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arXiv:2102.01981 · Journal of Mathematical Physics

Accessible information of a general quantum Gaussian ensemble

Alexander Semenovich Holevo

Published Sep 1, 2021• 9 citations
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Abstract

Accessible information, which is a basic quantity in quantum information theory, is computed for a general quantum Gaussian ensemble under certain “threshold conditions.” It is shown that the maximizing measurement is Gaussian, constituting a far-reaching generalization of optical heterodyning. This substantially extends the previous result concerning the gauge-invariant case even for a single bosonic mode. A simple sufficient condition is provided that implies the threshold condition for a general Gaussian ensemble. The tightness of the sufficient condition is demonstrated in the single-mode case.

DOI
10.1063/5.0048112

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•Aug 1, 2013•0 citations
Accessible information, which is a basic quantity in quantum information theory, is computed for a general quantum Gaussian ensemble under certain “threshold conditions.” It is shown that the maximizing measurement is Gaussian, constituting a far-reaching generalization of optical heterodyning. This substantially extends the previous result concerning the gauge-invariant case even for a single bosonic mode. A simple sufficient condition is provided that implies the threshold condition for a general Gaussian ensemble. The tightness of the sufficient condition is demonstrated in the single-mode case.
In this paper we prove two results related to the Gaussian optimizers conjecture for multimode bosonic system with gauge symmetry. First, we argue that the classical capacity of an arbitrary Gaussian observable is attained on a Gaussian ensemble of coherent states. This generalizes results previously known for heterodyne measurement in one mode. By using this fact and continuous variable version of ensemble-observable duality, we prove an old conjecture that accessible information of arbitrary Gaussian ensemble is attained on the multimode generalization of the heterodyne measurement.
In this paper, we consider the classical capacity problem for Gaussian measurement channels. We establish Gaussianity of the average state of the optimal ensemble in the general case and discuss the Hypothesis of Gaussian Maximizers concerning the structure of the ensemble. Then, we consider the case of one mode in detail, including the dual problem of accessible information of a Gaussian ensemble. Our findings are relevant to practical situations in quantum communications where the receiver is Gaussian (say, a general-dyne detection) and concatenation of the Gaussian channel and the receiver can be considered as one Gaussian measurement channel. Our efforts in this and preceding papers are then aimed at establishing full Gaussianity of the optimal ensemble (usually taken as an assumption) in such schemes.
Given a quantum state ensemble E\mathcal{E}E and quantum observable M,\mathcal{M},M, one can define the Shannon information I(E,M)I(\mathcal{E},\mathcal{M})I(E,M) and introduce the two fundamental information quantities. The accessible information A(E)A(\mathcal{E})A(E) of the ensemble E\mathcal{E}E is defined as supremum of I(E,M)I(\mathcal{E},\mathcal{M})I(E,M) over all observables M\mathcal{M}M . The information capacity C(M)C(\mathcal{M})C(M) of the observable M\mathcal{M}M is essentially supremum of I(E,M)I(\mathcal{E},\mathcal{M})I(E,M) over ensembles E\mathcal{E}E (where E\mathcal{E}E may be subject to certain additional constraints). Computation of these quantities presents important and rather involved optimization problems which are closely connected via the so called ensemble-observable duality. This paper is devoted to consideration of these two quantities for quantum Gaussian systems. In this case ensemble-observable duality admits quite an explicit description. The present paper surveys results recently obtained in these directions. It turns out that in both cases the maximizer is quantum Gaussian: the quantity I(E,M),I(\mathcal{E},\mathcal{M}),I(E,M), where E\mathcal{E}E is Gaussian ensemble, is maximized by a Gaussian observable, while I(E,M)I(\mathcal{E},\mathcal{M})I(E,M) for Gaussian observable M\mathcal{M}M is maximized by a Gaussian state ensemble. Thus, we have here still another confirmation of the famous ‘‘hypothesis of quantum Gaussian optimizers’’.
The pretty good measurement is a fundamental analytical tool in quantum information theory, giving a method for inferring the classical label that identifies a quantum state chosen probabilistically from an ensemble. Identifying and constructing the pretty good measurement for the class of bosonic Gaussian states is of immediate practical relevance in quantum information processing tasks. Holevo recently showed that the pretty good measurement for a bosonic Gaussian ensemble is a bosonic Gaussian measurement that attains the accessible information of the ensemble [IEEE Trans. Inf. Theory 66(9) (2020) 5634]. In this paper, we provide an alternate proof of Gaussianity of the pretty good measurement for a Gaussian ensemble of multimode bosonic states, with a focus on establishing an explicit and efficiently computable Gaussian description of the measurement. We also compute an explicit form of the mean square error of the pretty good measurement, which is relevant when using it for parameter estimation. Generalizing the pretty good measurement is a quantum instrument, called the pretty good instrument. We prove that the post-measurement state of the pretty good instrument is a faithful Gaussian state if the input state is a faithful Gaussian state whose covariance matrix satisfies a certain condition. Combined with our previous finding for the pretty good measurement and provided that the same condition holds, it follows that the expected output state is a faithful Gaussian state as well. In this case, we compute an explicit Gaussian description of the post-measurement and expected output states. Our findings imply that the pretty good instrument for bosonic Gaussian ensembles is no longer merely an analytical tool, but that it can also be implemented experimentally in quantum optics laboratories.
Abstract The present paper is devoted to investigation of the classical capacity of infinite-dimensional continuous variable quantum measurement channels. A number of usable conditions are introduced that enable us to apply previously obtained general results to specific models, in particular, to the multi-mode bosonic Gaussian measurement channels. An explicit formula for the classical capacity of the Gaussian measurement channel is obtained in this paper without assuming the global gauge symmetry, solely under certain ‘threshold condition’. The result is illustrated by the capacity computation for a one-mode squeezed-noise heterodyne measurement channel.
The classical capacity of quantum channels is the tight upper bound for the transmission rate of classical information. This is a quantum counterpart of the foundational notion of the channel capacity introduced by Shannon. Bosonic Gaussian quantum channels provide a good model for optical communications. In order to properly define the classical capacity for these quantum systems, an energy constraint at the channel input is necessary, as in the classical case. A further restriction to Gaussian input ensembles defines the Gaussian (classical) capacity, which can be studied analytically. It also provides a lower bound on the classical capacity and moreover, it is conjectured to coincide with the classical capacity. Therefore, the Gaussian capacity is a useful and important notion in quantum information theory. Recently, we have shown that the study of both the classical and Gaussian capacity of an arbitrary single mode Gaussian quantum channel can be reduced to the study of a particular fiducial channel. In this work we consider the Gaussian capacity of the fiducial channel, discuss its additivity and analyze its dependence on the channel parameters. In addition, we extend previously obtained results on the optimal channel environment to the single mode fiducial channel. In particular, we show that the optimal channel environment for the lossy, amplification, and phase conjugating channels is given by a pure quantum state if its energy is constrained.