IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 77

Sorelli et al.
ðw1r1 w0r0Þþ, where we have introduced the notation
Aþ for the positive part of the operator A, namely A
restricted to the subspace corresponding to its positive
eigenvectors. The latter can be expressed in term of the
operator absolute value jAj¼ ðAAÞ1=2
as Aþ ¼
ðjAjþ AÞ=2, therefore, we can write
Pe ¼
1 trjw1r1 w0r0j
2
:
(45)
We, therefore, have that the projector on the positive part
of w1r1 w0r0 is the quantum analogue of the classical
Bayesian decision rule (36),7 in the sense that it achieves
the minimal mean error probability given by (45), which
is known as the Helstrom bound [31].
We will be interested in evaluating the mean error
probability when we have access to M copies of the system
under study, in this case the Helstrom bound take the
form
Pe ¼
1 trjw1rM
2
1
w0rM
0 j
(46)
which is, however, often very difficult to calculate in
practice. Luckily, quantum versions of the Chernoff and
Bhattacharyya bounds exists, and their explicit forms
bear a striking resemblance with their classical analogues
[32], [33]
Pe eMQC
1
2
with
QC ¼log min0	s	1tr rs
and
QB ¼log tr
p
p
ffiffiffiffiffi
r0
ffiffiffiffiffi
r1
(49)
where for simplicity, we restricted ourselves to the case
w0 ¼ w1 ¼ 1=2. Interestingly enough, the Bhattacharyya
bound can be related to a lower bound for the mean error
probability [23]
Pe
1
2
p
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 eMQB
:
(50)
PF ¼
As in the classical case, the quantum Chernoff bound
is asymptotically tight, while the Bhattacharyya bound
(and the lower bound derived from it) is not as tight, but is
much simpler to calculate [32],[33]. For a Gaussian state
with known mean vector and covariance matrix, it is
always possible to obtain an analytical expression for
tr rs
0r1s
1
[34]. However, the optimization over s in (48)
7The reader should also notice that the classical likelihood-ratio test
can be rewritten as: decide H1ðH0Þ ifw1p1ðRÞ w0p0ðRÞ is positive
(negative).
MAY 2022
Z1
where we have denoted with p0ðLÞ the conditional probability
density of the likelihood ratio L under the assumption
that the hypothesis H0 is true. From (54), we notice
that, as opposed to the Bayesian decision threshold g,to
determine the Neyman-Pearson threshold one does not
need to make any assumption on the a priori probabilities
of the two hypotheses w0 and w1.
The performances of a Neyman-Pearson test are usually
evaluated by plotting the detection probability
IEEE A&E SYSTEMS MAGAZINE
77
p0ðLÞdL ¼ b
(54)
0r1s
1
(48)
and see that we can minimize F by assigning a point to the
decision region Z0 whenever the argument of the integral in
(52) is negative. This corresponds to a likelihood test [29]
LðRÞ¼
p1ðRÞ
p0ðRÞ
5
H1
H0
where the value of the threshold is determined by the
constraint condition
(53)
eMQB
1
2
Since
(47)
cannot always be carried out analytically. When this is the
case, we will resort to the Bhattacharyya bound in (49).
NEYMAN-PEARSONAPPROACH: THERECEIVINGOPERATING
CHARACTERISTIC
Classical Neyman-Pearson Approach
In the previous section, we described the Bayesian
decision strategy, which aims to minimize the mean error
probability. However, this strategy is not the most appropriate
for problems in which different types of error have
not the same importance, as it is the case in target detection.
In fact, especially if we think about military applications,
the damage produced by missing an enemy plane
can be much larger than that associated with a false alarm.
In such contests, it is more convenient to establish a value
of the false alarm probability that one can tolerate, PF ¼
b, and then minimize the miss probability PM, or equivalently
maximize the detection probability PD ¼ 1PM.
This is a constrained optimization problem that can be
solved by using Lagrange multipliers, namely by minimizing
the function [29]
F ¼ PM þPF b½
¼
Z0
R
Z0 p0ðRÞdR þ Z1 p0ðRÞdR ¼ 1, we can rewrite
R
this function as
F ¼ð1 bÞþ
Z
½p1ðRÞp0ðRÞ dR
Z0
(52)
Z " #
p1ðRÞdR þ
Z
Z1
p0ðRÞdR b :
(51)
IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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