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arXiv:2310.05889 · Nature Communications

Superadditive communication with the green machine as a practical demonstration of nonlocality without entanglement

Chaohan Cui, Jack T. Postlewaite, Babak N. Saif, Linran Fan, Saikat Guha

Published Apr 22, 2025• 18 citations

Abstract

Achieving the ultimate Holevo limit of optical communication capacity requires a joint-detection receiver that makes a collective quantum measurement over multiple modulated symbols. Such superadditivity—a higher communication rate than is achievable by symbol-by-symbol optical detection—is a special case of the well-known nonlocality without entanglement and has yet to be demonstrated. In this article, we propose and demonstrate the design of a joint-detection receiver, the Green Machine, that can achieve superadditivity. We build this receiver, experimentally obtain the transition probability matrix induced by the codebook-receiver pair, and deduce that its capacity surpasses that of any symbol-by-symbol receiver in the photon-starved regime for binary-phase-shift-keying (BPSK)modulation. Our Green Machine receiver can also significantly reduce the transmitter peak power requirement compared with the pulse-position modulation (the conventional modulation format used for deep space laser communication). We further demonstrate that the self-referenced phase makes it resilient to phase noise, e.g., atmospheric turbulence or platform vibrations. The so-called Green Machine is an old concept for a joint detection receiver that would allow superadditive optical communication capacity, but earlier designs are very hard to implement. Here, the authors propose a modified scheme and use it to demonstrate superadditive capacity with the BPSK Hadamard codewords.

DOI: 10.1038/s41467-025-59107-4

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Achieving the ultimate Holevo limit of optical communication capacity requires a joint-detection receiver that makes a collective quantum measurement over multiple modulated symbols. Such superadditivity—a higher communication rate than is achievable by symbol-by-symbol optical detection—is a special case of the well-known nonlocality without entanglement and has yet to be demonstrated. In this article, we propose and demonstrate the design of a joint-detection receiver, the Green Machine, that can achieve superadditivity. We build this receiver, experimentally obtain the transition probability matrix induced by the codebook-receiver pair, and deduce that its capacity surpasses that of any symbol-by-symbol receiver in the photon-starved regime for binary-phase-shift-keying (BPSK)modulation. Our Green Machine receiver can also significantly reduce the transmitter peak power requirement compared with the pulse-position modulation (the conventional modulation format used for deep space laser communication). We further demonstrate that the self-referenced phase makes it resilient to phase noise, e.g., atmospheric turbulence or platform vibrations. The so-called Green Machine is an old concept for a joint detection receiver that would allow superadditive optical communication capacity, but earlier designs are very hard to implement. Here, the authors propose a modified scheme and use it to demonstrate superadditive capacity with the BPSK Hadamard codewords.

For space-based laser communications, when the mean photon number per received optical pulse is much smaller than one, there is a large gap between communications capacity---measured in bits communicated per pulse---achievable with a receiver that detects (converts from optical to electrical domain) each modulated pulse one at a time, versus with the quantum-optimal joint-detection receiver that acts on a long codeword comprised of

n

modulated pulses; an effect often termed superadditive capacity. The action of this receiver cannot be described as the detection of each individual pulse, interspersed with classical feedforward and soft-information post-processing. In this paper, we consider the simplest scenario where a large superadditive capacity is known: a pure-loss channel with a coherent-state binary phase-shift keyed (BPSK) modulation. The two BPSK states can be mapped conceptually to two non-orthogonal states of a single qubit, described by an inner product that is a function of the mean photon number of each BPSK pulse. Using this map, we derive an explicit construction of the quantum circuit of a joint-detection receiver based on a recent idea of belief-propagation with quantum messages (BPQM) [arXiv:1607.04833]. We analyze this scheme rigorously and show that it achieves the quantum limit of minimum average error probability in discriminating

8

(BPSK) codewords of a length-

5

binary linear code with a tree factor graph. We quantify its performance improvement over the (Dolinar) receiver that optimally detects one pulse at a time. Our result suggests that a BPQM-receiver might attain the Holevo capacity, the quantum limit of classical communication capacity, of this BPSK-modulated pure-loss channel. This suggests a new application for a small, special-purpose, photonic quantum computer capable of so-called cat-basis universal qubit logic.

Abstract For space-based laser communications, when the mean photon number per received optical pulse is much smaller than one, there is a large gap between communications capacity achievable with a receiver that performs individual pulse-by-pulse detection, and the quantum-optimal “joint-detection receiver” that acts collectively on long codeword-blocks of modulated pulses; an effect often termed “superadditive capacity”. In this paper, we consider the simplest scenario where a large superadditive capacity is known: a pure-loss channel with a coherent-state binary phase-shift keyed (BPSK) modulation. The two BPSK states can be mapped conceptually to two non-orthogonal states of a qubit, described by an inner product that is a function of the mean photon number per pulse. Using this map, we derive an explicit construction of the quantum circuit of a joint-detection receiver based on a recent idea of “belief-propagation with quantum messages” (BPQM). We quantify its performance improvement over the Dolinar receiver that performs optimal pulse-by-pulse detection, which represents the best “classical” approach. We analyze the scheme rigorously and show that it achieves the quantum limit of minimum average error probability in discriminating 8 (BPSK) codewords of a length-5 binary linear code with a tree factor graph. Our result suggests that a BPQM receiver might attain the Holevo capacity of this BPSK-modulated pure-loss channel. Moreover, our receiver circuit provides an alternative proposal for a quantum supremacy experiment, targeted at a specific application that can potentially be implemented on a small, special-purpose, photonic quantum computer capable of performing cat-basis universal qubit logic.

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.

We propose a simple architecture based on multimode quantum memories for collective readout of classical information keyed using a pair coherent states, exemplified by the well-known binary phase shift keying format. Such a configuration enables demonstration of the superadditivity effect in classical communication over quantum channels, where the transmission rate becomes enhanced through joint detection applied to multiple channel uses. The proposed scheme relies on the recently introduced idea to prepare Hadamard sequences of input symbols that are mapped by a linear optical transformation onto the pulse position modulation format [Guha, S. Phys. Rev. Lett. 2011, 106, 240502]. We analyze two versions of readout based on direct detection and an optional Dolinar receiver which implements the minimum-error measurement for individual detection of a binary coherent state alphabet.

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.