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Retrodictive fidelities for pure state postselectors

John Jeffers, Supa Department Of Physics

Published Jan 1, 2006• 5 citations
OpenAlex

Abstract

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.

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Retrodictive fidelities for pure state postselectors

John Jeffers

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.

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In this paper we have described the retrodictive quantum formalism. We have defined the operator which describes preparation devices, and the retrodictive state, which is simply the normalised measurement POM element. These enable simple evaluation of preparation probabilities given the results of some later measurement event. We have shown how to apply the formalism to both closed and open quantum systems.

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High-Fidelity Quantum Logic Operations Using Linear Optical Elements

J. D. Franson, Michelle M. Donegan, Michael J. Fitch +2 more

Knill, Laflamme, and Milburn [Nature (London) 409, 46 ((2001))]] have shown that quantum logic operations can be performed using linear optical elements and additional ancilla photons. Their approach is probabilistic in the sense that the logic devices fail to produce an output with a failure rate that scales as 1/n, where n is the number of ancilla. Here we present an alternative approach in which the logic devices always produce an output with an intrinsic error rate that scales as 1/n(2), which may have several advantages in quantum computing applications.

arXiv:quant-ph/0202160•Physical Review Letters•Sep 4, 2002•132 citations
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.
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.
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.
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.
Knill, Laflamme, and Milburn [Nature (London) 409, 46 ((2001))]] have shown that quantum logic operations can be performed using linear optical elements and additional ancilla photons. Their approach is probabilistic in the sense that the logic devices fail to produce an output with a failure rate that scales as 1/n, where n is the number of ancilla. Here we present an alternative approach in which the logic devices always produce an output with an intrinsic error rate that scales as 1/n(2), which may have several advantages in quantum computing applications.