112
7 Device-Independent Quantum Cryptography
and similarly one gets
p(b|x, y) = p(b|y).
(7.24)
Note that another way to write the no-signaling constraint in (7.23) is
b p(a, b|
x, y) =
b p(a, b|x, y
) for every a, x, y and y
. The no-signaling constraints state
that the probability distribution of the outcomes of one party is independent of the
inputs of the other party.
The justification of the no-signaling constraints for spacelike separated parties
comes from the causality constraint of special relativity, according to which one
party cannot communicate with another party by sending a superluminal signal, i.e.
a signal that travels faster than the speed of light. Indeed, the constraints on the
probability distributions of a two-party Bell scenario imposed by causality coincide
with (7.23) and (7.24).
The no-signaling constraints are generalized as follows in an N -party Bell scenario
[30]:
a j
p(a 1 , . . . , a j , . . . , a N |x 1 , . . . , x j , . . . , x N )
=
a j
p(a 1 , . . . , a j , . . . , a N |x 1 , . . . , x
j , . . . , x N )
(7.25)
∀ j ∈ {1, . . . , N }, {a 1 , . . . , a N } \ {a j }, {x 1 , . . . , x j , x
j , . . . , x N },
stating that the probability distribution of the outcomes of any subset of parties is
independent of the inputs of the complementary set of parties. Note that the constraints in (7.25) explicitly express this statement only for subsets of one party each,
but the general statement can be deduced from (7.25) [22].
Interestingly, the multiparty no-signaling constraints (7.25) do not precisely capture the causality constraints for certain configurations of parties in the Minkowski
spacetime (when N ≥ 3). In other words, one can arrange the parties in spacetime
such that the constraints on their probability distributions purely derived from causality are a strict subset of the no-signaling constraints in (7.25) [22]. By labelling N S
the set of all possible correlations obeying the no-signaling constraints (7.25) and
analogously R the set of correlations obeying the causality constraints of relativity,
we have that: N S ⊂ R. We remark that this situation occurs only when the Bell
scenario is composed of N ≥ 3 parties, while for N = 2 we have that N S ≡ R as
stated above.
Considering again the bipartite Bell scenario, the set Q of quantum correlations
is defined by those probability distributions that can be expressed as:
p(a, b|x, y) = Tr
ρ AB M a|x ⊗ M b|y
,
(7.26)
where ρ AB is a quantum state on the joint Hilbert space H A ⊗ H B and M a|x , M b|y
are POVM elements (c.f. Sect. 2.2) relative to outcomes a, b given the inputs x, y.
7 Device-Independent Quantum Cryptography
and similarly one gets
p(b|x, y) = p(b|y).
(7.24)
Note that another way to write the no-signaling constraint in (7.23) is
b p(a, b|
x, y) =
b p(a, b|x, y
) for every a, x, y and y
. The no-signaling constraints state
that the probability distribution of the outcomes of one party is independent of the
inputs of the other party.
The justification of the no-signaling constraints for spacelike separated parties
comes from the causality constraint of special relativity, according to which one
party cannot communicate with another party by sending a superluminal signal, i.e.
a signal that travels faster than the speed of light. Indeed, the constraints on the
probability distributions of a two-party Bell scenario imposed by causality coincide
with (7.23) and (7.24).
The no-signaling constraints are generalized as follows in an N -party Bell scenario
[30]:
a j
p(a 1 , . . . , a j , . . . , a N |x 1 , . . . , x j , . . . , x N )
=
a j
p(a 1 , . . . , a j , . . . , a N |x 1 , . . . , x
j , . . . , x N )
(7.25)
∀ j ∈ {1, . . . , N }, {a 1 , . . . , a N } \ {a j }, {x 1 , . . . , x j , x
j , . . . , x N },
stating that the probability distribution of the outcomes of any subset of parties is
independent of the inputs of the complementary set of parties. Note that the constraints in (7.25) explicitly express this statement only for subsets of one party each,
but the general statement can be deduced from (7.25) [22].
Interestingly, the multiparty no-signaling constraints (7.25) do not precisely capture the causality constraints for certain configurations of parties in the Minkowski
spacetime (when N ≥ 3). In other words, one can arrange the parties in spacetime
such that the constraints on their probability distributions purely derived from causality are a strict subset of the no-signaling constraints in (7.25) [22]. By labelling N S
the set of all possible correlations obeying the no-signaling constraints (7.25) and
analogously R the set of correlations obeying the causality constraints of relativity,
we have that: N S ⊂ R. We remark that this situation occurs only when the Bell
scenario is composed of N ≥ 3 parties, while for N = 2 we have that N S ≡ R as
stated above.
Considering again the bipartite Bell scenario, the set Q of quantum correlations
is defined by those probability distributions that can be expressed as:
p(a, b|x, y) = Tr
ρ AB M a|x ⊗ M b|y
,
(7.26)
where ρ AB is a quantum state on the joint Hilbert space H A ⊗ H B and M a|x , M b|y
are POVM elements (c.f. Sect. 2.2) relative to outcomes a, b given the inputs x, y.