to the (i) existence of a single probe state allowing for optimal sensitivity for all parameters of interest, (ii) existence of a single measurement optimally extracting information from the probe state on all the parameters, and (iii) statistical independence of the estimated parameters. We consider the situation when these concerns present no obstacle and for every estimated parameter the variance obtained in the multiparameter scheme is equal to that of an optimal scheme for that parameter alone, assuming all other parameters are perfectly known. We call such models compatible. In establishing a rigorous theoretical framework for investigating compatibility, we clarify some ambiguities and inconsistencies present in the literature and discuss several examples to highlight interesting features of unitary and non-unitary parameter estimation, as well as deriving new bounds for physical problems of interest, such as the simultaneous estimation of phase and local dephasing.

True precision limits in quantum metrology

We show that quantification of the performance of quantum-enhanced measurement schemes based on the concept of quantum Fisher information (QFI) yields results that are asymptotically equivalent to those from the rigorous Bayesian approach, provided generic uncorrelated noise is present in the setup. At the same time, we show that for the problem of decoherence-free phase estimation this equivalence breaks down, and the achievable estimation uncertainty calculated within the Bayesian approach is larger by a factor of π than that predicted from the QFI even in the large prior knowledge (small parameter fluctuation) regime, where the QFI is conventionally regarded as a reliable figure of merit. We conjecture that an analogous discrepancy is present in the arbitrary decoherence-free unitary parameter estimation scheme, and propose a general formula for the asymptotically achievable precision limit. We also discuss protocols utilizing states with an indefinite number of particles, and show that within the Bayesian approach it is legitimate to replace the number of particles with the mean number of particles in the formulas for the asymptotic precision, which as a consequence provides another argument for proposals based on the properties of the QFI of indefinite particle number states leading to sub-Heisenberg precisions not being practically feasible.

Matrix product states for quantum metrology.

We demonstrate that the optimal states in lossy quantum interferometry may be efficiently simulated using low rank matrix product states. We argue that this should be expected in all realistic quantum metrological protocols with uncorrelated noise and is related to the elusive nature of the Heisenberg precision scaling in the asymptotic limit of a large number of probes.

Quantum interferometric measurements of temperature

We provide a detailed description of the quantum interferometric thermometer, which is a device that estimates the temperature of a sample from the measurements of the optical phase. We rigorously analyze the operation of such a device by studying the interaction of the optical probe system prepared in a single-mode Gaussian state with a heated sample modeled as a dissipative thermal reservoir. We find that this approach to thermometry is capable of measuring the temperature of a sample in the nanokelvin regime. Furthermore, we compare the fundamental precision of quantum interferometric thermometers with the theoretical precision offered by the classical idealized pyrometers, which infer the temperature from a measurement of the total thermal radiation emitted by the sample. We find that the interferometric thermometer provides a superior performance in temperature sensing even when compared with this idealized pyrometer. We predict that interferometric thermometers will prove useful for ultraprecise temperature sensing and stabilization of quantum optical experiments based on the nonlinear crystals and atomic vapors.

Incoherent on-off keying with classical and non-classical light.

We analyze the performance of on-off keying (OOK) and its restricted version pulse position modulation (PPM) over a lossy narrowband optical channel under the constraint of a low average photon number, when direct detection is used at the output. An analytical approximation for the maximum PPM transmission rate is derived, quantifying the effects of photon statistics on the communication efficiency in terms of the g((2)) second-order intensity correlation function of the light source. Enhancement attainable through the use of sub-Poissonian light is discussed.

On super-resolution imaging as a multiparameter estimation problem

We consider the problem of characterising the spatial extent of a composite light source using the superresolution imaging technique when the centroid of the source is not known precisely. We show that the essential features of this problem can be mapped onto a simple qubit model for joint estimation of a phase shift and a dephasing strength.

Phase noise in collective binary phase shift keying with Hadamard words.

We analyze the effect of phase fluctuations in an optical communication scheme based on collective detection of sequences of binary coherent state symbols using linear optics and photon counting. When the phase noise is absent, the scheme offers qualitatively improved nonlinear scaling of the spectral efficiency with the mean photon number in the low-power regime compared to individual detection. We show that this feature, providing a demonstration of superaddivitity of accessible information in classical communication over quantum channels, is preserved if random phases imprinted on transmitted symbols fluctuate around a reference fixed over the sequence length.

Classical capacity per unit cost for quantum channels

In most communication scenarios, sending a symbol encoded in a quantum state requires spending resources such as energy, which can be quantified by a cost of communication. A standard approach in this context is to quantify the performance of communication protocol by classical capacity, quantifying the maximal amount of information that can be transmitted through a quantum channel per single use of the channel. However, different figures of merit are also possible, and a particularly well-suited one is the classical capacity per unit cost, which quantifies the maximal amount of information that can be transmitted per unit cost. I generalize this concept to account for the quantum nature of the information carriers and communication channels and show that if there exist a symbol with cost equal to zero, e.g. a vacuum state, the capacity per unit cost can be expressed by a simple formula containing maximization of the relative entropy between two quantum states. This enables me to analyze the behavior of photon information efficiency for general communication tasks and show simple bounds on the capacity per unit cost in terms of quantities familiar from quantum estimation theory. I calculate also the capacity per unit cost for general Gaussian quantum channels.