Abstract
This chapter considers the principles of quantum information theory in conjunction with the principles of QM and QFT, as discussed in earlier chapters. Several recent approaches to quantum information theory will be addressed, such as those by B. Coecke and, more extensively, L. Hardy. Particular attention will, however, be given to the recent work by G. M. D’Ariano and co-workers, G. Chiribella and P. Perinotti. The approach to quantum information theory that they developed allowed D’Ariano and Perinotti to derive Dirac’s equation from the principles of quantum information alone, rather than, as Dirac did, by combining the principles of quantum theory and special relativity. The locality principle, however, plays a key role in this derivation, which, I suggest, may have important implications for fundamental physics. I will proceed as follows. After brief introductory remarks given in Sects. 7.1, 7.2 considers D’Ariano, Chiribella, and Perinotti’s program of finite-dimensional quantum theory (QFDT), based on these principles, and related work, especially that of Hardy, in terms of the operational language of circuits. Sect. 7.3 discusses D’Ariano and Perinotti’s derivation of Dirac’s equation.
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Notes
- 1.
By “theory” they refer, more in accord with the term “model” as used in this study, to the mathematical structure of quantum theory, rather than to its physical aspects.
- 2.
See Chiribella et al. (2011) for further references. For introductions to the subject of quantum information theory and further references, see Jaeger (2007) and Mermin (2007). See also J. A. Wheeler’s visionary manifesto (Wheeler 1990). Fuchs’s work more recently “mutated” to a somewhat different, if related, program, that of quantum Bayesianism or QBism, discussed in Chap. 4 (e.g., Fuchs et al. 2014 and further references there).
- 3.
Cf. (De Raedt et al. 2014), which pursues a logical-inference-based approach to deriving standard (infinite-dimensional) quantum mechanics, specifically by considering Schrödinger’s derivation of his equation as presented in his first paper (Schrödinger 1926b) considered earlier. Their argument, however, still relies on certain fundamental quantum principles, such as, expressly, “uncertainty about individual events,” referring, I surmise, to elementary individual quantum processes, for otherwise this uncertainty is found in classical statistical physics or chaos theory (De Raedt et al. 2014, p. 47). They are quite right to stress the role of logical inference in scientific reasoning, following Pólya (1954) and Jaynes (2003). The question of “the first principles” (the initial assumptions, postulates, and so forth) remains open, however. Thus, both Heisenberg’s and Schrödinger’s derivation of quantum mechanics depended on classical mechanics and, in Heisenberg, the mathematical correspondence principle, which the quantum-informational programs in question here want to avoid, as not sufficiently first-principle-like. Besides, as discussed in Chap. 2, neither Heisenberg’s nor Schrödinger’s “derivation” was free from guesses that, while, to be sure, accompanied by plausible reasoning and logical inferences, would be difficult to fully explain by them. On the other hand, the principal role of classical physics in both cases is unquestionable. See also the discussion of the role of the Hamiltonian formalism in QM and QFT in Chap. 6, Sect. 6.4.
- 4.
- 5.
It might be noted that, unlike von Neumann, Heisenberg and Dirac did not assume Hilbert spaces from the start either, but rather arrived at them with the help of, even if not quite derived them from, the fundamental principles they assumed.
- 6.
Bub’s article, cited earlier, also considers QM as a principle theory in Einstein’s sense in order to address the EPR-type experiments and quantum entanglement (Bub 2000).
- 7.
As explained earlier, this principle is different from that of classical causality (indeed already by virtue of the principle’s appeal to probability), while being consistent with locality. See also Pawlowski et al. (2009) for a suggestive quantum-informational concept of causality.
- 8.
In accordance with the definition given in Chap. 1, “postulate” may be a better term, because one can hardly have the self-evidence of “axioms” in such cases, but this is a secondary matter, which, as I said, does not affect the essence of Hardy’s argument itself.
- 9.
A remarkable precursor to this approach is Schwinger’s framework of “the algebra [of the symbols] of quantum measurements,” which in effect extends from Heisenberg’s approach and Bohr’s thinking, from which Schwinger borrows the terms “kinematic” and “symbolic,” respectively (Schwinger 1988, 2001). See Jaeger (2016).
- 10.
The information thus obtained is physically classical, but its architecture and mode of transmission are quantum; that is, it cannot be classically generated.
- 11.
Cf. (Hardy 2001, p. 26)
- 12.
On the other hand, the article provides “an analytical description of the QCA for the narrow-band states of quantum field theory in terms of a dispersive Schrödinger equation holding at all scales” (D’Ariano and Perinotti 2014, pp. 1, 4).
- 13.
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Plotnitsky, A. (2016). The Principles of Quantum Information Theory, Dirac’s Equation, and Locality Beyond Relativity. In: The Principles of Quantum Theory, From Planck's Quanta to the Higgs Boson. Springer, Cham. https://doi.org/10.1007/978-3-319-32068-7_7
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