IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 72

Detecting a Target With Quantum Entanglement
xðÞ¼ tr rDðÞ½
and to its Fourier transform
WðxÞ¼
Z
R2 N
ð2pÞ2 N expixTV
d2 N
(16)
Vð^qiÞVð^piÞ 1
(22)
xðÞ
(17)
which is a normalized, but generally nonpositive, quasiprobability
distribution known as the Wignerfunction [23].
The continuous variables x are the eigenvalues of the
quadrature operators, and they span a real 2N-dimensional
space known as phase space.3 Even though the Wigner
function itself is a quasi-probability distribution, its marginals
corresponding to measurable quadratures-like q,
p, and some linear combinations of their coordinates-are
proper measurement probabilities. For example, the probability
distribution of the q ¼ðx1;x3; ... ;x2N1Þ quadratures
can be obtained from the Wigner function by
integrating over all the p ¼ðx2;x4; .. . ;x2 NÞ quadratures
PðqÞ¼ WðxÞ
Z !
dx2n
YN
n¼1
:
(18)
A Wigner function can be characterized by the statistical
moments of the corresponding quantum state. In particular,
the first two moments are the mean vector
x ¼hxi¼ tr rxðÞ
and the covariance matrix V, whose elements are
defined as
Vi;j ¼ hfD^xi; D^xjgi
1
2
(20)
where the curly brackets denote the anticommutator
(f ^A; ^Bg¼ ^A ^Bþ ^B ^A), and D^xi ¼ ^xih^xii. The diagonal
elements of this matrix represent the covariances
of the quadrature operators Vii ¼ Vð^xiÞ¼ hðD^xiÞ2i¼
h^x2
i i h^xii2, while the off-diagonal elements quantify the
correlations between the different modes.
The fact that creation and annihilation operators do not
commute [see (8)] implies the following uncertainty relation
[23]:
V þ iV 0
(21)
which must be interpreted in the matrix sense, meaning
that all eigenvalues of the matrix V þ iV are larger than
zero. In particular, (21) tells us that
3To be more precise the phase space is the symplectic space formed
by R2 N equipped with the symplectic form V [23]. Such a mathematical
subtlety has several relevant consequences in quantum
optics, and some of them play quite an important role also in QI.
However, for the sake of simplicity, in this review, we will avoid to
refer explicitly to these rather involved mathematical details, but we
will refer the interested readers to the relevant literature [24].
72
which is the quantum optical analogue of the famous Heisenberg
uncertainty principle between position and
momentum in standard quantum mechanics [25], and tells
us that we cannot measure both quadratures qi and pi at
the same time with arbitrary precision.
A very important family of quantum states, which
comprise all states that we will consider in this manuscript,
is the one of Gaussian states [23]. These states are
fully determined by their mean vector and their covariance
matrix, and their Wigner function is given by the Gaussian
function
WðxÞ¼
expðxxÞTV1ðxxÞ=2
p
ffiffiffiffiffiffiffiffiffiffiffiffi
detV
hi
ð2pÞN
(23)
where det denotes the determinant.
The formalism described earlier is fairly general and
allows us to describe arbitrary multimode quantum states
of light. In the following, we will use it to introduce some
single-mode and two-mode quantum states, which are relevant
for QI.
Coherent states a.k.a. quasi classical states:
Let us now consider the so called coherent states jai.
(19)
These are single-mode states defined as the eigenstates of
the creation operator [20]
a^jai¼ ajai
(24)
with a ¼jajexpði'Þ a complex number. Equation (24) is
solved by the state [20]
jai¼ expjaj2=2
X1
n¼0
p jni
an
ffiffiffiffiffi
n!
(25)
and its Wigner function takes the Gaussian form in
(23) [23] with mean vector
x ¼ðq; pÞ¼ a þ a; a aðÞ=2 ¼jaj cos '; jaj sin 'ðÞ
(26)
and covariance matrix V ¼ 12, with 12 the 2
2 identity
matrix.
A coherent state is, thus, a superposition of infinitely
many photon-number states with a mean photon
number determined by the parameter a according to
h^ni¼ haj^ay^ajai¼ jaj2:
(27)
In particular, by setting a ¼ 0 in (25), we obtain
the vacuum state j0i. Therefore, every coherent states
has the same covariance matrix as the vacuum. Moreover,
this particular form of the covariance matrix V
saturates the uncertainty relation (22) [23]. As a consequence,
coherent states allow for the best precision in
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IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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