# Is there a way to compute how many hidden variables are needed to describe a given local state?

Consider a bipartite state $$\rho$$ that is local. By local I mean here that there is a local hidden variable model explaining the correlations resulting from measuring $$\rho$$. In other words, $$\rho$$ local means that there are probability distributions $$p_\lambda, p_\lambda(a|A),p_\lambda(b|B)$$ such that for all projective measurements $$\{\Pi^A_a\}_a, \{\Pi^B_b\}_b$$ we can write $$\langle \Pi^A_a\otimes\Pi^B_b, \rho\rangle =\int d\lambda \, p_\lambda \, p_\lambda(a|A)p_\lambda(b|B).$$ Nontrivial (as in, possibly entangled) examples of such local state are Werner states $$p|\Phi^+\rangle\!\langle\Phi^+|+(1-p)\frac{I}{4}$$ for $$\frac13, as shown e.g. in page 21 of (Brunner et al. 2013). The separability of Werner states is also discussed in What is a separable decomposition for the Werner state?.

For separable states (which are always ensured to be local), the question amounts to the "local hidden variable" in the separable decomposition of the state itself. We know that one can always find a separable decomposition for a bipartite state in $$\mathbb{C}^n\otimes \mathbb{C}^m$$ that uses at most $$n^2 m^2-1$$ local states (as discussed e.g. in this physics.SE post), so in this case the associated answer about separability is somewhat already known. At least as far as knowing the cardinality of the required set of values for the local hidden variable.

More generally, is there general technique, result, or reference discussing the "size" of the set of hidden variables required to provide such local explanations for a given quantum state? Of course, there is an immediate issue when asking about size, because a local model might employ a continuous hidden variable. Which makes my question about any result concerning whether a given local state can be described with a discrete, countably infinite, or uncountably infinite local hidden variable.

A related post about the locality of Werner states is How to show that Werner states produce correlations explainable via local hidden variable models?.

This is sort of the content of a lot of the research on the relationship between the contextuality of quantum theory and memory (space) costs of hidden variables theories. A few good starting points for this work are

• "Memory cost of quantum contextuality" by Kleinmann, Guhne, Portillo, Larson and Cabello arXiv:1007.3650
• "Optimal classical simulation of state-independent quantum contextuality" by Cabello, Gu, Guhne and Xu arXiv:1709.07372
• thanks a lot, I'll check the references. I have to admit though, the relation between (Bell) nonlocality and contextuality is not very clear in my mind. Can they be framed similarly?
– glS
Mar 21, 2022 at 9:30
• The relationship between contextuality and Bell nonlocality is indeed confusing, though there are many who I've talked to in foundations who really view contextuality as the more fundamental concern. As for a direct connection I think this paper has some interesting ideas arxiv.org/abs/1704.08223 Mar 22, 2022 at 16:51
• @dabacon this paper uses preparation contextuality; this is a nice paper indeed but 1) one must be careful when looking at literature because preparation noncontextuality is Spekkens proposal for contextuality, while Kleinmann, Guhne et all normally work with Kochen-Specker's notion of contextuality. The two are differently defined and are different in many subtle ways.
– R.W
Apr 11, 2022 at 9:05