The Kochen-Specker theorem is often described as ruling out "noncontextual" classical hidden variable theories. I understand the math behind the theorem, but I'm a little unclear on the exact, mathematically precise definition of "contextuality" that people use when discussing its implications.

This encyclopedia article and this blog post give nice overviews of the theorem. The encyclopedia entry defines "noncontextuality" to mean

If a QM system possesses a property (value of an observable), then it does so independently of any measurement context, i.e. independently of how that value is eventually measured ... a failure of NC [noncontextuality] could be understood in two ways. Either the value of an observable might be context-dependent, although the observable itself is not; or the value of an observable might be context-dependent, because the observable itself is. In either case, the independence from context of an observable implies that there is a correspondence of observables and operators. This implication of NC is what we will use presently in the derivation of FUNC. We will indeed assume that, if NC holds, this means that the observable — and thereby also its value — is independent of the measurement context, i.e. is independent of how it is measured. In particular, the independence from context of an observable implies that there is a 1:1 correspondence of observables and operators. This implication of NC is what we will use presently in the derivation of FUNC. Conversely, failure of NC will be construed solely as failure of the 1:1 correspondence.

I don't understand what this means concretely. One possible way to make this notion precise is:

(NC1) A set of distinct measurements will always yield the same measured values, regardless of the order in which they are performed.

This notion is clearly incompatible with the predictions of quantum mechanics: if you have a qubit in the $| \uparrow \rangle$ state and you measure both $\sigma^x$ and $\sigma^z$, then if you measure $\sigma^z$ first, you always get the value $+1$, while if you measure it second, you get $+1$ and $-1$ 50% of the time each. The fact that quantum mechanics rules out this particular notion of "noncontextuality" is pretty trivial and certainly does not merit a named theorem.

But as I understand it, the actual Kochen-Specker theorem seems to be ruling out a much broader notion of "noncontextuality":

(NC2) The measured value of an observable quantity is independent of the choice of complete measurement basis (as long as the measured observable itself is one of the basis vectors).

For example, in the example given in Section 3.2 of the article, if we denote the computational basis as $\{ | 1 \rangle, | 2 \rangle, | 3 \rangle, | 4 \rangle \}$ and we want to measure whether the observable corresponding to the projector $| 4 \rangle \langle 4 |$ has value 0 or 1, then (NC2) requires that the result that we get be independent of the choice of basis for the subspace orthogonal to $| 4 \rangle$ spanned by $\{ | 1 \rangle, | 2 \rangle, | 3 \rangle \}$. Under this definition, the "experimental context" of the measurement of $| 4 \rangle \langle 4 |$ is the choice of basis for the orthogonal subspace. The KS theorem straightforwardly rules out the possibility of (NC2) by means of an explicit construction.

The problem is that the theorem's setup is assuming that we always measure in a complete basis; i.e. we simultaneously (or sequentially) measure the values corresponding to a complete set of orthogonal projection operators that add up to the identity (a complete resolution of the identity). In this case, the KS theorem does indeed show that the measured values of the projection operators must be nontrivially correlated (beyond the classical requirement that the values add up to $1$).

But in practice, you usually don't measure over a complete basis. As a trivial example, any realistic experiment measuring the position $X$ cannot capture all the operator's eigenvalues, because the experimental apparatus must have finite spatial extent. (Indeed, in real particle-physics experiments, the huge array of detectors still misses the large majority of the particles.) But even putting aside the $X$ operator because of the complication that it acts on an infinite-dimensional Hilbert space, one could certainly imagine a partial measurement where one measures the value of the operator $| 4 \rangle \langle 4 |$ without measuring the value of any orthogonal projectors. (As the blog post mentions, this fails for a two-dimensional Hilbert space, because if $| \uparrow \rangle \langle \uparrow | + | \downarrow \rangle \langle \downarrow | = I$, then measuring the value of $| \uparrow \rangle \langle \uparrow |$ is equivalent to measuring the value of $| \downarrow \rangle \langle \downarrow |$.) See the Renninger negative-result experiment for further discussion. What is the "experimental context" of such an experiment? Does a failure to measure the values of orthogonal projectors count as an "experimental context?" Or is the resolution just that you could have measured over a complete basis and gotten unique values for a complete set of projectors?


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