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.
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.
Incompleteness and undecidability theorems have to be revised in view of quantum information and computation theory. qrt.tex 1 As has already been pointed out in Gödel’s centennial paper on the incompleteness af arithmetic [1], the classical undecidability theorems of formal logic [2] and the theory of computable functions [4, 5] are based on semantical pardoxes such as the liar [6] or Richard’s paradox. The method of diagonalization, which was first applied by Cantor for a proof of the undenumerability of real numbers [7], has been applied by Turing for a proof of the recursive undecidability of the halting problem [8]. The halting problem is the problem of whether or not an arbitrary algorithm terminates or produces a particular output and terminates. Assume that the halting problem is decidable. Turing [8] proved that this assumption yields a contradiction. To construct the contradiction, consider
Quantum information and computation may serve as a source of useful axioms and ideas for the quantum logic/quantum structures project of characterizing and classifying types of physical theories, including quantum mechanics and classical mechanics. The axiomatic approach of quantum structures may help isolate what aspects of quantum mechanics are responsible for what aspects of its greater-than-classical information processing power, and whether more general physical theories may escape some common limitations of classical and quantum theories. Also, by by helping us understand how existing quantum algorithms work, quantum structures analyses may suggest new quantum protocols exploiting general features of quantum mechanics. I stress the importance, for these matters, of understanding open and closed-system dynamics, and the structure of composite systems in general frameworks for operational theories, such as effect algebras, convex sets, and related structures.
Logical paradoxes and inconsistent information pose deep challenges in epistemology and the philosophy of logic. Classical systems typically handle contradictions only through external checks or by altering the logical framework, as in Tarski's hierarchy or paraconsistent logics. We propose a novel approach: a quantum circuit architecture that intrinsically enforces logical consistency during its unitary evolution. By encoding self-referential or contradictory propositions into a quantum state, the circuit uses interference effects to suppress inconsistent outcomes while preserving coherent ones. We demonstrate this with the Liar Paradox ("This statement is false"), showing that the quantum model naturally stabilizes truth values that would be paradoxical classically. The framework builds on orthomodular quantum logic treating logical propositions as subspace projectors and connects to belief revision and cognitive modeling by providing a physical mechanism for coherence restoration in epistemic states. This work bridges formal logic and quantum computation, suggesting that consistency can be embedded as a structural property of reasoning systems rather than imposed externally.
Quantum information theory is a wide area of investigation, which includes two main subjects: quantum computation and quantum communication. The success of quantum computation theory is due to a positive result: Shor's algorithm. The algorithm provides factorization of integers in a much more efficient way with respect to what is achieved not only by present-day computers, but even by the most powerful conceivable computers based on classical physics. On the other hand, the success of quantum communication is due to the intrinsic security of ideal quantum cryptographic protocols, and thus relies on a negative result, that is the impossibility of reading quantum information without scrambling it in an unrecoverable way. The no information without disturbance property spawns other no-go theorems, which represent some of the most famous and classic results in quantum information theory. Among these results one can certainly list the no-cloning theorem, stating the impossibility of copying quantum information. This impossibility is clearly a consequence of the impossibility of extracting information without disturbing it. Indeed, if we could copy exactly the state of a system, reproducing it on another system of the same type, then we would also be able to iterate this process, and produce as many copies as we want. We could then extract all the information we are interested in from the copies, keeping the original intact. This argument is so simple, but hides a very deep point about quantum information, and more generally about any type of information that can be purified. Suppose indeed that there exists a measurement allowing us to determine the state of a system, with absolute precision. We could then use such a measurement on the system, and once we know its exact state, we can in principle re-prepare the same state as many times as we want. If copying the state is forbidden within a theory, we must then conclude that, in that theory, measuring the state is forbidden, too! This is exactly the case in theories with purification, and in particular in quantum theory. In quantum theory, determining the state of a system is impossible. We can only have some information about what state our system is prepared in. The information may be accurate, but it will never be certain. This could be practically the case in classical theory, too.
Information is something that can be encoded in the state of a physical system, and a computation is a task that can be performed with a physically realizable device. Therefore, since the physical world is fundamentally quantum mechanical, the foundations of information theory and computer science should be sought in quantum physics. In fact, quantum information has weird properties that contrast sharply with the familiar properties of classical information. A quantum computer -- a new type of machine that exploits the quantum properties of infomation -- could perform certain typ$s of calculations far more efficiently than any foreseeable classical computer. To build a functional quantum computer will be an enormous technical challenge. New methods for quantum error correction are being developed that can help to prevent a quantum computer from crashing.
Information is something that can be encoded in the state of a physical system, and a computation is a task that can be performed with a physically realizable device. Therefore, since the physical world is fundamentally quantum mechanical, the foundations of information theory and computer science should be sought in quantum physics. In fact, quantum information has weird properties that contrast sharply with the familiar properties of classical information. A quantum computer-a new type of machine that exploits the quantum properties of information-could perform certain types of calculations far more efficiently than any foreseeable classical computer. To build a functional quantum computer will be an enormous technical challenge. New methods for quantum error correction are being developed that can help to prevent a quantum computer from crashing.