IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 78

Detecting a Target With Quantum Entanglement
PD ¼ 1PM ¼
Z1
as a function of the false alarm probability PF. The curve
PDðPFÞ is known as the receiver operating characteristic
(ROC), and can be used to determine the optimal operating
point of a given detector.
Analytical results for the integrals in (54) and (55) are
available when the probability densities p0=1ðRÞ are
Gaussian functions, as it will be the case in QI theory (see
the section " Gaussian QI " ). These results are exact in the
case of two Gaussians with different means but the
same variance, as well as in the opposite scenario of two
Gaussians of same mean and different variances [29]. In
the general case of Gaussians with different means a different
variances an approximation known as the extended
Van Trees receiving characteristic approximation is
available [35].
Quantum Neyman-Pearson Approach
Let us now discuss how the decision strategy discussed
earlier translate to the quantum case. As discussed
in the section " Bayesian Approach: the Chernoff bounds:
Classical Chernoff Bounds, " we consider the measurement
operators E0 and E1 such that E0 þE1 ¼ 1, which
define the false alarm, miss, and detection probabilities as
PF ¼ tr E1r0ðÞ
PM ¼ tr E0r1ðÞ
PD ¼ 1PM ¼ tr 1E0ðÞr1½¼ tr E1r1ðÞ:
(56a)
(56b)
(56c)
We now want to find the operator E1 that maximize
(56c) [or minimize (56b)] for a fixed value of the false
alarm probability PF ¼ b. By repeating the procedure that
led to (52) with the probability distributions replaced by
density matrices and the integrals replaced by traces, we
find that E1 must be the projector on the positive part of
ðÞr1r0 , with determined by the condition PF ¼ b
via (56a) [31] 8.
8It should be noted that it is not always possible to saturate the
condition PF ¼ tr½E1r0¼ b, with E1 the projector Pþ on
ðÞr1r0 þ by simply tuning. When this is not possible, one
can define E1 ¼ Pþ þP0; where P0 is the projector on the subspace
spanned by the eigenstates corresponding to vanishing eigenvalues
of r1r0ðÞ, i.e., the zero eigenspace. In this case, is
tuned to reach the largest value of tr½P0r0 b,and is then chosen
to reach the equality PF ¼ b [31]. This is particularly relevant
when dealing with states r0=1 with discrete spectra (e.g., qubits).
On the other hand, when dealing with states with a continuous spectrum
the zero eigenvalues subspace is a set of measure zero [31]. In
the following, we will apply this formula in the latter case, and we
will ignore the mathematical complications coming from the zero
eigenspaces.
78
ROC [31]
PD ¼
(p
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
PFð1 hÞ
1
þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1PFÞh
2
when 0 PF 1 h
when 1 h PF 1
:
(61)
This equation is valid only for pure states. In QI, we
will be interested in detecting a target located in a
region with a strong thermal background. Because of
this thermal background, all quantum states we will
consider will be statistical mixtures. A general quantum
framework to discriminate between two arbitrary (eventually
mixed) Gaussian states following the Neyman-
Pearson approach actually exists [36]. However, in the
following, to keep things simple, we will focus on specific
measurement strategies. In this context, the quantum
state discrimination problem is mapped into the
discrimination between two classical probability distributions
that we can tackle with the classical theory
described earlier. Moreover, we will see that under
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MAY 2022
Following this strategy one can determine a quantum
p1ðLÞdL
(55)
version of the ROC. To illustrate this idea, let us now consider
the binary discrimination between two pure states
r0 ¼jc0ihc0j and r1 ¼jc1ihc1j, which are in general
nonorthogonal hc1jc0i¼ n. To find the projector on the
positive part of r1r0ðÞ, we thus have to solve the
eigenvalue equation
ðÞjc1ihc1jjc0ihc0j jhi¼ hjhi:
(57)
Multiplying from the left by hc0j and hc1j, we obtain the
system of equations
1
2
nhc1jhiðh þÞhc0jhi¼ 0
ð1 hÞhc1jhi nhc0jhi¼ 0
which leads to the solutions
h1=0 ¼
with R ¼
ðÞ1 R;
q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1Þ2=4 þh
; and h ¼ 1jnj2:
(59)
It is easy to verify that h1 > 0 and h0 < 0. Accordingly,
the Neyman-Pearson criterion tells us to define E0=1 ¼
jh0=1ihh0=1j. Substituting into (56a) and (56c), we have
PF ¼jhc0jh1ij2 and PD ¼jhc1jh1ij2, where hc0=1jh1i
can be determined by substituting (59) into system (58).
Finally, one obtains the false alarm and detection probabilities
as functions of
PF ¼ðh1 hÞ=2R
PD ¼ðh1 þhÞ=2R:
(60a)
(60b)
Eliminating, we get an analytical expression for the
(58)
IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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