622
G. Chiribella and Swati
In this paper, we will focus on situations where the cause-effect relation between
two variables is reversible. In this case, the presence of a cause-effect relation
between variables A and B rules out the possibility of any cause-effect relation from
A to any other variable B that is independent of B. More precisely, if a process
D ∈ CPTP(A → B ⊗ B ) is such that its reduced process C = (I B ⊗ tr B ) ◦ D is
reversible, then D must be of the form D = C ⊗ β, where β ∈ St(B ) is some fixed
state of system B .
2.3 Discrimination of Causal Hypotheses
Consider the situation where an experimenter has access to a black box, implementing a quantum process with a given set of input systems and a given set of output
systems. The goal of the experimenter is to figure out the causal relations among
the systems involved in the process. For example, the black box could implement a
process with input system A and output system B, and the experimenter may want
to figure out whether the process induces a cause-effect relation.
In general, we will assume that the black box is guaranteed to satisfy one, and
only one, of k possible hypotheses (H i ) k
i=1 about the cause-effect relations occurring
between its inputs and outputs. No further information about the process is available
to the experimenter. To figure out which hypothesis is correct, the experimenter will
set up a test designed to probe the causal relations.
We will use the term causal hypotheses for hypotheses on the causal relations
induced by a given process. The problem of distinguishing between alternative
causal hypotheses will be called discrimination of causal hypotheses. To illustrate
the problem, we will focus on one basic instance: discover which of two variables
B and C is the effect of a given variable A. More specifically, we consider the
following alternative hypotheses:
H 1 : B is a causal intermediary for A, and C is uniformly random,
H 2 : C is a causal intermediary for A, and B is uniformly random.
For simplicity, we will assume that systems A, B, and C have the same dimension,
equal to d < ∞. In terms of the process C ∈ CPTP(A → (B ⊗ C)) the two
hypotheses correspond to the following statements:
H 1 : C has the form C(·) = U · U † ⊗ I/d, for some unitary operator U ∈ B(C d ),
H 2 : C has the form C(·) = I/d ⊗ V · V † , for some unitary operator V ∈ B(C d ).
In either case, the unitary operators U and V are unknown to the experimenter.
In general, every hypothesis on the causal relations between inputs and outputs
is in one-to-one correspondence with a subset of CPTP maps. In the above example,
the two hypotheses H 1 and H 2 correspond to the sets
H 1 =
U · U
†
⊗ I/d | U ∈ B(C
d ) , U
† U = UU
†
= I
(5)
G. Chiribella and Swati
In this paper, we will focus on situations where the cause-effect relation between
two variables is reversible. In this case, the presence of a cause-effect relation
between variables A and B rules out the possibility of any cause-effect relation from
A to any other variable B that is independent of B. More precisely, if a process
D ∈ CPTP(A → B ⊗ B ) is such that its reduced process C = (I B ⊗ tr B ) ◦ D is
reversible, then D must be of the form D = C ⊗ β, where β ∈ St(B ) is some fixed
state of system B .
2.3 Discrimination of Causal Hypotheses
Consider the situation where an experimenter has access to a black box, implementing a quantum process with a given set of input systems and a given set of output
systems. The goal of the experimenter is to figure out the causal relations among
the systems involved in the process. For example, the black box could implement a
process with input system A and output system B, and the experimenter may want
to figure out whether the process induces a cause-effect relation.
In general, we will assume that the black box is guaranteed to satisfy one, and
only one, of k possible hypotheses (H i ) k
i=1 about the cause-effect relations occurring
between its inputs and outputs. No further information about the process is available
to the experimenter. To figure out which hypothesis is correct, the experimenter will
set up a test designed to probe the causal relations.
We will use the term causal hypotheses for hypotheses on the causal relations
induced by a given process. The problem of distinguishing between alternative
causal hypotheses will be called discrimination of causal hypotheses. To illustrate
the problem, we will focus on one basic instance: discover which of two variables
B and C is the effect of a given variable A. More specifically, we consider the
following alternative hypotheses:
H 1 : B is a causal intermediary for A, and C is uniformly random,
H 2 : C is a causal intermediary for A, and B is uniformly random.
For simplicity, we will assume that systems A, B, and C have the same dimension,
equal to d < ∞. In terms of the process C ∈ CPTP(A → (B ⊗ C)) the two
hypotheses correspond to the following statements:
H 1 : C has the form C(·) = U · U † ⊗ I/d, for some unitary operator U ∈ B(C d ),
H 2 : C has the form C(·) = I/d ⊗ V · V † , for some unitary operator V ∈ B(C d ).
In either case, the unitary operators U and V are unknown to the experimenter.
In general, every hypothesis on the causal relations between inputs and outputs
is in one-to-one correspondence with a subset of CPTP maps. In the above example,
the two hypotheses H 1 and H 2 correspond to the sets
H 1 =
U · U
†
⊗ I/d | U ∈ B(C
d ) , U
† U = UU
†
= I
(5)