IEEE Aerospace and Electronic Systems Magazine - November 2020 - 42

Opportunities and Challenges of Quantum Radar

QUANTUM HYPOTHESIS DISCRIMINATION
In this section, we will briefly describe how certain
entangled photon states can be characterized by covariance matrices and how they are used for target detection.

COVARIANCE MATRIX OF PHOTON STATES
Consider a Gaussian entangled state as the one produced
in SPDC1 and let jCsi i denote the state of a system made
of a zero-mean Gaussian package of signal (s) and idler
(i) photon states. Formally, in an occupation number Fock
space, this state is given by
jCisi ¼

1
X

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Nsn

n¼0

ðNs þ 1Þnþ1

jnis jnii

ðSs Si À Cs2 ÞðSs Si À Ci2 Þ < ðSs2 þ Si2 Þ þ 2jCs Ci j À 1 (7)

(3)

where Ns is the average photon number and jnis and jnii
denote the photon state of n signal and n idler photons,
respectively [23], where the subindices i and s refer to the
idler and signal photons, respectively.
We can define a parameter as
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Ns

Ns þ 1

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1 À 2 j00i þ j11i:

(5)

This equation represents a quantum state made of 0 idler
and 0 signal photon states with probability % 1 À 2 , and
a state with 1 idler and 1 signal photon state with probability % 2 . In practice, % 10À2 [23] and therefore
Ns % 10À4 ( 1. That is, these entangled states are very
" diluted states " of signal and idler photons in the sense
that their average number is very small.
A general two-mode Gaussian state can be represented
through a Wigner-distribution covariance matrix that has
the following form:
0

Ss
1B
0
B
Gsi ¼ @
4 Cs
0

0
Ss
0
Ci

Cs
0
Si
0

1
0
Ci C
C
0 A
Si

(6)

given in terms of only four parameters [46]. In a sense, Si
terms can be understood to be related to the intensity of
the idler component, while Ss are related to the signal
photon intensity. On the other hand, Ci and Cs terms are
1

SPDC may not be the best or most optimal way to generate microwave photons. However, we can expect that the outgoing pair of
entangled photons will be described by the same type of mathematical expressions.

which is usually referred as Simon's criteria [47]. This is
the mathematical criterion for two photon Gaussian state
to be expressed as the product of two individual and independent photon states.
For simplicity, we can also define
f ðSs Si À Cs2 ÞðSs Si À Ci2 Þ À ðSs2 þ Si2 Þ À 2jCs Ci j þ 1
(8)

(4)

which can be used to parametrize a series expansion of the
Gaussian state. Thus, if ( 1, then
jCisi %

related to the degree of entanglement or correlation
between both components.
Mathematically, quantum entanglement is understood
as a characteristic of quantum states over two different
variables that cannot be separated as the product of two
individual states (one for each variable and independent of
the other). Thus, most criteria to measure the degree of
entanglement in a quantum system is related to a measure
of the separability of the state of the system as the product
of the state of the individual components. Following this
rationale, it can be shown that the necessary and sufficient
condition for entanglement between the signal and idler
photons in the Gaussian state can be simply written as

as a measure of entanglement/correlation between the
components of the system, and Simon's entanglement criteria reduces to f < 0. That is, the Gaussian system is
entangled if f < 0 and it is not entangled if f ! 0.
For the case under consideration, we have
S Si ¼ Ss ¼ 2Ns þ 1

(9)

and
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Cq Cs ¼ ÀCi ¼ 2 Ns ðNs þ 1Þ

(10)

which satisfies the inequality and, therefore, as expected,
jCisi is an entangled state. In addition, one can easily find
the approximate value of Cq for which the state showcases
minimum entanglement. It is found that this bound corresponds to approximately Cc ¼ 2Ns . Therefore, if Ns ) 1
then Cq % Cc and the entanglement is negligible. On the
other hand, if Ns ( 1 then Cq ) Cc and the state is
highly entangled. In other words, low brightness
(Ns ( 1) Gaussian states from SPDC showcase strong
nonclassical signatures and a high degree of entanglement [34] while high brightness (Ns ) 1) Gaussian states
from SPDC have minimal entanglement.

TARGET DETECTION
As discussed above, the signal photon state may be
bounced back toward the detector and be registered/measured or the detector may just measure noise photon states.
Therefore, we have two detection hypotheses:

IEEE A&E SYSTEMS MAGAZINE

NOVEMBER 2020

IEEE Aerospace and Electronic Systems Magazine - November 2020

Table of Contents for the Digital Edition of IEEE Aerospace and Electronic Systems Magazine - November 2020