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arXiv:1506.04831

A retrodiction paradox in quantum and classical optics

Andrea Aiello, J. P. Woerdman

Published Jun 16, 2015• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.1506.04831

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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.
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 the scientific enterprise of the twentieth century. The stunning success of quantum theory has led to many revolutionary inventions and its extraordinary concepts describe the heart of several important real world applications like transistors and lasers. The theory makes accurate predictions about a wide range of physical phenomena and has historically withstood the tests and scrutiny of every experimental investigation. However, inspite of the fact that quantum theory is widely regarded by the scientific establishment as the fundamental theory of nature and is immensly successful and useful, its conceptual framework makes many predictions which are difficult to comprehend “classically. The theory is, pardoxically, powerful and confusing at the same time. Quantum theory’s unusual predictions originate from its basic formailsm which involves concepts like probability amplitudes and the linear superposition principle(Dirac, 1947). The quantum view appears abstract and counterintuitive and at odds with classical perceptions. Many of the conceptual problems of quantum mechanics are encompassed in what is known as the quantum measurement problem(Peres, 1986; von Neumann, 1932; Wheeler & Zurek, 1983). Conventionally, the measurement paradox is supposedly ’resolved’ by forcing a notion of a sudden collapse of the state vector of the system being measured. However, the nature of this mechanism is at odds with the basic tenets of quantum mechanics and hence may lie outside its realm thus questioning the validity of the theory it self. Closely related with the problem of measurement in quantum mechanics is the question of its connection with the emergence of classicality and the elusive boundary between quantum and classical worlds. What is the connection between the ‘classical’ and the ‘quantum’ worlds? Is there a definite relationship? Are classical mechanics and quantum mechanics two mutually exclusive incompatible theories or are they two aspects of the same underlying philosophy? Classical objects are eventually composed of elements of the microworld which can be described quantum mechanically. So, how and where can there be a boundary between the two worlds? In the following section we begin by introducing the quantum measurement problem. In the next section we will discuss attempts to understand and explain away the underlying paradoxes in quantum theory as highlighted in the quantum measurement problem and the question of the quantum-classical connection. From among the various explanations that seek a resolution to the conceptual problems of quantum mechanics, we focus on the ‘environment induced decoherence theory’ an approach that employs the methods developed by several authors to analyse the quantum mechanics of Measurement in Quantum Mechanics: Decoherence and the Pointer Basis
Quantum mechanics is an immensely successful theory, occupying a unique position in the history of science. It has solved mysteries ranging from macroscopic superconductivity to the microscopic theory of elementary particles and has provided deep insights into the nature of vacuum on the one hand and the description of the nucleon on the other. Whole new fields such as quantum optics and quantum electronics owe their very existence to this body of knowledge. However, despite the stunning successes of quantum mechanics, there is no general agreement on the conceptual foundations and interpretation of the subject. The theory provides unambiguous information about the outcome of a measurement of a physical object. However, many feel that it does not provide a satisfactory answer to the nature of the “reality” we should attribute to the physical objects between the acts of measurement. The conceptual difficulty comes about because the wave function |ψ〉 is usually given by a coherent superposition of various distinguishable experimental outcomes. If we denote the collection of states that represent the possible outcomes of an experiment by |ψ j 〉, then |ψ〉 = ∑ j c j |ψ j 〉 where c j = 〈ψ j |ψ〉. The probability of the outcome |ψ j 〉 is P j =|c j | 2 . In the process of measurement, the so called collapse of the wave function takes place and a single, definite state |ψ i 〉 of the physical object is chosen. The difficulty comes about in the interpretation of the mechanism by which this definite state is chosen from amongst all the possible outcomes.
The problem of retrodiction is associated with basic questions, but for this very reason it is, unfortunately, quite difficult, even in classical physics. In quantum physics it is even more subtle. In quantum mechanics, a consistent definition of what is called a "system" leads to problems that do not exist in classical physics, and correspondingly the answers to the questions raised are less straightforward. This chapter discusses the conditions under which it is possible to speak of an "independent time evolution" of a system S and to study what kinds of predictions can legitimately be made when these conditions are fulfilled. It describes retroactive computation to investigate whether or not the assumed knowledge in question is really consistent with the standard quantum mechanical formalism. The concept of retrodiction is related to the notions of time reversal and of irreversibility. Both these notions play a central role in physics.
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.
Quantum paradoxes The next few chapters are devoted to resolving a number of quantum paradoxes in the sense of giving a reasonable explanation of a seemingly paradoxical result in terms of the principles of quantum theory discussed earlier in this book. None of these paradoxes indicates a defect in quantum theory. Instead, when they have been properly understood, they show us that the quantum world is rather different from the world of our everyday experience and of classical physics, in a way somewhat analogous to that in which relativity theory has shown us that the laws appropriate for describing the behavior of objects moving at high speed differ in significant ways from those of pre-relativistic physics. An inadequate theory of quantum measurements is at the root of several quantum paradoxes. In particular, the notion that wave function collapse is a physical effect produced by a measurement, rather than a method of calculation, see Sec. 18.2, has given rise to a certain amount of confusion. Smuggling rules for classical reasoning into the quantum domain where they do not belong and where they give rise to logical inconsistencies is another common source of confusion. In particular, many paradoxes involve mixing the results from incompatible quantum frameworks. Certain quantum paradoxes have given rise to the idea that the quantum world is permeated by mysterious influences that propagate faster than the speed of light, in conflict with the theory of relativity. They are mysterious in that they cannot be used to transmit signals, which means that they are, at least in any direct sense, experimentally unobservable.