The Daily Ant hosts a weekly series, Philosophy Phridays, in which real philosophers share their thoughts at the intersection of ants and philosophy. This is the twentieth contribution in the series and the first coauthored piece, jointly submitted by Eddy Chen (陈科名) and Isaac Wilhelm. Edited on Sunday, July 9, 2017.


From Ants to Quantum Non-Locality

Though much has been said about the amazing insects known as ants, their capacity to illustrate the novel and mysterious phenomenon of quantum non-locality is under-discussed. We hope to fill in the gap on this Philosophy Phriday.

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Our story centers around Alice, a brilliant myrmecologist (that is, an ant expert). She’s most famous for having discovered a colony of ants that appear to contradict the ordinary rules of logic and arithmetic.

To begin with, as Alice often says in public lectures, these ants are not ordinary at all: some are fast while others are slow; some love M&Ms while others hate M&Ms; some are politically liberal while others are conservative. In her (published) notebook, she records the six traits as follows:

F+: the ant is fast.       M+: the ant loves M&Ms.      L+: the ant is liberal.

F–: the ant is slow.      M–: the ant hates M&Ms.       L–: the ant is conservative.

A scientist by training, Alice is very strict about using the following clear, consistent terminology.

  1. The ‘type’ of a trait is the capital letter which it features. For example, the traits in the first column—being fast (F+) and being slow (F–)—belong to the same type: speed (F).
  2. The ‘value’ of a trait is its sign. For example, the value of F+ is ‘+’, and the value of L– is ‘–’.
    • One also says that a slow (F–) ant has value ‘–’ in F.
  3. Any ‘+’ trait and any ‘–’ trait are said to have ‘opposite values.’ For example, if Tom the ant is fast (F+) and Jerry the ant is slow (F–), then their F values are opposite. Similarly, if Tom is fast (F+) and Jerry hates M&Ms (M–), then Tom’s F value and Jerry’s M value are opposite too.
    • One also says that F+ and F– are ‘opposite traits in the type F’.

Alice’s notebook documents a series of startling discoveries about the ways in which these ants exemplify the six traits above. Alice’s first discovery is fairly innocuous.

(Every Trait) Every ant exemplifies exactly one trait in each type.

For example, Tom is either fast or slow (and not both), either loves or hates M&Ms (and not both), and is either liberal or conservative (and not both).

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Twin ants are (Opposite). Photo: Alex Wild

Alice’s second discovery concerns a certain correlation among their traits. In this particular colony, ants are always born in pairs. Call the ants that belong to the same pair ‘twin-ants’. As it turns out, the following fact always holds for ants that are twins.

(Opposite) For each pair of twin-ants, the twins have opposite traits in each type.

For example, suppose Tom and Jerry are twin-ants. Of course, because they are ants, each exemplifies one of the F traits, one of the M traits, and one of the L traits. According to (Opposite), if Tom is fast then Jerry is slow, and if Tom is slow then Jerry is fast. And similarly for the M and L traits.

Alice’s third discovery concerns a certain statistical correlation between traits in different types.

(Correlation) For any two distinct types of traits X and Y, exactly 25% of all pairs of twin-ants have the following ‘always opposite’ property: one twin’s value in type X is always opposite the other twin’s value in type Y.

For example, suppose that for types F and M, Tom and Jerry are among the 25% of twin-ant pairs that have the relevant ‘always opposite’ property. Then according to this instance of (Correlation), if Tom is fast (F+) then Jerry hates M&Ms (M–), and if Tom is slow (F–) then Jerry loves M&Ms (M+). Of course, it follows from (Correlation) that 75% of pairs fail to be like this. The reader can easily construct similar examples for the other two pairs of different traits: F and L, and M and L.

These three discoveries about ants are incontrovertible; Alice has checked them repeatedly. But as she soon realizes, they contradict the basic laws of arithmetic. So, as it turns out, she lives in an impossible world.

To derive the contradiction, Alice draws a table. Given (Every Trait), each ant in a twin-pair exemplifies exactly one F trait, one M trait, and one L trait. Given (Opposite), if the left twin is F+ then the right twin must be F–, and similarly for M and L. Therefore, for each pair of twin-ants, there are exactly eight (2x2x2) possibilities for the actual traits they may exhibit. Let’s label them Sorts (1) – (8).

Possible Assignments of Properties
Sorts Left Twin Right Twin Features
(1) F+, M+, L+ F–, M–, L– P, Q, S
(2) F+, M+, L– F–, M–, L+ Q
(3) F+, M–, L+ F–, M+, L– P
(4) F+, M–, L– F–, M+, L+ S
(5) F–, M+, L+ F+, M–, L– S
(6) F–, M+, L– F+, M–, L+ P
(7) F–, M–, L+ F+, M+, L– Q
(8) F–, M–, L– F+, M+, L+ P, Q, S

Alice then considers the following three features that each pair might have; she marks them on the right-most column:

(P) One twin’s F value is opposite the other twin’s L value.
(Q) One twin’s F value is opposite the other twin’s M value.
(S) One twin’s L value is opposite the other twin’s M value.

Each of these is a ‘feature’ which a pair of twin-ants may have. Of course, any pair of twin-ants need not satisfy all three features. A pair of twin-ants that have (P) may not have (Q). But some pairs (of Sorts (1) and (8)) do have more than one feature.

Let’s focus on these three features: (P), (Q), and (S). Since each one of the eight sorts exemplifies at least one of (P), (Q), and (S), 100% of the pairs in the colony have (P) or (Q) or (S).

However, this is not what (Correlation) tells us. According to (Correlation), exactly 25% of the pairs exemplify (P), exactly 25% of the pairs exemplify (Q), and exactly 25% of the pairs exemplify (S). So at most 75% of the pairs exemplify (P) or (Q) or (S).

To see why, note that if there were any overlap between the different sorts of twin-ant pairs––if some of the pairs that exemplify (P) also exemplified (Q), for example––then less than 75% of all pairs would exemplify at least one of the three features. And if there were no such overlap, then exactly 75% (= 25% + 25% + 25%) of all pairs would exemplify (P) or (Q) or (S).

So here’s the situation. According to (Opposite), 100% of the pairs exemplify (P) or (Q) or (S). But according to (Correlation), at most 75% of the pairs exemplify (P) or (Q) or (S). That leads to a contradiction, for it entails that the percentage of pairs that exemplify (P) or (Q) or (S) is at the same time less than or equal to 75% and equal to 100%, which would violate the basic rules of arithmetic (for it entails that 75% is bigger than or equal to 100%, an impossible inequality)!

Alice concludes that she lives in an impossible world.

Crematogaster

Impossible ants! Photo: Alex Wild

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     In the actual world, the physicist John Bell discovered an analogous situation. Parts of the impossible story about Alice have the same structure as the predictions of quantum mechanics: just replace the twin-ants with electrons that come in entangled pairs, and replace the traits F, M, and L with certain incompatible spin-observables.

Alice’s world is impossible; it has ants that contradict the laws of arithmetic. But our world is not: Bell did not discover that we live in an impossible world. He discovered that quantum predictions and a principle known as (Locality) lead to a contradiction. To streamline the argument, let’s first write down some propositions that will be used later. The first is the key assumption in the argument; the latter three are to be derived.

(Locality) If the space-time regions A and B are space-like separated, then events in A cannot influence events in B. (Assumed)

(Realism) The outcome of every quantum experiment is pre-determined by some variable λ. (Derived)

(Deterministic Constraint) The pre-determined value of the spin-observable of particle 1 is the opposite of the pre-determined value of the same spin-observable of particle 2.   [This is structurally the same as (Opposite) in Alice’s story.] (Derived)

(Statistical Constraint) For two incompatible spin-observables X and Y, exactly 25% of all pairs of entangled electrons e1 and e2 are such that the value of X exemplified by e1 is opposite the value of Y exemplified by e2.   [This is structurally the same as (Correlation) in Alice’s story.] (Derived)

(Here we assume that experiments have definite outcomes. Defenders of the Many-World Interpretation deny that. Can they save (Locality) here? Since it’s a tricky interpretive issue, we leave it to another time.)

Bell argues that we should reject (Locality). His argument proceeds in two parts:

Part I. Quantum mechanics and (Locality) jointly imply (Realism).
Part II. Quantum mechanics and (Realism) jointly imply (Deterministic Constraint) and (Statistical Constraint), which jointly imply a contradiction.

Putting the two parts together: quantum mechanics and (Locality) jointly imply a contradiction. Therefore, one of the two must be dropped. Given that the quantum predictions have been confirmed, Bell suggests dropping (Locality).

The argument in Part II is structurally the same as Alice’s derivation of a contradiction in the impossible ant story. If you followed Alice’s reasoning there, you can draw a similar table and reconstruct Bell’s argument. The argument in Part I derives from a famous paper by Einstein, Podolsky, and Rosen (EPR). Bell briefly mentioned it in an (often ignored) early paragraph in his paper.

Many physicists and philosophers have proposed ways of avoiding the contradiction between quantum mechanics and (Locality). Some suggest giving up (Realism): that doesn’t actually help, though, since (Realism) is derived rather than assumed. Some suggest giving up classical probability theory: that doesn’t help either, since the proof is based on empirical frequencies that obviously obey the classical axioms.

We suggest following Bell: (Locality) must go. Actual observations of quantum mechanical phenomenon show that events separated by arbitrarily vast distances can, in a certain precise sense, influence each other.

Quantum mechanics and (Locality) jointly imply a contradiction. Since quantum mechanics has been empirically confirmed to an extremely high degree, we have an empirical refutation of (Locality). Nature itself is non-local.

Moreover, we can learn quite a lot from ants.

(Many thanks to Tim Maudlin and Hans Westman for their feedback on Facebook that led to the edited version. Eddy Chen would like to thank the ANU philosophers, especially Alan Hájek, for the generous hospitality and helpful discussions.)

References:

Albert, David Z. (1992). Quantum Mechanics and Experience, Cambridge, MA: Harvard University Press.

Bell, John S. (1964). “On the Einstein Podolsky Rosen Paradox,” Physics 1 (3): 195–200.

Goldstein, Sheldon, Travis Norsen, Daniel Tausk, and Nino Zanghi (2011). “Bell’s Theorem,” in Scholarpedia, 6(10): 8378.

Maudlin, Tim (2011) Quantum Non-Locality and Relativity (Third Edition), Wiley-Blackwell.

Tumulka, Roderich (2016) “The Assumptions of Bell’s Proof,” in Mary Bell and Shan Gao (eds), Quantum Nonlocality and Reality: 50 Years of Bell’s Theorem, Cambridge, UK: Cambridge University Press.


EddyEddy Keming Chen is a Ph.D student in the department of philosophy at Rutgers University, New Brunswick. Recently he’s been thinking and writing about the metaphysics of the quantum wave function (forthcoming in The Journal of Philosophy), the connection between time’s arrow and self-locating (de se) uncertainty, the applicability of mathematics in the physical world, as well as infinite values and infinitarian paralysis in decision theory. 

 

Isaac

Isaac Wilhelm is a Ph.D. student in the philosophy department at Rutgers University, where he is also pursuing an M.S. in mathematics. He works on the structures of quantum theories, intrinsic properties in quantum mechanics, spacetime structure, biological mechanisms, and feminist philosophy (forthcoming in Philosophical Studies).