# How are Gleason's and Kochen-Specker's theorems related?

If, on the one hand, I were to paraphrase Gleason's theorem, it would loosely state that

if one can assign a truth value $$p_k$$ to each basis vector $$\vec{u}_k$$ such that $$\sum_k p_k = 1$$, then that assignment can only be produced by Born's rule using a density matrix $$\hat{\rho}$$.

while, on the other hand, Kochen-Specker states that

on the (3D) sphere, it is impossible to assign a truth value to each point among triples of points arising from orthogonal bases in such a way that those truth values behave as probabilities (namely, add up to one).

Based on the above, am I therefore to conclude that the link between Gleason and Kochen-Specker is that Kochen-Specker can only be violated by the Born's rule, i.e., by quantum mechanical assignments of truth values?

Please note: I don't have enough literacy in advanced algebra, so please keep the discussion intuitive and accessible to, say, a senior undergraduate, preferably starting from my (hopefully correct?) understanding of the theorems as I've stated them above. (This answer, for example, is currently beyond my grasp.)

• A direct consequence of Gleason's theorem is that when the Hilbert space $$H$$ has dimension at least three then there does not exist a value (aka frame) function $$v$$ that assigns only the values $$0$$ or $$1$$ to projections: $$P:H\to H\,.$$ Loosely speaking this means that it is not possible to assign true or false values to all elements of the Hilbert space that represent yes/no events (projections) in such a way that is compatible with their Boolean properties: e.g. $$v(P_1+P_2)=v(P_1)+v(P_2)$$ for $$P_1\bot P_2\,.$$

• The Kochen-Specker theorem is similar in spirit. For simplicity I reproduce only the case $$\dim(H)=4$$ here: There exist $$18$$ projections $$P_1,...,P_{18}$$ from which one can form $$9$$ bases of $$H\,,$$ \begin{align} \{P_1,P_2,P_3,P_4\}\,,\dots\,,\{P_{15},P_{16},P_{17},P_{18}\} \end{align} such that it is not possible to assign the values $$\{0,1\}$$ to the projections so that in each of those bases exactly one projection has value $$1\,.$$ (Each projection occurs in exactly two of the $$9$$ bases.) This theorem rules out a value function that is independent of the chosen measurement basis.

• Both theorems rule out non contextual hidden variables. I am not sure if your understanding of the Kochen-Specker theorem is correct. It deals with $$\{0,1\}$$-valuation, not just with summing to one.

A generalization of the Kochen-Specker theorem in $$\dim(H)=3$$ that looks fairly accessible you can find in

Voráček, V., Navara, M. Generalised Kochen–Specker Theorem in Three Dimensions. Found Phys 51, 67 (2021).

https://doi.org/10.1007/s10701-021-00476-3

• Gleason's theorem is originally how you stated it. In the wikipedia link I gave they apply a clever trick to draw the no-go conclusion. Essentially they argue that $v$ coming from a density matrix $\rho$ must be continuous on the unit-sphere. Since $v$ cannot assign one to both $P_1$ and $P_2$ when they are orthogonal this $v$ cannot just take the values $0$ or $1$ only. It must somewhere cross through the values in $(0,1)\,.$ Commented Feb 15, 2023 at 18:39