7.1 Bell’s Theorem
111
The last expression has the merit to show that it is optimal for the parties to choose
measurements such that the resulting CHSH expression, after being simplified, is
exclusively composed of correlators (expectation values) of stabilizers of the state
|
+
. This makes sense, since by definition the correlator of a stabilizer evaluated
on the stabilized state achieves the maximum value of 1.
The CHSH violation predicted by quantum theory has profound consequences,
as it implies that one of the three assumptions (7.3), (7.4) and (7.5) that led to the
derivation of the CHSH inequality does not hold for quantum theory. The quantum
theory we consider is the standard non-relativistic quantum theory, which describes
quantum systems and measurements in terms of tensor-product Hilbert spaces and
local Kraus operators acting on the corresponding Hilbert space. These features,
combined with the partial trace rule, ensure that the local statistics of a system only
depend on its reduced density operator. Therefore, no superluminal communication is
allowed between parties and in particular the conditions of parameter independence
(7.4) and of free will (7.5) are satisfied.
From the above argument, we conclude that quantum mechanics is a non-local
theory, i.e. it does not satisfy the locality assumption in (7.3), and its predictions
cannot be reproduced by any local theory. This concludes the proof of Bell’s theorem.
Another important consequence of Bell’s theorem are Bell inequalities. These can
seen as means to experimentally test if nature behaves according to local theories or
not. The numerous experiments demonstrating the violation of Bell inequalities have
proved beyond reasonable doubt that nature is non-local. In particular, we mention
the first successful results in this direction by Aspect et al. [26]. Recent and more
sophisticated experiments have confirmed the existence of non-local correlations in
loophole-free Bell tests [27–29], i.e. conducted without making any assumption that
could lead to a description of the non-local correlation through an LHV model.
7.2 Local, Quantum, No-Signaling and Causal Correlations
In this Section we wish to clarify the relations existing between different types of
correlations, starting from what we have seen in the proof of Bell’s theorem.
First of all, we can derive the so called no-signaling constraints [21, 22] from the
parameter-independence (7.4) and free-will (7.5) assumptions used in Bell’s theorem:
p(a|x, y) =
dλ p(a|x, y, λ)p(λ|x, y)
(7.5)
=
dλ p(a|x, y, λ)p(λ)
(7.4)
=
dλ p(a|x, λ)p(λ)
= p(a|x),
(7.23)
111
The last expression has the merit to show that it is optimal for the parties to choose
measurements such that the resulting CHSH expression, after being simplified, is
exclusively composed of correlators (expectation values) of stabilizers of the state
|
+
. This makes sense, since by definition the correlator of a stabilizer evaluated
on the stabilized state achieves the maximum value of 1.
The CHSH violation predicted by quantum theory has profound consequences,
as it implies that one of the three assumptions (7.3), (7.4) and (7.5) that led to the
derivation of the CHSH inequality does not hold for quantum theory. The quantum
theory we consider is the standard non-relativistic quantum theory, which describes
quantum systems and measurements in terms of tensor-product Hilbert spaces and
local Kraus operators acting on the corresponding Hilbert space. These features,
combined with the partial trace rule, ensure that the local statistics of a system only
depend on its reduced density operator. Therefore, no superluminal communication is
allowed between parties and in particular the conditions of parameter independence
(7.4) and of free will (7.5) are satisfied.
From the above argument, we conclude that quantum mechanics is a non-local
theory, i.e. it does not satisfy the locality assumption in (7.3), and its predictions
cannot be reproduced by any local theory. This concludes the proof of Bell’s theorem.
Another important consequence of Bell’s theorem are Bell inequalities. These can
seen as means to experimentally test if nature behaves according to local theories or
not. The numerous experiments demonstrating the violation of Bell inequalities have
proved beyond reasonable doubt that nature is non-local. In particular, we mention
the first successful results in this direction by Aspect et al. [26]. Recent and more
sophisticated experiments have confirmed the existence of non-local correlations in
loophole-free Bell tests [27–29], i.e. conducted without making any assumption that
could lead to a description of the non-local correlation through an LHV model.
7.2 Local, Quantum, No-Signaling and Causal Correlations
In this Section we wish to clarify the relations existing between different types of
correlations, starting from what we have seen in the proof of Bell’s theorem.
First of all, we can derive the so called no-signaling constraints [21, 22] from the
parameter-independence (7.4) and free-will (7.5) assumptions used in Bell’s theorem:
p(a|x, y) =
dλ p(a|x, y, λ)p(λ|x, y)
(7.5)
=
dλ p(a|x, y, λ)p(λ)
(7.4)
=
dλ p(a|x, λ)p(λ)
= p(a|x),
(7.23)