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arXiv:1207.6435 · Physical Review A

Reading boundless error-free bits using a single photon

Saikat Guha, Jeffrey H. Shapiro

Published Jun 6, 2013• 25 citations
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Abstract

We address the problem of how efficiently information can be encoded into and read out reliably from a passive reflective surface that encodes classical data by modulating the amplitude and phase of incident light. We show that nature imposes no fundamental upper limit to the number of bits that can be read per expended probe photon and demonstrate the quantum-information-theoretic trade-offs between the photon efficiency (bits per photon) and the encoding efficiency (bits per pixel) of optical reading. We show that with a coherent-state (ideal laser) source, an on-off (amplitude-modulation) pixel encoding, and shot-noise-limited direct detection (an overly optimistic model for commercial CD and DVD drives), the highest photon efficiency achievable in principle is about 0.5 bits read per transmitted photon. We then show that a coherent-state probe can read unlimited bits per photon when the receiver is allowed to make joint (inseparable) measurements on the reflected light from a large block of phase-modulated memory pixels. Finally, we show an example of a spatially entangled nonclassical light probe and a receiver design---constructible using a single-photon source, beam splitters, and single-photon detectors---that can in principle read any number of error-free bits of information. The probe is a single photon prepared in a uniform coherent superposition of multiple orthogonal spatial modes, i.e., a $W$ state. The code and joint-detection receiver complexity required by a coherent-state transmitter to achieve comparable photon efficiency performance is shown to be much higher in comparison to that required by the $W$-state transceiver, although this advantage rapidly disappears with increasing loss in the system.

DOI
10.1103/physreva.87.062306

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We address the problem of how efficiently information can be encoded into and read out reliably from a passive reflective surface that encodes classical data by modulating the amplitude and phase of incident light. We show that nature imposes no fundamental upper limit to the number of bits that can be read per expended probe photon and demonstrate the quantum-information-theoretic trade-offs between the photon efficiency (bits per photon) and the encoding efficiency (bits per pixel) of optical reading. We show that with a coherent-state (ideal laser) source, an on-off (amplitude-modulation) pixel encoding, and shot-noise-limited direct detection (an overly optimistic model for commercial CD and DVD drives), the highest photon efficiency achievable in principle is about 0.5 bits read per transmitted photon. We then show that a coherent-state probe can read unlimited bits per photon when the receiver is allowed to make joint (inseparable) measurements on the reflected light from a large block of phase-modulated memory pixels. Finally, we show an example of a spatially entangled nonclassical light probe and a receiver design---constructible using a single-photon source, beam splitters, and single-photon detectors---that can in principle read any number of error-free bits of information. The probe is a single photon prepared in a uniform coherent superposition of multiple orthogonal spatial modes, i.e., a WWW state. The code and joint-detection receiver complexity required by a coherent-state transmitter to achieve comparable photon efficiency performance is shown to be much higher in comparison to that required by the WWW-state transceiver, although this advantage rapidly disappears with increasing loss in the system.
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.
The readout of a classical memory can be modelled as a problem of quantum channel discrimination, where a decoder retrieves information by distinguishing the different quantum channels encoded in each cell of the memory (Pirandola 2011 Phys. Rev. Lett. 106 090504). In the case of optical memories, such as CDs and DVDs, this discrimination involves lossy bosonic channels and can be remarkably boosted by the use of nonclassical light (quantum reading). Here we generalize these concepts by extending the model of memory from single-cell to multi-cell encoding. In general, information is stored in a block of cells by using a channel-codeword, i.e. a sequence of channels chosen according to a classical code. Correspondingly, the readout of data is realized by a process of 'parallel' channel discrimination, where the entire block of cells is probed simultaneously and decoded via an optimal collective measurement. In the limit of a large block we define the quantum reading capacity of the memory, quantifying the maximum number of readable bits per cell. This notion of capacity is nontrivial when we suitably constrain the physical resources of the decoder. For optical memories (encoding bosonic channels), such a constraint is energetic and corresponds to fixing the mean total number of photons per cell. In this case, we are able to prove a separation between the quantum reading capacity and the maximum information rate achievable by classical transmitters, i.e. arbitrary classical mixtures of coherent states. In fact, we can easily construct nonclassical transmitters that are able to outperform any classical transmitter, thus showing that the advantages of quantum reading persist in the optimal multi-cell scenario.
We provide a summary of the classical information capacity of single-mode free-space optical communication for pure-loss channels. We compare the capacities afforded by structured transmitters and receivers to that of the ultimate communication capacity dictated by the quantum nature of light, and we draw the following conclusions. The ultimate capacity can be achieved with classical coherent states (i.e., ideal laser light), but the capacity-achieving receiver (measurement) is yet to be determined. In photon-starved pure-loss channels, binary phase modulation in combination with the optimal receiver is near-capacity achieving, and more importantly, it is superior to on-off keying with either the optimal receiver (as yet to be determined) or with a photon-counter. Heterodyne detection approaches the ultimate capacity at high mean photon numbers.
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.
It has been shown recently S. Pirandola [Phys. Rev. Lett. 106, 090504 (2011)] that entangled light with Einstein-Podolsky-Rosen correlations retrieves information from digital memory better than any classical light. In identifying this, a model of digital memory with each cell consisting of a reflecting medium with two reflectivities (each memory cell encoding the binary numbers 0 or 1) is employed. The readout of binary memory essentially corresponds to discrimination of two bosonic attenuator channels characterized by different reflectivities. The model requires an entire mathematical paraphernalia of a continuous variable Gaussian setting for its analysis when arbitrary values of reflectivities $0\ensuremath{\le}{r}_{0},{r}_{1}\ensuremath{\le}1$ are considered. Here we restrict ourselves to a basic quantum readout mechanism with two different families of non-Gaussian entangled states of light, in which the binary channels to be discriminated are (i) ideal memory characterized by reflectivity r1=1{r}_{1}=1r1​=1 (identity channel) and (ii) a thermal noise channel---where the signal light illuminating the memory location gets completely lost (r0=0{r}_{0}=0r0​=0) and only a white thermal noise hitting the upper side of the memory reaches the decoder. We compare the quantum reading efficiency of two families of non-Gaussian entangled light [($m,{m}^{\ensuremath{'}}$) family of path-entangled photon states and entangled state obtained by mixing a single photon with coherent light in a 50:50 beam splitter] with any classical source of light in this model. We identify that the classes of non-Gaussian entangled transmitters studied here offer significantly better reading performance than any classical transmitters of light in the regime of low signal intensity. We also demonstrate that the ($m,{m}^{\ensuremath{'}}$) family of entangled light exhibits better reading performance than NOON states.
Coherent states achieve the Holevo capacity of a pure-loss channel when paired with an optimal measurement, but a physical realization of this measurement scheme is as of yet unknown, and it is also likely to be of high complexity. In this paper, we focus on the photon-counting measurement and study the photon and dimensional efficiencies attainable with modulations over classical- and nonclassical-state alphabets. We analyze two binary-modulation architectures that improve upon the dimensional versus photon efficiency tradeoff achievable with the state-of-the-art coherent-state on-off keying modulation. We show that at high photon efficiency these architectures achieve an efficiency tradeoff that differs from the best possible tradeoff--determined by the Holevo capacity--by only a constant factor. The first architecture we analyze is a coherent-state transmitter that relies on feedback from the receiver to control the transmitted energy. The second architecture uses a single-photon number-state source.