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G. Chiribella and Swati
quantum systems are often called quantum variables. We will denote quantum
systems by Roman letters, such as A, B, . . . , and the corresponding Hilbert spaces
as H A , H B , . . . , respectively. We will use the shorthand notation St(A) := St(H A ).
Let A and B be two quantum systems, and let H A and H B be the corresponding
Hilbert spaces. We will denote by A ⊗ B the composite system consisting of
subsystems A and B, corresponding to the tensor product Hilbert space H A⊗B =
H A ⊗ H B . The partial trace over the Hilbert space H A will be denoted as tr A .
A quantum process with input A and output B is described by a linear, completely
positive, trace-preserving map C : T (H A ) → T (H B ), mapping input states
ρ ∈ St(A) into output states C(ρ) ∈ St(B). Linear, completely positive and tracepreserving maps will be sometimes abbreviated as CPTP maps. The set of CPTP
maps from T (H A ) to T (H B ) will be denoted by CPTP(A → B).
2.2 Cause-Effect Relations Induced by Quantum Processes
Let A and B be two quantum systems.
Definition 1 We say that a process C ∈ CPTP(A → B) induces a cause-effect
relation from A to B if and only if C is non-constant. If this is the case, then we say
that A is a cause for B, and that B is an effect of A.
In other words, a process induces a causal relation from A to B if and only if
changing the state of system A can induce changes in the state of system B. The
ability to induce changes serves as a witness of the fact that A is a cause for B.
Definition 1 can be generalized to processes involving multiple inputs and
outputs.
Definition 2 We say that a bipartite process D ∈ CPTP((A ⊗ A ) → (B ⊗ B ))
induces a cause-effect relation from A to B if and only if there exists at least one
state α ∈ StH (A ) such that the reduced process
D α ∈ CPTP(A → B) ,
ρ → tr B [D(ρ ⊗ α)]
(1)
is non-constant. If this is the case, then we say that A is a cause for B, and that B is
an effect of A.
Note that a process C ∈ CPTP(A → B) may not induce a cause-effect relation
from A to B, but may still be the reduced process of some other process D ∈
CPTP((A ⊗ A ) → (B ⊗ B )) that does induce a cause-effect relation from A
to B. This observation shows that the presence of a cause-effect relation, as defined
in Definitions 1 and 2, is a property of the process under consideration, and not
just of the variables A and B. In other words, a cause-effect relation that is actually
present may not be detected by inspecting the process from A to B alone.
The notion of “process inducing a cause-effect relation” provided in Definitions 1
and 2 is rather weak, because it allows the influence of the cause on the effect to be
G. Chiribella and Swati
quantum systems are often called quantum variables. We will denote quantum
systems by Roman letters, such as A, B, . . . , and the corresponding Hilbert spaces
as H A , H B , . . . , respectively. We will use the shorthand notation St(A) := St(H A ).
Let A and B be two quantum systems, and let H A and H B be the corresponding
Hilbert spaces. We will denote by A ⊗ B the composite system consisting of
subsystems A and B, corresponding to the tensor product Hilbert space H A⊗B =
H A ⊗ H B . The partial trace over the Hilbert space H A will be denoted as tr A .
A quantum process with input A and output B is described by a linear, completely
positive, trace-preserving map C : T (H A ) → T (H B ), mapping input states
ρ ∈ St(A) into output states C(ρ) ∈ St(B). Linear, completely positive and tracepreserving maps will be sometimes abbreviated as CPTP maps. The set of CPTP
maps from T (H A ) to T (H B ) will be denoted by CPTP(A → B).
2.2 Cause-Effect Relations Induced by Quantum Processes
Let A and B be two quantum systems.
Definition 1 We say that a process C ∈ CPTP(A → B) induces a cause-effect
relation from A to B if and only if C is non-constant. If this is the case, then we say
that A is a cause for B, and that B is an effect of A.
In other words, a process induces a causal relation from A to B if and only if
changing the state of system A can induce changes in the state of system B. The
ability to induce changes serves as a witness of the fact that A is a cause for B.
Definition 1 can be generalized to processes involving multiple inputs and
outputs.
Definition 2 We say that a bipartite process D ∈ CPTP((A ⊗ A ) → (B ⊗ B ))
induces a cause-effect relation from A to B if and only if there exists at least one
state α ∈ StH (A ) such that the reduced process
D α ∈ CPTP(A → B) ,
ρ → tr B [D(ρ ⊗ α)]
(1)
is non-constant. If this is the case, then we say that A is a cause for B, and that B is
an effect of A.
Note that a process C ∈ CPTP(A → B) may not induce a cause-effect relation
from A to B, but may still be the reduced process of some other process D ∈
CPTP((A ⊗ A ) → (B ⊗ B )) that does induce a cause-effect relation from A
to B. This observation shows that the presence of a cause-effect relation, as defined
in Definitions 1 and 2, is a property of the process under consideration, and not
just of the variables A and B. In other words, a cause-effect relation that is actually
present may not be detected by inspecting the process from A to B alone.
The notion of “process inducing a cause-effect relation” provided in Definitions 1
and 2 is rather weak, because it allows the influence of the cause on the effect to be