Fast Tests for Probing the Causal Structure of Quantum Processes
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arbitrarily small. A stronger notion arises when the cause-effect relation is faithful,
in the following sense:
Definition 3 We say that a process C ∈ CPTP(A → B) induces a faithful causeeffect relation from A to B if and only if C is correctable, meaning that there exists
another process R ∈ CPTP(B → A) such that
R ◦ C = I A ,
(2)
where I A ∈ CPTP(A → A) is the identity process. When this is the case, we say
that B is a causal intermediary of A.
Intuitively, variable B is a causal intermediary for variable A if all the possible
causal influences of A can be reconstructed from B. In other words, every process
from A to a third variable B must factorize into the process from A to B, followed
by some process from B to B . This intuition is formalized by the following
proposition:
Proposition 1 The process C ∈ CPTP(A → B) induces a faithful cause-effect
relation from A to B if and only if, for every quantum system B , and for every
process E ∈ CPTP(A → B ) there exists a process D ∈ CPTP(B → B ) such that
E = D ◦ C .
(3)
Proof Suppose that C induces a faithful cause-effect relation. Then, let R ∈
CPTP(B → A) be the process that inverts C, namely R ◦ C = I A . Then, for
every system B and every process E ∈ CPTP(A → B ), one has E = D ◦ C, with
D := E ◦ R.
Conversely, suppose that, for every quantum system B and for every process
E ∈ CPTP(A → B ), there exists a process D ∈ CPTP(B → B ) such that
E = D ◦ C. In particular, one can pick B ≡ A and E ≡ I A , in which case the
condition E = D ◦ C implies that C is correctable, and therefore induces a faithful
cause-effect relation.
In general, the presence of a faithful cause-effect relation from A to B does not
imply that the causal influences of A propagate exclusively through B. For example,
quantum secret sharing protocols, such as those presented in Ref. [20], provide
examples of processes where a given cause can have multiple causal intermediaries.
The situation is much simpler when the cause-effect relation is reversible, in the
following sense:
Definition 4 We say that a process C ∈ CPTP(A → B) induces a reversible causeeffect relation between A and B if and only if C is reversible, meaning that there
exists another process R ∈ CPTP(B → A) such that
R ◦ C = I A
and
C ◦ R = I A ,
(4)
where I X ∈ CPTP(X → X) is the identity process on system X ∈ {A, B}.
621
arbitrarily small. A stronger notion arises when the cause-effect relation is faithful,
in the following sense:
Definition 3 We say that a process C ∈ CPTP(A → B) induces a faithful causeeffect relation from A to B if and only if C is correctable, meaning that there exists
another process R ∈ CPTP(B → A) such that
R ◦ C = I A ,
(2)
where I A ∈ CPTP(A → A) is the identity process. When this is the case, we say
that B is a causal intermediary of A.
Intuitively, variable B is a causal intermediary for variable A if all the possible
causal influences of A can be reconstructed from B. In other words, every process
from A to a third variable B must factorize into the process from A to B, followed
by some process from B to B . This intuition is formalized by the following
proposition:
Proposition 1 The process C ∈ CPTP(A → B) induces a faithful cause-effect
relation from A to B if and only if, for every quantum system B , and for every
process E ∈ CPTP(A → B ) there exists a process D ∈ CPTP(B → B ) such that
E = D ◦ C .
(3)
Proof Suppose that C induces a faithful cause-effect relation. Then, let R ∈
CPTP(B → A) be the process that inverts C, namely R ◦ C = I A . Then, for
every system B and every process E ∈ CPTP(A → B ), one has E = D ◦ C, with
D := E ◦ R.
Conversely, suppose that, for every quantum system B and for every process
E ∈ CPTP(A → B ), there exists a process D ∈ CPTP(B → B ) such that
E = D ◦ C. In particular, one can pick B ≡ A and E ≡ I A , in which case the
condition E = D ◦ C implies that C is correctable, and therefore induces a faithful
cause-effect relation.
In general, the presence of a faithful cause-effect relation from A to B does not
imply that the causal influences of A propagate exclusively through B. For example,
quantum secret sharing protocols, such as those presented in Ref. [20], provide
examples of processes where a given cause can have multiple causal intermediaries.
The situation is much simpler when the cause-effect relation is reversible, in the
following sense:
Definition 4 We say that a process C ∈ CPTP(A → B) induces a reversible causeeffect relation between A and B if and only if C is reversible, meaning that there
exists another process R ∈ CPTP(B → A) such that
R ◦ C = I A
and
C ◦ R = I A ,
(4)
where I X ∈ CPTP(X → X) is the identity process on system X ∈ {A, B}.