Fast Tests for Probing the Causal Structure of Quantum Processes
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other by a quantum process (timelike configuration, corresponding to a cause-effect
relation directed from one system to the other). Remarkably, the authors found out
that certain quantum correlations can distinguish between these two situations, thus
defying the classical motto “correlation does not imply causation.”
The fact that quantum correlations can be witnesses of causal relationships
suggests that quantum measurements could offer more powerful ways to identify
causal relations compared to their classical counterpart. In a recent work [19], Chiribella and Ebler showed that quantum features like entanglement and superposition
can lead to speedups in the identification of various types of causal relations. In
particular, they addressed the following question: given a variable, which variable
out of a list of candidate variables, is the effect of it? For simplicity, they focussed
on the basic scenario where the cause-effect relation is induced by a reversible
process. Classically, it turns out that the minimum probability of error is p C
err (N) =
1/[2d N −1 ], where d is the dimension of the quantum systems associated with the
given variables, and N is the number of times the variables are probed. This means
that the error probability decays exponentially as p C
err (N) ≈ 2 −R C N , with decay rate
R C = log 2 d. In stark contrast, Chiribella and Ebler showed that the error probability
for quantum strategies decays quadratically faster, with an exponential decay rate
equal to 2 log 2 d. Similar advantages arise in the task of identifying the presence of
a causal link between two variables, and in the task of identifying the cause of a
given effect.
This paper reviews the framework and the results of Ref. [19] in a way that
is suitable for non-specialists. The paper is organized as follows: in Sect. 2, we
provide preliminary notions that will be used later in the paper. In Sect. 3, we
review the problem of identifying the effect of a given variable, and we provide
the minimum error probability and its decay rate for classical strategies. Then, we
provide quantum strategies that achieve a speedup over their classical counterpart
(Sect. 4). In Sect. 5, we generalize the above results for the case of multiple
hypotheses and provide some applications. Finally, we conclude in Sect. 6 by
discussing directions of future research.
2 Preliminaries
In this section we review and expand the framework of Ref. [19], providing some
additional definitions that help clarifying the different types of causal relations
induced by quantum processes.
2.1 Notation
For a given Hilbert space H, we denote by B(H) the space of bounded operators on
H, by T (H) the space of trace-class operators, and by St(H) := {ρ ∈ T (H) | ρ
0 , tr[ρ] = 1} the convex set of density operators. In the context of causal inference,
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other by a quantum process (timelike configuration, corresponding to a cause-effect
relation directed from one system to the other). Remarkably, the authors found out
that certain quantum correlations can distinguish between these two situations, thus
defying the classical motto “correlation does not imply causation.”
The fact that quantum correlations can be witnesses of causal relationships
suggests that quantum measurements could offer more powerful ways to identify
causal relations compared to their classical counterpart. In a recent work [19], Chiribella and Ebler showed that quantum features like entanglement and superposition
can lead to speedups in the identification of various types of causal relations. In
particular, they addressed the following question: given a variable, which variable
out of a list of candidate variables, is the effect of it? For simplicity, they focussed
on the basic scenario where the cause-effect relation is induced by a reversible
process. Classically, it turns out that the minimum probability of error is p C
err (N) =
1/[2d N −1 ], where d is the dimension of the quantum systems associated with the
given variables, and N is the number of times the variables are probed. This means
that the error probability decays exponentially as p C
err (N) ≈ 2 −R C N , with decay rate
R C = log 2 d. In stark contrast, Chiribella and Ebler showed that the error probability
for quantum strategies decays quadratically faster, with an exponential decay rate
equal to 2 log 2 d. Similar advantages arise in the task of identifying the presence of
a causal link between two variables, and in the task of identifying the cause of a
given effect.
This paper reviews the framework and the results of Ref. [19] in a way that
is suitable for non-specialists. The paper is organized as follows: in Sect. 2, we
provide preliminary notions that will be used later in the paper. In Sect. 3, we
review the problem of identifying the effect of a given variable, and we provide
the minimum error probability and its decay rate for classical strategies. Then, we
provide quantum strategies that achieve a speedup over their classical counterpart
(Sect. 4). In Sect. 5, we generalize the above results for the case of multiple
hypotheses and provide some applications. Finally, we conclude in Sect. 6 by
discussing directions of future research.
2 Preliminaries
In this section we review and expand the framework of Ref. [19], providing some
additional definitions that help clarifying the different types of causal relations
induced by quantum processes.
2.1 Notation
For a given Hilbert space H, we denote by B(H) the space of bounded operators on
H, by T (H) the space of trace-class operators, and by St(H) := {ρ ∈ T (H) | ρ
0 , tr[ρ] = 1} the convex set of density operators. In the context of causal inference,