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Journal of the Optical Society of America A

Characterizing the information capacity of volume holograms with the Holevo bound

Roberto Zecca, David R. Smith, Daniel L. Marks

Published Apr 29, 2019• 0 citations
Publisher

Abstract

We show that Holevo's inequality upper-bounds the information capacity of a volume hologram without requiring a specification of the implementation of the hologram or the measurements to be made on the field scattered by the hologram. We find that, in the weakly scattering limit, the information capacity is not determined by the number of possible configurations of the holographic medium but only by the specification of the incident fields in the medium volume, which are determined at the volume boundary. By treating a volume hologram as a quantum communication channel, we establish a correspondence between the reconstruction of a hologram with multiple incident and scattered waves and the measurement of a quantum state with an operator-valued measure. We determine a bound on the probability of communications error, which describes the degree to which the incident fields may be distinguished by measurements of the scattered waves or, in other words, the crosstalk between the incident fields.

DOI
10.1364/josaa.36.000930

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We show that Holevo's inequality upper-bounds the information capacity of a volume hologram without requiring a specification of the implementation of the hologram or the measurements to be made on the field scattered by the hologram. We find that, in the weakly scattering limit, the information capacity is not determined by the number of possible configurations of the holographic medium but only by the specification of the incident fields in the medium volume, which are determined at the volume boundary. By treating a volume hologram as a quantum communication channel, we establish a correspondence between the reconstruction of a hologram with multiple incident and scattered waves and the measurement of a quantum state with an operator-valued measure. We determine a bound on the probability of communications error, which describes the degree to which the incident fields may be distinguished by measurements of the scattered waves or, in other words, the crosstalk between the incident fields.
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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.
The classical capacity of a noisy optical channel is the maximum rate at which information can be transmitted reliably, given a constraint on the mean input energy. For the ubiquitous bosonic Gaussian channels, which include standard models of loss and noise in fibre and free-space optical links, determining this rate requires addressing whether entanglement between channel uses or non-Gaussian signals can outperform Gaussian encodings. More precisely, the question is whether the energy-constrained Holevo information is additive across arbitrarily many channel uses and coincides with its Gaussian counterpart. We answer these fundamental questions in the affirmative for every single-mode bosonic Gaussian channel at every input energy. Namely, we prove that independently prepared Gaussian states with Gaussian-distributed random displacements achieve the capacity, including in the low-energy regime where squeezing the signals competes with encoding information in their displacement. Our proof connects an entanglement inequality to a sharp, additive lower bound on output entropy. We then characterise communication above capacity: under a strict total photon-number cutoff, the probability of correct decoding decreases exponentially with the number of channel uses, and we derive explicit bounds on the decay exponent. When only the mean photon number is constrained, a simple energy-rescaling formula determines exactly the entire asymptotic rate--error trade-off: the capacity at mean photon number NNN and allowed average decoding error εεε equals the vanishing-error capacity at mean photon number N/(1−ε)N/(1-ε)N/(1−ε). Together, these results identify the ultimate limits of classical communication for some of the most fundamental models of optical communication.
We present a family of easily computable upper bounds for the Holevo (information) quantity of an ensemble of quantum states depending on a reference state (as a free parameter). These upper bounds are obtained by combining probabilistic and metric characteristics of the ensemble. We show that an appropriate choice of the reference state gives tight upper bounds for the Holevo quantity which in many cases improve the estimates existing in the literature. We also present an upper bound for the Holevo quantity of a generalized ensemble of quantum states with finite average energy depending on the metric divergence of an ensemble. In the case of a multi-mode quantum oscillator, this upper bound is tight for large energy. Upper bounds for the Holevo capacity of finite-dimensional quantum channels depending on metric characteristics of the channel output are obtained.
The capacity CCC of a communication channel is the maximum rate at which information can be transmitted without error from the channel's input to its output. The authors review quantum limits on the capacity that can be achieved with linear bosonic communication channels that have input power PPP. The limits arise ultimately from the Einstein relation that a field quantum at frequency fff has energy E=hfE=hfE=hf. A single linear bosonic channel corresponds to a single transverse mode of the bosonic field---i.e., to a particular spatial dependence in the plane orthogonal to the propagation direction and to a particular spin state or polarization. For a single channel the maximum communication rate is ${C}_{\mathrm{WB}}=(\frac{\ensuremath{\pi}}{\mathrm{ln}2})\sqrt{\frac{2P}{3h}}$ bits/s. This maximum rate can be achieved by a "number-state channel," in which information is encoded in the number of quanta in the bosonic field and in which this information is recovered at the output by counting quanta. Derivations of the optimum capacity CWB{C}_{\mathrm{WB}}CWB​ are reviewed. Until quite recently all derivations assumed, explicitly or implicitly, a number-state channel. They thus left open the possibility that other techniques for encoding information on the bosonic field, together with other ways of detecting the field at the output, might lead to a greater communication rate. The authors present their own general derivation of the single-channel capacity upper bound, which applies to any physically realizable technique for encoding information on the bosonic field and to any physically realizable detection scheme at the output. They also review the capacities of coherent communication channels that encode information in coherent states and in quadrature-squeezed states. A three-dimensional bosonic channel can employ many transverse modes as parallel single channels. An upper bound on the information flux that can be transferred down parallel bosonic channels is derived.