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IEEE International Symposium on Information Theory (ISIT)

Achieving the Holevo capacity of a pure state classical-quantum channel via unambiguous state discrimination

Masahiro Takeoka, Hari Krovi, Saikat Guha

Published Jul 1, 2013• 12 citations
Publisher

Abstract

We prove that the ultimate channel capacity, the Holevo bound, for sending classical data on a quantum channel (the so-called classical-quantum, or cq channel) can be achieved for a pure-state cq channel by decoding codewords via a collective quantum measurement based on unambiguous state discrimination (USD). In cq communication theory, the channel decoder acts directly on the modulated codeword waveform in the quantum (viz., electromagnetic or optical) domain, and it is known that collective measurements on long codeword blocks are needed to attain the Holevo capacity, which is strictly larger than the Shannon capacity of the classical channel induced by any specific measurement choice on each channel use. The USD measurement based channel decoder we propose, can distinguish finite blocklength codeword quantum states unambiguously (i.e., an incorrect codeword is never chosen) provided one allows for a finite probability of obtaining an inconclusive (erasure) outcome. We show that the probability of the inconclusive outcome goes to zero for asymptotically long codewords whenever the code rate is below the Holevo bound. The USD channel decoder is an addition to a small list of other collective measurements known to achieve the Holevo capacity (such as, the square root measurement, the minimum probability of error measurement, the sequential decoding measurement, and the quantum successive cancellation decoder for the cq polar code). A structured optical receiver design is not known yet for any of these decoders. What makes the USD decoder special is that there is no classical analogue to truly unambiguous discrimination (say, of samples drawn from a set of probability distributions). Secondly, the erasures-only decoding of USD is likely to result in a better channel reliability function. Finally, the USD measurement seems more likely to lead naturally to a structured optical receiver design and implementation.

DOI
10.1109/isit.2013.6620209

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We prove that the ultimate channel capacity, the Holevo bound, for sending classical data on a quantum channel (the so-called classical-quantum, or cq channel) can be achieved for a pure-state cq channel by decoding codewords via a collective quantum measurement based on unambiguous state discrimination (USD). In cq communication theory, the channel decoder acts directly on the modulated codeword waveform in the quantum (viz., electromagnetic or optical) domain, and it is known that collective measurements on long codeword blocks are needed to attain the Holevo capacity, which is strictly larger than the Shannon capacity of the classical channel induced by any specific measurement choice on each channel use. The USD measurement based channel decoder we propose, can distinguish finite blocklength codeword quantum states unambiguously (i.e., an incorrect codeword is never chosen) provided one allows for a finite probability of obtaining an inconclusive (erasure) outcome. We show that the probability of the inconclusive outcome goes to zero for asymptotically long codewords whenever the code rate is below the Holevo bound. The USD channel decoder is an addition to a small list of other collective measurements known to achieve the Holevo capacity (such as, the square root measurement, the minimum probability of error measurement, the sequential decoding measurement, and the quantum successive cancellation decoder for the cq polar code). A structured optical receiver design is not known yet for any of these decoders. What makes the USD decoder special is that there is no classical analogue to truly unambiguous discrimination (say, of samples drawn from a set of probability distributions). Secondly, the erasures-only decoding of USD is likely to result in a better channel reliability function. Finally, the USD measurement seems more likely to lead naturally to a structured optical receiver design and implementation.
Unambiguous state discrimination (USD) measurements are attractive because outcomes are either marked as conclusive (i.e., error free) or inconclusive (i.e., erased). We study affine filtering measurements, a structured variant of USD for decoding classical linear codes over pure-state classical-quantum channels, where a conclusive outcome identifies an affine subspace containing the transmitted codeword and an inconclusive outcome is treated as an erasure. For a group-covariant indexing of pure-state codewords, we show that the optimal design of affine filtering measurements is a semidefinite program that can be reduced to a linear program via character-based diagonalization. We use the resulting measurement to build a quantum decoding framework for local codes, and we demonstrate (via simulations on regular LDPC codes from Gallager ensembles using single parity check local constraints) that affine filtering based decoding can outperform symbol-wise USD and symbol-wise pretty good measurement based decoding methods on i.i.d. pure-state channels. In an independent and concurrent work, Buzet and Chailloux study similar fine-grained USD measurements for symmetric families of states. Their focus is on the code-agnostic setting whereas our focus is on code-aware constructions and decoding.
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's 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's 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.
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.
Attaining the ultimate (Holevo) limit to the classical capacity of a quantum channel requires the receiver to make joint measurements over long code-word blocks. For a pure-state channel, we show that the Holevo limit can be attained by a receiver that uses a multisymbol unitary transformation on the quantum code word followed by separable projective measurements. We show a concatenated coding and joint-detection architecture to approach the Holevo limit. We then construct some of the first concrete examples of codes and structured joint-detection receivers for the lossy bosonic channel, which can achieve fundamentally higher (superadditive) capacity than conventional receivers that detect each modulation symbol individually. We thereby pave the way for research into codes and structured receivers for reliable communication data rates approaching the Holevo limit.
We present a decoding procedure to transmit classical information in a quantum channel which, saturating asymptotically the Holevo bound, achieves the optimal rate of the communication line. In contrast to previous proposals, it is based on performing a sequence of projective yes-no measurements which in NNN steps determines which codeword was sent by the sender (NNN being the number of the codewords). Our analysis shows that, as long as NNN is below the limit imposed by the Holevo bound, the error probability can be sent to zero asymptotically in the length of the codewords.
When classical information is sent over a channel with quantum-state modulation alphabet, such as the free-space optical (FSO) channel, attaining the ultimate (Holevo) limit to channel capacity requires the receiver to make joint measurements over long codeword blocks. In recent work, we showed a receiver for a pure-state channel that can attain the ultimate capacity by applying a single-shot optical (unitary) transformation on the received codeword state followed by simultaneous (but separable) projective measurements on the single-modulation-symbol state spaces. In this paper, we study the ultimate tradeoff between photon efficiency and spectral efficiency for the FSO channel. Based on our general results for the pure-state quantum channel, we show some of the first concrete examples of codes and laboratory-realizable joint-detection optical receivers that can achieve fundamentally higher (superadditive) channel capacity than receivers that physically detect each modulation symbol one at a time, as is done by all conventional (coherent or direct-detection) optical receivers.