On the description of quantum correlations by means of a one-particle density operator
We develop an approach to the description of processes of the creation of correlations and the propagation of initial correlations in large particlequantum systems by means of a one-particle density operator that is a solution of the generalized quantum kinetic equation with initial correlations.Mor...
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| author | Gerasimenko, V. I. Gerasimenko, V. I. |
| author_facet | Gerasimenko, V. I. Gerasimenko, V. I. |
| author_institution_txt_mv | [
{
"author": "V. I. Gerasimenko",
"institution": "Institute of Mathematics of NAS of Ukraine"
}
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| description | We develop an approach to the description of processes of the creation of correlations and the propagation of initial correlations in large particlequantum systems by means of a one-particle density operator that is a solution of the generalized quantum kinetic equation with initial correlations.Moreover, mean field asymptotic behavior of the constructed correlation operators of the system state is established. |
| first_indexed | 2026-08-04T01:00:29Z |
| format | Article |
| fulltext |
Збiрник праць Iн-ту математики НАН України 2017, том 14, № 1, 116–127
УДК 517.956.223
V. I. Gerasimenko
(Institute of Mathematics of NAS of Ukraine, Kyiv)
gerasym@imath.kiev.ua
On the description of quantum
correlations by means of a one-particle
density operator
Dedicated to Prof. Yu. B. Zelinskii on the occasion of his 70th birthday
We develop an approach to the description of processes of the creation
of correlations and the propagation of initial correlations in large particle
quantum systems by means of a one-particle density operator that is a
solution of the generalized quantum kinetic equation with initial correlati-
ons. Moreover, mean field asymptotic behavior of the constructed correlati-
on operators of the system state is established.
Розвинуто пiдхiд до опису процесiв народження кореляцiй та поши-
рення початкових кореляцiй у квантових системах багатьох частинок
за допомогою одночастинкового оператора густини, який є розв’язком
узагальненого квантового кiнетичного рiвняння з початковими коре-
ляцiями. Крiм того, встановлено асимптотичну поведiнку побудованих
кореляцiйних операторiв стану системи у наближеннi самоузгодженого
поля.
1. Introduction. As known, the marginal correlation operators give
an equivalent approach to the description of the evolution of states of large
particle quantum systems in comparison with marginal density operators.
The physical interpretation of marginal correlation operators is that the
macroscopic characteristics of fluctuations of mean values of observables
are determined by them on the microscopic level [1, 2].
Traditionally marginal correlation operators are introduced by means
of the cluster expansions of the marginal density operators [3]. In article [4]
c© V. I. Gerasimenko, 2017
On the description of quantum correlations 117
we developed an approach based on the definition of the marginal correlati-
on operators within the framework of dynamics of correlations governed
by the von Neumann hierarchy [5]. As a result of which it is establi-
shed that the marginal correlation operators governed by the hierarchy of
nonlinear evolution equations, known as the quantum nonlinear BBGKY
(Bogolyubov–Born–Green–Kirkwood–Yvon) hierarchy, are represented in
the form of series expansions over the number of particles of subsystems
which generating operators are the corresponding-order cumulants of the
groups of nonlinear operators of the von Neumann hierarchy for a sequence
of correlation operators [5].
In this paper we consider the problem of the rigorous description of the
evolution of states of large particle quantum systems within the framework
of a one-particle (marginal) density operator that is a solution of the
generalized quantum kinetic equation with initial correlations. We remark
that initial states specified by correlations are typical for the condensed
states of many-particle systems in contrast to their gaseous state [1, 6].
Moreover, in the paper mean field asymptotic behavior of processes of
the creation of correlations and the propagation of initial correlations in
large particle quantum systems is established.
We note that the conventional approach to the problem of the descri-
ption of the propagation of initial chaos [7], i.e. in case of initial states
specified by a one-particle density operator without correlation operators,
is based on the consideration of an asymptotic behavior of a solution of
the quantum BBGKY hierarchy for marginal density operators constructed
within the framework of the perturbation theory [8–10].
2. Preliminaries: marginal correlation operators. Let
the space H be a one-particle Hilbert space, then the n-particle space
Hn = H⊗n is a tensor product of n Hilbert spaces H. We adopt the
usual convention that H⊗0 = C. The Fock space over the Hilbert space H
we denote by FH =
⊕∞
n=0Hn. A self adjoint operator fn defined on the
n-particle Hilbert space Hn = H⊗n will be also denoted by the symbol
fn(1, . . . , n).
Let L1(Hn) be the space of trace class operators
fn ≡ fn(1, . . . , n) ∈ L1(Hn)
that satisfy the symmetry condition: fn(1, . . . , n) = fn(i1, . . . , in) for arbi-
trary (i1, . . . , in) ∈ (1, . . . , n), and equipped with the norm:
‖fn‖L1(Hn) = Tr1,...,n|fn(1, . . . , n)|,
118 V. I. Gerasimenko
where Tr1,...,n are partial traces over 1, . . . , n particles. We denote
by L1
0(Hn) the everywhere dense set of finite sequences of degenerate
operators with infinitely differentiable kernels with compact supports.
On the space L1(FH) = ⊕∞n=0L
1(Hn) of sequences f = (f0, f1, . . . ,
fn, . . .) of trace class operators fn ∈ L1(Hn) and f0 ∈ C it is defined the
following nonlinear one-parameter mapping
G(t; 1, . . . , s | f)
.
= (1)∑
P: (1,...,s)=
⋃
j Xj
A|P|(t, {X1}, . . . , {X|P|})
∏
Xj⊂P
f|Xj |(Xj), s ≥ 1,
where the symbol
∑
P: (1,...,s)=
⋃
j Xj
means the sum over all possible parti-
tions P of the set (1, . . . , s) into |P| nonempty mutually disjoint subsets
Xj , the set ({X1}, . . . , {X|P|}) consists from elements of which are subsets
Xj ⊂ (1, . . . , s), i.e., |({X1}, . . . , {X|P|})| = |P|. The generating operator
A|P|(t) of expansion (1) is the |P|th-order cumulant of the groups of
operators defined by the following expansion
A|P|(t, {X1}, . . . , {X|P|})
.
= (2)∑
P′ : ({X1},...,{X|P|})=
⋃
k Zk
(−1)|P
′
|−1(|P
′
| − 1)!
∏
Zk⊂P′
G∗|θ(Zk)|(t, θ(Zk)),
where θ is the declusterization mapping: θ({X1}, . . . , {X|P|})
.
= (1, . . . , s),
and on the space L1(Hn) the one-parameter mapping G∗n(t) is defined by
the formula
R1 3 t 7→ G∗n(t)fn
.
= e−itHnfne
itHn . (3)
In (3) the operator Hn is the Hamiltonian of a system of n parti-
cles, obeying Maxwell–Boltzmann statistics, and we use units, where
h = 2π~ = 1 is a Planck constant and m = 1 is the mass of particles.
The inverse group to the group G∗n(t) we denote by (G∗n)−1(t) = G∗n(−t).
On its domain of the definition the infinitesimal generator N ∗n of the group
of operators (3) is determined in the sense of the strong convergence of the
space L1(Hn) by the operator
lim
t→0
1
t
(
G∗n(t)fn − fn
)
= −i (Hnfn − fnHn)
.
= N ∗nfn, (4)
that has the structure: N ∗n =
∑n
j=1N ∗(j) + ε
∑n
j1<j2=1N ∗int(j1, j2), where
the operatorN ∗(j) is a free motion generator of the von Neumann equation
On the description of quantum correlations 119
[3], the operator N ∗int is defined by means of the operator of a two-body
interaction potential Φ by the formula:
N ∗int(j1, j2)fn
.
= −i (Φ(j1, j2)fn − fnΦ(j1, j2)),
and we denote a scaling parameter by ε > 0.
The evolution of all possible states of large particle quantum systems,
obeying the Maxwell–Boltzmann statistics, can be described by means of
the sequence G(t) = (I,G1(t), G2(t), . . . , Gs(t), . . .) ∈ L1(FH) of margi-
nal correlation operators governed by the hierarchy of nonlinear evoluti-
on equations known as the quantum nonlinear BBGKY hierarchy [1]. If
G(0) = (I,G0,ε
1 (1), . . . , G0,ε
s (1, . . . , s), . . .) is a sequence of initial margi-
nal correlation operators, then a nonperturbative solution of the Cauchy
problem of the quantum nonlinear BBGKY hierarchy is represented by a
sequence of the following operators [4]:
Gs(t, 1, . . . , s) = (5)
=
∞∑
n=0
1
n!
Trs+1,...,s+n A1+n(t; {1, . . . , s}, s+ 1, . . . , s+ n | G(0)),
s ≥ 1,
where the generating operator A1+n(t; {1, . . . , s}, s+1, . . . , s+n | G(0)) of
series expansion (5) is the (1 +n)th-order cumulant of groups of nonlinear
operators (1) of the von Neumann hierarchy for correlation operators:
A1+n(t; {1, . . . , s}, s+ 1, . . . , s+ n | G(0))
.
= (6)∑
P: ({1,...,s},s+1,...,s+n)=
⋃
kXk
(−1)|P|−1 ( |P| − 1)!G(t; θ(X1) | . . .
G(t; θ(X|P|) | G(0)) . . . ) , n ≥ 0,
and the composition of mappings (1) of the corresponding noninteracting
groups of particles we denote by the symbol
G(t; θ(X1) | . . .G(t; θ(X|P|) | G(0)) . . .).
We remark that nonperturbative solution (5) of the quantum nonli-
near BBGKY hierarchy is transformed to the solution represented be
perturbation (iteration) series as a result of the application of analogs
of the Duhamel equation to cumulants (2) of the groups of operators (3).
120 V. I. Gerasimenko
In case of initial states specified in terms of a one-particle (marginal)
density operator and correlation operators the evolution of all possible
states of large particle quantum systems can be described in an equivalent
way within the framework of a one-particle density operator governed by
the kinetic equation, i.e. without any approximations.
3. A main result: marginal correlation functionals of
the state. We shall consider the case of initial states specified by a one-
particle marginal density operator with correlations, namely, initial states
specified by the following sequence of marginal correlation operators:
G(c) =
(
I,G0,ε
1 (1), gε2(1, 2)
2∏
i=1
G0,ε
1 (i), . . . , gεn(1, . . . , n)
n∏
i=1
G0,ε
1 (i), . . .
)
, (7)
where the operators gεn(1, . . . , n) ≡ gεn ∈ L1
0(Hn), n ≥ 2, are specifi-
ed the initial correlations. We remark that such assumption about ini-
tial states is intrinsic for the kinetic description of many-particle systems.
On the other hand, initial data (7) is typical for the condensed states of
large particle quantum systems, for example, the equilibrium state of the
Bose condensate satisfies the weakening of correlation condition with the
correlations which characterize the condensed state [1, 6].
For initial states specified in terms of a one-particle density operator
and correlation operators (7) the evolution of states given in the framework
of the sequence G(t) = (I,G1(t), . . . , Gs(t), . . .) of marginal correlation
operators (5) can be described by means of the sequence
G(t | G1(t)) = (I,G1(t), G2(t | G1(t)), . . . , Gs(t | G1(t)) , . . .)
of marginal correlation functionals: Gs(t, 1, . . . , s | G1(t)), s ≥ 2, with
respect to the one-particle correlation operator G1(t) governed by the ki-
netic equation.
In this case the marginal correlation functionals Gs(t | G1(t)), s ≥ 2,
are defined with respect to the one-particle (marginal) density operator
G1(t, 1) =
∞∑
n=0
1
n!
Tr2,...,1+n A1+n(t, 1, . . . , n+ 1)× (8)
×
∑
P : (1, . . . , n+ 1) =
⋃
iXi
∏
Xi⊂P
gε|Xi|(Xi)
n+1∏
i=1
G0,ε
1 (i),
On the description of quantum correlations 121
where the generating operator A1+n(t) is the (1 + n)− th order cumulant
(2) of the groups of operators (3), and these functionals are represented
by the series expansions:
Gs
(
t, 1, . . . , s | G1(t)
)
=
∞∑
n=0
1
n!
Trs+1,...,s+nGs+n
(
t, θ({1, . . . , s}), (9)
s+ 1, . . . , s+ n
) s+n∏
i=1
G1(t, i), s ≥ 2,
where the (s+n)th-order generating operator Gs+n(t), n ≥ 0, of this series
is determined by the following expansion
Gs+n
(
t, θ({1, . . . , s}), s+ 1, . . . , s+ n
)
= (10)
= n!
n∑
k=0
(−1)k
n∑
n1=1
. . .
n−n1−...−nk−1∑
nk=1
1
(n− n1 − . . .− nk)!
×
×Ăs+n−n1−...−nk
(t, θ({1, . . . , s}), s+ 1, . . . , s+ n− n1 − . . .− nk)×
×
k∏
j=1
∑
Dj : Zj =
⋃
lj
Xlj
,
|Dj | ≤ s+ n− n1 − · · · − nj
1
|Dj |!
s+n−n1−...−nj∑
i1 6=...6=i|Dj |=1
∏
Xlj
⊂Dj
1
|Xlj |!
×
×Ă1+|Xlj
|(t, ilj , Xlj ).
In formula (10) the sum over all possible dissections [15] of the linearly
ordered set Zj ≡ (s + n − n1 − . . . − nj + 1, . . . , s + n − n1 − . . . − nj−1)
on no more than s+ n− n1 − . . .− nj linearly ordered subsets we denote
by
∑
Dj :Zj=
⋃
lj
Xlj
and the (s+ n) th-order scattering cumulant is defined
by the formula
Ăs+n(t, θ({1, . . . , s}), s+ 1, . . . , s+ n) =
= As+n(t, 1, . . . , s+ n)gεs+n(1, . . . , s+ n)
s+n∏
i=1
A−11 (t, i),
where the operator gεs+n(1, . . . , s + n) is specified initial correlations (7),
and notations accepted above were used. We adduce simplest examples of
122 V. I. Gerasimenko
generating operators (10):
Gs(t, θ({1, . . . , s})) = Ăs(t, θ({1, . . . , s})) =
= As(t, 1, . . . , s))g
ε
s(1, . . . , s)
s∏
i=1
A−11 (t, i),
Gs+1(t, θ({1, . . . , s}), s+ 1) =
= As+1(t, 1, . . . , s+ 1)gεs+1(1, . . . , s+ 1)
s+1∏
i=1
A−11 (t, i)−
−As(t, 1, . . . , s)gεs(1, . . . , s)
s∏
i=1
A−11 (t, i)
s∑
j=1
A2(t, j, s+ 1)×
×gε2(j, s+ 1)A−11 (t, j)A−11 (t, s+ 1).
A method of the construction of marginal correlation functionals (9) is
based on the application of kinetic cluster expansions [3] to the generating
operators of series (5). If ‖G1(t)‖L1(H) < e−(3s+2), then for arbitrary t ∈ R
series expansion (9) converges in the norm of the space L1(Hs).
We emphasize that marginal correlation functionals (9) describe the
all possible correlations generated by dynamics of large particle quantum
systems with initial correlations by means of a one-particle density
operator.
Now we establish the evolution equation for one-particle (marginal)
density operator (8). As a result of the differentiation over time variable of
the operator represented by series (8) in the sense of the norm convergence
of the space L1(H), then due to the application of the kinetic cluster
expansions [13] to the generating operators of obtained series expansion,
for one-particle density operator (8) we derive the following identity:
∂
∂t
G1(t, 1) = N ∗(1)G1(t, 1) + εTr2N ∗int(1, 2)G1(t, 1)G1(t, 2) + (11)
+εTr2N ∗int(1, 2)G2
(
t, 1, 2 | G1(t)
)
,
where the second part of the collision integral in equality (11) is determi-
ned in terms of the marginal correlation functional represented by series
expansions (9) in case of s = 2. This identity we treat as the quantum
kinetic equation and we refer to this evolution equation as the generalized
quantum kinetic equation with initial correlations.
On the description of quantum correlations 123
We emphasize that the coefficients in an expansion of the collision
integral of the non-Markovian kinetic equation (11) are determined by
the operators specified initial correlations (7).
On the space L1(H) for the Cauchy problem of the established
generalized quantum kinetic equation with initial correlations the following
statement is true.
Theorem 1. If ‖G0,ε
1 ‖L1(H) < (e(1 + e9))−1, a global in time solution
of the Cauchy problem of kinetic equation (11) is determined by series
expansion (8). For initial data G0,ε
1 ∈ L1
0(H) it is a strong solution and for
an arbitrary initial data it is a weak solution.
The proof of this existence theorem is similar to the proof in the case
of the generalized quantum kinetic equation given in [15].
4. On a propagation of initial correlations in a mean
field limit. Further we establish the mean field asymptotic behavior
of constructed marginal correlation functionals (9) in case of initial states
specified by the one-particle density operator with correlations (7).
We assume the existence of a mean field limit of initial one-particle
density operator in the following sense
lim
ε→0
∥∥εG0,ε
1 − g01
∥∥
L1(H)
= 0, (12)
and initial correlations as follows:
lim
ε→0
∥∥gεn − gn∥∥L1(Hn)
= 0, n ≥ 2. (13)
Let us observe that for arbitrary finite time interval for an asymptoti-
cally perturbed first-order cumulant of groups of operators (3), i.e. for
strongly continuous groups (3), the following equality is valid
lim
ε→0
∥∥∥G∗s (t, 1, . . . , s)fs −
s∏
j=1
G∗1 (t, j)fs
∥∥∥
L1(Hs)
= 0.
As a result of this fact for the (s+n)−th order cumulants of asymptotically
perturbed groups of operators (3) the following equalities are true:
lim
ε→0
∥∥∥ 1
εn
As+n(t, 1, . . . , s+ n)fs+n
∥∥∥
L1(Hs+n)
= 0, s ≥ 2. (14)
124 V. I. Gerasimenko
In consequence of the validity of equalities (14) for one-particle density
operator (8) the following mean field limit theorem holds.
Theorem 2. If conditions (12), (13) holds, then for series expansion ( 8)
the equality is true
lim
ε→0
∥∥εG1(t)− g1(t)
∥∥
L1(H)
= 0,
where for finite time interval the limit one-particle density operator g1(t)
is given by the following norm convergent series on the space L1(H)
g1(t, 1) = (15)
∞∑
n=0
t∫
0
dt1 . . .
tn−1∫
0
dtn Tr2,...,n+1G∗1 (t− t1, 1)N ∗int(1, 2)
2∏
j1=1
G∗1 (t1 −
−t2, j1) . . .
n∏
in=1
G∗1 (tn − tn, in)
n∑
kn=1
N ∗int(kn, n+ 1)
n+1∏
jn=1
G∗1 (tn, jn)×
×
∑
P : (1, . . . , n+ 1) =
⋃
iXi
∏
Xi⊂P
g|Xi|(Xi)
n+1∏
i=1
g01(i).
In series expansion (15) the operator N ∗int(j1, j2) is defined according to
formula (4) and the group of operators G∗1 (t) is defined by (3). For bounded
interaction potentials series (15) is norm convergent on the space L1(H)
under the condition that: t < t0 ≡ (2 ‖Φ‖L(H2)‖g01‖L1(H))
−1.
According to Theorem 2, for marginal correlation functionals (9) the
following limit theorem holds.
Theorem 3. Under conditions (12), (13) on initial state (7) there exists
a mean field limit of marginal correlation functionals (9) in the following
sense:
lim
ε→0
∥∥∥εsGs(t, 1, . . . , s | G1(t)
)
− gs
(
t, 1, . . . , s | g1(t)
)∥∥∥
L1(Hs)
= 0, s ≥ 2,
where the limit marginal correlation functionals gs
(
t | g1(t)
)
, s ≥ 2, are
represented by the expansions:
gs
(
t, 1, . . . , s | g1(t)
)
=
On the description of quantum correlations 125
=
s∏
i1=1
G∗1 (t, i1)gs(1, . . . , s)
s∏
i2=1
(G∗1 )−1(t, i2)
s∏
j=1
g1(t, j), (16)
and, respectively, the limit one-particle density operator g1(t) is represented
by series expansion (15).
The proof of these statements is based on the validity of equality (14)
for cumulants of asymptotically perturbed groups of operators (3) and the
explicit structure of the generating operators of series expansions (9) of
marginal correlation functionals and series expansion (8).
We remark that limit marginal correlation functionals (15), (16) are
a solution of the Cauchy problem of the quantum Vlasov hierarchy of
nonlinear evolution equations [2], which describes a mean field asymptotic
behavior of marginal correlation operators in case of arbitrary initial states,
namely,
∂
∂t
gs(t, 1, . . . , s) =
s∑
i=1
N ∗(i)gs(t, 1, . . . , s) +
+ Trs+1
s∑
i=1
N ∗int(i, s+ 1)
(
gs+1(t, 1, . . . , s+ 1) +
+
∑
P : (1, . . . , s+ 1) = X1
⋃
X2,
i ∈ X1; s+ 1 ∈ X2
g|X1|(t,X1)g|X2|(t,X2)
)
,
gs(t)
∣∣
t=0
= g0s , s ≥ 1,
where we used notations similar to accepted above.
It should be noted that limit marginal correlation functionals (16)
describe the process of the evolution of correlations of large particle
quantum systems by means of a one-particle density operator in a mean
field approximation.
Similar to the derivation of kinetic equation (11) we establish that
the one-particle density operator represented by series expansion (15) is a
solution of the Cauchy problem of the Vlasov-type quantum kinetic equati-
on with initial correlations:
∂
∂t
g1(t, 1) = N ∗(1)g1(t, 1) + (17)
+Tr2N ∗int(1, 2)
2∏
i1=1
G∗1 (t, i1)(g2(1, 2) + I)
2∏
i2=1
(G∗1 )−1(t, i2)g1(t, 1)g1(t, 2),
g1(t)|t=0 = g01 , (18)
126 V. I. Gerasimenko
and consequently, for pure states we derive the Hartree-type equation with
initial correlations. We point out that equation (17) is the non-Markovian
quantum kinetic equation.
Thus, we established that a mean field behavior of processes of the
creation of correlations and the propagation of initial correlations in large
particle quantum systems are governed by kinetic equation (17).
5. Conclusion. The concept of quantum kinetic equations in case
of initial states specified in terms of a one-particle density operator and
correlation operators (11), for instance, the initial correlation operators,
characterizing the condensed states [1, 6] or their influence on ultrafast
relaxation processes in plasmas [11], was considered.
This paper dealt with a quantum system of a non-fixed, i.e. arbi-
trary but finite, number of identical (spinless) particles obeying Maxwell–
Boltzmann statistics. The obtained results can be extended to large parti-
cle quantum systems of bosons and fermions like in paper [5].
In case of pure states the quantum Vlasov-type kinetic equation with
initial correlations (17) can be reduced to the Gross–Pitaevskii-type kinetic
equation. It was also established that in this case mean field dynamics
does not create new correlations except of those that generating by initial
correlations (16).
We note that in papers [12,13] two other approaches to the description
of the propagation of initial correlations of large particle quantum systems
in a mean field scaling limit where developed. In paper [12] the process of
the propagation of initial correlations was proved within the framework of
the description of the evolution by means of marginal observables [14] and
in paper [13] it was established by another method in terms of marginal
density functionals with respect to a one-particle density operator governed
by the generalized quantum kinetic equation [15].
The developed approach to the derivation of the quantum Vlasov-type
kinetic equation with initial correlations (17) from underlying dynamics
governed by the generalized quantum kinetic equation with initial correlati-
ons (11) enables to construct the higher-order corrections to the mean field
evolution of large particle quantum systems.
References
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| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
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| language | English |
| last_indexed | 2026-08-04T01:00:29Z |
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| spelling | oai:trim.imath.kiev.ua:article-1082018-02-13T13:58:16Z On the description of quantum correlations by means of a one-particle density operator Gerasimenko, V. I. Gerasimenko, V. I. We develop an approach to the description of processes of the creation of correlations and the propagation of initial correlations in large particlequantum systems by means of a one-particle density operator that is a solution of the generalized quantum kinetic equation with initial correlations.Moreover, mean field asymptotic behavior of the constructed correlation operators of the system state is established. Інститут математики НАН України 2017-04-25 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/108 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 14 No. 1 (2017): Vol. 14 No. 1 (2017): Analysis and Applications; 116–127 Сборник Трудов Института математики НАН Украины; Том 14 № 1 (2017): Том 14 № 1 (2017): Анализ и приложения; 116–127 Збірник Праць Інституту математики НАН України; Том 14 № 1 (2017): Аналіз та застосування; 116–127 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/108/97 Авторське право (c) 2017 Праці Інституту математики НАН України |
| spellingShingle | Gerasimenko, V. I. Gerasimenko, V. I. On the description of quantum correlations by means of a one-particle density operator |
| title | On the description of quantum correlations by means of a one-particle density operator |
| title_full | On the description of quantum correlations by means of a one-particle density operator |
| title_fullStr | On the description of quantum correlations by means of a one-particle density operator |
| title_full_unstemmed | On the description of quantum correlations by means of a one-particle density operator |
| title_short | On the description of quantum correlations by means of a one-particle density operator |
| title_sort | on the description of quantum correlations by means of a one-particle density operator |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/108 |
| work_keys_str_mv | AT gerasimenkovi onthedescriptionofquantumcorrelationsbymeansofaoneparticledensityoperator AT gerasimenkovi onthedescriptionofquantumcorrelationsbymeansofaoneparticledensityoperator |