Consistent use of paradoxes in deriving constraints on the dynamics of physical systems and of no-go theorems

Abstract

The classical methods used by recursion theory and formal logic to block paradoxes do not work in quantum information theory. Since quantum information can exist as a coherent superposition of the classical "yes" and "no" states, certain tasks which are not conceivable in the classical setting can be performed in the quantum setting. Classical logical inconsistencies do not arise, since there exist fixed point states of the diagonalization operator. In particular, closed timelike curves need not be eliminated in the quantum setting, since they would not lead to any paradoxical outcome controllability. Quantum information theory can also be subjected to the treatment of inconsistent information in databases and expert systems. It is suggested that any two pieces of contradicting information are stored and processed as coherent superposition. In order to be tractable, this strategy requires quantum computation. paradox.texThis letter introduces two novel features of quantum information theory. Physically, it is shown how quantum information allows the consistent implementation of nonlocal correlations. Technically, a diagonalization operator is used to compute consistent fixed point solutions to classical "paradoxical" tasks. The implications for quantum recursion theory [1] and algorithmic information theory [2] as well as for database applications will only be shortly sketched.Classical information theory (e.g., [3]) is based on the bit as fundamental atom. This classical bit, henceforth called cbit, is in one of two classical states.

On the Brightness of the Thomson Lamp: A Prolegomenon to Quantum Recursion Theory

“…As has already been argued [24,25,26,27], |ψ + = (1/ √ 2) (|0 + |1 ) is the quantum fixed point state of the "not" operator, which is essential for diagonal arguments, as X|ψ + = |ψ + . Thus in quantum recursion theory, the diagonal argument consistently goes through without leading to a contradiction, as h(A(A)) yielding |ψ + still allows a consistent response of A by a coherent superposition of its halting and non-halting states.…”

“…One might still put forward that, in accordance with our operationalization of the average brightness, there is a 50:50 probability that it is in state "on" or "off." Just as for the Thomson lamp discussed above, one needs to be careful in defining the output of an accelerated Turing machine [28,24,25,26,27,38,39,19].…”

“…The method of diagonalization presents an important technique of recursion theory [21,22,23]. Some aspects of its physical realizability have already been discussed by the author [24,25,26,27]. In what follows we shall investigate some further physical issues related to "accelerated" agents and the operationalizability of the diagonalization operations in general.…”

The diagonalization method in quantum recursion theory

“…|ψ + does not give rise to inconsistencies [38]: If agent A hands over the fixed point state |ψ + to the diagonalization operator D, the same state |ψ + is recovered. Stated differently, as long as the output of the "halting algorithm" to input A(A) is |ψ + , diagonalization does not change it.…”

Physics and Metaphysics Look at Computation

“…x * does not give rise to inconsistencies (75). If agent A hands over the fixed point state x * to the diagonalization operator D, the same state x * is recovered.…”