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1 Optical channel with direct detections, also known as Poisson channel, is well studied. With direction detections, only the intensity of the optical signal is measured and used as the carrier of information. In coherent detection, the phase of optical signal is also utilized. Constrained by optical devices, it is hard to directly measure this phase. However, it is possible to mix the input signal with a local reference signal to create output that is phase dependent. Generating such local signals can be based on the instantaneous receiver knowledge, and updated from time to time. We study the channel capacity of such channel, and make connection to the recent development in feedback channels. We are particularly interested in the low SNR regime, and present a capacity result based on a new scaling law.

In this paper, we propose an open-loop unequal-error-protection querying policy based on superposition coding for the noisy 20 questions problem. In this problem, a player wishes to successively refine an estimate of the value of a continuous random variable by posing binary queries and receiving noisy responses. When the queries are designed non-adaptively as a single block and the noisy responses are modeled as the output of a binary symmetric channel the 20 questions problem can be mapped to an equivalent problem of channel coding with unequal error protection (UEP). A new non-adaptive querying strategy based on UEP superposition coding is introduced whose estimation error decreases with an exponential rate of convergence that is significantly better than that of the UEP repetition coding introduced by Variani et al., [2]. With the proposed querying strategy, the rate of exponential decrease in the number of queries matches the rate of a closed-loop adaptive scheme where queries are sequentially designed with the benefit of feedback. Furthermore, the achievable error exponent is significantly better than that of random block codes employing equal error protection. Index TermsNoisy 20 questions problem, estimation, superposition coding, unequal error protection, error exponents. I. INTRODUCTIONConsider a noisy 20 questions game between a player and an oracle. The objective of the player is to estimate the value of a continuous target variable X ∼ unif[0, 1]. The player asks binary queries to the oracle who knows the value of X, and receives a noisy version of the oracle's correct answers transmitted through a binary symmetric channel with flipping probability ∈ (0, 1/2), denoted BSC( ). The central question addressed here is: What is the optimal sequence of queries to estimate the value of X with a minimum estimation error at a fixed number of querying? This general setup of noisy 20 questions game and the optimal query design problem is of broad interest, arising in various areas, including active learning The problem of optimal query design for the noisy 20 questions game can be categorized into two main approaches, adaptive vs. non-adaptive designs. In each approach, the sequence of queries is designed by a controller that may either use feedback (adaptive 20 questions) or operate open-loop (non-adaptive 20 questions) to formulate the sequence of questions. For the adaptive case, the controller uses noisy answers to previous questions to determine the next question posed to the oracle. For the non-adaptive case, on the other hand, the controller designs the sequence of queries ahead of time without access to future answers of the oracle. In general, the use of feedback in the adaptive design provides an information advantage, allowing a better error rate of convergence, but at the cost of higher query design complexity and the need for a feedback channel.Previous studies on optimal query design for the noisy 20 questions problem often sought to design queries that acquire observations minimizing the...

Abstract-The maximum rate at which classical information can be reliably transmitted per use of a quantum channel strictly increases in general with N , the number of channel outputs that are detected jointly by the quantum joint-detection receiver (JDR). This phenomenon is known as superadditivity of the maximum achievable information rate over a quantum channel. We study this phenomenon for a pure-state classicalquantum (cq) channel and provide a lower bound on CN /N , the maximum information rate when the JDR is restricted to making joint measurements over no more than N quantum channel outputs, while allowing arbitrary classical error correction. We also show the appearance of a superadditivity phenomenonof mathematical resemblance to the aforesaid problem-in the channel capacity of a classical discrete memoryless channel (DMC) when a concatenated coding scheme is employed, and the inner decoder is forced to make hard decisions on N -length inner codewords. Using this correspondence, we develop a unifying framework for the above two notions of superadditivity, and show that for our lower bound to CN /N to be equal to a given fraction of the asymptotic capacity C of the respective channel, N must be proportional to V /C 2 , where V is the respective channel dispersion quantity.Index Terms-Pure-state classical input-quantum output (cq) channel, Holevo capacity, superadditivity of capacity, joint measurement, concatenated codes. I. BACKGROUND AND MOTIVATIONHow many classical bits per channel use can be reliably communicated over a quantum channel? This has been a central question in quantum information theory in an effort to understand the intrinsic limit on the classical capacity of physical quantum channels such as optical fiber or free-space optical channels. The Holevo limit of a quantum channel is an upper bound to the Shannon capacity of the classical channel induced by pairing the quantum channel with any specific transmitted states, modulation format, and the choice of a receiver measurement [3], [4]. The Holevo limit is in principle also an achievable information rate, and is known for several important practical channels, such as the lossy-noisy bosonic channel [5], [6]. However, a receiver that attains the Holevo capacity, must in general make joint (collective) measurements over long codeword blocks. Such measurements cannot be Hye Won Chung (hyechung@umich.edu) was with the EECS department at MIT and is currently with the EECS department at the University of Michigan. Lizhong Zheng (lizhong@mit.edu) is with the EECS department at MIT. Saikat Guha (sguha@bbn.com) is with the Quantum Information Processing realized by detecting single modulation symbols followed by classical post processing.The phenomenon that a joint-detection receiver (JDR) is able to yield a higher information rate (in error-free bits communicated per use of the quantum channel) than what is possible by any single-symbol receiver measurement is known as superadditivity in the classical capacity of a quantum channel [7], [8]. We wo...

In this paper, we propose an open-loop unequal-error-protection querying policy based on superposition coding for the noisy 20 questions problem. In this problem, a player wishes to successively refine an estimate of the value of a continuous random variable by posing binary queries and receiving noisy responses. When the queries are designed non-adaptively as a single block and the noisy responses are modeled as the output of a binary symmetric channel the 20 questions problem can be mapped to an equivalent problem of channel coding with unequal error protection (UEP). A new non-adaptive querying strategy based on UEP superposition coding is introduced whose estimation error decreases with an exponential rate of convergence that is significantly better than that of the UEP repetition coding introduced by Variani et al., [2]. With the proposed querying strategy, the rate of exponential decrease in the number of queries matches the rate of a closed-loop adaptive scheme where queries are sequentially designed with the benefit of feedback. Furthermore, the achievable error exponent is significantly better than that of random block codes employing equal error protection. Index TermsNoisy 20 questions problem, estimation, superposition coding, unequal error protection, error exponents.