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Retrodictive quantum optical state engineering

K. L. Pregnell, David Thomas Pegg

Published Oct 28, 2005• 10 citations
OpenAlex

Abstract

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 the a preparation event. Retrodictive states, which are associated with a measurement event and propagate backwards in time, are also useful, however. In this paper we show 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. We show how the probability of successful state generation can be optimised by adjusting the multiport device and also examine a versatile configuration that is useful for generating a range of states.

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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 the a preparation event. Retrodictive states, which are associated with a measurement event and propagate backwards in time, are also useful, however. In this paper we show 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. We show how the probability of successful state generation can be optimised by adjusting the multiport device and also examine a versatile configuration that is useful for generating a range of states.
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.
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.
The usual formalism of quantum mechanics is predictive with state vectors being assigned on the basis of past preparation procedures. The retrodictive formalism, in which some state vectors are assigned on the basis of future measurement results is less usual, although it has been known for some time. Although at first sight this formalism may appear to be anticausal and subject to paradoxes, if used carefully it still predicts the same measurable correlations as the more usual approach despite different assignments of state vectors to states in the interval between preparation and measurement of the system. Here we use a procedure which involves retrodiction to examine some quantum optical processes, including a recent state truncation process. The collapse of the state vector occurs at a different time from that for the usual approach underlining the difficulty in regarding the collapse as a real physical process.
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.
We propose two experimental schemes for producing coherent-state superpositions which approximate different nonclassical states conditionally in traveling optical fields. Although these setups are constructed of a small number of linear optical elements and homodyne measurements, they can be used to generate various photon number superpositions in which the number of constituent states can be higher than the number of measurements in the schemes. We determine numerically the parameters to achieve maximal fidelity of the preparation for a large variety of nonclassical states, such as amplitude squeezed states, binomial states, squeezed cat states, and various photon number superpositions. The proposed setups can generate these states with high fidelities and with success probabilities that can be sufficient for practical applications.