# Are locality and separability two distinct notions?

Is there any difference between locality and separability in quantum mechanics, or do they mean the same thing? It seems authors do not always agree.

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Here is my personal point of view about these different notions.

Locality / Causality / Setting independence

The notion of locality (or causality) means that local measurements cannot be influenced by distant systems, for instance by distant measurement apparatus.

Let Alice and Bob be $2$ spatially distant observers, they are doing local measurements with apparatus $a$ and $b$, and they obtain outcomes $A$ and $B$. There exist a joint probability for outcomes : $p(A, B |a,b;\lambda)$.

$\lambda$ is here a parameter which could be seen as a hypothetic additional parameter, used with a density $\rho(\lambda)$, that is : $p(A, B |a,b) =\int d\lambda ~\rho(\lambda)~p(A, B |a,b;\lambda)$

Locality means that if Alice makes a local measurement, the apparatus $b$ has no influence on it, it could be expressed as : $$p(A|a,b;\lambda) = p(A,a;\lambda)\tag{1}$$ where : $p(A|a,b;\lambda) = \sum_Bp(A, B |a,b;\lambda)$

Separability /Reality / Outcome independence

The notion of separability means that, if we consider a system made of $2$ sub-systems, the system is separable if we can attribute physical properties to each sub-system. Each sub-system has its own reality. But the sub-systems could be correlated. This can be written :

$$p(A|B,a;\lambda) = p(A,a;\lambda)\tag{2}$$

Locality + Reality

If you have a local realist theory, then you may show, with probabilities laws and the 2 above equations that :

$$p(A B|a,b;\lambda) = p(A,a;\lambda)p(B,b;\lambda)\tag{3}$$

This is precisely the context of classical theories.

Quantum mechanics

Quantum mechanics does not respect outcome independence $(2)$property, so, there is a quantum non-separability, or said it differently, quantum mechanics is incompatible with realism.

Of corse, quantum mechanics is compatible with locality, so it respects setting independence $(1)$

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Maybe I misread what you meant, but I'd like to add that causality and locality are not the same thing (though they are very much related). –  Malabarba Aug 8 '13 at 12:41
@BruceConnor : Right. Systems separared by a space-like interval cannot have a causal relation. So, if $b$ is not "local" to $a$, you cannot have a causality relation (at equal times) between $a$ and $b$. People often use the term non-local for quantum mechanics, and they (incorrectly) think about a instantaneous information speed, and so a instantaneous possible causality between space-like separated events, which is false, of corse. –  Trimok Aug 8 '13 at 16:56

There is no general agreement on these terms.

Bell in 1964 did seem to use 'locality' as Trimok answered, $p(A|a,b,\lambda) = p(A|a,\lambda)$.

Jarret in 1984 explicitly defined it this way.

Bell in 1964 used 'separability' interchangeably with 'locality'.

Don Howard in 1985 used 'separability' to mean outcome independence as Trimok answered, but personally this usage doesn't make sense to me.

It is not at all standard to use 'causality' and 'locality' interchangeably.

Here are a list of similar concepts, but the details depend on the author:

locality ~ separability ~ setting independence = parameter independence

causality ~ determinism ~ hidden variables ~ outcome independence

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