Explicit capacity-achieving receivers for optical communication and quantum reading

An important practical open question has been to design explicit, structured optical receivers that achieve the Holevo limit in the contexts of optical communication and “quantum reading.” The Holevo limit is an achievable rate that is higher than the Shannon limit of any known optical receiver. We demonstrate how a sequential decoding approach can achieve the Holevo limit for both of these settings. A crucial part of our scheme for both settings is a non-destructive “vacuum-or-not” measurement that projects an n-symbol modulated codeword onto the n-fold vacuum state or its orthogonal complement, such that the post-measurement state is either the n-fold vacuum or has the vacuum removed from the support of the n symbols' joint quantum state. The sequential decoder for optical communication requires the additional ability to perform multimode optical phase-space displacements - realizable using a beamsplitter and a laser, while the sequential decoder for quantum reading also requires the ability to perform phase-shifting (realizable using a phase plate) and online squeezing (a phase-sensitive amplifier)

Explicit receivers for pure-interference bosonic multiple access channels

The pure-interference bosonic multiple access channel has two senders and one receiver, such that the senders each communicate with multiple temporal modes of a single spatial mode of light. The channel mixes the input modes from the two users pairwise on a lossless beamsplitter, and the receiver has access to one of the two output ports. In prior work, Yen and Shapiro found the capacity region of this channel if encodings consist of coherent-state preparations. Here, we demonstrate how to achieve the coherent-state Yen-Shapiro region (for a range of parameters) using a sequential decoding strategy, and we show that our strategy outperforms the rate regions achievable using conventional receivers. Our receiver performs binary-outcome quantum measurements for every codeword pair in the senders' codebooks. A crucial component of this scheme is a non-destructive "vacuum-or-not" measurement that projects an n-symbol modulated codeword onto the n-fold vacuum state or its orthogonal complement, such that the post-measurement state is either the n-fold vacuum or has the vacuum removed from the support of the n symbols' joint quantum state. This receiver requires the additional ability to perform multimode optical phase-space displacements which are realizable using a beamsplitter and a laser.Comment: v1: 9 pages, 2 figures, submission to the 2012 International Symposium on Information Theory and its Applications (ISITA 2012), Honolulu, Hawaii, USA; v2: minor change

Second-order coding rates for pure-loss bosonic channels

A pure-loss bosonic channel is a simple model for communication over free-space or fiber-optic links. More generally, phase-insensitive bosonic channels model other kinds of noise, such as thermalizing or amplifying processes. Recent work has established the classical capacity of all of these channels, and furthermore, it is now known that a strong converse theorem holds for the classical capacity of these channels under a particular photon number constraint. The goal of the present paper is to initiate the study of second-order coding rates for these channels, by beginning with the simplest one, the pure-loss bosonic channel. In a second-order analysis of communication, one fixes the tolerable error probability and seeks to understand the back-off from capacity for a sufficiently large yet finite number of channel uses. We find a lower bound on the maximum achievable code size for the pure-loss bosonic channel, in terms of the known expression for its capacity and a quantity called channel dispersion. We accomplish this by proving a general "one-shot" coding theorem for channels with classical inputs and pure-state quantum outputs which reside in a separable Hilbert space. The theorem leads to an optimal second-order characterization when the channel output is finite-dimensional, and it remains an open question to determine whether the characterization is optimal for the pure-loss bosonic channel.Comment: 18 pages, 3 figures; v3: final version accepted for publication in Quantum Information Processin

Strong converse for the classical capacity of the pure-loss bosonic channel

This paper strengthens the interpretation and understanding of the classical capacity of the pure-loss bosonic channel, first established in [Giovannetti et al., Physical Review Letters 92, 027902 (2004), arXiv:quant-ph/0308012]. In particular, we first prove that there exists a trade-off between communication rate and error probability if one imposes only a mean-photon number constraint on the channel inputs. That is, if we demand that the mean number of photons at the channel input cannot be any larger than some positive number N_S, then it is possible to respect this constraint with a code that operates at a rate g(\eta N_S / (1-p)) where p is the code's error probability, \eta\ is the channel transmissivity, and g(x) is the entropy of a bosonic thermal state with mean photon number x. We then prove that a strong converse theorem holds for the classical capacity of this channel (that such a rate-error trade-off cannot occur) if one instead demands for a maximum photon number constraint, in such a way that mostly all of the "shadow" of the average density operator for a given code is required to be on a subspace with photon number no larger than n N_S, so that the shadow outside this subspace vanishes as the number n of channel uses becomes large. Finally, we prove that a small modification of the well-known coherent-state coding scheme meets this more demanding constraint.Comment: 18 pages, 1 figure; accepted for publication in Problems of Information Transmissio

Polar coding to achieve the Holevo capacity of a pure-loss optical channel

In the low-energy high-energy-efficiency regime of classical optical communications - relevant to deep-space optical channels - there is a big gap between reliable communication rates achievable via conventional optical receivers and the ultimate (Holevo) capacity. Achieving the Holevo capacity requires not only optimal codes but also receivers that make collective measurements on long (modulated) codeword waveforms, and it is impossible to implement these collective measurements via symbol-by-symbol detection along with classical postprocessing [1], [2]. Here, we apply our recent results on the classical-quantum polar code [3] - the first near-explicit, linear, symmetric-Holevo-rate achieving code - to the lossy optical channel, and we show that it almost closes the entire gap to the Holevo capacity in the low photon number regime. In contrast, Arikan\u27s original polar codes, applied to the DMC induced by the physical optical channel paired with any conceivable structured optical receiver (including optical homodyne, heterodyne, or direct-detection) fails to achieve the ultimate Holevo limit to channel capacity. However, our polar code construction (which uses the quantum fidelity as a channel parameter rather than the classical Bhattacharyya quantity to choose the good channels in the polar-code construction), paired with a quantum successive-cancellation receiver - which involves a sequence of collective non-destructive binary projective measurements on the joint quantum state of the received codeword waveform - can attain the Holevo limit, and can hence in principle achieve higher rates than Arikan\u27s polar code and decoder directly applied to the optical channel. However, even a theoretical recipe for construction of an optical realization of the quantum successive-cancellation receiver remains an open question. © 2012 IEEE

Towards efficient decoding of classical-quantum polar codes

Known strategies for sending bits at the capacity rate over a general channel with classical input and quantum output (a cq channel) require the decoder to implement impractically complicated collective measurements. Here, we show that a fully collective strategy is not necessary in order to recover all of the information bits. In fact, when coding for a large number N uses of a cq channel W, N I(W_acc) of the bits can be recovered by a non-collective strategy which amounts to coherent quantum processing of the results of product measurements, where I(W_acc) is the accessible information of the channel W. In order to decode the other N (I(W) - I(W_acc)) bits, where I(W) is the Holevo rate, our conclusion is that the receiver should employ collective measurements. We also present two other results: 1) collective Fuchs-Caves measurements (quantum likelihood ratio measurements) can be used at the receiver to achieve the Holevo rate and 2) we give an explicit form of the Helstrom measurements used in small-size polar codes. The main approach used to demonstrate these results is a quantum extension of Arikan's polar codes.Comment: 21 pages, 2 figures, submission to the 8th Conference on the Theory of Quantum Computation, Communication, and Cryptograph

Joint source-channel coding for a quantum multiple access channel

Suppose that two senders each obtain one share of the output of a classical, bivariate, correlated information source. They would like to transmit the correlated source to a receiver using a quantum multiple access channel. In prior work, Cover, El Gamal, and Salehi provided a combined source-channel coding strategy for a classical multiple access channel which outperforms the simpler "separation" strategy where separate codebooks are used for the source coding and the channel coding tasks. In the present paper, we prove that a coding strategy similar to the Cover-El Gamal-Salehi strategy and a corresponding quantum simultaneous decoder allow for the reliable transmission of a source over a quantum multiple access channel, as long as a set of information inequalities involving the Holevo quantity hold.Comment: 21 pages, v2: minor changes, accepted into Journal of Physics

Quantum reading under a local energy constraint

Nonclassical states of light play a central role in many quantum information protocols. Their quantum features have been exploited to improve the readout of information from digital memories, modelled as arrays of microscopic beam splitters [S. Pirandola, Phys. Rev. Lett. 106, 090504 (2011)]. In this model of quantum reading, a nonclassical source of light with Einstein-Podolski-Rosen correlations has been proven to retrieve more information than any classical source. In particular, the quantum-classical comparison has been performed under a global energy constraint, i.e., by fixing the mean total number of photons irradiated over each memory cell. In this paper we provide an alternative analysis which is based on a local energy constraint, meaning that we fix the mean number of photons per signal mode irradiated over the memory cell. Under this assumption, we investigate the critical number of signal modes after which a nonclassical source of light is able to beat any classical source irradiating the same number of signals.Comment: REVTeX. Published versio