Electron spin was derived by the use of EPI in Chapter 4, in the context of the Dirac equation. This was for the scenario of a single particle. Here, by comparison, we treat spin for a two-particle scenario, that of the well-known EPR–Bohm experiment. A general two-particle PDF can exhibit correlation. We show that the well-known probabilities of spin-pair combinations in the EPR–Bohm experiment follow simply from EPI; and that these do obey certain well-known correlation effects. See also a related derivation by S. Luo (2002). Information J was previously regarded as the “bound” information (Chapter 3), in the sense of the amount of information that is bound to the unobserved source phenomenon. Here we find that it can also represent a degree of information entanglement between two particles : At the EPI solution, the information J for the observed particle equals the information I for the un observed one. Or, the source of information for the observed particle is the unobserved one. A final point of interest will be the use, in Sec. 12.4, of the game corollary | I | = min . (Eq. (3.21)) to determine a vital free parameter of the problem. This approach is also used in Chapter 4 to establish the appropriate number N of amplitude functions to describe various quantum particles, and in Chapter 15 to determine the unknown exponent of a power law for cancer growth. All parameters in the preceding are unitless, as is required for the use of Eq. (3.21). See further discussion in Sec. 12.10.
Electron spin was derived by the use of EPI in Chapter 4, in the context of the Dirac equation. This was for the scenario of a single particle. Here, by comparison, we treat spin for a two-particle scenario, that of the well-known EPR–Bohm experiment. A general two-particle PDF can exhibit correlation. We show that the well-known probabilities of spin-pair combinations in the EPR–Bohm experiment follow simply from EPI; and that these do obey certain well-known correlation effects. See also a related derivation by S. Luo (2002). Information J was previously regarded as the “bound” information (Chapter 3), in the sense of the amount of information that is bound to the unobserved source phenomenon. Here we find that it can also represent a degree of information entanglement between two particles : At the EPI solution, the information J for the observed particle equals the information I for the un observed one. Or, the source of information for the observed particle is the unobserved one. A final point of interest will be the use, in Sec. 12.4, of the game corollary | I | = min . (Eq. (3.21)) to determine a vital free parameter of the problem. This approach is also used in Chapter 4 to establish the appropriate number N of amplitude functions to describe various quantum particles, and in Chapter 15 to determine the unknown exponent of a power law for cancer growth. All parameters in the preceding are unitless, as is required for the use of Eq. (3.21). See further discussion in Sec. 12.10.
We now have under our belts several interpretations of quantum mechanics, each requiring, or anyhow advocating, some understanding of quantum probabilities. In the next two chapters, I will consider some connections among various interpretations of quantum probabilities and non-locality. I do so in the context of the well-known EPR–Bohm experiment, though it is worth emphasizing at the start that non-locality is very likely the rule rather than the exception for quantum-mechanical systems. Entanglement of systems occurs not only in the confines of a laboratory, but also in the course of quite typical interactions among quantum-mechanical systems. Nonetheless, the EPR-Bohm experiment shines a bright light on the phenomenon of non-locality, and is therefore the most useful context in which to explore the relation between probability and non-locality. In this chapter, I consider models of the EPR–Bohm experiment that deliver probabilities for the various outcomes given the initial state of the pair of particles. In the next chapter, I consider fully dynamical models, i.e., ones that provide a dynamics for the complete state of the pair of particles as well as probabilities for various outcomes based on these complete states. The EPR–Bohm experiment The EPR–Bohm experiment is well known, but some observations about it are important for later.
Review and preview In chapters 6 and 7, I described some links between locality and the treatment of probabilities in models of the EPR–Bohm experiment. From chapter 6, the main lesson was that, given the strict correlations of quantum mechanics, factorizability of the two-time probabilities is equivalent to two-time determinism. I also noted there that an adequate factorizable theory – one that avoids Bell's theorem – is possible only if λ-independence fails. In chapter 7, I argued that weak symmetry, local determination, and independent evolution together entail deterministic results and weak deterministic transitions. In chapter 8, I discussed the relationships among the locality conditions of chapter 6 and 7, Lorentz-invariance, and ‘metaphysical’ locality, especially local causality. I have by no means tried to give general answers to the questions raised there. Indeed, part of my thesis is that there is no general answer to be had. Nonetheless, one can use the results of chapters 6 and 7 to investigate the status of various interpretations. Most obviously, they can be used to evaluate whether a given theory is local: if a theory can be shown to violate one or more of the types of determinism entailed by the locality conditions of chapters 6 and 7, then that theory must be non-local in some sense.
Introduction In preceding chapters, EPI has been used as a computational procedure for establishing the physical laws governing various measurement scenarios. We showed, by means of the optical measurement model of Sec. 3.8, that EPI is, as well, a physical process that is initiated by a measurement. Specifically, it arises out of the interaction of the measuring instrument's probe particle with the object under measurement. This perturbs the system probability amplitudes, which perturbs the informations I and J , etc., as indicated in Figs. 3.3 and 3.4. The result is that EPI derives the phenomenon's physics as it exists at the input space to the measuring device. We also found, in Sec. 3.8, the form of the phenomenon's physics at the output to the measuring instrument. This was given by Eq. (3.51) for the output probability amplitude function. The analysis in Sec. 3.8 was, however, severely limited in dimensionality. A one-dimensional analysis of the measurement phenomenon was given. A full, covariant treatment would be preferable, i.e., where the space-time behaviors of all probability amplitudes were determined. Such an analysis will be given next. It constitutes a covariant quantum theory of measurement . This covariant theory will be developed from an entirely different viewpoint than that in Sec. 3.8. The latter was an analysis that focussed attention upon the probe–particle interaction and the resulting wave propagation through the instrument. The covariant approach will concentrate, instead, on the meaning of the acquired data to the observer as it reflects upon the quantum state of the measured particle. Hence, whereas the previous analysis was purely physical in nature, this one will be “knowledge-based” as well as physical.
Introduction We will now turn to quantum entanglements and their contemporary, possibly practical, interest. Entanglements, first discussed in the E.P.R. paradox (Einstein, Podolsky, Rosen, 1935) are a perplexing nonlocal feature of quantum mechanics. This was immediately and succinctly discussed by Schrödinger (1935). The long history of this apparent paradox is outlined in the wonderful book of Jammer (1974). We will not focus on the central issue of “hidden variables” and their resolution by the Bell inequalities (Bell, 1964) and the test, nor the failure, of these in experiment (see Fry, 1998). The distinctive quantum nature of entanglements has led to two quantum effects which we will discuss: quantum information teleportation (Zeilenger, 1998) and quantum computation by means of entangled states. A nice, recent, elementary introduction to the latter is in the Los Alamos reports of James and Kwiat (2002). The quantum correlations or entanglements are sensitive to environmental destruction. This was pointed out early by Zurek, who termed this “decoherence.” The loss of coherence may occur on a short time scale. Recent discussions have been given by Zurek (2002, 2003). We have already given a theoretical example early in Chapter 2. It is pertinent to discuss this here, since it is a property of opensystem quantum master equations that is a central part of our study in this book. In a sense, the decoherence of correlations has turned out to be a “practical” application of these theoretical notions. Are there remedies for unwanted decoherence? Quantum error correction is a possibility. This will be mentioned also. Entanglements: foundations Following the reading of the E.P.R. paper, Schrödinger (1935b) quickly repeated the argument from quantum theory, but more generally. He introduced the term “entanglement” to describe what he seemed to agree was a curious, if not unacceptable, property. Let us follow his point of view. Given a state ψ ( x, y ) of a composite system of particles x and y formed in their mutual interaction. ψ ( x , y ) is not a product state.
The unknown amplitude law q(x) defining an observed effect may be found using the principle of Extreme Physical Information. EPI is derived as follows. The observations follow an information flow J --> I, with J the information intrinsic to the source and I the Fisher information level in its data, obeying (i) I=4 Integral dx q' -squared. Here q'= dq/dx and p(x) = q(x)-squared is the probability. It was previously shown, using L. Hardy's 5 axioms defining physics, that I = max. Therefore, its variation (ii) delta I = 0. Note that I is generic, obeying (i) for all source effects, whereas J is specific to the particular effect. Hence, rather than having form (i), J obeys (iii) J = Integral dx j[q(x),s(x)] with j some function of its arguments and s(x) a known source, such as of mass, biological fitness, etc. Information I decreases under any irreversible operation such as measurement, so that I l.e. J or, equivalently, I = kJ where 0 l.e. k l.e. 1. Then the variation delta I = k delta J so that property (ii) gives (iv) delta J = 0 as well. Then combining (ii) and (iv), delta(I - J) = 0. Or, I - J = L = extremum. What kind of extremum? Eqs. (i) and (iii) give (v) L=4q'^2 - j[q(x),s(x)]. Differentiating (v), (d^2 L )/(dq'^2) = +8. Then by the Legendre condition the extremum is a minimum. The unknown source effect obeys (vi) I - J= minimum, EPI.