624
G. Chiribella and Swati
p
C
err (N) =
1
2d N −1 .
(7)
This implies the optimal decay rate is
R C = log d.
(8)
A result by Hayashi [22] implies that the decay rate cannot be improved by
considering more general sequential strategies such as those in Fig. 1.
4 Quantum Advantages
Classical random variables can be regarded as quantum systems that have lost
coherence with respect to a privileged orthonormal basis of “classical states.”
But what if the coherence is preserved? In the following we will show that the
possibility of probing processes through coherent superpositions of classical inputs
can enhance our ability to identify the correct causal hypothesis.
4.1 A Benefit of Quantum Coherence
The three classical random variables A, B, and C considered in the previous section
can be considered as the decohered versions of three quantum systems A, B, and
C with d-dimensional Hilbert spaces. Similarly, a permutation of the values of the
random variable A according to an element π of the symmetric group S(d) can
be regarded as the decohered version of the unitary permutation operator U π =
d
i=1 |π(i) where {|i d
i=1 is the orthonormal basis representing the classical
states.
In this scenario, the two “classical” causal hypotheses become
H 1 : C has the form C(·) = U π · U †
π ⊗ I/d, for some permutation π ∈ S(d),
H 2 : C has the form C(·) = I/d ⊗ U σ · U †
σ , for some permutation σ ∈ S(d).
To distinguish between these two hypotheses, the experimenter could prepare N
probes in the superposition state
|e 0 =
1
√
d
d−1
i=0
|i ,
(9)
which is invariant under permutations. Thus, the unknown process C yields either
|e 0 0 | ⊗
I
d
⊗N
or
I
d ⊗ |e 0 0 |
⊗N
depending on which causal hypothesis is
correct.
G. Chiribella and Swati
p
C
err (N) =
1
2d N −1 .
(7)
This implies the optimal decay rate is
R C = log d.
(8)
A result by Hayashi [22] implies that the decay rate cannot be improved by
considering more general sequential strategies such as those in Fig. 1.
4 Quantum Advantages
Classical random variables can be regarded as quantum systems that have lost
coherence with respect to a privileged orthonormal basis of “classical states.”
But what if the coherence is preserved? In the following we will show that the
possibility of probing processes through coherent superpositions of classical inputs
can enhance our ability to identify the correct causal hypothesis.
4.1 A Benefit of Quantum Coherence
The three classical random variables A, B, and C considered in the previous section
can be considered as the decohered versions of three quantum systems A, B, and
C with d-dimensional Hilbert spaces. Similarly, a permutation of the values of the
random variable A according to an element π of the symmetric group S(d) can
be regarded as the decohered version of the unitary permutation operator U π =
d
i=1 |π(i) where {|i d
i=1 is the orthonormal basis representing the classical
states.
In this scenario, the two “classical” causal hypotheses become
H 1 : C has the form C(·) = U π · U †
π ⊗ I/d, for some permutation π ∈ S(d),
H 2 : C has the form C(·) = I/d ⊗ U σ · U †
σ , for some permutation σ ∈ S(d).
To distinguish between these two hypotheses, the experimenter could prepare N
probes in the superposition state
|e 0 =
1
√
d
d−1
i=0
|i ,
(9)
which is invariant under permutations. Thus, the unknown process C yields either
|e 0 0 | ⊗
I
d
⊗N
or
I
d ⊗ |e 0 0 |
⊗N
depending on which causal hypothesis is
correct.