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AIP conference proceedings

Postselection Based on Retrodiction from the Results of Inefficient Detection

John Jeffers

Published Jan 1, 2004• 0 citations
Publisher

Abstract

We show that retrodictive quantum mechanics provides a simple method of calculating the probability that a postselecting device produces the required output state. A retrodictive equivalence between optical attenuators and amplifiers allows us to simplify the calculation.
DOI
10.1063/1.1834459

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Given the result of a measurement on the output of a quantum optical communication channel, we show how to calculate a retrodictive state at the input. This state can be used by the receiver to determine the probability that any one of a given set of states was selected by the transmitter. We establish the remarkably simple result that retrodicting the prepared input signal for an attenuating (amplifying) channel corresponds to predicting the measured output signal for an amplifying (attenuating) channel.
The transformation that an attenuator makes on the state of an optical field is the time reverse of that of an amplifier. Thus predicting the output state for an amplifier is equivalent to retrodicting the input state of an attenuator. We explore the consequences of this equivalence for simple optical quantum communication channels. One counterintuitive consequence is that the mean number of photons sent into an amplifier as retrodicted from a measurement of the number of output photons does not include the contribution of the amplifier noise.
Two measures of fidelity are proposed for postselecting devices, the retrodictive conditional probability that the state in the measurement arm is the one indicated by the detectors, and the probability that the device produces the state that it would produce if working perfectly. The first is the natural quantity that one wishes to maximize to improve the device operation. The second corresponds more closely with the accurate operation of the device than the more usual overlap-based fidelity. The results are particularly applicable to the types of state preparation and logic gate operation in linear optical quantum computing.
Two measures of fidelity are proposed for postselecting devices, the retrodictive conditional probability that the state in the measurement arm is the one indicated by the detectors, and the probability that the device produces the state that it would produce if working perfectly. The first is the natural quantity that one wishes to maximise to improve the device operation. The second corresponds more closely with the accurate operation of the device than the more usual overlap-based fidelity. The results are particularly applicable to the types of state preparation and logic gate operation in linear optical quantum computing.