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G. Chiribella and Swati
(a permutation in the classical case, or a unitary gate in the quantum case) and that
all the variables have sample space of cardinality d in the classical case, or Hilbert
space of dimension d in the quantum case.
Classically, the problem is to find the random variable A i such that B is a
permutation of A i , with i ∈ {1, · · · , m}. In the simplest case, when the permutation
is known, the cause can be determined without error by interrogating the unknown
process log d (m) times.
In the quantum case, Ref. [19] showed that, if the unitary operator inducing the
causal relation is known, then there exists a test that achieves error probability
p err (N) =
m − 1
d 2N + m − 1
.
(24)
This means that one can get an error probability smaller than any desired > 0 by
using N = =(1 + )(log d m)/2 interrogations, which is approximately half of the
number of classical interrogations.
In the case where the dependency between cause and effect is unknown (arbitrary
permutation in the classical case, or arbitrary unitary operator in the quantum case),
the analysis is more complex. However, it is still possible to show that quantum
strategies can identify the cause using only N = =(1+)(log d m)/2 interrogations.
6 Discussion and Conclusions
In this paper, we reviewed the framework and the results of Ref. [19], which showed
that quantum features such as coherence and entanglement offer advantages in
detecting cause-effect relations induced by reversible processes.
In the problem of identifying the effect of a given variable, it was shown that
entanglement between the probes and an external reference system can double the
rate at which the error probability decays. For classical random variables with d
possible values, the decay rate is log d. For quantum systems of dimension d, the
decay rate is 2 log d. As it turns out, the value 2 log d is the ultimate limit posed
by quantum theory to the problem of identifying the causal intermediary of a given
variable.
Interestingly, both the classical and quantum decay rates can be expressed as
log dim St R (A) where St R (A) is the vector space spanned by linear combinations
of states of system A. For classical systems, the states are probability distributions
over the sample space, and the dimension of the corresponding vector space is d.
For quantum systems, the states are density matrices, and the dimension of the
vector space is d 2 . It would be interesting to study the problem of causal hypothesis
discrimination in toy theories with higher dimensional state spaces, such as quantum
theory on quaternionic Hilbert spaces [28] or the quartic toy theory proposed by
˙
Zyczkowski in Ref. [29].
G. Chiribella and Swati
(a permutation in the classical case, or a unitary gate in the quantum case) and that
all the variables have sample space of cardinality d in the classical case, or Hilbert
space of dimension d in the quantum case.
Classically, the problem is to find the random variable A i such that B is a
permutation of A i , with i ∈ {1, · · · , m}. In the simplest case, when the permutation
is known, the cause can be determined without error by interrogating the unknown
process log d (m) times.
In the quantum case, Ref. [19] showed that, if the unitary operator inducing the
causal relation is known, then there exists a test that achieves error probability
p err (N) =
m − 1
d 2N + m − 1
.
(24)
This means that one can get an error probability smaller than any desired > 0 by
using N = =(1 + )(log d m)/2 interrogations, which is approximately half of the
number of classical interrogations.
In the case where the dependency between cause and effect is unknown (arbitrary
permutation in the classical case, or arbitrary unitary operator in the quantum case),
the analysis is more complex. However, it is still possible to show that quantum
strategies can identify the cause using only N = =(1+)(log d m)/2 interrogations.
6 Discussion and Conclusions
In this paper, we reviewed the framework and the results of Ref. [19], which showed
that quantum features such as coherence and entanglement offer advantages in
detecting cause-effect relations induced by reversible processes.
In the problem of identifying the effect of a given variable, it was shown that
entanglement between the probes and an external reference system can double the
rate at which the error probability decays. For classical random variables with d
possible values, the decay rate is log d. For quantum systems of dimension d, the
decay rate is 2 log d. As it turns out, the value 2 log d is the ultimate limit posed
by quantum theory to the problem of identifying the causal intermediary of a given
variable.
Interestingly, both the classical and quantum decay rates can be expressed as
log dim St R (A) where St R (A) is the vector space spanned by linear combinations
of states of system A. For classical systems, the states are probability distributions
over the sample space, and the dimension of the corresponding vector space is d.
For quantum systems, the states are density matrices, and the dimension of the
vector space is d 2 . It would be interesting to study the problem of causal hypothesis
discrimination in toy theories with higher dimensional state spaces, such as quantum
theory on quaternionic Hilbert spaces [28] or the quartic toy theory proposed by
˙
Zyczkowski in Ref. [29].