It is known that quantum fidelity, as a measure of the closeness of two quantum states, is operationally equivalent to the minimal overlap of the probability distributions of the two states over all possible positive-operator-valued measures (POVM's); the POVM realizing the minimum is optimal. We consider the ability of homodyne detection to distinguish two single-mode Gaussian states and investigate to what extent it is optimal in this information-theoretic sense. We completely identify the conditions under which homodyne detection makes an optimal distinction between two single-mode Gaussian states of the same mean and show that, if the Gaussian states are pure, they are always optimally distinguished.
It is known that quantum fidelity, as a measure of the closeness of two quantum states, is operationally equivalent to the minimal overlap of the probability distributions of the two states over all possible positive-operator-valued measures (POVM's); the POVM realizing the minimum is optimal. We consider the ability of homodyne detection to distinguish two single-mode Gaussian states and investigate to what extent it is optimal in this information-theoretic sense. We completely identify the conditions under which homodyne detection makes an optimal distinction between two single-mode Gaussian states of the same mean and show that, if the Gaussian states are pure, they are always optimally distinguished.
Quantum fidelity is a measure to quantify the closeness between two quantum states. In an operational sense, it is defined as the minimal overlap between the probability distributions of measurement outcomes and the minimum is taken over all possible positive-operator valued measures (POVMs). Quantum fidelity has been investigated in various scientific fields, but the identification of associated optimal measurements has often been overlooked despite its great importance both for fundamental interest and practical purposes. We find here the optimal POVMs for quantum fidelity between multimode Gaussian states in a closed analytical form. Our general finding is applied for selected single-mode Gaussian states of particular interest and we identify three types of optimal measurements: an excitation-number-resolving detection, a projection onto the eigenbasis of operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}+\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}$, and a quadrature variable detection, each of which corresponds to distinct types of single-mode Gaussian states. We also show the equivalence between optimal measurements for quantum fidelity and those for quantum parameter estimation when two arbitrary states are infinitesimally close. It is applied for simplifying the derivations of quantum Fisher information and the associated optimal measurements, exemplified by displacement, phase, squeezing, and loss parameter estimation using Gaussian states.
Are Gaussian measurements enough to distinguish between Gaussian states? Here, we tackle this question by focusing on the max-relative entropy as an operational distinguishability metric. Given two general multimode Gaussian states, we derive a condition, based on their covariance matrices, that completely determines whether or not there exists an optimal Gaussian measurement achieving the max-relative entropy. When the condition is satisfied, we find this optimal measurement explicitly. When the condition is not met, there is a strict gap between the distinguishability achievable by Gaussian measurements and the unconstrained max-relative entropy in which all measurements are allowed. We illustrate our results in the single-mode setting, and show examples of states for which this gap can be made arbitrarily large, revealing novel instances of Gaussian data hiding.
Quantum state discrimination plays a central role in quantum information and communication. For the discrimination of optical quantum states, the two most widely adopted measurement techniques are photon detection, which produces discrete outcomes, and homodyne detection, which produces continuous outcomes. While various protocols using photon detection have been proposed for optimal and near-optimal discrimination between two coherent states, homodyne detection is known to have higher error rates, with its minimum achievable error rate often referred to as the Gaussian limit. In this work, we demonstrate that, despite the fundamental differences between discretely labelled and continuously labelled measurements, continuously labelled non-Gaussian measurements can also achieve near-optimal coherent state discrimination. We design two discrimination protocols that surpass the Gaussian limit: one using non-Gaussian unitary operations with homodyne detection, and another based on orthogonal polynomials. Our results show that photon detection is not required for near-optimal coherent state discrimination and that we can achieve error rates close to the Helstrom bound at low energies with continuously labelled measurements. We also find that our schemes maintain an advantage over the photon detection-based Kennedy receiver for a moderate range of coherent state amplitudes.
In this work, we consider the estimation of single-mode Gaussian states using four different measurement schemes, namely, (1) homodyne measurement, (2) heterodyne measurement, (3) sequential measurement scheme, and (4) the Arthurs-Kelly measurement scheme, with a view to compare their relative performance. To this end, we work in the phase space formalism, specifically at the covariance matrix level, which provides an elegant and intuitive way to explicitly carry out the involved calculations. We show that the optimal performance of the Arthurs-Kelly and sequential measurement schemes is equal to the heterodyne measurement. While the heterodyne measurement outperforms the homodyne measurement in the mean estimation of squeezed state ensembles, the homodyne measurement outperforms the heterodyne measurement for variance estimation of squeezed state ensembles up to a certain range of the squeezing parameter. We then modify the Hamiltonian in the Arthurs-Kelly measurement scheme, such that the two meters can have correlations and show that the optimal performance is achieved when the meters are uncorrelated. We expect that these results will be useful in various quantum information and quantum communication protocols.
The description of Gaussian noise for single-mode quantum fields is briefly reviewed and applied to calculate the maximal information properties of three quantum communication channels degraded by Gaussian noise. These channels are based on (i) heterodyne detection of coherent states, (ii) homodyne detection of squeezed states, and (iii) photodetection of number states. It is found that each channel can outperform the others for a given noise level, where the optimum channel is essentially determined by the product of the average noise and signal energies.
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 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 $M\ensuremath{\rightarrow}\ensuremath{\infty}$.