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Mathematica Slovaca

Quantum teleportation and quantum epistemic semantics

Enrico G. Beltrametti, Maria L. Dalla Chiara, Roberto Giuntini, Giuseppe Sergioli

Published Dec 1, 2012• 13 citations
Publisher

Abstract

Abstract 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.

DOI
10.2478/s12175-012-0069-y

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Abstract 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.
Quantum computation theory has inspired new forms of quantum logic, called quantum computational logics, where formulas are supposed to denote pieces of quantum information, while logical connectives are interpreted as special examples of quantum logical gates. The most natural semantics for these logics is a form of holistic semantics, where meanings behave in a contextual way. In this framework, the concept of quantum probability can assume different forms. We distinguish an absolute concept of probability, based on the idea of quantum truth, from a relative concept of probability (a form of transition-probability, connected with the notion of fidelity between quantum states). Quantum information has brought about some intriguing epistemic situations. A typical example is represented by teleportation-experiments. In some previous works we have studied a quantum version of the epistemic operations “to know”, “to believe”, “to understand”. In this article, we investigate another epistemic operation (which is informally used in a number of interesting quantum situations): the operation “being probabilistically informed”.
We construct a formal framework for investigating epistemic and temporal notions in the context of distributed quantum computation. While we rely on structures developed earlier, we stress that our notion of quantum knowledge makes sense more generally in any agent-based model for distributed quantum systems. Several arguments are given to support our view that an agent's possibility relation should not be based on the reduced density matrix, but rather on local classical states and local quantum operations. In this way, we are able to analyse distributed primitives such as superdense coding and teleportation, obtaining interesting conclusions as to how the knowledge of individual agents evolves. We show explicitly that the knowledge transfer in teleportation is essentially classical, in that eventually, the receiving agent knows that its state is equal to the initial state of the sender. The relevant epistemic statements for teleportation deal with this correlation rather than with the actual quantum state, which is unknown throughout the protocol.
Does quantum holism give rise to new forms of abstract holistic semantics? This question can be discussed in the framework of quantum computational logics, non-standard versions of quantum logic that have been suggested by the theory of logical gates in quantum computation. The basic idea of the quantum computational semantics is the following: meanings of sentences are identified with quantum information quantities, represented by pure or mixed states of special quantum systems. In particular cases, the meaning of a sentence may correspond to an entangled state. On this basis, one can formally describe some typical holistic semantic situations, where the meaning of the whole determines the meanings of its parts, but generally not vice versa.
This paper applies to quantum systems a modelling for the logic of knowledge, originally developed for reasoning about distributed systems, but since then applied to game theory, computer security and artificial intelligence. A formal model of quantum message passing systems is developed and the question of how one might define the semantics of a modal operator for knowledge in this model is considered. It is argued that there are at least two plausible semantics, depending on whether the agents are permitted to make use of their quantum state in determining what they know, and on whether one is dealing with single instances of quantum systems, or ensembles. The framework is illustrated using a number of examples from the quantum computing literature, including protocols for quantum key distribution and teleportation. 1
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 α is identified with a quantum information quantity, represented by a density operator of a Hilbert space, whose dimension depends on the logical complexity of α. At the same time, the logical connectives of the language are interpreted as operations defined in terms of quantum logical gates. Standard quantum computational models can be described as special cases of Fock space models, where the meaning of any sentence is localized in a precise sector of a Fock space ℱ. From an intuitive point of view, the increasing number of particles described in the different sectors of ℱ can be interpreted as increasing information.
This paper applies to quantum systems a modelling for the logic of knowledge, originally developed for reasoning about distributed systems, but since then applied to game theory, computer security and artificial intelligence. A formal model of quantum message passing systems is developed and the question of how one might define the semantics of a modal operator for knowledge in this model is considered. It is argued that there are at least two plausible semantics, depending on whether the agents are permitted to make use of their quantum state in determining what they know, and on whether one is dealing with single instances of quantum systems, or ensembles. The framework is illustrated using a number of examples from the quantum computing literature, including protocols for quantum key distribution and teleportation.