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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

The Dolinar receiver in an information theoretic framework

Baris I. Erkmen, Kevin M. Birnbaum, Bruce E. Moision, Samuel J. Dolinar

Published Aug 30, 2011• 7 citations

Abstract

Optical communication at the quantum limit requires that measurements on the optical field be maximally informative, but devising physical measurements that accomplish this objective has proven challenging. The Dolinar receiver exemplifies a rare instance of success in distinguishing between two coherent states: an adaptive local oscillator is mixed with the signal prior to photodetection, which yields an error probability that meets the Helstrom lower bound with equality. Here we apply the same local-oscillator-based architecture with an information-theoretic optimization criterion. We begin with analysis of this receiver in a general framework for an arbitrary coherent-state modulation alphabet, and then we concentrate on two relevant examples. First, we study a binary antipodal alphabet and show that the Dolinar receiver's feedback function not only minimizes the probability of error, but also maximizes the mutual information. Next, we study ternary modulation consisting of antipodal coherent states and the vacuum state. We derive an analytic expression for a near-optimal local-oscillator feedback function, and, via simulation, we determine its photon information efficiency (PIE). We provide the PIE versus dimensional information efficiency (DIE) trade-off curve and show that this modulation and the our receiver combination performs universally better than (generalized) on-off keying plus photon counting, although, the advantage asymptotically vanishes as the bits-per-photon diverges towards infinity.

DOI: 10.1117/12.893158

Related papers

On approaching the ultimate limits of photon-efficient and bandwidth-efficient optical communication

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Technical Program

Dolinar's receiver is an architecture for distinguishing between two possible coherent states, using a photon detector and a local signal which may depend on past detector measurements.The optimal local signal satisfies very favorable properties: it is independent of the time horizon, and the resulting error probability is independent of the measurements.It was also shown that the same signal is optimal in the sense of maximizing the mutual information between the identity of the state and the measurements.In this work we show that the same signal is optimal for the optimization of the expected value of a wide class of objective functions.Our proof is based entirely on convex optimization and functional analysis, without resorting to any "quantum" arguments.

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Several proposed techniques for distinguishing between optical coherent states are analyzed under a physically realistic model of photodetection. Quantum error probabilities are derived for the Kennedy receiver, the Dolinar receiver, and the unitary rotation scheme proposed by Sasaki and Hirota for subunity detector efficiency. Monte Carlo simulations are performed to assess the effects of detector dark counts, dead time, signal processing bandwidth, and phase noise in the communication channel. The feedback strategy employed by the Dolinar receiver is found to achieve the Helstrom bound for subunity detection efficiency and to provide robustness to these other detector imperfections making it more attractive for laboratory implementation than previously believed.

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Part of the difficulties in implementing communication in quantum information stems from the fragility of Schrödinger's cat-like superpositions. A recent experiment in quantum optics by Cook et al (2007 Nature 446 774) has proved the feasibility of a feedback-mediated quantum measurement for discriminating between optical coherent states under photodetection. Minimizing the error in receiver measurement over all possible POVMs leads to the so-called quantum error probability or 'Helstrom bound', and CMG measurements validate the theoretical prediction by Helstrom, Dolinar and Geremia concerning this bound. In this work, we present some preliminary theoretical and numerical explorations concerning the properties of the Helstrom bound in binary (or multibinary) communication involving non-Poissonian or nonlinear coherent states.

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Optical communication at the quantum limit requires that measurements on the optical field be maximally informative, but devising physical measurements that accomplish this objective has proven challenging. The Dolinar receiver exemplifies a rare instance of success in distinguishing between two coherent states: an adaptive local oscillator is mixed with the signal prior to photodetection, which yields an error probability that meets the Helstrom lower bound with equality. Here we apply the same local-oscillator-based architecture with an information-theoretic optimization criterion. We begin with analysis of this receiver in a general framework for an arbitrary coherent-state modulation alphabet, and then we concentrate on two relevant examples. First, we study a binary antipodal alphabet and show that the Dolinar receiver's feedback function not only minimizes the probability of error, but also maximizes the mutual information. Next, we study ternary modulation consisting of antipodal coherent states and the vacuum state. We derive an analytic expression for a near-optimal local-oscillator feedback function, and, via simulation, we determine its photon information efficiency (PIE). We provide the PIE versus dimensional information efficiency (DIE) trade-off curve and show that this modulation and the our receiver combination performs universally better than (generalized) on-off keying plus photon counting, although, the advantage asymptotically vanishes as the bits-per-photon diverges towards infinity.

It is well known that ideal free-space optical communication at the quantum limit can have unbounded photon information efficiency (PIE), measured in bits per photon. High PIE comes at a price of low dimensional information efficiency (DIE), measured in bits per spatio-temporal-polarization mode. If only temporal modes are used, then DIE translates directly to bandwidth efficiency. In this paper, the DIE vs. PIE tradeoffs for known modulations and receiver structures are compared to the ultimate quantum limit, and analytic approximations are found in the limit of high PIE. This analysis shows that known structures fall short of the maximum attainable DIE by a factor that increases linearly with PIE for high PIE. The capacity of the Dolinar receiver is derived for binary coherent-state modulations and computed for the case of on off keying (OOK). The DIE vs. PIE tradeoff for this case is improved only slightly compared to OOK with photon counting. An adaptive rule is derived for an additive local oscillator that maximizes the mutual information between a receiver and a transmitter that selects from a set of coherent states. For binary phase-shift keying (BPSK), this is shown to be equivalent to the operation of the Dolinar receiver. The Dolinar receiver is extended to make adaptive measurements on a coded sequence of coherent state symbols. Information from previous measurements is used to adjust the a priori probabilities of the next symbols. The adaptive Dolinar receiver does not improve the DIE vs. PIE tradeoff compared to independent transmission and Dolinar reception of each symbol.

Dolinar's receiver is an architecture for distinguishing between two possible coherent states, using a photon detector and a local signal which may depend on past detector measurements. The optimal local signal satisfies very favorable properties: it is independent of the time horizon, and the resulting error probability is independent of the measurements. It was also shown that the same signal is optimal in the sense of maximizing the mutual information between the identity of the state and the measurements. In this work we show that the same signal is optimal for the optimization of the expected value of a wide class of objective functions. Our proof is based entirely on convex optimization and functional analysis, without resorting to any "quantum" arguments.

We study the problem of designing optical receivers to discriminate between multiple coherent states using coherent processing receivers---i.e., one that uses arbitrary coherent feedback control and quantum-noise-limited direct detection---which was shown by Dolinar to achieve the minimum error probability in discriminating any two coherent states. We first derive and reinterpret Dolinar's binary-hypothesis minimum-probability-of-error receiver as the one that optimizes the information efficiency at each time instant, based on recursive Bayesian updates within the receiver. Using this viewpoint, we propose a natural generalization of Dolinar's receiver design to discriminate

M

coherent states, each of which could now be a codeword, i.e., a sequence of

N

coherent states, each drawn from a modulation alphabet. We analyze the channel capacity of the pure-loss optical channel with a general coherent-processing receiver in the low-photon number regime and compare it with the capacity achievable with direct detection and the Holevo limit (achieving the latter would require a quantum joint-detection receiver). We show compelling evidence that despite the optimal performance of Dolinar's receiver for the binary coherent-state hypothesis test (either in error probability or mutual information), the asymptotic communication rate achievable by such a coherent-processing receiver is only as good as direct detection. This suggests that in the infinitely long codeword limit, all potential benefits of coherent processing at the receiver can be obtained by designing a good code and direct detection, with no feedback within the receiver.

Dolinar's receiver is an architecture for distinguishing between two possible coherent states, using a photon detector and a local signal which may depend on past detector measurements.The optimal local signal satisfies very favorable properties: it is independent of the time horizon, and the resulting error probability is independent of the measurements.It was also shown that the same signal is optimal in the sense of maximizing the mutual information between the identity of the state and the measurements.In this work we show that the same signal is optimal for the optimization of the expected value of a wide class of objective functions.Our proof is based entirely on convex optimization and functional analysis, without resorting to any "quantum" arguments.

Several proposed techniques for distinguishing between optical coherent states are analyzed under a physically realistic model of photodetection. Quantum error probabilities are derived for the Kennedy receiver, the Dolinar receiver, and the unitary rotation scheme proposed by Sasaki and Hirota for subunity detector efficiency. Monte Carlo simulations are performed to assess the effects of detector dark counts, dead time, signal processing bandwidth, and phase noise in the communication channel. The feedback strategy employed by the Dolinar receiver is found to achieve the Helstrom bound for subunity detection efficiency and to provide robustness to these other detector imperfections making it more attractive for laboratory implementation than previously believed.

Part of the difficulties in implementing communication in quantum information stems from the fragility of Schrödinger's cat-like superpositions. A recent experiment in quantum optics by Cook et al (2007 Nature 446 774) has proved the feasibility of a feedback-mediated quantum measurement for discriminating between optical coherent states under photodetection. Minimizing the error in receiver measurement over all possible POVMs leads to the so-called quantum error probability or 'Helstrom bound', and CMG measurements validate the theoretical prediction by Helstrom, Dolinar and Geremia concerning this bound. In this work, we present some preliminary theoretical and numerical explorations concerning the properties of the Helstrom bound in binary (or multibinary) communication involving non-Poissonian or nonlinear coherent states.