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arXiv:math/9911150

Machines, Logic and Quantum Physics

David Deutsch, ARTUR K. EKERT, Rossella Lupacchini

Published Nov 19, 1999• 4 citations
arXivHTMLPDF

Abstract

Though the truths of logic and pure mathematics are objective and independent of any contingent facts or laws of nature, our knowledge of these truths depends entirely on our knowledge of the laws of physics. Recent progress in the quantum theory of computation has provided practical instances of this, and forces us to abandon the classical view that computation, and hence mathematical proof, are purely logical notions independent of that of computation as a physical process. Henceforward, a proof must be regarded not as an abstract object or process but as a physical process, a species of computation, whose scope and reliability depend on our knowledge of the physics of the computer concerned.

DOI
10.48550/arxiv.math/9911150

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Though the truths of logic and pure mathematics are objective and independent of any contingent facts or laws of nature, our knowledge of these truths depends entirely on our knowledge of the laws of physics. Recent progress in the quantum theory of computation has provided practical instances of this, and forces us to abandon the classical view that computation, and hence mathematical proof, are purely logical notions independent of that of computation as a physical process. Henceforward, a proof must be regarded not as an abstract object or process but as a physical process, a species of computation, whose scope and reliability depend on our knowledge of the physics of the computer concerned.
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We present a procedure which allows us to recover classical and nonclassical logical structures as \emph{concrete logics} associated with physical theories expressed by means of classical languages. This procedure consists in choosing, for a given theory T\mathcal TT and classical language L\mathcal LL expressing T\mathcal TT, an observative sublanguage LLL of L\mathcal LL with a notion of truth as correspondence, introducing in LLL a derived and theory-dependent notion of \emph{C-truth} (\emph{true with certainty}), defining a \emph{physical preorder} induced by C-truth, and finally selecting a set of sentences that are \emph{verifiable} (or \emph{testable}) according to T\mathcal TT, on which a \emph{weak complementation} is induced by T\mathcal TT. The triple consisting of the set of verifiable sentences, physical order and weak complementation is then the desired concrete logic. By applying our procedure we recover a classical logic as the concrete logic associated with classical mechanics and standard quantum logic as the concrete logic associated with quantum mechanics. We also show that our alternative view of standard quantum logic, which can be constructed in a purely formal way, can be provided with a physical meaning by adopting a recent interpretation of quantum mechanics that reinterprets quantum probabilities as conditional on detection rather than absolute. Our results then show that some nonstandard logics can be obtained as mathematical structures formalizing the properties of different notions of verifiability in different physical theories. More generally, they strongly support the idea that many nonclassical logics can coexist without conflicting with classical logic (\emph{global pluralism}), for they formalize metalinguistic notions that do not coincide with the notion of truth (described by Tarski's truth theory).
Einstein initially objected to the probabilistic aspect of quantum mechanics—the idea that God is playing at dice. Later he changed his ground, and focussed instead on the point that the Copenhagen Interpretation leads to what Einstein saw as the abandonment of physical realism. We argue here that Einstein’s initial intuition was perfectly sound, and that it is precisely the fact that quantum mechanics is a fundamentally probabilistic theory which is at the root of all the controversies regarding its interpretation. Probability is an intrinsically logical concept. This means that the quantum state has an essentially logical significance. It is extremely difficult to reconcile that fact with Einstein’s belief, that it is the task of physics to give us a vision of the world apprehended sub specie aeternitatis. Quantum mechanics thus presents us with a simple choice: either to follow Einstein in looking for a theory which is not probabilistic at the fundamental level, or else to accept that physics does not in fact put us in the position of God looking down on things from above. There is a widespread fear that the latter alternative must inevitably lead to a greatly impoverished, positivistic view of physical theory. It appears to us, however, that the truth is just the opposite. The Einsteinian vision is much less attractive than it seems at first sight. In particular, it is closely connected with philosophical reductionism.
The forcing process from mathematical logic offers a promising framework for studying the feasibility of quantum computation on a practical scale, since decoherence is a serious concern and, so far, questions of control, communication, and the implementation of operations that are important for a working computational system have received less attention than mathematical research on algorithms and basic physical investigations of creating simple gates and storing mixed states. Using forcing in this way is a new application of areas of model theory in which propositions and predicates take values in a lattice. Takeuti develops set theory for any universe built on a Boolean algebra generated by commutable projection operators on a Hilbert space, each such universe being a elementary topos of set- valued sheaves on the algebra, which thus is the lattice of truth values for the topos. Since it has a natural numbers object, it thus supports the general forcing method of Scedrov, but the idea of forcing is demonstrated by an example in the simpler context of partially ordered sets. Stout's lamination construction assembles toposes into objects with truth-value lattices that are orthomodular, like the lattice of all projection operators on Hilbert space, and clarifies some difficulties identified by Takeuti in the case where truth values are noncommutable operators. A substantial body of existing sheaf and topos theory thus is potentially relevant to quantum computation, and further work may provide guidance for system development.