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Journal of Optics B Quantum and Semiclassical Optics

Retrodiction in quantum optics

David Thomas Pegg, Stephen Mark Barnett

Published Aug 1, 1999• 39 citations
Publisher

Abstract

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.

DOI
10.1088/1464-4266/1/4/314

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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.
Quantum mechanics represents one of the greatest triumphs of human intellect and, undoubtedly, is the most successful physical theory we have to date. However, since its foundation about a century ago, it has been uninterruptedly the center of harsh debates ignited by the counterintuitive character of some of its predictions. The subject of one of these heated discussions is the so-called "retrodiction paradox", namely a deceptive inconsistency of quantum mechanics which is often associated with the "measurement paradox" and the "collapse of the wave function"; it comes from the apparent time-asymmetry between state preparation and measurement. Actually, in the literature one finds several versions of the retrodiction paradox; however, a particularly insightful one was presented by Sir Roger Penrose in his seminal book \emph{The Road to Reality}. Here, we address the question to what degree Penrose's retrodiction paradox occurs in the classical and quantum domain. We achieve a twofold result. First, we show that Penrose's paradox manifests itself in some form also in classical optics. Second, we demonstrate that when information is correctly extracted from the measurements and the quantum-mechanical formalism is properly applied, Penrose's retrodiction paradox does not manifest itself in quantum optics.
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.
Prediction, of course, is the making of statements about the likelihood of future events. retrodiction (or postdiction) is the making of statement about past events. In these lectures we shall study the rules governing retrodictive probabilities both in the quantum and classical domains. In doing so we develop a powerful tool with which to interpret physical phenomena and both analyse and interpret experiments. We use retrodictive methods to analyze a number of phenomena in quantum optics including the familiar photon antibunching, entanglement and ghost imaging and the perhaps less familiar ideas of quantum scissors, quantum computation in the past a single-particle analysis of two-particle interference. Quantum retrodiction is a quantitative theory and the familiar techniques for evolving quantum states and modelling open systems are applicable to retrodictive problems, but only with the introduction of some novel features. We conclude with a few remarks on more philosophical issues related to the arrow of time and wavefunction "collapse" in the measurement process. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
An attempt is made within the framework of the accepted quantum physics to achieve the maximum parallelism between prediction (inference of the future observational data from the present ones) and retrodiction (inference of the past observational data from the present ones). To implement this program, it is shown that the "retrodictive state function" (extrapolation of the present data to the past) can be just as useful as the ordinary "predictive state function" (extrapolation of the present data to the future). This leads to a formalism in which time-reversal becomes a linear transformation and double time-reversal becomes a ccc-number. In spite of this formal symmetry, it can be shown that the actual success of a retrodiction depends on the satisfaction of an additional condition which is not required in prediction, and which is not always fulfilled. From the same point of view, a logical loophole is pointed out in the indiscriminate application of the HHH-theorem to the past. The so-called irreversibility of observation is interpreted in terms of the decrease of "information" in the process of inference.
In the Jaynes–Cummings model a two-level atom interacts with a single-mode electromagnetic field. Quantum mechanics predicts collapses and revivals in the probability that a measurement will show the atom to be excited at various times after the initial preparation of the atom and field. In retrodictive quantum mechanics we seek the probability that the atom was prepared in a particular state given the initial state of the field and the outcome of a later measurement on the atom. Although this is not simply the time reverse of the usual predictive problem, we demonstrate in this paper that retrodictive collapses and revivals also exist. We highlight the differences between predictive and retrodictive evolutions and describe an interesting situation where the prepared state is essentially unretrodictable.
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.