# Meaning updating of density matrices

arXiv: https://arxiv.org/abs/2001.00862 · HTML: https://mlsanity.com/paper/2001.00862/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Jan 2020.

## Built on by (most cited first, 9 of 9)

- [Compositionality as we see it, everywhere around us](https://mlsanity.com/paper/2110.05327/) (2021)
- [Conversational Negation using Worldly Context in Compositional Distributional Semantics](https://mlsanity.com/paper/2105.05748/) (2021)
- [Grammar Equations](https://mlsanity.com/paper/2106.07485/) (2021)
- [Composing Conversational Negation](https://mlsanity.com/paper/2107.06820/) (2021)
- [Distilling Text into Circuits](https://mlsanity.com/paper/2301.10595/) (2023)
- [How to make qubits speak](https://mlsanity.com/paper/2107.06776/) (2021)
- [Talking Space: inference from spatial linguistic meanings](https://mlsanity.com/paper/2109.06554/) (2021)
- [Density Matrices for Metaphor Understanding](https://mlsanity.com/paper/2408.11846/) (2024)
- [Quantum Methods for Managing Ambiguity in Natural Language Processing](https://mlsanity.com/paper/2504.00040/) (2025)

## Abstract (arXiv)

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.