Probabilities and Epistemic Operations in the Logics of Quantum Computation (original) (raw)

Quantum teleportation and quantum epistemic semantics

Mathematica Slovaca, 2012

Quantum information gives rise to some puzzling epistemic problems that can be interestingly investigated from a logical point of view. A characteristic example is represented by teleportation phenomena, where knowledge and actions of observers (epistemic agents) play a relevant role. By abstracting from teleportation, we propose a simplified semantics for a language that consists of two parts: 1)the quantum computational sub-language, whose sentences α represent pieces of quantum information (which are supposed to be stored by some quantum systems)2)the classical epistemic sub-language, whose atomic sentences have the following forms: agent a has a probabilistic information about the sentence α; agent a knows the sentence α.Interestingly enough, some conceptual difficulties of standard epistemic logics can be avoided in this framework.

Probabilities in the logic of quantum propositions

arXiv: Quantum Physics, 2018

In quantum logic, i.e., within the structure of the Hilbert lattice imposed on all closed linear subspaces of a Hilbert space, the assignment of truth values to quantum propositions (i.e., experimentally verifiable propositions relating to a quantum system) is unambiguously determined by the state of the system. So, if only pure states of the system are considered, can a probability measure mapping the probability space for truth values to the unit interval be assigned to quantum propositions? In other words, is a probability concept contingent or emergent in the logic of quantum propositions? Until this question is answered, the cause of probabilities in quantum theory cannot be completely understood. In the present paper it is shown that the interaction of the quantum system with its environment causes the irreducible randomness in the relation between quantum propositions and truth values.

Quantum Theory as a Relevant Framework for the Statement of Probabilistic and Many-Valued Logic

Based on ideas of quantum theory of open systems we propose the consistent approach to the formulation of logic of plausible propositions. To this end we associate with every plausible proposition diagonal matrix of its likelihood and examine it as density matrix of relevant quantum system. We are showing that all logical connectives between plausible propositions can be represented as special positive valued transformations of these matrices. We demonstrate also the above transformations can be realized in relevant composite quantum systems by quantum engineering methods. The approach proposed allows one not only to reproduce and generalize results of wellknown logical systems (Boolean, Lukasiewicz and so on) but also to classify and analyze from unified point of view various actual problems in psychophysics and social sciences.

Probabilistic Logics in Quantum Computation

New Challenges to Philosophy of Science, 2013

The quantum computation process may be summarized as follows: first an initial state of a physical system is provided as the input. Then, it evolves according to the elementary operations (quantum gates) that are performed on it. Finally, the access to the information content of the resulting state is possible via the measurement operation that provides one of the possible results. In this note we describe probabilistictype semantics for propositional logics designed to describe e↵ective procedure based on measurement processes.

A first-order epistemic quantum computational semantics with relativistic-like epistemic effects

Fuzzy Sets and Systems, 2016

Quantum computation has suggested new forms of quantum logic, called quantum computational logics. In these logics wellformed formulas are supposed to denote pieces of quantum information: possible pure states of quantum systems that can store the information in question. At the same time, the logical connectives are interpreted as quantum logical gates: unitary operators that process quantum information in a reversible way, giving rise to quantum circuits. Quantum computational logics have been mainly studied as sentential logics (whose alphabet consists of atomic sentences and of logical connectives). In this article we propose a semantic characterization for a first-order epistemic quantum computational logic, whose language can express sentences like "Alice knows that everybody knows that she is pretty". One can prove that (unlike the case of logical connectives) both quantifiers and epistemic operators cannot be generally represented as (reversible) quantum logical gates. The "act of knowing" and the use of universal (or existential) assertions seem to involve some irreversible "theoretic jumps", which are similar to quantum measurements. Since all epistemic agents are characterized by specific epistemic domains (which contain all pieces of information accessible to them), the unrealistic phenomenon of logical omniscience is here avoided: knowing a given sentence does not imply knowing all its logical consequences.

Quantum probability: a reliable tool for an agent or a reliable source of reality?

Synthese, 2019

In this paper we attempt to analyze the concept of quantum probability within quantum computation and quantum computational logic. While the subjectivist interpretation of quantum probability explains it as a reliable predictive tool for an agent in order to compute measurement outcomes, the objectivist interpretation understands quantum probability as providing reliable information of a real state of affairs. After discussing these different viewpoints we propose a particular objectivist interpretation grounded on the idea that the Born rule provides information about an intensive realm of reality. We then turn our attention to the way in which the subjectivist interpretation of probability is presently applied within both quantum computation and quantum computational logic. Taking as a standpoint our proposed intensive account of quantum probability we discuss the possibilities and advantages it might open for the modeling and development of both quantum computation and quantum computational logic.

A Quantum Computational Semantics for Epistemic Logical Operators. Part I: Epistemic Structures

International Journal of Theoretical Physics, 2013

By using the abstract structures investigated in the first Part of this article, we develop a semantics for an epistemic language, which expresses sentences like "Alice knows that Bob does not understand that π is irrational". One is dealing with a holistic form of quantum computational semantics, where entanglement plays a fundamental role; thus, the meaning of a global expression determines the contextual meanings of its parts, but generally not the other way around. The epistemic situations represented in this semantics seem to reflect some characteristic limitations of the real processes of acquiring information. Since knowledge is not generally closed under logical consequence, the unpleasant phenomenon of logical omniscience is here avoided.

A Quantum Computational Semantics for Epistemic Logical Operators. Part II: Semantics

2016

By using the abstract structures investigated in the first Part of this article, we develop a semantics for an epistemic language, which expresses sentences like "Alice knows that Bob does not understand that PI is irrational". One is dealing with a holistic form of quantum computational semantics, where entanglement plays a fundamental role, thus, the meaning of a global expression determines the contextual meanings of its parts, but generally not the other way around. The epistemic situations represented in this semantics seem to reflect some characteristic limitations of the real processes of acquiring information. Since knowledge is not generally closed under logical consequence, the unpleasant phenomenon of logical omniscience is here avoided.

Logics from Quantum Computation

International Journal of Quantum Information, 2005

The theory of logical gates in quantum computation has suggested new forms of quantum logic, called quantum computational logics. The basic semantic idea is the following: the meaning of a sentence is identified with a quregister (a system of qubits) or, more generally, with a mixture of quregisters (called qumix ). In this framework, any sentence α of the language gives rise to a quantum tree: a kind of quantum circuit that transforms the quregister (qumix) associated to the atomic subformulas of α into the quregister (qumix) associated to α. A variant of the quantum computational semantics is represented by the quantum holistic semantics, which permits us to represent entangled meanings. Physical models of quantum computational logics can be built by means of Mach-Zehnder interferometers.