Unitary quantum theory, having no Born Rule, is non-probabilistic . Hence the notorious problem of reconciling it with the unpredictability and appearance of stochasticity in quantum measurements. Generalizing and improving upon the so-called ‘decision-theoretic approach’, I shall recast that problem in the recently proposed constructor theory of information— where quantum theory is represented as one of a class of superinformation theories , which are local , non-probabilistic theories conforming to certain constructor-theoretic conditions. I prove that the unpredictability of measurement outcomes (to which constructor theory gives an exact meaning) necessarily arises in superinformation theories. Then I explain how the appearance of stochasticity in (finitely many) repeated measurements can arise under superinformation theories. And I establish sufficient conditions for a superinformation theory to inform decisions (made under it) as if it were probabilistic, via a Deutsch–Wallace-type argument—thus defining a class of decision-supporting superinformation theories. This broadens the domain of applicability of that argument to cover constructor-theory compliant theories. In addition, in this version some of the argument's assumptions, previously construed as merely decision-theoretic, follow from physical properties expressed by constructor-theoretic principles.
Unitary quantum theory, having no Born Rule, is non-probabilistic . Hence the notorious problem of reconciling it with the unpredictability and appearance of stochasticity in quantum measurements. Generalizing and improving upon the so-called ‘decision-theoretic approach’, I shall recast that problem in the recently proposed constructor theory of information— where quantum theory is represented as one of a class of superinformation theories , which are local , non-probabilistic theories conforming to certain constructor-theoretic conditions. I prove that the unpredictability of measurement outcomes (to which constructor theory gives an exact meaning) necessarily arises in superinformation theories. Then I explain how the appearance of stochasticity in (finitely many) repeated measurements can arise under superinformation theories. And I establish sufficient conditions for a superinformation theory to inform decisions (made under it) as if it were probabilistic, via a Deutsch–Wallace-type argument—thus defining a class of decision-supporting superinformation theories. This broadens the domain of applicability of that argument to cover constructor-theory compliant theories. In addition, in this version some of the argument's assumptions, previously construed as merely decision-theoretic, follow from physical properties expressed by constructor-theoretic principles.
We propose a theory of information expressed solely in terms of which transformations of physical systems are possible and which are impossible—i.e. in constructor-theoretic terms. It includes conjectured, exact laws of physics expressing the regularities that allow information to be physically instantiated. Although these laws are directly about information, independently of the details of particular physical instantiations, information is not regarded as an a priori mathematical or logical concept, but as something whose nature and properties are determined by the laws of physics alone. This theory solves a problem at the foundations of existing information theory, namely that information and distinguishability are each defined in terms of the other. It also explains the relationship between classical and quantum information, and reveals the single, constructor-theoretic property underlying the most distinctive phenomena associated with the latter, including the lack of in-principle distinguishability of some states, the impossibility of cloning, the existence of pairs of variables that cannot simultaneously have sharp values, the fact that measurement processes can be both deterministic and unpredictable, the irreducible perturbation caused by measurement, and locally inaccessible information (as in entangled systems).
A major problem facing no-collapse interpretations of quantum mechanics in the tradition of Everett is how to understand the probabilistic axiom of quantum mechanics (the Born rule) in the context of a deterministic theory in which every outcome of a measurement occurs. Deutsch claims to derive a decision-theoretic analogue of the Born rule from the non-probabilistic part of quantum mechanics and some non-probabilistic axioms of classical decision theory, and hence concludes that no probabilistic axiom is needed. I argue that Deutsch’s derivation begs the question.
Constructor theory is a proposal to extend quantum information theory beyond both quantum theory and computation, to cover more general machines than programmable computers -- called constructors. It consists of newly conjectured physical principles that can be expressed as constraints on what tasks are possible, what are impossible, and why. These principles also determine the repertoire of the universal constructor, which is a programmable machine that can perform all physically possible tasks. The principles of constructor theory have novel physical content that supplements current dynamical laws, leading to new predictions for experimental tests. In this paper, we review the main experimental proposals to test the principles of constructor theory and discuss their implications for existing theories of physics and for their successors.
ABSTRACT: David Deutsch has a forthcoming article called “Quantum Theory of Probability and Decisions " in which he claims to derive the standard probabilistic interpretation of the wavefunction from non-probabilistic considerations. In this note, I try to develop his argument in a classical context, to see whether the same ideas can generate probabilistic conclusions from nonprobabilistic assumptions in a more general way. While my answer is negative, the attempt pinpoints the parts of Deutsch’s quantum mechanical argument that must rely on special features of the formalism (assuming that his proof is correct). 1. No Probabilities In, No Probabilities Out? You cannot derive an ‘ought ’ from an ‘is’. And I believe that it was Popper (1959) who once said, in response to those who think that randomness can be defined nonprobabilistically, that one cannot derive a probabilistic conclusion from non-probabilistic premises. Yet this is exactly what Deutsch (to appear) claims for quantum mechanical probabilities. Historically in the development of quantum theory, the Schrödinger
In a recent paper, Deutsch claims to derive the ‘probabilistic predictions of quantum theory’ from the ‘non–probabilistic axioms of quantum theory’ and the ‘nonprobabilistic part of classical decision theory.’ We show that his derivation includes a crucial hidden assumption that vitiates the force of his argument. Furthermore, we point out that in classical decision theory a standard set of non–probabilistic axioms is already sufficient to endow possible outcomes with a natural probability structure. Within that context we argue that Gleason's theorem, relying on fewer assumptions than Deutsch, provides a compelling derivation of the quantum probability law.
Issues of control and causation are central to the Quantum Theory of Computation. Yet there is no place for them in fundamental laws of Physics when expressed in the prevailing conception, i.e., in terms of initial conditions and laws of motion. This thesis aims at arguing that Constructor Theory, recently proposed by David Deutsch to generalise the quantum theory of computation, is a candidate to provide a theory of control and causation within Physics. To this end, I shall present a physical theory of information that is formulated solely in constructor-theoretic terms, i.e., in terms of which transformations of physical systems are possible and which are impossible. This theory solves the circularity at the foundations of existing information theory; it provides a unifying relation between classical and quantum information, revealing the single property underlying the most distinctive phenomena associated with the latter: the unpredictability of the outcomes of some deterministic processes, the lack of distinguishability of some states, the irreducible perturbation caused by measurement and the existence of locally inaccessible information in composite systems (entanglement). This thesis also aims to investigate the restrictions that quantum theory imposes on copying-like tasks. To this end, I will propose a unifying, picture-independent formulation of the no-cloning theorem. I will also discuss a protocol to accomplish the closely related task of transferring perfectly a quantum state along a spin chain, in the presence of systematic errors. Furthermore, I will address the problem of whether self-replication (as it occurs in living organisms) is compatible with Quantum Mechanics. Some physicists, notably Wigner, have argued that this logic is in fact forbidden by Quantum Mechanics, thus claiming that the latter is not a universal theory. I shall prove that those claims are invalid and that the logic of self-replication is, of course, compatible with Quantum Mechanics.