Fast Tests for Probing the Causal Structure of Quantum Processes
625
The probability of error can be computed by taking advantage of the symmetry
of the problem, which implies that the worst case error probability is equal to the
average error probability when the prior probability for the two hypotheses H 1 and
H 2 is uniform. In this case, Helstrom’s theorem [23] states that the minimum error
probability in distinguishing between the states ρ 1 and ρ 2 , given with uniform a
priori probability, is
p err,ave (ρ 1 , ρ 2 ) =
1 −
1
2 ρ 1 − ρ 2 1
2
,
(10)
where 1 := tr
√
G † G denotes the trace norm of a generic trace-class operator
G ∈ T (H). Applied to our problem, Helstrom’s theorem yields the error probability
p
coh
err =
1
2d N .
(11)
By comparison with the classical error probability (7), we can see that coherence
provides a reduction of the error probability by a factor d. This is a relatively
minor improvement, as it does not affect the decay rate. Still, it is an interesting
first illustration of how quantum effects can affect the discrimination of causal
hypotheses.
4.2 General Quantum Scenario
Let us move on now to the general quantum scenario, where the relation between
cause and effect is described by an arbitrary unitary operator (not necessarily a
permutation operator, as in the previous subsection).
At first, this problem appears to be much more challenging for our experimenter,
since the dependence between cause and effect can be any arbitrary element of the
special unitary group SU(d). Surprisingly, however, the same error probability as in
Eq. (11) can be achieved.
Let us consider the case where N is an integer multiple of d, say N = d t for
some non-negative integer t. In this case, a universal quantum strategy with error
probability (11) is to divide the N probes into groups of d, and to prepare each
group of probes in SU(d) singlet state
|S d =
1
√
d!
k 1 ,··· ,k d
k 1 ···k d |k 1 , · · · , k d ,
(12)
where k 1 ···k d represents the totally antisymmetric tensor and the summation extends
over all the vectors in the computational basis.
Each of the d particles in each group is then fed into one use of the unknown
process. Since the singlet state is invariant under unitary transformations, the
625
The probability of error can be computed by taking advantage of the symmetry
of the problem, which implies that the worst case error probability is equal to the
average error probability when the prior probability for the two hypotheses H 1 and
H 2 is uniform. In this case, Helstrom’s theorem [23] states that the minimum error
probability in distinguishing between the states ρ 1 and ρ 2 , given with uniform a
priori probability, is
p err,ave (ρ 1 , ρ 2 ) =
1 −
1
2 ρ 1 − ρ 2 1
2
,
(10)
where 1 := tr
√
G † G denotes the trace norm of a generic trace-class operator
G ∈ T (H). Applied to our problem, Helstrom’s theorem yields the error probability
p
coh
err =
1
2d N .
(11)
By comparison with the classical error probability (7), we can see that coherence
provides a reduction of the error probability by a factor d. This is a relatively
minor improvement, as it does not affect the decay rate. Still, it is an interesting
first illustration of how quantum effects can affect the discrimination of causal
hypotheses.
4.2 General Quantum Scenario
Let us move on now to the general quantum scenario, where the relation between
cause and effect is described by an arbitrary unitary operator (not necessarily a
permutation operator, as in the previous subsection).
At first, this problem appears to be much more challenging for our experimenter,
since the dependence between cause and effect can be any arbitrary element of the
special unitary group SU(d). Surprisingly, however, the same error probability as in
Eq. (11) can be achieved.
Let us consider the case where N is an integer multiple of d, say N = d t for
some non-negative integer t. In this case, a universal quantum strategy with error
probability (11) is to divide the N probes into groups of d, and to prepare each
group of probes in SU(d) singlet state
|S d =
1
√
d!
k 1 ,··· ,k d
k 1 ···k d |k 1 , · · · , k d ,
(12)
where k 1 ···k d represents the totally antisymmetric tensor and the summation extends
over all the vectors in the computational basis.
Each of the d particles in each group is then fed into one use of the unknown
process. Since the singlet state is invariant under unitary transformations, the