IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 74

Detecting a Target With Quantum Entanglement
B
B
V ¼ @
A 0 C 0
0 A 0C
C 0 B 0
0C 0 B
1
C
C
A
of which (33) is a particular case. We have seen earlier
that the correlations between the modes are encoded in the
off-diagonal elements of the covariance matrix. Therefore,
it is reasonable to expect that it exists a criterion that tells
us if a Gaussian state is entangled or not by comparing the
off-diagonal terms of the covariance matrix with the diagonal
ones. For a covariance matrix in the form (34), such
an entanglement criterion reads [26], [27]
Figure 2.
Wigner distributions ofa thermal state rT with mean photon number
NT ¼ 3.
ji¼
X1
n¼1
s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðNsÞn
C>
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1ABþAB
:
(35)
ðNs þ 1Þnþ1jni1jni2
(32)
where Ns is the mean photon number in the two modes,
i.e., h^n1i¼h^n2i¼ Ns. The state (32) is also a Gaussian
state [23] with mean vectorx ¼ð0; 0; 0; 0Þ and covariance
matrix
V ¼ @Cq
B
B
S 0 Cq
0 S 0Cq
0Cq
0 S
1
C
C
0 S 0 A
p
(33)
where S ¼ 2Ns þ 1 represents the variances of the quadratures
of the two modes, while Cq ¼ 2
In particular, in the case A ¼ B ¼ S ¼ 2Ns þ 1, we have
C> Cc ¼ 2Ns, where Cc represents the largest value of
the correlations C that can be obtained by classical means,
i.e., without entanglement. From the expressions for Cq
and Cc, we see that the two-mode squeezed vacuum is
always entangled, but also that the correlation-enhancement
enabled by entanglement (Cq >Cc) becomes less
and less important as we increase the number of signal
photons Ns. We will see that this fact plays a fundamental
role in understanding the quantum advantage in QI. It is
worth mentioning that the uncertainty principle (21)
imposes a bound on the off diagonal terms of the covariance
matrix (34), which in the case of A ¼ B ¼ S ¼
2Ns þ 1 reduces to C
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
NsðNs þ 1Þ
indicates the correlations between the two modes.
In Figure 3, we plotted the marginals of the two-mode
squeezed vacuum Wigner functions (note that different
panels have different color scales). In the two top panels,
we see that the probability distributions for the quadratures
of a single mode are identical to those of a thermal
state (compare with Figure 2), and therefore reveal no correlation.
On the other hand, in the bottom row, we see that
the probability distributions Pðp1;p2Þ and Pðq1;q2Þ are
strongly squeezed along a specific direction. In particular,
the q quadratures are correlated, while the p quadratures
are anticorrelated, we are, therefore, in presence of strong
phase-sensitive cross-correlations. The name two-mode
squeezed vacuum stems from these cross-correlations, and
in the following, we will see that the strength of these correlations
surpass what is allowed by classical physics.
To this goal, let us now make a few comments on the
covariance matrices of two-mode Gaussian states. In particular,
in QI, we will always deal with matrices ofthe form
74
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
S2 1
¼ 2Ns
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ 1=Ns
,
which means that for a fixed value of S, i.e., a fixed number
of photons Ns
vacuum achieves the largest possible values of the correlations
C.
Let us conclude this section by saying that the twomode
squeezed vacuum is the entangled state, which is
most commonly generated in physics laboratories. It is
normally produced using a nonlinear process called spontaneous
parametric down conversion (SPDC) in which a
pump beam with frequency vP and wave vector kP is converted
in correlated pairs of photons signal and idler
modes characterized by the frequencies vS=I and the wave
vectors kS=I (see Figure 4) [21]. For such a process to happen
it is necessary to respect at the same time energy and
momentum conservation. The latter is often referred to in
this context asphase matching condition, which is verified
only within a limited frequency range known as the phase
matching bandwidth. SPDC can be observed at optical frequencies
in nonlinear crystals, also known as optical
parametric amplifiers (OPAs) [21], as well as at microwave
frequencies, where the analogue of an OPA is a
superconducting circuit known as JPA [28].
IEEE A&E SYSTEMS MAGAZINE
MAY 2022
in the signal, the two-mode squeezed
(34)
IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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