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Journal of Information and Optimization Sciences

Detection of Quantum Signals by a Mismatched Receiver

Carl W. Helstrom

Published Jan 1, 1980• 3 citations
Publisher

Abstract

A quantum receiver is modelled as a harmonic oscillator imperfectly matched to the incident signal-bearing field. At the beginning of its interaction with the field the oscillator is primed by being placed in a predetermined quantum state. It is shown how to calculate the state of this oscillator at the end of the observation interval both in the presence and in the absence of an incident coherent signal. As examples the error probabilities of a binary on-off communication system with such a mismatched receiver are calculated (a) for a signal of known phase, the receiver primed with a two-photon coherent state, and (b) for a signal of random phase, the receiver primed with a number eigenstate. In both cases the error probability depends on the mean number of externally available signal photons when the number of photons in the priming state is very large, rather than on the smaller number that would actually be absorbed by the mismatched receiver unprimed.

DOI
10.1080/02522667.1980.10698656

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A quantum receiver is modelled as a harmonic oscillator imperfectly matched to the incident signal-bearing field. At the beginning of its interaction with the field the oscillator is primed by being placed in a predetermined quantum state. It is shown how to calculate the state of this oscillator at the end of the observation interval both in the presence and in the absence of an incident coherent signal. As examples the error probabilities of a binary on-off communication system with such a mismatched receiver are calculated (a) for a signal of known phase, the receiver primed with a two-photon coherent state, and (b) for a signal of random phase, the receiver primed with a number eigenstate. In both cases the error probability depends on the mean number of externally available signal photons when the number of photons in the priming state is very large, rather than on the smaller number that would actually be absorbed by the mismatched receiver unprimed.
When the phase of the transmittance between a unimodal transmitter and a unimodal receiver is uniformly distributed over(0,2\pi), the density operator of the received signal field commutes with the photon number operator, and the optimum detector of the signal counts photons. A simple formula is given for the probability generating function of the number of photons in terms of the quasi-probability distribution of the transmitted signal. It is used to determine the average error probability of an on-off binary communication system transmitting two-photon coherent-state signals, received in the presence of thermal noise.
The proposed quantum receiver in this innovation partitions each binary signal interval into two unequal segments: a short "pre-measurement" segment in the beginning of the symbol interval used to make an initial guess with better probability than 50/50 guessing, and a much longer segment used to make the high-sensitivity signal detection via field-cancellation and photon-counting detection. It was found that by assigning as little as 10% of the total signal energy to the pre-measurement segment, the initial 50/50 guess can be improved to about 70/30, using the best available measurements such as classical coherent or "optimized Kennedy" detection.
Suppressing the Errors Due to Mode Mismatch for MMM-Ary PSK Quantum Receivers Using Photon-Number-Resolving Detector
A quantum-communication-theoretical approach to the homodyne-direct receiver is presented and used to obtain the quantum detection operators and average probability of error for any binary quantum-state signal. The effect of the quantum efficiency η of a photodetector used in the receiver is taken into account. Which quantum states have been received is determined by comparing the read-out of the receiver with a certain decision level; so the dependences of the detection operators and average probability of error on a decision level are also considered. As a quantum-state signal, the binary phaseshift keyed signal including thermal noise is investigated in detail, and the condition necessary for the average probability of error of the homodyne-direct receiver to be smaller than the standard quantum limit (the short noise limit) of the signal detection is obtained.
A method is given for calculating the minimum average error probability attainable in deciding between a mixed state and a pure state of a quantum system. It is applied, first, to the detection of a noisy coherent pulse represented by a state with a Gaussian P-representation and, second, to the detection of a signal represented at the transmitter by a two-photon coherent state (TCS). In both problems the field of the receiver is free of background noise and, when no signal was transmitted, is in the ground state.