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.
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.
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.
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.
Laser-light (coherent-state) modulation is sufficient to achieve the ultimate (Holevo) capacity of classical communication over a lossy and noisy optical channel, but requires a receiver that jointly detects long modulated code words with highly nonlinear quantum operations, which are near-impossible to realize using current technology. We analyze the capacity of the lossy-noisy optical channel when the transmitter uses coherent-state modulation but the receiver is restricted to a general quantum-limited Gaussian receiver, i.e., one that may involve arbitrary combinations of Gaussian operations [passive linear optics: beam splitters and phase shifters; second-order nonlinear optics (or active linear optics): squeezers, along with homodyne or heterodyne detection measurements] and any amount of classical feedforward within the receiver. Under these assumptions, we show that the Gaussian receiver that attains the maximum mutual information is either homodyne detection, heterodyne detection, or time sharing between the two, depending upon the received power level. In other words, our result shows that to exceed the theoretical limit of conventional coherent optical communication, one has to incorporate non-Gaussian, i.e., third- or higher-order nonlinear operations in the receiver. Finally we compare our Gaussian receiver limit with experimentally feasible non-Gaussian receivers and show that in the regime of low received photon flux, it is possible to overcome the Gaussian receiver limit by relatively simple non-Gaussian receivers based on photon counting.
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.
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.
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).