# How are contextuality and locality related?

I know how quantum non-locality is defined in Wikipedia

quantum nonlocality is a characteristic of some measurements made at a microscopic level that contradict the assumptions of local realism found in classical mechanics. Despite consideration of hidden variables as a possible resolution of this contradiction, some aspects of entangled quantum states have been demonstrated irreproducible by any local hidden variable theory

[Wiki] and how quantum contextuality is defined in Wikipedia

Quantum contextuality is a foundational concept in quantum mechanics stating that the outcome one observes in a measurement is dependent upon what other measurements one is trying to make. More formally, the measurement result of a quantum observable is dependent upon which other commuting observables are within the same measurement set.

[Wiki]

I however can't grasp how they are related or even if they are related at all. Can anyone help me with a high level explanation?

Locality, specifically Bell locality, is the hypothesis that the outcome of a given measurement is pre-determined by information that was present within its past light-cone.

(Beware that the word "locality" is also used with different meanings, such as Einstein locality, which refers to the hypothesis that observables localized in spacelike-separated regions commute with each other. Quantum field theory satisfies Einstein locality but not Bell locality. Einstein locality is enough to preclude faster-than-light communication.)

Non-contextuality is the hypothesis that the outcome of one measurement does not depend on whatever else is also being measured.

(Bell) locality implicitly assumes non-contextuality for space-like separated measurements but not necessarily for others.

The CHSH inequality can be derived using either Bell locality or just non-contextuality. A derivation from non-contextuality is shown in https://physics.stackexchange.com/a/446977. Nature does not respect the CHSH inequality (or other Bell inequalities), and quantum theory correctly predicts the observed violations.

• Following up on your statement "(Bell) locality can be regarded as a special case of non-contextuality": What about state-independent demonstrations of contextuality (e.g., Peres-Mermin square)? Since they're state independent, that would mean that even disentangled states could exhibit contextuality, and hence even nonlocality. But that surely can't be right. Am I misunderstanding something? – Tfovid Oct 21 '20 at 20:41
• @Tfovid After re-reading what I wrote, I think calling it a "special case" wasn't the right way to say it. I should have simply said that (Bell) locality assumes non-contextuality for space-like separated measurements but not necessarily for others. I edited the answer to fix the incorrect wording. Thanks! – Chiral Anomaly Oct 21 '20 at 23:50
• Does this mean that, conversely, contextuality for space-like measurements implies nonlocality? (If so, the crux of my confusion lies with the fact that contextuality can be desmonstrated in a state-independent fashion, i.e., even with disentagled states. But surely, those can't exhibit nonlocality. Hence the paradox in my head.) – Tfovid Oct 22 '20 at 5:25
• @Tfovid Whatever "nonlocality" means, suppose that (1) contextuality for space-like measurements implies nonlocality, (2) that only entangled states can exhibit nonlocality, and (3) that Peres-Mermin is a state-independent demonstration of contextuality. I don't see any paradox here, because Peres-Mermin doesn't say anything about which observables are space-like separated. To get a paradox, we'd need to show that the Peres-Mermin observables can be arranged in space-time in such a way that makes those three statements contradict each other. Is that possible? – Chiral Anomaly Oct 22 '20 at 14:52
• By "is that possible?", do you mean experimentally? My point is that the PM square is assembled from measurements at two (possibly space-like) points. If we use disentangled states, we'd then have (3) and hence (1), and thus finally contradicting (2). – Tfovid Oct 22 '20 at 16:27