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OPENNESS OF MANY-ELECTRON QUANTUM
SYSTEMS FROM THE GENERALIZED PAULI
EXCLUSION PRINCIPLE
1
ROMIT CHAKRABORTY
THE MAZZIOTTI GROUP
THE UNIVERSITY OF CHICAGO
THE PAULI EXCLUSION PRINCIPLE
•  No two fermions can occupy the
same quantum state (Pauli, 1925)
•  Fermion occupation numbers must
lie between 0 and 1
0 ≤ ni ≤ 1
•  Comes from the skew-symmetry of
the N-fermion wave function (Dirac,
Heisenberg, 1926)
1A.
J. Coleman, Rev. Mod. Phys. 35, 668 (1963)
Wolfgang Pauli
2
numbers. Coleman showed that the Pauli exclusion principle applied to the natural occupation numbers imposes
necessary and sufficient ensemble N -representability conditions on the 1-RDM, that the eigenvalues of the 1-RDM
(1)
must lie between 0 and 1 [6].
While the Pauli conditions of the 1-RDM are complete
nberg [4] showed
ensemble N -representability conditions, additional consymmetry of the
ditions on the 1-RDM are required to ensure that it arises
•  A general from
N-fermion
pure state
outermatrix
product of the Nthe integration
of is
anexpressible
N -fermion by
purethe
density
eneral N -fermion
fermion wave function
ermion ensemble
N
D(1, 2, .., N ; 1̄, 2̄, .., N̄ ) = Ψ(1, 2, .., N )Ψ∗(1̄, 2̄, .., N̄)
(4)
, .., N )Ψ∗i (1̄, 2̄, .., N̄) where the N D can be spectrally resolved in terms of
the single N -fermion wave function Ψ(1̄, 2̄, .., N̄ ). These
(2)
additional 1-RDM constraints are known as pure N at sum to unity,
representability conditions or generalized Pauli condictions, and each
tions [6–11]. The pure N -representability conditions
coordinates of a
of the 1-RDM depend only on its natural occupation
ensemble density
numbers [6], and hence, we will use the terms N ns save one yields
representability of the 1-RDM and N -representability of
(1-RDM)
the 1-RDM spectrum, interchangeably. Smith showed
that pairwise degeneracy of occupation numbers are suffi)d2d3..dN. (3)
cient to ensure pure N -representability of the 1-RDM [7].
Furthermore, he showed that such degeneracy occurs nat1-RDM must be
urally in even-N quantum systems with time-reversal
positive semidefsymmetry. In 1972 Borland and Dennis reported pure
obey
additional
1A. J. Coleman, Rev. Mod. N
-representability
conditions for active space of three
Phys.
35, 668 (1963)
ed otherwise, the
upation numbers
en 0 and 1
DEFINITIONS
3
3
arXiv:1404.5223v1 [physi
numbers.
Coleman
showed
that the atomic
Pauli exclusion
prin-[2],
by Pauli
in 1925
to explain
transitions
ed otherwise, the tulated
ues of the 1-R
ciple
appliedplays
to thea natural
numbers
imposes
principle
key roleoccupation
in predicting
electronic
conupation numbers this
necessaryof
and
sufficient
N -representability
con-the
numbers. Cole
en 0 and 1
figurations
atoms
and ensemble
molecules.
Stated otherwise,
ditions
on the says
1-RDM,
eigenvalues
of the 1-RDM
ciple applied t
Pauli
principle
thatthat
thethe
fermion
occupation
numbers
(1)
lie betweensystem
0 and 1must
[6]. lie between 0 and 1
necessary and
λi must
of a quantum
While the Pauli conditions of the 1-RDM are complete
ditions on the
nberg [4] showed
0 ≤ λi conditions,
≤ 1.
ensemble N -representability
additional con-(1)
must lie betw
symmetry of the
ditions on the 1-RDM are required to ensure that it arises
the P
Subsequent
work
Dirac
and Heisenberg
[4]matrix
showed of the While
•  A general
N-fermion
pure by
state
by
outer
product
Nfrom
the integration
of is
anexpressible
N[3]
-fermion
purethe
density
eneral N -fermion
ensemble N -r
fermionthat
wave
function
this
principle arises from the antisymmetry of the
ermion ensemble
ditions on the
fermion
wave function.
N
∗
D(1, 2, .., N ; 1̄, 2̄, .., N̄ ) = Ψ(1, 2, .., N )Ψ (1̄, 2̄, .., N̄)
from the integ
As discussed by von Neumann [5], a general N -fermion
(4)
, .., N )Ψ• ∗i (A
1̄, general
2̄, .., N̄)
quantum
state
expressible
byexpressible
anresolved
N -fermion
ensemble
N
N-fermion
quantum
system
is
by
N-fermion
ensemble
where
the
Dis can
be
spectrally
in an
terms
of
N
matrix
density density
matrix
D(1, 2, .., N
the single
N -fermion wave function Ψ(1̄, 2̄, .., N̄ ). These
(2)
! are known as pure N additional 1-RDM constraints
N
at sum to unity, D(1, 2, .., N ; 1̄, 2̄, .., N̄) =
N )Ψ∗icondi(1̄, 2̄, .., N̄) where the N D
i Ψi (1, 2, ..,Pauli
representability conditions or w
generalized
ctions, and each
tions [6–11]. The pure N i-representability conditions
the single N -f
coordinates of a
(2)
of the 1-RDM depend only on its natural occupation
additional 1-R
ensemble density where wi are non-negative weights that sum to unity,
numbers [6], and hence, we will use the terms N representabilit
ns save one yields Ψ (1, 2, .., N ) are N -fermion wave functions, and each
i representability of the 1-RDM and N -representability of
tions [6–11].
(1-RDM)
number
denotes
the spatial
and spin coordinates
of a
the 1-RDM
spectrum,
interchangeably.
Smith showed
of the 1-RDM
fermion.
Integration
of the
N -fermionnumbers
ensemble
density
that pairwise
degeneracy
of occupation
are suffi)d2d3..dN. (3)
numbers [6],
cient over
to ensure
pure N -representability
of thesave
1-RDM
[7].
matrix
the coordinates
of all fermions
one yields
representabili
Furthermore,
hereduced
showed that
such matrix
degeneracy
occurs natthe
one-fermion
density
(1-RDM)
1-RDM must be
the 1-RDM s
urally in even-N
quantum systems with time-reversal
"
positive semidefthat pairwise
symmetry.
Borland
Dennis
reported pure(3)
1
N
D(1,
2, .., Nand
; 1̄, 2,
.., N )d2d3..dN.
D(1; 1̄) = In 1972
obey
additional
1A. J. Coleman, Rev. Mod. N
cient to4ensure
-representability
conditions for active space of three
Phys.
35, 668 (1963)
DEFINITIONS
3
arXiv:1404.5223v1 [p
arXiv:1404.5223v1 [physi
numbers.
Coleman
showed
that the atomic
Pauli exclusion
prin1-RDM
kn
ed otherwise, the tulated
appliedare
to th
Pauli
in 1925
explain
transitions
Pauli by
principle
says
thatto
the
fermion occupation
numbers[2], ciple
ciple applied to the natural occupation numbers imposes
ues of and
the suffi
1-R
upation numbers this
playssystem
a key role
predicting
con- necessary
λiprinciple
of a quantum
mustinlie
between 0electronic
and 1
necessary and sufficient ensemble N -representability conen 0 and 1
ditions
on the Cole
1-R
numbers.
figurations
of
atoms
and
molecules.
Stated
otherwise,
the
ditions on the 1-RDM, that
the
of the 1-RDM
0≤λ
1.
(1)
i ≤eigenvalues
must
lie applied
between t0
ciple
Pauli
principle
says
that
the
fermion
occupation
numbers
(1)
must lie between 0 and 1 [6].
While
the Pauli
necessary
and
work
by Dirac
[3] and
Heisenberg
[4] showed
λi Subsequent
of a quantum
system
must
lie between
0 and
1
While
Pauli conditions
of the
are complete
nberg [4] showed
ensemble
-repre
ditionsNon
the
that
thisthe
principle
arises from
the 1-RDM
antisymmetry
of the
ensemble
N
-representability
conditions,
additional
consymmetry of the
thebetw
1-R
(1) ditions
fermion wave function.0 ≤ λi ≤ 1.
mustonlie
ditions
on
the
1-RDM
are
required
to
ensure
that
it
arises
from the integratio
As discussed
bystate
von Neumann
[5], abygeneral
N -fermion
•  A general
N-fermion
pure
is
expressible
the
outer
product
of
Nthe P
workisbyexpressible
Dirac
andanHeisenberg
[4]matrix
showed the While
from
the integration
of an N[3]
-fermion
pure
density
eneral N -fermion Subsequent
quantum
state
by
N
-fermion
ensemble
fermion wave
function
N -r
this
principle
arises from the antisymmetry of the Nensemble
ermion ensemble that
density
matrix
D(1, 2, .., N ; 1̄,
ditions
on the
N
∗
fermion
wave
function.
!
D(1, 2, .., N ; 1̄, 2̄, .., N̄ ) = Ψ(1, 2, .., N )Ψ (1̄, 2̄,∗.., N̄)
N
fromthe
theNinteg
wi[5],
Ψi (1,
.., N )ΨN
1̄, 2̄,(4)
.., N̄ ) where
2, .., N ;by
1̄, 2̄,
.., N̄
)=
AsD(1,
discussed
von
Neumann
a 2,
general
i (-fermion
D ca
∗
, .., N )Ψ• i (A
1̄, general
2̄, .., N̄)
N
i
N-fermion
quantum
system
is
by
N-fermion
ensemble
quantum
state
expressible
byexpressible
anresolved
N -fermion
ensemble
where
the
Dis can
be
spectrally
in an
terms
of
the
single N -ferm
(2)
density density
matrix
N
the single
N -fermion wave function Ψ(1̄, 2̄, .., N̄ ). These
matrix
additional
D(1,1-RDM
2, .., N
(2)
where wi are non-negative weights that sum to unity,
additional 1-RDM constraints
are
known
as
pure
N
representability co
!wave functions, and each
at sum to unity, N Ψi (1, 2, .., N ) are N -fermion
∗
Ψi (1, 2, ..,Pauli
N )Ψicondi(1̄, 2̄, .., N̄)
D(1,
2, .., N ; 1̄, 2̄,conditions
.., N̄) = or w
representability
generalized
tions
[6–11].
N
where
the Th
D
number denotes the spatial andi spin
coordinates
of a
ctions, and each
of
the
1-RDM
de
tions
[6–11].
The
pure
N
-representability
conditions
fermion. Integration of the iN -fermion ensemble density
coordinates of a
the single N -f
(2)
numbers [6], and
of
the
1-RDM
depend
only
on
its
natural
occupation
matrix
over
the coordinates
of all fermions
save
one yields save one
ensemble•  density
additional 1-R
Integration
of the
N-fermion
wave
function
over
all
co-ordinates
representability
of
numbers
[6],
and
hence,
we
will
use
the
terms
N
where
w
are
non-negative
weights
that
sum
to
unity,
i
the one-fermion
reduceddensity
densitymatrix
matrixor(1-RDM)
ns save oneyields
yieldsthe one-electron
reduced
the
1-RDM
representabilit
representability
1-RDM wave
and Nfunctions,
-representability
Ψi (1,
2, .., N ) are
-fermion
and of
each the 1-RDM spect
" ofNthe
(1-RDM)
tions [6–11].
the 1 1-RDM
interchangeably.
Smith showed
N
number
denotes
the
spatial
coordinates
of a that pairwise dege
D(1,
2, .., Nand
; 1̄, 2,spin
.., N )d2d3..dN
. (3)
D(1;
1̄) =spectrum,
of tothe
1-RDM
cient
ensure
pur
that pairwise degeneracy of occupation numbers are suffi)d2d3..dN. (3) fermion. Integration of the N -fermion ensemble density Furthermore,
numbers he
[6],sh
cient
to ensure
pure N
-representability
thesave
1-RDM
[7].
matrix
coordinates
ofmatrix,
all fermions
one
Like over
the
Nthe
-fermion
density
theof1-RDM
mustyields
be
urally
in even-N
representabili
Furthermore,
showed
that such
degeneracy
occurs
natHermitian,he(ii)
normalized,
and
(iii) positive
semidefone-fermion
reduced
density
matrix
(1-RDM)
1-RDM must be the(i)
symmetry.
In 197
urally
even-N
quantum
withobey
time-reversal
the 1-RDM
s
inite. in
However,
1-RDMsystems
must also
additional
" the
positive semidefN
-representability
symmetry.
In 1972
Borland and Dennis reported pure
that pairwise
1
N
obey
additional
D(1,
2,
..,
N
;
1̄,
2,
..,
N
)d2d3..dN.
(3)
D(1;
1̄)
=
fermions
in5 six or
1A. J. Coleman, Rev. Mod. N
-representability
conditions for active space of three
Phys.
35, 668 (1963)
cient
to
ensure
3
of numerical calcu
DEFINITIONS
arXiv:1404.5223v1 [p
arXiv:1404.5223v1 [physi
numbers.
Coleman
showed
that the atomic
Pauli exclusion
prin-[2], ciple
1-RDM
aretokn
ed otherwise, the tulated
applied
t
Pauli
in
1925
tothe
explain
transitions
Pauliby
principle
says
that
fermion occupation
numbers
ciple
applied
to
the
natural
occupation
numbers
imposes
ues of the
upation numbers thisλprinciple
and 1-R
suffi
plays system
a key role
in lie
predicting
con- necessary
must
between 0electronic
and 1
i of a quantum
necessary and sufficient ensemble N -representability conen 0 and 1
ditions
on theCole
1-R
numbers.
figurations
of
atoms
and
molecules.
Stated
otherwise,
the
ditions on the 1-RDM, that
eigenvalues
of the 1-RDM
0 ≤the
λi ≤
1.
(1)
must
lieapplied
betweent
ciple
Pauli
principle
says
that
the
fermion
occupation
numbers
(1)
must lie between 0 and 1 [6].
While the Paul
necessary
and
work
by Dirac
[3]lie
andbetween
Heisenberg
[4] showed
λi ofSubsequent
a quantum
system
must
0 and
1
While
theprinciple
Pauli conditions
of the
are complete
nberg [4] showed
ensemble
ditions N
on-repr
the
that
this
arises from
the1-RDM
antisymmetry
of the
ensemble
N
-representability
conditions,
additional
consymmetry of the
1-R
0 ≤ λi ≤ 1.
(1) ditions
fermion wave function.
muston
liethe
betw
ditions
on
the
1-RDM
are
required
to
ensure
that
it
arises
from the integrat
As discussed
by
vonisNeumann
[5],by
a general
N -fermion
•  A general
N-fermion
pure
state
expressible
the
outer
product
of
Nthe P
work
by
Dirac
and
Heisenberg
showed the While
from
the integration
of an N[3]
-fermion
pure
density[4]
matrix
eneral N -fermion Subsequent
quantum
state
is
expressible
by
an
N
-fermion
ensemble
fermion wave
function
ensemble N -r
this
principle
N
ermion ensemble thatdensity
matrix arises from the antisymmetry of the
D(1, 2, .., N ; 1̄
ditions
on the
N
∗
fermion
wave
function.
!
D(1, 2, .., N ; 1̄, 2̄, .., N̄ ) = Ψ(1, 2, .., N )Ψ (1̄, 2̄, ..,
N̄)
∗
N
fromthe
the Ninteg
w[5],
2, .., N )Ψ
(1̄, 2̄,
2, .., Nby
; 1̄, von
2̄, .., Neumann
N̄ ) =
i Ψi (1,
As D(1,
discussed
a general
Ni -fermion
(4).., N̄ ) where
D c
∗
, .., N )Ψ• i (A
1̄, general
2̄, .., N̄)
N
i
N-fermion
quantum
system
is
by
N-fermion
ensemble
quantum
state
expressible
byexpressible
anresolved
N -fermion
ensemble
where
the
Dis can
be
spectrally
in an
terms
of
the single N -ferm
(2)
density density
matrix
N
the single
N -fermion wave function Ψ(1̄, 2̄, .., N̄ ). These
matrix
additional
D(1, 1-RDM
2, .., N
(2)
where wi are non-negative weights that sum to unity,
additional 1-RDM constraints
are
known
as
pure
N
representability c
! wave functions, and each
at sum to unity, N Ψi (1, 2, .., N ) are N -fermion
∗
N )Ψicondi(1̄, 2̄, .., N̄)tions
D(1,
2, .., N ; 1̄, 2̄,conditions
.., N̄) = or w
representability
generalized
[6–11].
i Ψi (1, 2, ..,Pauli
where
the NTD
number denotes the spatial and
spin coordinates
of a
ctions, and each
of the 1-RDM d
tions
[6–11]. The pure N -representability conditions
fermion. Integration of thei N -fermion ensemble density
coordinates of a
the single N -f
(2)
numbers [6], an
ofmatrix
the 1-RDM
depend
only
on
its
natural
occupation
over
the coordinates
of all fermions
save
one yields save one
ensemble•  density
additional 1-R
Integration
of the
N-fermion
wave
function
over
all
co-ordinates
representability
o
numbers
[6],
and
hence,
we
will
use
the
terms
N
where
w
are
non-negative
weights
that
sum
to
unity,
i
the one-fermion
reduced
densitymatrix
matrix
(1-RDM)
ns save oneyields
yieldsthe one-electron
reduced
density
or
the
1-RDM
representabilit
representability
1-RDM wave
and Nfunctions,
-representability
Ψi (1,
2, .., N ) are"ofNthe
-fermion
and of
each the 1-RDM spec
(1-RDM)
tions [6–11].
the 11-RDM
Smith showed
N spatial
number
denotes
and
spin
of a that pairwise deg
D(1,interchangeably.
2, .., N
; 1̄, 2,
.., N coordinates
)d2d3..dN
. (3)
D(1;
1̄) spectrum,
= the
of the
1-RDM
cient
to ensure
pu
that pairwise degeneracy of occupation numbers are suffi)d2d3..dN. (3) fermion. Integration of the N -fermion ensemble density Furthermore,
he s
numbers [6],
cient
to
ensure
pure Ndensity
-representability
of 1-RDM
thesave
1-RDM
[7].be
matrix
the
coordinates
all fermions
one
Likeover
the
N
-fermion
matrix,
the
must
•  Eigenfunctions
of
the
1-RDM
are theof
natural
orbitals
while
its yields
eigenvalues
are
urally
in even-N
representabili
Furthermore,
he(ii)
showed
that such
degeneracy
occurs
nat(i)
Hermitian,
normalized,
(iii) positive
semidefnatural
occupation
numbers
one-fermion
reduced
densityand
matrix
(1-RDM)
1-RDM must
be the
symmetry.
In 19
urally
even-N
quantum
systems
withobey
time-reversal
the 1-RDM
s
inite. inHowever,
the
1-RDM
must also
additional
"
positive semidefN
-representabilit
symmetry.
In 1972
Borland and Dennis reported pure
that pairwise
1
N
obey
additional
D(1,
2,
..,
N
;
1̄,
2,
..,
N
)d2d3..dN.
(3)
D(1;
1̄)
=
fermions
in
6 six or
1A. J. Coleman, Rev. Mod. N
-representability
conditions for active space of three
Phys.
35, 668 (1963)
cient
to
ensure
3
of numerical calc
DEFINITIONS
N-REPRESENTABILITY
Ensemble
1-RDM is derivable from the Nfermion ensemble density matrix
The Pauli Exclusion Principle is
necessary and sufficient for
ensemble N-representability of
the 1-RDM
0 ≤ ni ≤ 1
(Coleman, 1963)
1A.
J. Coleman, Rev. Mod. Phys. 35, 668 (1963)
7
N-REPRESENTABILITY
Ensemble
Pure
1-RDM is derivable from the Nfermion ensemble density matrix
1-RDM arises from the pure
density matrix
The Pauli Exclusion Principle is
necessary and sufficient for
ensemble N-representability of
the 1-RDM
Generalized Pauli Conditions are
pure N-representability
conditions on the 1-RDM
0 ≤ ni ≤ 1
(Coleman, 1963)
1A.
J. Coleman, Rev. Mod. Phys. 35, 668 (1963)
More stringent and complicated
than the Pauli condition
Defines a convex polytope in
the space of natural
occupations
8
GENERALIZED PAULI CONSTRAINTS
Constraints
Feasible Set
ni ≥ ni+1,∀ i ∈ {1..(r −1)}
Pauli
0 ≤ ni ≤ 1
r
∑n = N
i
i=1
defines convex set 1E(N,r )
2R.
3A.
E. Borland and K. Dennis, J. Phys. B 5, 7 (1972)
A. Klyachko, J. Phys. Conf. Ser. 36, 72 (2006)
1E
(3,6)
9
GENERALIZED PAULI CONSTRAINTS
Constraints
Feasible Set
ni ≥ ni+1,∀ i ∈ {1..(r −1)}
Pauli
0 ≤ ni ≤ 1
r
∑n = N
i
i=1
defines convex set 1E(N,r )
Generalized
Pauli
r
0 ≤ ni ≤ 1, ∑ ni = N,
i=1
r
ij
i
j
i=1
defines convex set 1P(N,r )
3A.
5
E. Borland and K. Dennis, J. Phys. B 5, 7 (1972)
A. Klyachko, J. Phys. Conf. Ser. 36, 72 (2006)
(3,6)
ni ≥ ni+1,∀ i ∈ {1..(r −1)}
∑A n ≥ b
2R.
1E
1P
(3,6)
Feasible sets in the Borland
Dennis setting
10
FEASIBLE SETS
Occupation numbers
1.0000
1.0000
0.9000
0.9996
0.6000
0.9996
0.3000
0.0004
0.1000
0.0004
0.1000
0.0000
Both sum to 3
Both obey the Pauli principle
Which one of these sets come from the
wave function?
11
GENERALIZED PAULI EXCLUSION
Occupation numbers
1.0000
1.0000
0.9000
0.9996
0.6000
0.9996
0.3000
0.0004
0.1000
0.0004
0.1000
0.0000
Spectrum of occupations
in Li
12
OPEN QUANTUM SYSTEMS
• 
Generalized Pauli constraints are necessary for conditions for a pure
quantum state
A closed system
13
OPEN QUANTUM SYSTEMS
• 
Generalized Pauli constraints are necessary for conditions for pure
quantum states
An open system
• 
Violation of Generalized Pauli conditions provide a sufficient condition for
the openness of a many-electron quantum system
• 
These conditions can be used to study the interaction of a quantum
system with its environment
14
SUFFICIENT CONDITIONS FOR
OPENNESS
6R.
• 
Spectra can be represented by an N-fermion wavefunction (pure) if and only if
they lie inside the pure set 1P(N,r)
• 
Spectra can be represented by an N-fermion density matrix if and only if they lie
inside the ensemble set 1E(N,r)
• 
Spectra in the outside the pure set but inside the ensemble set (1E(N,r)\1P(N,r))
cannot be represented by an N-fermion wavefunction
• 
Violation of Generalized Pauli Conditions are sufficient to certify openness of a
many-electron quantum system form sole knowledge of the 1-RDM
Chakraborty, D. A. Mazziotti Phys. Rev. A, 91, 010101(R) (2015)
15
DEGREE OF OPENNESS
6R.
• 
The nature and extent of spectral deviation from Generalized Pauli condtions
can be used to quantify the degree of openness in an interacting quantum
system
• 
We use a euclidean metric (pure distance) to quantify deviation from the
facets of the pure set (polytope) defined by the Generalized Pauli conditions
Chakraborty, D. A. Mazziotti Phys. Rev. A, 91, 010101(R) (2015)
16
DEGREE OF OPENNESS
6R.
• 
The nature and extent of spectral deviation from Generalized Pauli condtions
can be used to quantify the degree of openness in an interacting quantum
system
• 
We use a euclidean metric (pure distance) to quantify deviation from the
facets of the pure set (polytope) defined by the Generalized Pauli conditions
• 
Pure distance is written as the Sequential Quadratic Program:
Chakraborty, D. A. Mazziotti Phys. Rev. A, 91, 010101(R) (2015)
17
DEGREE OF OPENNESS
• 
The nature and extent of spectral deviation from Generalized Pauli condtions
can be used to quantify the degree of openness in an interacting quantum
system
• 
We use a euclidean metric (pure distance) to quantify deviation from the
facets of the pure set (polytope) defined by the Generalized Pauli conditions
• 
Pure distance is written as the Sequential Quadratic Program:
! !
Pure Distance = min min
σ
n
−p
!
j
p
r
such that ∑ pi = N
i=1
pi ≥ pi+1∀ i ∈ [1, r −1]
0 ≤ pi ≤ 1∀ i ∈ [1, r]
r
∑A n = b
ji
6R.
i=1
i
j
Chakraborty, D. A. Mazziotti Phys. Rev. A, 91, 010101(R) (2015)
18
DEGREE OF OPENNESS
• 
The nature and extent of spectral deviation from Generalized Pauli conditions
can be used to quantify the degree of openness in an interacting quantum
system
• 
We use a euclidean metric (pure distance) to quantify deviation from the
facets of the pure set (polytope) defined by the Generalized Pauli conditions
• 
Pure distance is written as the Sequential Quadratic Program:
! !
Pure Distance = min min
σ
n
−p
!
j
p
r
such that ∑ pi = N
i=1
pi ≥ pi+1∀ i ∈ [1, r −1]
0 ≤ pi ≤ 1∀ i ∈ [1, r]
r
∑A n = b
ji
6R.
i=1
i
j
Chakraborty, D. A. Mazziotti Phys. Rev. A, 91, 010101(R) (2015)
Ø  σ is positive (negative) when the spectrum is
inside (outside) the pure set
Ø  Minimum distance to the facets of the ensemble
set 1E(N,r) and Slater point calculated for
comparison
Ø  Pure ≤ Ensemble ≤ Slater by definition
Ø  Pinned (quasi-pinned) if constraints are saturated
(close to being saturated)
19
ting
antenThe computations
presented
reveal methods
that the functional
entanglement
within
chromophores.
Two here
advanced
in
ules, called
subsystems achieve their efficiencies by a quantum mechanism
1
sity-matrix
group
[17]
and including
two-electron
reducedThe renormalization
light.
similar to that of
the full
system
the roles
of entanglement and environmental noise. We assess each subsystem’s
mplexes to
12,13 as
8, 19],2 have recently
shown that
networks
of conjugated
bonds
based on
entanglement
by a global
entanglement
measure
ency. For
14!19
of these chrothe and
squared
Euclideanare
distance.
a classical
acene
sheets [20],
chlorophyll
associatedThe
withstudy
polyradical
mophore subsystems gives more information about the role of
y transfers
adequately
described
without a strongly
correlated
each chromophore
in the energy
transfermany-electron
in the whole FMO
ion until it
20!31
This more in-depth
understanding can
provide
complex.
•  Environmental
interactions
in photosynthetic
energy
transfer
perimental
insight to other antennae complexes
and ultimately
m coherent
•  The Fenna-Mathews-Olson
complex
(FMO) be applied to
create synthetic solar cells that rival the efficiency of nature.
Matthews!
We consider a single monomer of the FMO complex with M
acteria has
Figurechromophore
1. Populations ofsites
chromophores
and the reaction center for (A) the subsystem 123 and (B) the full FMO system 1234
with the 1%3
Hamiltonian
electronic of the FMO
complex transfers energy more efficiently from chromophore 1 to the reaction center (sink) than the full FMO syste
es and over
M
þ !
! þ
þ !
^elements
pω
σl þ
σl Þ
ð1Þ
¼
j σdensity
j, l ðσ
Fkl are theH
of
the
matrix pv
D in
thej basis
set σ
ofj wave
stronger
than any classical correlation.” Par
j σj þ
j
¼
1
j6
¼
l
harvesting functions {Ψk}, and L^ is the Lindblad operator, which accounts
system become entangled when the total dens
y we mean for interactions +of the!M chromophores with the environment. At
system cannot be expressed as a product of the
where
σ
(σ
)
creates
(annihilates)
a
single
excitation
on
chroj
j
ple efficient t = 0 we initialize the density matrix with a single excitation
for the parts.12,13,34 Global entanglement12,13,3
the
mophore
j. The
of each
chromophore
is pω
by the squared Euclidean distance be
on either
site site
1 or energy
6. Because
the Hamiltonian
does
notj, whilemeasured
tion center. (exciton)
.
coupling
constant
between
the
pair
of
chromophores
j
and
l
is
pv
j,l
matrix
F and the nearest classical density matr
mix singly excited wave functions with other wave functions, the set
nae’s chro•  {Ψ
Dynamics
in
the
FMO
complex
The
time
evolution
of
the
system’s
density
matrix
is
governed
k} only includes M wave functions formed from sequentially
antennae
by thea single
quantum
Liouville
equation
considering
excitation
on each
of the M chromophores.
σðFÞ ¼ ∑ jFij % ξij j2
where
each
or correlated
Each
theofseven
i, j
is theofsum
three chromophores
operators that in
The chromophores.
Lindblad operator32,30
d dephasing,i dissipation,
a chromo- account for
^in D'
^and
to the(left),
reaction
center
D electron
¼ ! ½H,
þ L
ðDÞloss
ally
treated as a single
a two-state
model
and
yet the ð2Þ
monomer
dt
p
On the basis of M wave functions, in which ea
(sink)
or
chromoted from chlorophyll molecules with many strongly correlated electrons has a single excitation on one of the M ch
where
is the density
matrix
^diss ðDÞ
^sink ðDÞ
^D
þ L
ð4Þ
¼L
deph ðDÞ þ L
re 3, which L^ðDÞ
measure corresponds to the sum of the sq
ch
of
the
chromophores
by
a
correlated
N
-electron
model
by
Lipkin,
diagonal elements of F, or the sum of the
re we show where D ¼
Fkl jΨk æÆΨl j
ð3Þ
concurrences.
The squared Euclidean distance
there
exist
k, l
tration by K. ^Naftchi-Ardebili,
The University of Chicago, 2011. Used
20 chrom
if the excitations (excitons) on the
ð5Þ
ncies
close P.LRebentrost,
deph ðDÞ ¼ α ∑ 2ÆkjDjkæjkæÆkj % fjkæÆkj, Dg
5M. Mohseni,
S. Lloyd, and A. Aspuru-Guzik, J. Chem. Phys. 129, 174106 (2008)
entangled. Both the squared Euclidean distance
k
PHOTOSYNTHETIC ENERGY
TRANSFER
∑
∑
∑
TRAJECTORY OF THE FMO
•  The three chromophore subsystem has similar efficiency to the full 7-chromophore
network
•  We are able to visualize the time dependent trajectory in the space of natural orbital
occupations
Closed
Trajectory in population space with
femtosecond resolution. Points in green
lie inside the pure set
6R.
Chakraborty, D. A. Mazziotti Phys. Rev. A, 91, 010101(R) (2015)
21
CHARACTERIZED TRAJECTORY IN THE FMO
•  Environmental noise increases the size of the set of 1-RDMs accessible to a quantum
system facilitating in the transfer of excitation to the reaction center
Open
Trajectory in population
space
Points in green lie inside the pure set and points in red are outside
6R.
Chakraborty, D. A. Mazziotti Phys. Rev. A, 91, 010101(R) (2015)
22
Table 2 reports not only the minimum distance to a facet of
the pure 1-RDM set (pure distance) but also the harmonic,
geometric, and arithmetic means of the Euclidean distances to
each facet of the set. While the harmonic mean is similar to
the minimum distance, the geometric and arithmetic means
provide information about the distance to the majority of fac-
seven excited sta
ence of electron
seven excited sta
RDMs of these st
The pure distanc
states 3 and 6 are
and 6 have the g
•  Assuming single excitations, the 7x7 matrix of site and coupling energies, on
diagonalization, gives seven stationary state wavefunctions
have the highest
Table 3. Minimum distances to the pure, ensemble, and Slater boundaexcited states. Int
ries for the ground and excited states of the seven-site FMO complex are
•  We use theshown.
euclidean distance metric to study the nature of pinning in the FMO
state 2, and state
and 6 have the
Euclidean distances
degenerate pairs.
21
FMO
State
Energy (cm )
Pure
Ensemble
Slater
in the FMO comp
plexity of excited
Ground
0
0.00
0.0000
0.0000
0.0000
Excited
1
223.74
0.0000
0.0005
0.8876
quasi-pinned, or b
STATIONARY STATES OF THE FMO
2
101.97
0.0000
0.0048
3
120.96
0.0020
0.0085
4
268.37
0.0000
0.0141
5
307.13
0.0000
0.0001
6
332.00
0.0003
0.0003
7
513.32
0.0000
0.0003
Ø  Ground state corresponds to a Slater determinant
0.8528
0.9958
0.9246
0.9320
0.9784
0.8797
Because the ground state in this model is a Slater determinant, its
Slater, ensemble, and pure distances vanish. In contrast, the Slater
distances of all seven excited states are nonzero (! 0:85), reflecting the
presence of electron correlation. The ensemble distances of all seven
8R. Chakraborty, D. A. Mazziotti Int. J. Quantum Chem. 116, 784-790 (2016)
excited states are also nonzero. The pure distances, however, for all the
Conclusions
The role of the g
chemistry of excit
tions to three-, fo
ecules. While th
boundary of the
excited-state 2-RD
23
ensemble N-repre
Table 2 reports not only the minimum distance to a facet of
the pure 1-RDM
1-RDM set (pure
(pure distance) but
but also the
the harmonic,
the
the pure
pure 1-RDM set
set (pure distance)
distance) but also
also the harmonic,
harmonic,
geometric,
and
arithmetic
means
of
the
Euclidean
distances
to
geometric,
geometric, and
and arithmetic
arithmetic means
means of
of the Euclidean distances to
each facet of
of the set.
set. While the
the harmonic mean is similar to
each
each facet
facet of the
the set. While
While the harmonic
harmonic mean is similar to
the
minimum
distance,
the
geometric
and arithmetic means
the
the minimum
minimum distance,
distance, the
the geometric
geometric and arithmetic means
provide information
information about
about the
the distance
distance to
to the
the majority
provide
provide
information
about
the
distance
majority of
of facfac-
seven excited sta
ence of electron
ence of electron
seven excited sta
seven excited sta
RDMs of these st
RDMs of these st
The pure distance
The pure distanc
states
states 33 and
and 66 are
are
and
and 66 have
have the
the gg
•  Assuming single excitations, the 7x7 matrix of site and coupling energies, on
have
diagonalization,
gives
sevendistances
stationary
state
wavefunctions
have the
the highest
highest
Table 3.
3. Minimum
Minimum
to the
the
pure,
ensemble, and Slater
Table
pure,
Table 3.
Minimum distances
distances to
to the
pure, ensemble,
ensemble, and
Slater boundaboundaexcited
ries for
for the
the ground
ground and
and excited
excited states
states of
of the
the seven-site
seven-site FMO
ries
excited states.
states. Int
Int
ries
for
the
ground
and
excited
states
of
the
seven-site
FMO complex
complex are
are
•  We use theshown.
euclidean
distance
metric
to
study
the
nature
of
pinning
in
the
FMO
shown.
state
shown.
state 2,
2, and
and state
state
and
and 66 have
have the
the h
Euclidean
Euclidean
distances
Euclidean distances
degenerate
degenerate pairs.
pairs.
21
21
21
FMO
State
Energy
(cm
)
Pure
Ensemble
Slater
FMO
State
Energy
Pure
in
FMO
State
Energy (cm
(cm ))
Pure
Ensemble
Slater
in the
the FMO
FMO comp
comp
plexity
Ground
0.00
0.0000
Ground
000
0.00
0.0000
0.0000
0.0000
plexity of
of excited
excited
Ground
0.00
0.0000
0.0000
0.0000
Excited
1
223.74
0.0000
0.0005
0.8876
Excited
11
223.74
0.0000
Excited
223.74
0.0000
0.0005
0.8876
quasi-pinned,
quasi-pinned, or
or bb
STATIONARY STATES OF THE FMO
222
33
44
55
66
77
101.97
101.97
101.97
120.96
120.96
268.37
268.37
307.13
307.13
332.00
332.00
513.32
513.32
0.0000
0.0000
0.0000
0.0020
0.0020
0.0000
0.0000
0.0000
0.0000
0.0003
0.0003
0.0000
0.0000
0.0048
0.0048
0.0085
0.0141
0.0001
0.0003
0.0003
0.8528
0.8528
0.9958
0.9246
0.9320
0.9784
0.8797
Ø  Ground state corresponds to a Slater determinant
Ø  Excited states are correlated
Because
Because the
the ground
ground state
state in
in this
this model
model is
is aa Slater determinant, its
Slater,
Slater, ensemble,
ensemble, and
and pure
pure distances
distances vanish.
vanish. In contrast, the Slater
distances
distances of
of all
all seven
seven excited
excited states
states are
are nonzero
nonzero (! 0:85), reflecting the
presence of
of electron
electron correlation.
correlation. The
The ensemble
ensemble distances of all seven
8R. Chakraborty, D.presence
A. Mazziotti Int. J. Quantum Chem. 116, 784-790 (2016)
excited
excited states
states are
are also
also nonzero.
nonzero. The
The pure
pure distances,
distances, however, for all the
Conclusions
Conclusions
The role
role of
of the
the gg
chemistry of excit
tions to
to three-,
three-, fo
fo
ecules. While
While th
th
boundary of the
excited-state 2-RD
2-RD
excited-state
24
ensemble N-repre
N-repre
ensemble
Table
Table 2
2 reports
reports not
not only
only the
the minimum
minimum distance
distance to
to aa facet
facet of
of
the pure 1-RDM set (pure distance)
distance) but
but also
also the
the harmonic,
harmonic,
the pure 1-RDM set (pure distance) but also the harmonic,
geometric, and arithmetic means
means of the
the Euclidean distances
distances to
geometric, and arithmetic means of
of the Euclidean
Euclidean distances to
to
each
facet
of
the
set.
While
the
harmonic
mean
is
similar
to
each facet of the set. While the
the harmonic
harmonic mean
mean is
is similar
similar to
to
the
geometric and
and arithmetic
arithmetic means
means
the minimum
minimum distance,
distance, the
the geometric
geometric
and
arithmetic
means
provide
distance to
to the
the majority
majority of
of facfacprovide information
information about
about the
the distance
distance
to
the
majority
of
fac-
seven
seven excited
excited ss
ence of electro
electr
ence
ence of
of electr
seven excited
seven
seven excited
excited
RDMs
of these
these
RDMs
RDMs of
of
these
The pure
pure distan
dista
The
The
pure
dista
states 333and
and666
states
states
and
and 666 have
have the
th
and
and
have
the
•  Assuming single excitations, the 7x7 matrix of site and coupling energies, on
have the
the high
high
have
have
the
high
Table
Minimum
the pure,
pure,
ensemble, and
and Slater
Slater boundaboundaTable
to
ensemble,
diagonalization,
sevendistances
stationary
state
wavefunctions
Table 3.
3.gives
Minimum
distances
to the
the
pure,
ensemble,
and
Slater
boundaexcited states.
states.
excited
ries
states of
of the
the seven-site
seven-site FMO
FMO complex
complex are
are
ries
excited
states.
ries for
for the
the ground
ground and
and excited
excited states
states
of
the
seven-site
FMO
complex
are
shown.
shown.
state 2,2, and
andsta
st
state
•  We use theshown.
euclidean distance metric to study the nature of pinning in the FMO state 2, and sta
and 666 have
have th
th
and
and
have
th
Euclidean
distances
Euclidean
distances
Euclidean distances
degenerate pai
pa
degenerate
degenerate
pai
21
21
FMO
State
Pure
Ensemble
Slater
FMO
Energy
Pure
Ensemble
Slater
in the
the FMO
FMO co
co
in
FMO
State
Energy (cm
(cm21)))
Pure
Ensemble
Slater
in
the
FMO
co
plexity of
of excit
exci
plexity
Ground
00
0.0000
0.0000
0.0000
Ground
0.00
0.0000
0.0000
0.0000
plexity
of
excit
Ground
0.00
0.0000
0.0000
0.0000
Excited
11
0.0000
0.0005
0.8876
Excited
223.74
0.0000
0.0005
0.8876
Excited
223.74
0.0000
0.0005
0.8876
quasi-pinned,oo
quasi-pinned,
STATIONARY STATES OF THE FMO
22
33
4
5
6
7
101.97
101.97
120.96
268.37
307.13
332.00
513.32
0.0000
0.0000
0.0000
0.0020
0.0020
0.0000
0.0000
0.0000
0.0000
0.0003
0.0003
0.0000
0.0000
0.0048
0.0048
0.0048
0.0085
0.0085
0.0141
0.0141
0.0001
0.0001
0.0003
0.0003
0.0003
0.0003
0.8528
0.8528
0.8528
0.9958
0.9958
0.9246
0.9246
0.9320
0.9320
0.9784
0.9784
0.8797
0.8797
Because
state in this
is
aa Slater
Because
the
this
model
isdeterminant
Slater determinant,
determinant, its
its
the ground
Ø  Ground
state corresponds
to model
a Slater
Slater,
distances
Slater,
ensemble,
distances vanish.
vanish. In
In contrast,
contrast, the
the Slater
Slater
ensemble,
Ø  Excited
statesand
arepure
correlated
distances
of allensemble
seven excited
states
distances
states are
are nonzero
nonzero (!
(! 0:85),
0:85), reflecting
reflecting the
the
Ø  Nonzero
distances
presence of
of electron correlation.
presence
correlation. The
The ensemble
ensemble distances
distances of
of all
all seven
seven
states are also nonzero. The
excited states
excited
The pure
pure distances,
distances, however,
however, for
for all
all the
the
statesInt.except
excited
statesChem.
3 and
are
to
8R. Chakraborty, D.
excited
states
and
are zero
zero
to arbitrary
arbitrary digits
digits of
of
A. Mazziotti
J. Quantum
116,66784-790
(2016)
precision.
precision.
Conclusion
Conclusions
The
The role
role of
of the
th
chemistry
chemistryof
ofex
e
tions
tions to
to three-,
threeecules.
ecules. While
While
boundary
boundary of
of t
excited-state
excited-state 22
ensemble
ensembleN-rep
N-re
25
RDMs
RDMs to
to lie
lie ini
the pure 1-RDM set (pure distance) but also the harmonic,
geometric, and arithmetic means of the Euclidean distances to
each facet of the set. While the harmonic mean is similar to
the minimum distance, the geometric and arithmetic means
provide information about the distance to the majority of fac-
ence of electron
seven excited sta
RDMs of these s
The pure distanc
states 3 and 6 are
and 6 have the g
have the highes
•  Assuming single
excitations,
7x7 matrix
site and
coupling
energies,
on
Table
3. Minimumthe
distances
to theofpure,
ensemble,
and Slater
boundadiagonalization, gives seven stationary state wavefunctions
excited states. In
ries for the ground and excited states of the seven-site FMO complex are
shown.
state 2, and state
•  We use the euclidean distance metric to study the nature of pinning in the FMO
and 6 have the
Euclidean distances
degenerate pairs.
21
FMO
State
Energy (cm )
Pure
Ensemble
Slater
in the FMO comp
plexity of excited
Ground
0
0.00
0.0000
0.0000
0.0000
Excited
1
223.74
0.0000
0.0005
0.8876
quasi-pinned, or b
STATIONARY STATES OF THE FMO
2
3
4
5
6
7
101.97
120.96
268.37
307.13
332.00
513.32
0.0000
0.0020
0.0000
0.0000
0.0003
0.0000
0.0048
0.0085
0.0141
0.0001
0.0003
0.0003
0.8528
0.9958
0.9246
0.9320
0.9784
0.8797
ground
state in this
a Slater determinant, its
Ø  Because
Groundthe
state
corresponds
to amodel
Slaterisdeterminant
Ø  Slater,
Excited
states are
ensemble,
andcorrelated
pure distances vanish. In contrast, the Slater
Ø  distances
Nonzeroofensemble
distances
all seven excited
states are nonzero (! 0:85), reflecting the
Ø  presence
Quasi-degenerate
states areThe
quasi-pinned
of electron correlation.
ensemble distances of all seven
excited states are also nonzero. The pure distances, however, for all the
8R. Chakraborty, D. A.excited
states
states
3 and
are zero
to arbitrary digits of
Mazziotti
Int. J.except
Quantum
Chem.
116, 6784-790
(2016)
Conclusions
The role of the g
chemistry of excit
tions to three-, fo
ecules. While th
boundary of the
excited-state 2-RD
ensemble N-repre
26
RDMs to lie insi
the pure 1-RDM set (pure distance) but also the harmonic,
geometric, and arithmetic means of the Euclidean distances to
each facet of the set. While the harmonic mean is similar to
the minimum distance, the geometric and arithmetic means
provide information about the distance to the majority of fac-
ence of electron
seven excited sta
RDMs of these s
The pure distanc
states 3 and 6 are
and 6 have the g
have the highes
•  Assuming single
excitations,
7x7 matrix
site and
coupling
energies,
on
Table
3. Minimumthe
distances
to theofpure,
ensemble,
and Slater
boundadiagonalization, gives seven stationary state wavefunctions
excited states. In
ries for the ground and excited states of the seven-site FMO complex are
shown.
state 2, and state
•  We use the euclidean distance metric to study the nature of pinning in the FMO
and 6 have the
Euclidean distances
degenerate pairs.
21
FMO
State
Energy (cm )
Pure
Ensemble
Slater
in the FMO comp
plexity of excited
Ground
0
0.00
0.0000
0.0000
0.0000
Excited
1
223.74
0.0000
0.0005
0.8876
quasi-pinned, or b
STATIONARY STATES OF THE FMO
2
3
4
5
6
7
101.97
120.96
268.37
307.13
332.00
513.32
0.0000
0.0020
0.0000
0.0000
0.0003
0.0000
0.0048
0.0085
0.0141
0.0001
0.0003
0.0003
0.8528
0.9958
0.9246
0.9320
0.9784
0.8797
ground
state in this
a Slater determinant, its
Ø  Because
Groundthe
state
corresponds
to amodel
Slaterisdeterminant
Ø  Slater,
Excited
states are
ensemble,
andcorrelated
pure distances vanish. In contrast, the Slater
Ø  distances
Nonzeroofensemble
distances
all seven excited
states are nonzero (! 0:85), reflecting the
Ø  presence
Quasi-degenerate
states areThe
quasi-pinned
of electron correlation.
ensemble distances of all seven
excited states are also nonzero. The pure distances, however, for all the
8R. Chakraborty, D. A.excited
states
states
3 and
are zero
to arbitrary digits of
Mazziotti
Int. J.except
Quantum
Chem.
116, 6784-790
(2016)
Conclusions
The role of the g
chemistry of excit
tions to three-, fo
ecules. While th
boundary of the
excited-state 2-RD
ensemble N-repre
27
RDMs to lie insi
ENVIRONMENTAL EFFECTS
•  We use the pure distance as a metric to study the interaction of a quantum system
with its environment
•  Pure distance is positive or negative based whether the spectrum of occupations is
inside or outside the pure set respectively
Ø  Pure distance is ≥ 0 at all times for the closed
system
Ø  Becomes positive when sites 1 and 2 become
maximally entangled
28
ENVIRONMENTAL EFFECTS
•  Pure distance as a metric to study the interaction of a quantum system with its
environment
6R.
Chakraborty, D. A. Mazziotti Phys. Rev. A, 91, 010101(R) (2015)
29
ENVIRONMENTAL EFFECTS
•  Pure distance as a metric to study the interaction of a quantum system with its
environment
•  Pure distance is positive or negative based whether the spectrum of occupations is
inside or outside the pure set respectively
6R.
Chakraborty, D. A. Mazziotti Phys. Rev. A, 91, 010101(R) (2015)
30
ENVIRONMENTAL EFFECTS
•  Pure distance as a metric to study the interaction of a quantum system with its
environment
•  Pure distance is positive or negative based whether the spectrum of occupations is
inside or outside the pure set respectively
Ø  Pure distance is ≥ 0 at all times for the closed
system
Ø  Becomes positive when sites 1 and 2 become
maximally entangled
31
ENVIRONMENTAL EFFECTS
•  We use the pure distance as a metric to study the interaction of a quantum system
with its environment
•  Pure distance is positive or negative based whether the spectrum of occupations is
inside or outside the pure set respectively
Ø  Pure distance is ≥ 0 at all times for the closed Ø  Pure distance is ≤ 0 at most times
system
Ø  Spectrum enters the pure set when sites 1 and
Ø  Becomes positive when sites 1 and 2 become
2 become maximally entangled
32
maximally entangled
ANTISYMMETRY AND ENERGY
MINIMIZATION
•  Environmental noise increases the size of the set of 1-RDMs accessible to a quantum
system facilitating in the transfer of excitation to the reaction center
•  Information about the interaction of a many-electron quantum system with its
environment is encoded in the 1-RDM which scales polynomially with system size
Closed
6R.
Chakraborty, D. A. Mazziotti Phys. Rev. A, 91, 010101(R) (2015)
Open
33
OPENNESS OF QUANTUM SYSTEMS FROM
GENERALIZED PAULI CONDITIONS
•  Generalized Pauli constraints ensure orbital occupations arise from pure quantum
states
•  Give sufficient conditions for openness of a many-electron quantum system from
sole knowledge of the 1-RDM
•  Can be used to quantify the degree of openness in many-electron quantum
systems
•  Highlight interplay between antisymmetry and energy minimization photosynthetic
energy transfer
•  Emphasize the coaction of entanglement and decoherence in quantum
information transport
6R.
7R.
Chakraborty, D. A. Mazziotti Phys. Rev. A, 91, 010101(R) (2015)
Chakraborty, D. A. Mazziotti Int. J. Quantum Chem. 2015
34
REFERENCES
1.  A. J. Coleman, Rev. Mod. Phys. 35, 668 (1963)
2.  R. E. Borland and K. Dennis, J. Phys. B 5, 7 (1972)
3.  A. A. Klyachko, J. Phys. Conf. Ser. 36, 72 (2006)
4.  C. Schilling, M Christandl, D. Gross, Phys, Rev. A. (2013)
5.  M. Mohseni, P. Rebentrost, S. Lloyd, and A. Aspuru-Guzik, J. Chem. Phys. 129, 174106 (2008)
6.  R. Chakraborty, D. A. Mazziotti, Phys. Rev. A, 89, 042505 (2014)
7.  R. Chakraborty, D. A. Mazziotti Phys. Rev. A, 91, 010101(R) (2015)
8.  R. Chakraborty, D. A. Mazziotti Int. J. Quantum Chem., 115, 1305-1310 (2015)
9.  R. Chakraborty, D. A. Mazziotti Int. J. Quantum Chem. 116, 784-790 (2016)
10. R. Chakraborty, D. A. Mazziotti, 2016, Manuscript being prepared
35
THANKS!
The Mazziotti Group
36