Entanglement as a Semantic Resource

Abstract

The characteristic holistic features of the quantum theoretic formalism and the intriguing notion of entanglement can be applied to a field that is far from microphysics: logical semantics. Quantum computational logics are new forms of quantum logic that have been suggested by the theory of quantum logical gates in quantum computation. In the standard semantics of these logics, sentences denote quantum information quantities: systems of qubits (quregisters) or, more generally, mixtures of quregisters (qumixes), while logical connectives are interpreted as special quantum logical gates (which have a characteristic reversible and dynamic behavior). In this framework, states of knowledge may be entangled, in such a way that our information about the whole determines our information about the parts; and the procedure cannot be, generally, inverted. In spite of its appealing properties, the standard version of the quantum computational semantics is strongly "Hilbert-space dependent". This Found Phys (2010) 40: 1494-1518 1495 certainly represents a shortcoming for all applications, where real and complex numbers do not generally play any significant role (as happens, for instance, in the case of natural and of artistic languages). We propose an abstract version of quantum computational semantics, where abstract qumixes, quregisters and registers are identified with some special objects (not necessarily living in a Hilbert space), while gates are reversible functions that transform qumixes into qumixes. In this framework, one can give an abstract definition of the notions of superposition and of entangled pieces of information, quite independently of any numerical values. We investigate three different forms of abstract holistic quantum computational logic.

Holistic logical arguments in quantum computation

“…This meaning determines the contextual meanings of all subformulas of α (from the whole to the parts!). It may happen 4 See [5] and [1].…”

“…Interestingly enough, a suggestion, in this direction, comes from a formal semantics of music. 5 Consider the case of musical modulations, whose characteristic role is creating tonality-changes in a given composition. The starting point may be a precise tonality (say, C major), which is followed by a situation of tonal ambiguity, where different tonalities co-exist in a form that seems to behave like a quantum superposition.…”

Entanglement and Quantum Logical Gates. Part II.

“…5 This approach can be naturally transformed into a pseudo-unitary semantics, where the same connectives are interpreted as pseudo-unitary gates. Let us briefly recall the basic concepts of the unitary semantics.…”

“…Accordingly, the gate-tree of α can be naturally identified with the following gate-sequence: (1,1,1) . 5 See [5]. 6 We recall that for any m, n ≥ 1 and for any density operator ρ of the Hilbert space H This procedure can be naturally generalized to any α, whose gate-tree will be indicated by D G α n−1 , .…”

“…We consider here a holistic version of the quantum computational semantics, where entanglement can be used as a "semantic resource" [5]. Generally, the meaning of a compound formula determines the contextual meanings of its parts (and not the other way around, as happens in the case of most compositional semantic approaches).…”