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Physical Review A

Retrodiction for quantum optical communications

Stephen Mark Barnett, David Thomas Pegg, John Jeffers, Ottavia Jedrkiewicz, R. Loudon

Published Jul 20, 2000• 49 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.62.022313

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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.
Retrodictive quantum states are states that propagate backwards in time from a measurement event. Although retrodictive quantum mechanics appears to be very different from the usual predictive formalism, in that propagation of states into the past appears to violate causality, this is not so. Indeed causality is not manifest in the time direction of propagation of quantum states at all. Instead, causality is ensured by the different normalization conditions applied to the preparation and measurement device operators. It is this difference that introduces the arrow of time into quantum mechanics. Retrodictive states are useful for applications such as measurement, predictive quantum state engineering and quantum communication. Here we show how any optical retrodictive state that can be expressed to a good approximation in a finite‐dimensional Hilbert space can be generated from predictive coherent states, a lossless multiport device and photodetectors. The composition of the retrodictive state can be controlled by adjusting the input predictive coherent states. This allows, for example, projection synthesis for an optical state to be achieved with the exotic reciprocal binomial reference state replaced by a simple coherent state.
Although it has been known for some time that quantum mechanics can be formulated in a way that treats prediction and retrodiction on an equal footing, most attention in engineering quantum states has been devoted to predictive states, that is, states associated with a preparation event. Retrodictive states, which are associated with a measurement event and propagate backwards in time, are also useful, however. In this paper it is shown how any retrodictive state of light that can be written to a good approximation as a finite superposition of photon number states can be generated by an optical multiport device. The composition of the state is adjusted by controlling predictive coherent input states. It is shown how the probability of successful state generation can be optimized by adjusting the multiport device and also a versatile configuration that is useful for generating a range of states is examined.
Although it has been known for some time that quantum mechanics can be formulated in a way that treats prediction and retrodiction on an equal footing, most attention in engineering quantum states has been devoted to predictive states, that is, states associated with a preparation event. Retrodictive states, which are associated with a measurement event and propagate backwards in time, are also useful, however. In this paper it is shown how any retrodictive state of light that can be written to a good approximation as a finite superposition of photon number states can be generated by an optical multiport device. The composition of the state is adjusted by controlling predictive coherent input states. It is shown how the probability of successful state generation can be optimized by adjusting the multiport device and also a versatile configuration that is useful for generating a range of states is examined.