Gaussian Quantum Systems and Kahler Geometrical Structure
MYKOLA YAREMENKO
Department of Partial Differential Equations,
The National Technical University of Ukraine
“Igor Sikorsky Kyiv Polytechnic Institute”
UKRAINE
Abstract: - In this article, we study the phase-space distribution of the quantum state as a framework to describe
the different properties of quantum systems in continuous-variable systems. The natural approach to quantum
systems is given the Gaussian Wigner representation, to unify the description of bosonic and fermionic
quantum states, we study the structure of the Kahler space geometry as the geometry generated by three forms
under the agreement conditions depended on the nature of the state bosonic or fermionic. Multimode light is
studied, and we established that the Fock space vacuum corresponds to a certain homogeneous Gaussian state.
Keywords: - Systems Theory, Wigner function, Fock space, Kahler space, photon, boson, fermion, Gaussian
state, Maxwell equations.
Received: May 9, 2022. Revised: January 20, 2023. Accepted: February 17, 2023. Published: March 7, 2023.
1 Introduction (Some Definitions and
Notations)
The central limit theorem establishes that a sum of
numbers of the independent and identically
distributed random variables, which variances are
finite, will approach a normal distribution as the
number of variables will grow. This statement has
many different variations with slightly different
conditions on random variables, colloquially
speaking, the central limit theorem maintains that
the properties of the normalized sums have a
tendency to normalize [1, 3, 25, 26]. From a
mathematical perspective, this theorem highlights
the impotence of Gaussian (or normal) distributions,
from a physical viewpoint, the gaussian states play a
central role in the theory of Bose gases and the
formalism of the theory of optical coherency. The
central limit theorem warrants the Gaussian theory a
prominent place in the quantum information theory
of continuous variables [29-35].
The general form of the Gaussian
probability density function is
11
exp 2
2
2
xm
ux
, where
m
is
its mean, mode, and median,
is a standard
deviation. Thus, the Gaussian states are completely
defined by their mean-field and covariance matrix.
Let
a
be a vector in the phase space with the
symmetric bilinear form
ab
g
, the Wigner function
for the bosonic Gaussian states is
1
det exp 2
ab
ab
g
Wg
.
Now, to clarify our considerations, let us
introduce some notations and definitions. We will
assume that a set of
n
identical particles is
described by a quantum state vector
in a
reflexive Banach space
B
. The joint state of
n
particles can be determined by the classical tensor
product
,..., n
1
, where
i
is a state vector
for the
i
-th particle.
Definition 1. A linear operator
:A
BB
on a reflexive Banach space is said
to be adjoint to the linear operator
:ABB
if
,,y Ax A y x
holds for all
xB
and all
y
B
.
Definition 2. A linear operator
:ABB
on a reflexive Banach space is said to be strictly
Hermitian if the following equality
,,A y Ax y x
holds for all
xB
and all
y
B
.
The permutation
n
S
is defined by
strictly Hermitian operator
P
according to the
following formula
... ... .
nn
PP
11
(1)
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This permutation guarantees the invariance of the
observable physics of the identical particle with the
same internal attributes.
The span
,..., n
span u u
1
of a set of
vectors
,..., n
uu
1
is the set of all linear
combinations of these vectors
,...,
... : ,...,
n
n n n
span u u
u u K
1
1 1 1
, (2)
where
K
is a field over which the vector space is
considered.
Assuming that the particles are identical,
postulating the invariance under the permutation
gives us that the state vector is either fully
symmetric (Bosons) or fully antisymmetric
(Fermions) relative to these permutations, and a
single particle is symmetric. So, the natural
condition to demand is
nn
P
(3)
for Bosons or
nn
P
(4)
for Fermions.
The first quantization is a description of a
n
-
particles system. We consider the Boson case. The
Banach space
n
s
B
that describes
n
-Bosons system
is a subspace of the Banach space
n
B
, which
consists of all linear combinations of vectors such
that
nn
P
, and can be written as
... ...
n
nn
S
11
, (5)
so
... :
n
s
ni
clos span
1
B
B
, (6)
where closure is understood in the topology
generated by the norm of the Banach space.
Let us denote
n
-particles Boson Banach
system by
n
s
B
, the direct sum of such systems is
....
s s s
0 1 2
B B B B
.
(7)
The component
s0
B
describes the vacuum state
with the single state
0
.
Pure separable states of Bosons (Fermions)
can be described by the following formula
...
0 1 2
, (8)
where
ii
s
B
are vectors from
i
-th Banach
space. Now, to define the state
, the formula
(8) must be completed by the normalization
requirement
1
. (9)
Creation and annihilation operators will be
denoted as
†
ˆ
a
and
ˆ
a
, the operator
†
ˆ
a
creates and
ˆ
a
deletes particles. The creation operator
†
ˆ
a
can be
defined by
†
ˆ0
...
a
0
12
, (10)
correctly defined all
B
. The annihilation
operator
ˆ
a
can be defined as the conjugation of the
operator
†
ˆ
a
with the condition
ˆ00a
.
Creation and annihilation operators are the
generators of the algebra of observables, which
provides a unique representation of the algebra. The
canonical commutation relation on the Fock space is
given by
†
ˆˆ
,|aa
, which holds
all vectors
B
and
B
in the single-particle
Banach space
B
.
A basis in the Fock space can be constructed
as follows. Let set
V
be a basis in single-particle
Banach space
B
then the basis in the Fock space
consist of all possible Fock states, which can be
formed by generating particles in vectors of
V
.
Particles in the vacuum can be created by the
creation operator
†
ˆ
a
as
† † †
ˆ ˆ ˆ
... ... 0
nn
a a a
1 1 2
,
which generates a certain Fock space, the whole
Fock space can be obtained as a direct sum of all
such Fock spaces by definition
ˆ0
...
a
0
12
,
(10)
correctly defined all
B
.
2 The Classical Model of Multimode
Light and Its Generalization
The light propagates as a wave, which is regulated
by Maxwell equations. A vector field
,u r t
1
is
called a mode of the electromagnetic field. The
Maxwell equations yield the following equations
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1,0u r t
ct
2
1
22
, (11)
,0u r t
1
, (12)
1,1
V
d r u r t
V
2
31
, (13)
where
V
is a volume containing whole considering
physical system.
Taking
,u r t
1
as a first element, we can
construct an orthogonal mode basis
,
m
u r t
with orthogonality condition
1,,
m n mn
V
d r u r t u r t
V
3
. (14)
The modes
,
m
u r t
fashion a basis for
the representation of any solution to the Maxwell
equations in the form of a series
,,
mm
m
E r t u r t
, (15)
m
are the complex amplitudes, which is
convenient to present in the form of the sum of the
real (amplitude quadrature) and imaginary (phase
quadrature) components
xp
m m m
E iE
. (16)
The space of all solutions to the Maxwell equations
constitutes a mode space with the basis
,
m
u r t
.
The series
,
mm
m
u r t
has finite numbers of
summands since a vector
m
u
(we will omit
arguments, where it is possible, notation
independent of any representation) consists of zeros
except for one at the m-th position.
In Hilbert spaces, there is a unitary operator
U
, which defines the unitary transformation from
one basis to another basis such that
,,
m km k
k
u r t U v r t
,
(17)
†
,,
m km k
k
v r t U u r t
, (18)
the first formula can be rewritten in the form
k
m m k
k
u U v
.
The infinite-dimensional matrix
,,
m km k
k
u r t U v r t
is such that
1,,
km k k
V
U d r v r t u r t
V
3
. (19)
The expansion of the electric field of the new basis
can be written as
,,
kk
k
E r t v r t
, (20)
where
k km m
m
U
. Since the unitary
transformation
U
is arbitrary, the mode basis can
be chosen in accordance with the optical process,
for instance, spatial or frequency Hermite-Gauss
modes.
3 Quantum Representation of
Multimode Light
Let
†
ˆm
a
be a set of creation operators and
U
be a
unitary operator with matrix
k
m
U
so a new set of
operators
†
ˆm
b
can be written as
††
ˆˆ
k
m m k
k
b U a
(21)
or in the form
ˆ
ˆk
k m k
k
a U b
. (22)
Since
U
is a unitary operator, we have
†
ˆˆ
,
m k mk
bb
, (23)
and a positive electric field has the following
representation
ˆ
ˆ,,
k m k
k
E r t f b u r t
1
, (24)
where
ˆm
b
is the one-photon annihilation operator in
the mode
,
k
u r t
, such that
k
m k m
k
fU
22
2
11
. (25)
Since mode
k
u
associated with a creation
operator
†
ˆk
a
, the new set of modes relative to the
plane wave basis is
1k
m k m k
k
m
u U u
f
1
1
. (26)
Let us assume that a mode basis is established
then the general quantum light state
can be
written as
... ...
... ... : ... : ...
n
n
k k n n
kk
C k u k u
1
1
11
,
(27)
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where
†
ˆ0
:!
n
k
k
nk
k
nn
n
Ua
ku k
.
Intrinsic properties of the state of the
multimode light are those properties that are
invariant relatively to the choice of the mode basis.
The intrinsic properties are:
1. Structural properties, which are solely
determined by the class of the quantum
system such as composition, set of the
observable, the action (Hamiltonian) of
the system.
2. Conditional properties are solely
determined by the preparation of the
system. For instance, let the particle
possess a spin
1
2
, then, we can
prepare the state with spin projection to
z
- axis equal to
2
, from these
assumptions arises no contradictions
since there is the value of
z
.
3. Classical properties.
Let
be a mixed state and
n
a minimal
span on
n
modes
,...,
1n
uu
. The coherency matrix
1
mk
is
†
ˆˆ
,
1mk
mk aa
, (28)
and elements of
1
mk
for
mn
and
kn
equal to zero, so that matrix
1
mk
composed of a
square
nn
non-zero diagonal matrix. The number
n
of modes relates to the intrinsic properties of the
quantum system and coincides with the rank of the
matrix of coherency. The given state coincides with
a vacuum for all
kn
and
†
ˆˆ
,0
kk
aa
for
kn
.
Let
mk
1
be a coherency matrix
corresponding to the annihilation operators
ˆk
b
of
the arbitrary mode basis
k
v
, so that
†
ˆˆ
,
mk
mk bb
1
. Since the matrix
mk
1
is
Hermitian there is a unitary operator
U
that
transforms
mk
1
into diagonal form
†,..., ,0,0,....
n
U U Diag k k
11
(29)
and the transformation of the creation operators in
the vector form
††
ˆ
ˆ
c Ub
. The matrix
†
UU1
can be presented as
†
† T T † T
,..., ,0,0,....
ˆˆ ˆˆ
, , .
n
U U Diag k k
Ub b U c c
11
(30)
So, from the well-known result of linear
algebra that a Hermitian matrix can be transformed
by a unitary operator to the diagonal form, we have
obtained that by the diagonalization of the
coherency matrix one can obtain the simplest
representation of the given quantum state. The
principal eigenvalues correspond with the
magnitude of energy of the modes.
4 Exemplar, Gaussian States
The electric field of light is a quantum observable
ˆ,E r t
that can be presented as
ˆˆ
ˆ,,
2
1mm
mm
m
x ip
E r t u r t
, (31)
where
1
m
are electric fields of single-photon;
ˆm
x
and
ˆm
p
are quadrature operators, which must
satisfy the Heisenberg inequality
ˆˆ1xp
and
canonical commutation condition
ˆˆ
,2
m k mk
xp
.
An observable
ˆ
qu
can be defined according to
the formula
,...,
ˆ ˆ ˆ
2 1 2
1k k k k
kn
q u u x u p
(32)
for any
2n
uR
.
The characteristic function
for
quadrature
ˆ
qu
is defined as
,....
ˆˆ
exp
ˆˆ
!
kk
k
tr i q u
itr q u
k
0
(33)
for any
R
. The distribution of the probability
can be defined as
1exp
2R
p z d i z
. (34)
Let set
,...,
1n
uu
is such that
ˆˆ
,0
mk
q u q u
holds for all
m
and
,k
the
characteristic function
defines as
ˆˆ
exptr i q u
(35)
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where
ˆ ˆ ˆ
,...,
1n
q u q u q u
and the vector
...
11 nn
uu
.
The inverse Fourier transformation
T
1exp
22
2n
n
R
W z d i z
(36)
is called the Wigner function.
The Gaussian quantum state is the state, in
which the Wigner function has a Gaussian form
T
11
exp 2
2n
Wz
zz
Det
1
,
(37)
where
is the covariance matrix and
is the
displacement vector with the property
Tˆˆ
u tr q u
. (38)
The Gaussian state is invariant relative to the
symplectic transformation
SL
, which means that
the Gaussian state remains Gaussian under
symplectic transformation.
The value
1
Det
is called the purity
P
of
a Gaussian state. The covariance matrix transforms
as
T
SS
(39)
where
S SL
. For a gaussian state to be pure, it is
necessary and sufficient that its covariance matrix
was a positive symplectic matrix so that
T
SS
,
the symplectency of the covariance matrix
guarantees the purity of the state.
We assume that symplectic space is
2n
R
equipped with the symplectic form determined by
a nonsingular, skew-symmetric matrix in the form
,...,
0 -1
,10
1in
(40)
so that
is immune to the orthogonal
transformations.
Definition 3. The set of all completely
positive maps from one Gaussian state to another
Gaussian state, which preserves trace, is called a
Gaussian channel
G
.
The Gaussian channel
G
maps the
displacement vector
and covariance matrix
as
follows
T
:N
G Z Z
(41)
:P
GZ
, (42)
where the matrix
Z
is a transform and reshaping of
the covariance matrix, the matrix
N
is Gaussian
noise and vector
P
is additional displacement in
phase space. The Gaussian channel
G
transforms
as
TT
ˆ
:exp
1
ˆ
exp 2
PN
G iq
iq Z i
(43)
and the mapping of the Wigner function
T
:
1
exp 2.
2
k
P N P
k
RN
G W z
xx
d x W Z z x Det
2
1
1
(44)
The matrices
Z
and
N
must satisfy the
following condition
T0
Ni iZ Z
, (45)
which guarantees
TN
ZZ
will be the well-
defined covariance matrix.
Next, tet us consider a mixed state as a
statistical ensemble of pure states with a density
matrix as follows
ˆk k k
k
p
, (46)
where
k
is a pure state and
k
p
is a fraction of
the ensemble for each
k
. Let the variance of the
pure state
k
be
ˆ
2
kqu
and
ˆ
2qu
be the
variance of the mixed state.
The Heisenberg inequality yields the
following estimation
ˆˆ
ˆˆ
ˆˆ 1.
k k k
k
k i k i
ki
q u q u
p q u q u
p p q u q u
22
2 2 2
22
(47)
However, Jensen's inequality renders the estimation
ˆˆ
.
22
kk
ki
q u p q u
(48)
The terms
ˆˆ
22
k i k i
p p q u q u
in (47)
show that the mixed state can only saturate
Heisenberg’s inequality when the state is pure so
only pure Gaussian states saturate Heisenberg’s
inequality. Thus, the Heisenberg inequality can be
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saturated if and only if the covariance matrix is
symplectic. The covariance matrix is symplectic.
5 Kahler Space
Now, let us add in our consideration the metric
structure of the physical space-time continuum. A
Kahler manifold is a Riemannian manifold equipped
with a symplectic structure and with a complex
structure. The Kahler structure provides the
mathematical framework for the unification of the
description of bosonic and fermionic states with the
Wigner function in the Gaussian form.
Bosons and fermions can be described by a
vector
,xp
of
2n
-dimensional phase space
and an adjoint vector of observables
v
. The
Riemannian structure is presented by the symmetric
covariant metric tensor
ab
g
, its contravariant form
ab
G
such that
cb b
ac a
gG
. The symplectic
structure is given by a symplectic form
ab
and its
adjoint
ab
. The complex structure is presented by
linear form on the phase space as follows
cb b
ac a
gJ
.
The essential difference between the description
of bosonic and fermionic states is hidden in the
geometric structure of the space of the observables.
To describe the bosonic state, the adjoint to phase
space is equipped with the symplectic structure
ab
and the phase space with its dual form
ab
under
the condition
ac a
cb b
. In order to describe the
fermions state, the phase space is metricized by
positive form
ab
G
and on adjoint space metric
ab
g
.
For arbitrary Gaussian state
, we can
write
ˆ ˆ ˆ ˆ
1.
22
a b a b
ab ab
i
G
The bosonic system is commutative and the
symplectic form is defined independently from a
specific state, the canonic commutation relations are
ˆˆ
,
a b ab
i
.
The fermionic system is anti-commutative and
the metric does not depend on the state, and the
canonic anticommutation relations are
ˆˆ
,
a b ab
G
.
Let us consider the classical bosonic state with
one degree of freedom, so
,xp
. The creation
and annihilation operators are
†1
2
a x ip
and
1
2
a x ip
. The Gaussian state is
defined as such that satisfies the equation
0a
. The Bogolubov transformation is giving
†
††
.
a a a
a a a
The communication relations are
††
, , 1a a a a
, where
and
such that
1
22
. Thus, the Bogolubov transformation
can be presented in the form
cosh
sinh .
exp i r
exp i r
Assume an initial state is
and state after the
Bogolubov transformation is denoted by
, so
the Bogolubov transformation from
†
,aa
to
†
,aa
induce linear mapping
ba
X
on the vector
space spanned by
a
, such that
b a b
a
X
. From
the invariancy of the commutation relations, for a
symplectic
, we deduce the following condition
Tab ab
XX
.
Let us denote an operator of correlation as
,
ab a b
G
then we have
b ab
ab a cd
cd
G X G X X GX
TT
, which gives
the value of the expectation of the operator
a
in
the state
after transformation.
The lineal Bogolubov transformation can be
represented by a symplectic matrix
cos cosh cos sinh
sin sinh sin cosh
sin cosh sin sinh
cos cosh cos sinh ,
X r r
X r r
X r r
X r r
1
1
2
1
1
2
2
2
assuming that the initial state corresponds with
1G
, we obtain
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cosh 2 cos sinh 2
sin sinh
sin sinh
cosh 2 cos sinh 2 .
G r r
Gr
Gr
G r r
11
12
21
22
Next, we are going to consider the Gaussian
state in the case of two fermions. Similar to the
bosons, let the creation operator
†i
a
creates a
fermion in a quantum state
,i
which is described by
i
, and the annihilation operator creates the
corresponding antiparticle. The fermionic operators
are defined as
†
1
2
i i i
x a a
and
†
2
i i i
i
p a a
.
The anti-communication relations are
,,
i k ik i k
x x p p
and
,0
ik
xp
. The
matrix
G
in the basis
,xp
is an identity
matrix
1G
. The Gaussian state
is given by
the anti-symmetric correlation operator as
,
ab a b
i
,
if the state
is annihilated by
i
a
,
ab
is
symplectic and we have
0
0
ab I
I
.
The pair of different Gaussian states can be
defined as
0
ii
a
and
0
ii
a
. The
Bogolubov transformation is a linear mapping
†
,
ii
aa
into
†
,
ii
aa
(here the parentheses
,
denotes a set). The requirement for the
preservation of the anti-commutation relation, we
have
TT
b ab
ab a cd
cd
G X G X X GX
,
where
a a c
c
X
. Then the transformation of the
anti-symmetric correlator is
ab
ab XX T
.
In the case of single pair, let us define the linear
Bogolubov mapping as
†
††
.
a a a
a a a
From the preserving anti-communication
relation
†
,
ii
aa
, we obtain the following
conditions
1
22
and
†
0i
aa
2
.
These conditions lead to the conclusion that the
creation and annihilation operators interchange
under Bogolubov transformation in the sense
†
aa
.
For two pairs of creation and annihilation
operators
†
,
11
aa
and
†
,
22
aa
of fermions, we
have
†
††
,
a a a
a a a
1 1 2
2 1 2
which corresponds to the Gaussian states
0
ii
a
and
0
ii
a
. The linear Bogolubov
transformation can be represented in the
parametrized form as
cos
sin .exp i
The mapping
c
into
c
can be represented by
the symplectic matrix
cos
sin cos
0
sin sin
sin cos
cos
sin sin
0
X
X
X
X
X
X
X
X
1
1
2
1
3
1
4
1
1
2
2
2
3
2
4
2
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0
sin sin
cos
sin cos
sin sin
0
sin cos
cos .
X
X
X
X
X
X
X
X
1
3
2
3
3
3
4
3
1
4
2
4
3
4
4
4
The anticommutation relation for the fermionic
quantum systems is given by the formula
ˆˆ
,
ab a b
G
, form
ab
G
is the symmetric metric
on the adjoint to phase space. This transformation
satisfies the condition
Tab
ab
G X GX
since this
transformation continuously reaches identity
transformation. The creation operator changes on
annihilation operator at
2
and annihilation on
creation operators so that
††
,,a a a a
1 2 2 1
,
when
2
and
0
from one Gaussian state
to the different Gaussian state
.
The pure Gaussian state
(bosonic and
fermionic) can be described by the linear complex
structure
as
1ˆ
2
a a c
cc
i
under the
condition of homogeneity of the Gaussian state for
fermions.
Thus, the structure of the Kahler space is
completely defined by the linear complex structure
synchronically with the symplectic correlator
ab
in the case of bosonic state or by the metric
ab
G
for
the fermions, for the bosons, the metric is defined as
ab a cb
c
G
, or for fermions, the correlator is
given by
ab a cb
cG
. Then, we can calculate
the covariance matrix
1
ˆˆ
,2
a b ab ab
Gi
.
The Fock space vacuum corresponds to the
homogeneous Gaussian state. Assume
and
%
are pair of Gaussian states, there is the corresponded
Fock space vacuum representation, if and only if the
Hilbert-Schmidt norm
HS
is bounded.
References:
[1] G. Adesso, S. Ragy, and A. R. Lee,
Continuous variable quantum information:
Gaussian states and beyond, Open
Systems & Information Dynamics 21
(2014) 1440001.
[2] W. Arendt, H. Vogt, and J. Voigt, Form
Methods for Evolution Equations. Lecture
Notes of the 18th International Internet
seminar, version: 6 March (2019).
[3] A. Ashtekar and P. Singh, Loop Quantum
Cosmology: A Status Report, Class.
Quant. Grav. 28 (2011) 213001.
[4] Budde C. and Landsman K. A bounded
transform approach to self-adjoint
operators: functional calculus and
affiliated von Neumann algebras. Ann.
Funct. Anal. 7, 3 (2016), 411–420.
[5] C. Batty, A. Gomilko, and Y. Tomilov,
Product formulas in functional calculi for
sectorial operators. Math. Z. 279, 1-2
(2015), 479–507.
[6] S. Clark, Sums of operator logarithms. Q.
J. Math. 60, 4 (2009), 413–427.
[7] F. Colombo, G. Gentili, I. Sabadini,
D.C. Struppa, Noncommutative functional
calculus: unbounded operators, preprint,
(2007).
[8] R. DeLaubenfels, Automatic extensions of
functional calculi. Studia Math. 114, 3
(1995), 237–259.
[9] L. D’Alessio, Y. A. Kafri, and M. Rigol,
From quantum chaos and eigenstate
thermalization to statistical mechanics and
thermodynamics, Advances in Physics 65
(2016) 239.
[10] N. Dungey, Asymptotic type for sectorial
operators and an integral of fractional
powers. J. Funct. Anal. 256, 5 (2009),
1387–1407.
[11] N. Dupuis, L. Canet, A. Eichhorn, and
others, The nonperturbative functional
renormalization group and its
applications, Physics Reports 910, 1–114
(2021).
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2023.22.15
Mykola Yaremenko
E-ISSN: 2224-2678
167
Volume 22, 2023
[12] P. Chalupa, T. Schafer, M. Reitner, and
others, Fingerprints of the Local Moment
Formation and its Kondo Screening in the
Generalized Susceptibilities of Many-
Electron Problems, Phys. Rev. Lett. 126,
056403 (2021).
[13] F. Krien, A.I. Lichtenstein, and G.
Rohringer, Fluctuation diagnostic of the
nodal/antinodal dichotomy in the Hubbard
model at weak coupling: A parquet dual
fermion approach, Phys. Rev. B 102,
235133 (2020).
[14] T. Schafer and A. Toschi, How to read
between the lines of electronic spectra: the
diagnostics of fluctuations in strongly
correlated electron systems, Journal of
Physics: Condensed Matter (2021).
[15] L. D. Re and G. Rohringer, Fluctuations
diagnostic of the spin susceptibility: Neel
ordering revisited in dmft (2021),
arXiv:2104.11737.
[16] G. Rohringer, A. Valli, and A. Toschi,
Local electronic correlation at the two-
particle level, Phys. Rev. B 86, 125114
(2012).
[17] G. Rohringer, H. Hafermann, A. Toschi,
and others, Diagrammatic routes to
nonlocal correlations beyond dynamical
mean-field theory, Rev. Mod. Phys. 90,
025003 (2018).
[18] N. Wentzell, G. Li, A. Tagliavini, C.
Taranto, G. Rohringer, K. Held, A.
Toschi, and S. Andergassen, High-
frequency asymptotics of the vertex
function: Diagrammatic parametrization
and algorithmic implementation, Phys.
Rev. B 102, 085106 (2020).
[19] T. Eisner, B. Farkas, M. Haase, and R.
Nagel, Operator Theoretic Aspects of
Ergodic Theory. Vol. 272 of Graduate
Texts in Mathematics. Springer, Cham,
(2015).
[20] Eugenio Bianchi, Lucas Hackl, Nelson
Yokomizo “Linear growth of the
entanglement entropy and the
Kolmogorov-Sinai rate,” Journal for High
Energy Physics (2018) 2018: 25.
[21] M. Haase, The Functional Calculus for
Sectorial Operators. Vol. 169 of Operator
Theory: Advances and Applications.
Birkh¨auser Verlag, Basel, (2006).
[22] M. Haase, Functional analysis. An
Elementary Introduction. Vol. 156 of
Graduate Studies in Mathematics.
American Mathematical Society,
Providence, RI, (2014).
[23] Lucas Hackl, Robert C. Myers “Circuit
complexity of free fermions,” Journal for
High Energy Physics (2018) 2018: 139.
[24] Krien F., Valli A., and Capone M. Single-
boson exchange decomposition of the
vertex function, Phys. Rev. B 100, 155149
(2019).
[25] J. Nokkala, R. Martínez-Peña, G. L.
Giorgi, V. Parigi, M. C. Soriano, and R.
Zambrini, Gaussian states of continuous-
variable quantum systems provide
universal and versatile reservoir
computing, Commun. Physics 4, 53
(2021).
[26] E. Martin-Martnez, D. Aasen and A.
Kempf, Processing quantum information
with the relativistic motion of atoms,
Phys. Rev. Lett. 110 (2013) 160501
[1209.4948].
[27] M. Reed and B. Simon, Methods of
Modern Mathematical Physics I.
Functional analysis. Second edition.
Academic Press, Inc. [Harcourt Brace
Jovanovich, Publishers], New York,
(1980).
[28] D. R. M. Arvidsson-Shukur, Yunger N.
Halpern, H. V. Lepage, A. A. Lasek, C. H.
W. Barnes, and S. Lloyd, Quantum
advantage in postselected metrology, Nat.
Commun. 11, 3775 (2020).
[29] K. Schmudgen, Unbounded Self-adjoint
Operators on Hilbert Space. Vol. 265 of
Graduate Texts in Mathematics. Springer,
Dordrecht, (2012).
[30] T. Shi, E. Demler, and J. I. Cirac,
Variational study of fermionic and
bosonic systems with non-gaussian states:
Theory and applications, Annals of
Physics (2017).
[31] P. Woit, Quantum theory, groups, and
representations: An introduction.
Springer, 2017.
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2023.22.15
Mykola Yaremenko
E-ISSN: 2224-2678
168
Volume 22, 2023
[32] D. Vilardi, P. M. Bonetti, and W.
Metzner, Dynamical functional
renormalization group computation of
order parameters and critical temperatures
in the two-dimensional Hubbard model,
Phys. Rev. B 102, 245128 (2020).
[33] P. M. Bonetti, Accessing the ordered
phase of correlated Fermi systems: Vertex
bosonization and mean-field theory within
the functional renormalization group,
Phys. Rev. B 102, 235160 (2020).
[34] G. Adesso, S. Ragy, and A. R. Lee,
Continuous variable quantum information:
Gaussian states and beyond, Open Syst.
Inf. Dynamics 21, 1440001 (2014).
[35] M. Qin, T. Sch¨afer, S. Andergassen, P.
Corboz, and E. Gull, The Hubbard model:
A computational perspective (2021).
[36] M. Walschaers, N. Treps, B. Sundar, L. D.
Carr, and V. Parigi, Emergent complex
quantum networks in continuous-variables
non-gaussian states, arXiv:2012. 15608
[quant-ph] (2021).
[37] M.I. Yaremenko, Calderon-Zygmund
Operators and Singular Integrals, Applied
Mathematics & Information Sciences:
Vol. 15: Iss. 1, Article 13, (2021).
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2023.22.15
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E-ISSN: 2224-2678
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