Meaning updating of density matrices
arXiv:2001.00862
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, from Leifer and Spekkens. Unfortunately, neither of these satisfies any desirable algebraic properties, nor are internal to the meaning category. By passing to double density matrices we do get an elegant internal diagrammatic update mechanism. We also show that (commutative) spiders can be cast as an instance of the Leifer-Spekkens update mechanism. 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 on quantum hardware (a.k.a. QNLP), for which exponential space-gain and quadratic speed-up have previously been identified.
24 pages, many figures
References in corpus (7)
- The Frobenius anatomy of word meanings I: subject and object relative pronouns
- Quantum Algorithms for Compositional Natural Language Processing
- A Study of Entanglement in a Categorical Framework of Natural Language
- The Frobenius anatomy of word meanings II: possessive relative pronouns
- Compositional Distributional Semantics with Compact Closed Categories and Frobenius Algebras
- Completeness of the ZX-calculus for Pure Qubit Clifford+T Quantum Mechanics
- Dual Density Operators and Natural Language Meaning