Meaning updating of density matrices

ABSTRACT

The DisCoCat model of natural language meaning assigns meaning to a sentence given: (i) the meanings of its words, and, (ii) its grammatical structure. The recently introduced DisCoCirc model extends this to text consisting of multiple sentences. While in DisCoCat all meanings are fixed, in DisCoCirc each sentence updates meanings of words. In this paper we explore different
update mechanisms for DisCoCirc, in the case where meaning is encoded in density matrices—which come with several advantages as compared to vectors.

Our starting point are two non-commutative update mechanisms, borrowing one from quantum foundations research [34, 35], and the other one from [9, 36]. Unfortunately, neither of these satisfies any desirable algebraic properties, nor are internal to the meaning category. By passing to double density matrices [2, 51] we do get an elegant internal diagrammatic update mechanism.

We also show that (commutative) spiders can be cast as an instance of the update mechanism of [34, 35]. This result is of interest to quantum foundations, as it bridges the work in Categorical Quantum Mechanics (CQM) with that on conditional quantum states. Our work also underpins implementation of text-level Natural Language Processing (NLP) on quantum hardware, for which exponential space-gain and quadratic speed-up have previously been identified.

Bob Coecke, Konstantinos Meichanetzidis

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