Thirty-Seven Dimensions That Are Not Space

Recent headlines have announced an experiment in which light “exists in 37 dimensions.” A reader who has just followed Kaluza, Klein and the eleven-dimensional theory could be forgiven for thinking the headline meant 37 dimensions of space. On the evidence of the paper's abstract it does not, and the difference is worth a chapter.

Two meanings of “dimension”

In Kaluza-Klein theory, a dimension is a direction of spacetime. You can point along it, and a particle can move in it. In quantum mechanics, the dimension of a system is the number of distinct outcomes a measurement can have, the dimension of its Hilbert space. A single spin-½ particle lives in a two-dimensional space of states. A quantum system with 37 distinguishable outcomes lives in a 37-dimensional state space. Both are called “dimension,” and neither tells you anything about the other.

What the experiment did

The arXiv paper is “Exploring the boundary of quantum correlations with a time-domain optical processor” by Zheng-Hao Liu and colleagues (arXiv:2208.07794). By its abstract, the team built a GHZ-type paradox from quantum contextuality, the property that the outcome of a measurement cannot be assigned independently of which other measurements are made alongside it. They report that the events can be covered by contexts of number three, which saturates the lower bound allowed by quantum theory, and they implemented it on a fibre-optic platform that uses time-multiplexed pulses of coherent light, high-speed modulation, convolution and homodyne detection. They report that the quantum prediction was recovered.

[STATUS]The abstract mentions “high-dimensional Hilbert space”; the figure 37 comes from secondary coverage of the published version, which I have not read, and I have not confirmed which journal and year. The New Scientist article (the headline that prompted this chapter) could not be opened in this workflow, so nothing here is drawn from it. Secondary coverage is not a source for the claim; the paper is.

What it does and does not say

It demonstrates a feature of quantum theory in a state space of high dimension. It does not bear on whether space has extra dimensions. A paragraph of caution belongs in any popular account: a number of dimensions in one sense says nothing about the number in the other. What the two stories share is a habit of thought: when a measurement or a motion has more outcomes than the ordinary picture allows, it is often because some structure was hidden in a direction you were not looking at.

That sentence is analogy, labelled as such. It is not a shared invariant, in the sense this series uses the term.