ENTANGLEMENT IN SMALL
SELF-CONTAINED QUANTUM FRIDGES
NICOLAS BRUNNER, RALPH SILVA,
PAUL SKRZYPCZYK, MARCUS HUBER
NOAH LINDEN & SANDU POPESCU
SINGAPORE AUG 2013
3-QUBIT FRIDGE
i [ H0 + H
ee Hamiltonian and place3-QUBIT
an interaction
between the
FRIDGE
ubits. This interaction takes the form
Hint = g (|010i h101| + |101i h010|) .
(2)
We require that this interaction Hamiltonian couples
nly states within a degenerate subspace of the free
amiltonian, such that the coupling constant g can
e taken to be arbitrarily small while still producing
hanges in the steady state behaviour of the refrigeror. In this regime, where g ⌧ Ej , the eigenvalues
nd eigenstates remain governed by H0. The above
quirements thus impose
that E2 = E1 + E3 so that
DESIGN
e states |010i and |101i , connected via the interacon Hamiltonian, become degenerate in energy.
Finally, each qubit is taken to be in contact with a
eparate thermal reservoir. The temperatures of the
As shown in
lytically for all
takes the form
where g is a
upon all param
and temperatu
with a single o
The important
be shown that
g > 0. In this c
ary state that is
ature TS < TC
erator tends to
FIG. 1. Schematic diagram of the quantum refrigerator. The
i [ Heach
+H
fridge
contains
three
qubits
(inside
the
0each
fridge
contains
three
qubits
(inside
theyellow
yellowcircle),
circle),
in weak
thermal
contact
(wiggly
lines)
with
aabath
atat aadifin weak
thermal
contact
(wiggly
lines)
with
bath
difee Hamiltonian
and
place
an
interaction
between
the
3-QUBIT
FRIDGE
ferent
temperature.
qubits
interact
interferent
temperature.
The
qubits
interactvia
viathe
theweak
weak
inter-in
ubits.
This
interaction
takesThe
the
form
As shown
action
Hamiltonian
H intH,int
which
couples
degenerate
action
Hamiltonian
, which
couplesthe
thetwo
two
degenerate
lytically
for all
levels
The
lower
H
= |gi010
010
i |h101
101
|i ,+ depicted
i h010|)by.bythe
(2)
(|and
int| 010
levels
i and
|101
i|,101
depicted
thearrows.
arrows.
The
lower
takes
the
form
qubitqubit
(purple)
is the
object
to to
bebe
cooled.
t equilibrium,
(purple)
is the
object
cooled.AAt
equilibrium, itit
We require
that
this interactionTHamiltonian
couples
reaches
a temperature
other
reaches
a temperatureS T<S <TCT. CThe
. The
othertwo
twoqubits
qubits(red
(red
nly and
statesblue)
within
degenerate
subspace
of the free to heat baths at
areaare
the
machine
qubits,
connected
and blue)
the
machine
qubits,
connected to heat baths at
amiltonian,
such that
the coupling
where g is a
temperatures
TR and
TH T
. H . constant g can
temperatures
TR and
e taken to be arbitrarily small while still producing
upon all param
hanges in the steady state behaviour of the refrigerand temperatu
or. In this regime, where g ⌧ Ej , the eigenvalues
with a single o
freefree
Hamiltonian
andand
place
interaction
the
Hamiltonian
place
interaction
between
the
nd eigenstates
remain governed
by
Han
The
above between
The
important
0.an
qubits.
This
interaction
quirements
thus
impose
that E2 takes
= takes
E1 +the
E
soform
that
be shown that
qubits.
This
interaction
the
DESIGN
3 form
e states |010i and |101i , connected via the interacg > 0. In this c
on Hamiltonian,INTERACTION
Hbecome
i h101
|energy.
+| +|101
i hi010
(2)
(|g010
=degenerate
i hin101
|101
h010|)ary
|). . state that
(2)is
(|010
intH=
int g
ature TS < TC
Finally, each qubit is taken to be in contact with a
erator
tends to
eparate
reservoir.
The
temperatures
of the
We
require
this
interactionHamiltonian
Hamiltonian
couples
Wethermal
require
thatthat
this
interaction
couples
FIG. 1. Schematic diagram of the quantum refrigerator. The
i [ Heach
+H
fridge
contains
three
qubits
(inside
the
0each
fridge
contains
three
qubits
(inside
theyellow
yellowcircle),
circle),
in weak
thermal
contact
(wiggly
lines)
with
aabath
atat aadifin weak
thermal
contact
(wiggly
lines)
with
bath
difee Hamiltonian
and
place
an
interaction
between
the
3-QUBIT
FRIDGE
ferent
temperature.
qubits
interact
interferent
temperature.
The
qubits
interactvia
viathe
theweak
weak
inter-in
ubits.
This
interaction
takesThe
the
form
As shown
action
Hamiltonian
H intH,int
which
couples
degenerate
action
Hamiltonian
, which
couplesthe
thetwo
two
degenerate
lytically
for all
levels
The
lower
H
= |gi010
010
i |h101
101
|i ,+ depicted
i h010|)by.bythe
(2)
(|and
int| 010
levels
i and
|101
i|,101
depicted
thearrows.
arrows.
The
lower
takes
the
form
qubitqubit
(purple)
is the
object
to to
bebe
cooled.
t equilibrium,
(purple)
is the
object
cooled.AAt
equilibrium, itit
We require
that
this interactionTHamiltonian
couples
reaches
a temperature
other
reaches
a temperatureS T<S <TCT. CThe
. The
othertwo
twoqubits
qubits(red
(red
nly and
statesblue)
within
degenerate
subspace
of the free to heat baths at
areaare
the
machine
qubits,
connected
and blue)
the
machine
qubits,
connected to heat baths at
amiltonian,
such that
the coupling
where g is a
temperatures
TR and
TH T
. H . constant g can
temperatures
TR and
e taken to be arbitrarily small while still producing
upon all param
hanges in the steady state behaviour of the refrigerand temperatu
or. In this regime, where g ⌧ Ej , the eigenvalues
with a single o
freefree
Hamiltonian
andand
place
interaction
the
Hamiltonian
place
interaction
between
the
nd eigenstates
remain governed
by
Han
The
above between
The
important
0.an
qubits.
This
interaction
quirements
thus
impose
that E2 takes
= takes
E1 +the
E
soform
that
be shown that
qubits.
This
interaction
the
DESIGN
3 form
e states |010i and |101i , connected via the interacg > 0. In this c
on Hamiltonian,INTERACTION
Hbecome
i h101
|energy.
+| +|101
i hi010
(2)
(|g010
=degenerate
i hin101
|101
h010|)ary
|). . state that
(2)is
(|010
intH=
int g
ature TS < TC
Finally, each qubit is taken to be in contact with a
BIAS
COOLING
erator
tends to
eparate
thermal
reservoir.
The
temperatures
of
the
require
this
interactionHamiltonian
Hamiltonian
couples
We We
require
thatthat
this
interaction
couples
We require that this interaction Hamiltonian couples
only states within a degenerate subspace of the free
THEthat
MODEL
Hamiltonian, such
the coupling constant g can
be taken to be arbitrarily small while still producing
changes in the steady state
behaviour
of the refrigerFREE
HAMILTONIAN
ator. In this regime, where g ⌧ Ej , the eigenvalues
and eigenstates remain governed by H0. The above
WITH
requirements thus impose
that E2 = E1 + E3 so that
the states |010i and |101i , connected via the interaction Hamiltonian, become degenerate in energy.
Finally, each qubit is taken to be in contact with a
separate thermal reservoir. The temperatures of the
reservoirs are denoted by TC (cold), TR (room), and
TH (hot), for qubits 1, 2 and 3 respectively. The thermal contact between each qubit and bath is governed
by Linbladian dissipative dynamics, which we model
here using a simple reset model, the justification of
which we shall comment on briefly. In this model,
with probability pi dt per time dt, qubit i is reset to the
thermal state t , at the temperature of its bath, while
whe
upon
and
with
The
be sh
g>
ary s
atur
erato
Ar
ertie
close
refer
insp
a ful
direc
no e
ferent temperature. The qubits interact via the we
We require thataction
this interaction
couples
HamiltonianHamiltonian
Hint , which couples
the two de
only states within
a |degenerate
subspace
of by
thethe
free
levels
010i and |101
i , depicted
arrows. Th
THE
MODEL
Hamiltonian, such
the is
coupling
g canAt equilib
whe
qubit that
(purple)
the objectconstant
to be cooled.
a temperature
TS still
< TCproducing
. The other two
qu
be taken to be reaches
arbitrarily
small while
upon
blue)state
are
the
machine of
qubits,
connected to and
heat
changes in the and
steady
behaviour
the refrigerFREE
HAMILTONIAN
temperatures
ator. In this regime,
whereTgR and
⌧ ETH, .the eigenvalues
with
j
and eigenstates remain governed by H0. The above
The
WITH
requirements thus impose
that E2 = E1 + E3 so that
be sh
and place
an interaction
betw
the states |010ifree
andHamiltonian
|101i , connected
via the
interacg>
qubits.
This
interactionintakes
the form
INTERACTION
tion Hamiltonian,
become
degenerate
energy.
ary s
atur
Finally, each qubit is taken
in icontact
H int =togbe
h101| + with
|101iah010|)
.
(|010
erato
separate thermal reservoir. The temperatures of the
reservoirs are denoted
by that
TC (cold),
TR (room),Hamiltonian
and
Ar
We require
this interaction
TH (hot), for qubits
2 and
3 respectively.
The subspace
therertie
only 1,
states
within
a degenerate
of
mal contact between
each qubit
andthat
baththe
is governed
close
Hamiltonian,
such
coupling constan
by Linbladian dissipative
which
we model
be taken todynamics,
be arbitrarily
small
while stillrefer
pro
here using a simple
reset
model,
thestate
justification
of of the
inspr
changes
in the
steady
behaviour
which we shallator.
comment
briefly.where
In this
model,
a ful
In thison
regime,
g ⌧
Ej , the eigen
with probability
pi dteigenstates
per time dt,remain
qubit i governed
is reset to by
the H0. direc
and
The
thermal state t , at the temperature of its bath, while
no e
this regime, w here g ⌧ ferent
Ej , the
eigenvalues
ith a via
single
o
temperature.
The qubitsw
interact
the we
We require thataction
this interaction
Hamiltonian
couples
H
,
which
couples
the two de
enstates remain
governed by
HHamiltonian
.
The
above
The
important
int
0
only states within
a |degenerate
subspace
of by
thethe
free
levels
010
i
and
|
101
i
,
depicted
arrows.
Tht
ments thus impose
that
E
=
E
+
E
so
that
be
show
n
that
THE
MODEL
2
3
1 the is
Hamiltonian, such
coupling
g canAt equilib
whe
qubit that
(purple)
the objectconstant
to be cooled.
s | 010i and |be
101taken
i , connected
via
the
interacg. The
> 0.
In this
ca
a temperature
TS still
< TCproducing
other
two
qu
to be reaches
arbitrarily
small
while
upon
blue)
are
the
machine of
qubits,
connected
to and
heat
miltonian, become
degenerate
instate
energy.
state that
is
changes
in the and
steady
behaviour
theary
refrigerFREE
HAMILTONIAN
strong
events. It isSstraigh
temperatures
T
and
T
.
R
H
ature T < with
TC .
ator.
In
this
regime,
where
g
⌧
E
,
y, each qubit is taken to be in contact w ithj athe eigenvalues
tion
of
motion
fortends
the
ref
strong
events.
and eigenstates
remain governed
above
The
erator
to
thermal reservoir.
The temperatures
of by
theH0. The
WITH
dissipation
[17],
wofhich
requirements
thus impose
that
E2and
= E1 + E
so that
beis
sh
motion
3 tion
s are
denoted
by
T
(cold),
T
(room),
Around
theCa
R
C
strong
events.
It states
is
toiderive
the
equaandthe
place
an interaction
betw
strong
events.
It straightforward
is straightforward
derive
equathe
|010ifree
andHamiltonian
|101
,toconnected
via
the
interacg [17
>
ter
equation
dissipation
, for
qubits
1,
2
and
3
respectively.
The
thererties
of
r
for
qubits.
This
interaction
tion
ofof
motion
forHamiltonian,
thethe
refrigerator
using
thisthis
model
ofenergy.
tion
motion
for
refrigerator
using
model
ofthe form S ary s
INTERACTION
tion
become
degenerate
intakes
ter equation
actdissipation
betw
een
each
qubit
and
bath
is
governed
close
to
the
Car
[17],
which
is
given
by
the
following
Masdissipation
[17],
which
is
given
by
the
following
MasE2
atur
Finally, each qubit is taken
togbe
in icontact
with
ah010|) .
∂r
H
=
010
h
101
|
+
|
101
i
(|
int
terterequation
adian
dissipative
dynamics,
w hich wThe
e model
refrig
equation
E3 thermalEreservoir.
erato
= − refer
i [ Hof0 to
+∂r as
H int
,r
2
separate
temperatures
the
∂tinteraction
− iAr
[EH
ng a simple reset
model,
the
justifi
ofTR (room),
inspection
reservoirs
are denoted
by that
Tcation
(cold),
and = of
We
require
this
Hamiltonian
C
∂r∂r
∂t
THERMALISATION
1
eE
shall
comment
on
briefl
y.
In
this
model,
a
fully
(hot),
for
3(Tr
The subspace
ther-separab
ertie
H
,+ronly
]Â
+Â
(and
t iTr
⌦
r ) (3)
= =− i−T
[ HHi [0H
+0 +
Hint
,int
rE]qubits
p1,
( tp2ii ⌦
rrespectively.
)i (−ra)r degenerate
)− (3)
states
within
of
i
i
1
∂t ∂t
i i each
mal contact
qubit
andthe
bath
is direct
governed
close
T
bability
p
dt
per
time
dt,
i
is
reset
to
product
w
here
t
=
r
|
0
i
h
0
|
+
(
1|

RESET
QUBIT
TOqubit
THERMAL
STATE
Rbetween
Hamiltonian,
such
that
the
coupling
constan
i
i
i
i
TH
T
where
t
=
r
R
i refer
i
−
E
/ hile
Ti which
by
Linbladian
dissipative
dynamics,
we
model
i
be
taken
to
be
arbitrarily
small
while
still
pro
state
t
,
at
the
temperature
of
its
bath,
w
no
entangleme
e
)
.
In
general
one
w
− Ei / Ti ) . In ge
i t t=i =
where
0| (+1a−
(1
−
r
)|
1
i
h
1
|
with
r
=
1/
(
1
+
where
rhere
0i|h+
r
)|
1
i
h
1
|
with
r
=
1/
(
1
+
e
i
i
i
i
i |r0ii|i0hiusing
i
i
i
simple
reset
model,
the
justification
of of the
insp
changes
in the
steady
state
behaviour
r
−
E
/
T
−
E
/
T
ther
times
it
evolves
unitarily
according
to
interestingly,
th
i
i
i
i
ditional
terms
in
equation
. In
general
one
would
expect
there
to
be
ade e ) . )In
general
one
would
expect
there
to
be
additional
terms
which we shallator.
comment
on
briefly.where
In this
model,
a ful
In
this
regime,
g
⌧
E
,
the
eigen
j w ithin
bined
H
amiltonian
H
+
H
.
That
is,
in
this
region
0
ditional
terms
equation
corresponding
to
dissiditional
terms
in in
equation
(3),int
corresponding
to dissipative
dynamics
ondynam
with
probability
p(3),
per
time
dt,remain
qubit
i governed
is resetpative
to by
the
and
Hqubi
The
i dteigenstates
0. direc
hermalization
events
are
befrom
rare
but
Carnot
dynamics
qubit
j originating
from
the
comdynamics
onon
qubit
originating
the
comTpative
TCtjtaken
Cpative
thermal
state
, at
thetotemperature
of
its
bath,
whilepoint.
no eT
this regime, w here g ⌧ ferent
Edissipation
eigenvalues
winteract
ith
a via
single
of
temperature.
The
qubitsis
the
we
j , the
[17],
which
given
by
the
We require thataction
this interaction
Hamiltonian
couples
H
,
which
couples
the two de
enstates remain
governed by
HHamiltonian
.
The
above
The
important
int
0
teraequation
only states within
degenerate
subspace
of by
thethe
free
levels
|
010
i
and
|
101
i
,
depicted
arrows.
Tht
ments thus impose
that
E
=
E
+
E
so
that
be
show
n
that
THE
MODEL
2
3
1 the is
Hamiltonian, such
coupling
g canAt equilib
whe
qubit that
(purple)
the objectconstant
to be cooled.
sE2| 010i and |be
101taken
i , connected
via
the
interacg. The
> 0.
In this
ca
TS still
< TCproducing
other
two
qu
∂ra temperature
to be reaches
arbitrarily
small
while
upon
= the
− iHAMILTONIAN
[ H0 + H
,ary
rrefriger]connected
+state
pthat
⌦
blue)
are
machine
qubits,
miltonian, become
degenerate
instate
energy.
is
int
i (tot and
iheat
Â
changes
in the and
steady
behaviour
of
the
FREE
∂t
strong
events. It is
Sstraigh
temperatures
T
and
T
.
i
R
H
ature
T
< with
TC .
ator.
In
this
regime,
where
g
⌧
E
,
the
eigenvalues
y, each qubit is taken to be in contact w ithj a
tion
of
motion
fortends
the
ref
strong
events.
and eigenstates
remain governed
by
H0. The
above
The
erator
to
thermal reservoir.
The
temperatures
of
the
T
where
ti =
r i |E02i i=h0E| 1++(E1[17],
−tion
r ithat
)|
1hich
imotion
| wi
R
i h1be
WITH
dissipation
w
issh
requirements
thus
impose
that
so
of
3
s are
denoted
by
T
(cold),
T
(room),
and
Around
theCa
R
−
E
/
T
C
strong
events.
It states
is
the
equaandthe
place
anwould
interaction
betw
strong
events.
It straightforward
is straightforward
derive
equai to
i ) .derive
e Hamiltonian
general
one
expect
the
|010ifree
and
|101
i ,toIn
connected
via
the
interacg [17
>
ter
equation
dissipation
, for
qubits
1,
2
and
3
respectively.
The
thererties
of
r
for
qubits.
This
interaction
form S ary s
tion
ofof
motion
forHamiltonian,
thethe
refrigerator
using
thisthis
model
ofenergy.
tion
motion
for
refrigerator
using
ofthe(3),
INTERACTION
tion
become
degenerate
intakes
ditional
terms
in model
equation
correspo
ter
equation
actdissipation
betw
een
each
qubit
and
bath
is
governed
close
to
the
Car
[17],
which
is
given
by
the
following
Masdissipation
[17],
which
is
given
by
the
following
MasE2
atur
Finally, each qubit
is taken
togbe
in
contact
with
ah010|) .
pative
dynamics
on
qubit
j
originating
∂r
H
=
010
i
h
101
|
+
|
101
i
(|
int
terterequation
adian
dissipative
dynamics,
w hich wThe
e model
refer
as
refrig
equation
E3 thermalEreservoir.
erato
=
−
i
[
H
+
H
,r
2
separate
temperatures
of0 to
the
∂r
int
bination
of
the
interaction
Hamiltonia
∂tinteraction
− iAr
[EH
ng a simple reset
model,
the
justifi
ofTR (room),
inspection
reservoirs
are denoted
by that
Tcation
(cold),
and = of
We
require
this
Hamiltonian
C
∂r∂r
∂t6
sipative
dynamics
on
qubit
i
=
j.
In
THERMALISATION
1
eE
shall
comment
on
briefl
y.
In
this
model,
a
fully
separab
(hot),
for
2
and
3
respectively.
The
therertie
H
,+ronly
]Â
+Â
p
(
t
⌦
Tr
(
r
)
−
r
)
(3)
= =− i−T
[ HHi [0H
+0 +
Hint
,int
rE]qubits
p1,
(
t
⌦
Tr
(
r
)
−
r
)
(3)
states
i a degenerate subspace of
i ii iwithin
i
1
∂t ∂t
one
may
expect
each
qubit
to
be
effecti
i i each
mal contact
between
qubit
andthe
bath
is direct
governed
close
T
bability
p
dt
per
time
dt,
qubit
i
is
reset
to
product
w
here
t
=
r
|
0
i
h
0
|
+
(
1|

RESET
QUBIT
TO
THERMAL
STATE
R
Hamiltonian,
such
that
the
coupling
constan
i
i
i
i
quantum
refrigerator.
The
TH
T
where
t
=
r
R
i refer
i
−
E
/baths
Ti which
by
Linbladian
dissipative
dynamics,
we
model
with
all
three
due
to
the
intera
i
be
taken
to
be
arbitrarily
small
while
still
pro
state
t
,
at
the
temperature
of
its
bath,
w
hile
no
entangleme
e
)
.
In
general
one
w
− Ei / Ti ) . In ge
i t t=i =
e the
yellow
circle),
where
h+0| (+1a−
(1
−
r
)|
1
i
h
1
|
with
r
=
1/
(
1
+
where
rhere
0i|each
r
)|
1
i
h
1
|
with
r
=
1/
(
1
+
e
i
i
i
i
i |r0ii|i0hiusing
i
i
i
simple
reset
model,
the
justification
of
insp
changes
in
the
steady
state
behaviour
of
the
r
nian
[21,
22].
However,
when
focusing
−
E
/
T
−
E
/
T
ther
times
it
evolves
unitarily
according
to
interestingly,
th
i
i
i
i
ines)
at a difditional
terms
in
equation
.bath
In
general
one
would
expect
there
to
be
ade ewith) .a)In
general
one
would
expect
there
to
be
additional
terms
which we shallator.
comment
on
briefly.where
In this
model,
a ful
In
this
regime,
g
⌧
E
,
the
eigen
j
where
p
⇡
g
⌧
E
,
these
additional
bined
H
amiltonian
H
+
H
.
That
is,
in
this
region
w
ithin
WEAK
COUPLING
REGIME
teract
via the
weak
inter0
ditional
terms
equation
corresponding
to
dissii pative
iis resetpative
ditional
terms
in in
equation
(3),int
corresponding
to dissidynamics
ondynam
with
probability
p(3),
per
time
dt,remain
qubit
i governed
to by
the
and
Hqubi
The
i dteigenstates
0. direc
hermalization
events
are
be
rare
but
Carnot
point.
ouples
thedynamics
two
degenerate
is
approximately
gpi ⌧
gnocaeT
dynamics
qubit
j originating
from
the
comonon
qubit
originating
from
the
comTpative
TCtjtaken
Cpative
thermal
state
,strength
at
thetotemperature
of
its
bath,
while
Here
our
focus
ison
on the
stationary
state
(i.e.
long
H
ere
our
focus
is
the
stationary
state
(i.e.
qubit
(purple)
is the
object
to be
(red
reaches
a
temperature
T
<
T
bination
of
the
interaction
Hamiltonian
and
the
disS
C
reaches
a
temperature
T
<
T
.
term
behaviour)
of
the
refrigerator,
r
,
which
satisfies
S
C
S
term
behaviour)
of
the
refrigerator,
r
,
w
hich
sat
hs at strong
events. Itdynamics
is straightfor
ward
to derive
the
equaS machine
sipative
on
qubit
i
6
=
j.
In
other
words,
and
blue)
are
the
qubits,
and
blue)
are
the
machine
qubits
˙
r
=
0
i.e.
SOLVING
THE
MODEL
Sof motion for the refrigerator using this model of
temperatures
TR and
TH .
r˙S = tion
0
i.e.
one
may
expect
each
qubit
to
be
effectively
in
contact
temperatures
T
and
T
.
dissipation [17], which is given by the following Mas-
or. The
R
H
withi [all
three
baths
due to
the
interaction
HamiltoH
+
H
,
r
]
=
p
(
t
⌦
Tr
(
r
)
−
r S) . (4)
), each ter equation
0
S
S
int
i i
i
MASTER EQUATION
nian
[21,H22].
on
the
regime
H0 +
r S] = when
t
⌦
Tr
(
r
)
−
r
)
.
t a dif- i [∂r
i pi (focusing
S
S
int ,However,
i
i
free
Hamiltonian
and
place an
the
= − i [p
Hi0 +⇡H g
r ] +E
p
(
t
⌦
Tr
(
r
)
−
r
)
(3)
where
,
these
additional
effects,
whose
int ,⌧
i
i
i
Â
interi
i qubits. This interaction takes t
∂t
i
As shown
in [18],
this
equation
solved
anaenerate
strength
is approximately
gpi ⌧ gcan
canbe
be
safely
ne- an
free
H
amiltonian
and
place
lower where
t i = r i |for
0This
i i h0all
| +setup
( 1 − r i is
)|1depicted
iDISSIPATOR
| with
r ischematically
= (LINDBLAD)
1/ ( 1 + Thein
glected.
Fig.
1.
lytically
values
of
parameters.
solution
i h1the
H
=
g
010
i
h
101| +
(|
A
s
show
n
in
[18],
this
equation
can
be
solved
int
−
E
/
T
i
i ) . In general one qubits.
e
would
expect
there
to
be
adum,
it
This
interaction
takes
(2)
Here
our
focus is on the stationary state (i.e. long
takes
theSTATE
form
ditional
terms
in
equation
(3), corresponding
to dissiSTEADY
ts
(red
lytically
for
all
values
of
the
parameters.
The
sol
We
require
that
this
interactio
term
behaviour)
the refrigerator,
r S, which satisfies
pative
dynamics
on qubitof
j originating
from the comaths
at
ples
states
within
a degenera
takesbination
the
+ gs
(5)
r˙S =form
i.e.interactionr Hamiltonian
of 0the
and
the
disS = t 1t 2t 3only
H
=
g
|
010
101
(
int
free sipative dynamics on qubit i 6= j. In Hamiltonian,
other words,
such that the co
one
may expect
each
to ]be
effectively
in
contact
be
taken
ig
[ H0is
+ aHqubit
,
r
=
p
(
t
⌦
Trgs
r Sbe
) −arbitrarily
r S) . (4)sma
can
where
dimensionless
parameter
depending
S
int
i
i
i (to
Â
The
TRACELESS
MATRIX
r
=
t
t
t
+
3 Hamilto1i 2changes
S therequire
with all three baths dueWe
to
interaction
that
this
interacti
each
in the
steady
state
be
cing
upon
all
parameters
of
the
model
(namely
p
,
g,
E
,
i
i
nian
[21,
22].
However,
when
focusing
on
the
regime
aen
dif-the
ator.
Inithin
this
regime,
where g
gerandAs
T
),
and
s
is
a
traceless
matrix
where
ptemperatures
g ⌧ Ei , these
additional
effects,
whose
only
states
w
a
degener
nterC,R,H
i ⇡shown
in [18],POPULATIONS
this equation
can
be AND
solved
anaw here
g ≈isis
aBETWEEN
dimensionless
parameter
depen
BIAS
OF
|010>
|101>
and
eigenstates
remain
gover
erate
strength
approximately
gp
⌧
g
can
be
safely
nei
ues
with
a
single
off-diagonal
term
(see
[18]
for
details).
lytically
for all
values
of the
the parameters.
The
solution
H amiltonian,
such
that
the
ower
glected.
This setup
is depicted
schematically
in Fig.
1. (namely
requirements
thus
impose
tha
upon
all
parameters
of
model
p
,
i
(2)
bove
The
important
property
of
the
solution
is
that
it
can
m, it
takes
form
Here
our the
focus
is on the stationary state
long|010i and |101i , con
the(i.e.
states
Â
Â
ih
i
CARNOT
hown in [18], this AROUND
equation can
be solvedPOINT
anay for all values of the parameters. The solution
e form STEADY STATE
r S = t 1t 2t 3 + gs
(5)
BIASparameter
TRACELESS
MATRIX
g is a dimensionless
depending
l parameters of the model (namely pi , g, Ei ,
mperatures TC,R,H ), and s is a traceless matrix
single off-diagonal term (see [18] for details).
portant property of the solution is that it can
wn that the refrigerator cools qubit 1 whenever
n this case, onefinds that qubit 1 is in a statione that is diagonal, with corresponding temperS < T . Moreover, the efficiency of the refrigC
ted to heat baths ati
the refrigerator cools qubit
1 whenever
r˙S =
0 i.e.
case, onefindsthat
qubit 1 isincan
aCARNOT
stationhown
in [18], this AROUND
equation
be solvedPOINT
anaitemper[ H0The
+ Hsolution
diagonal,
withofcorresponding
int , r S] = Â pi ( t i ⌦Tr i (
ysfor
all values
the parameters.
Moreover, the efficiency of the refrigi
Ce. form
STEADY
STATE
ction
between
o the Carnot
limitthe
in the limit g ! 0.
m
As
shown
in
[18],
this
equation
can
Carnot point.rLet
usfirst
discuss
theprop=
t
t
t
+
gs
(5)
1 2 3
S
lytically
for all values of the paramete
r those refrigerators which
are operating
|)
BIAS
TRACELESS
MATRIX
gh010
is efficiency
a .dimensionless
parameter
depending
takes
the form
arnot
–(2)
which
from
hereon
we
l parameters
of the
theCarnot
modelpoint.
(namely
igerators
around
From pi , g, Ei ,
miltonian
couples
mperatures
Tclear
and
CARNOT
POINT
r S = t 1t 2t 3 + gs
Eq. (5), it is
that
fors gis=a traceless
0, r S is matrix
C,R,H ),
space
of asthe
single
off-diagonal
termother
(see than
[18] the
for details).
ble state,
it isfree
nothing
portant
property
of each
thewhere
solution
it can
g
g can
g isisthat
a dimensionless
param
t ofconstant
thermal
state
for
qubit. Hence,
wn
that
the refrigerator
cools
qubit
1 whenever of the model (n
ent
is present
at the Carnot
point.
e still
producing
upon
allMore
parameters
n this
case,
onefinds
qubit
is in a stationthis
remainsthat
true
for
a1small
ur
ofstatement
the
refrigerand
temperatures
TC,R,H ), and s is a
thatset
is diagonal,
corresponding
of all r S inwith
thewith
vicinity
of the temper,eSthe
the
eigenvalues
a single
off-diagonal term (see
< TCall. Moreover,
the
efficiency
of the refrigThus
refrigerators
which
are highly
y H0. The above
The important property of the soluti
ted to heat baths ati
the refrigerator cools qubit
1 whenever
r˙S =
0 i.e.
case, onefindsthat
qubit 1 isincan
aCARNOT
stationhown
in [18], this AROUND
equation
be solvedPOINT
anaitemper[ H0The
+ Hsolution
diagonal,
withofcorresponding
int , r S] = Â pi ( t i ⌦Tr i (
ysfor
all values
the parameters.
Moreover, the efficiency of the refrigi
Ce. form
STEADY
STATE
ction
between
o the Carnot
limitthe
in the limit g ! 0.
m
As
shown
in
[18],
this
equation
can
Carnot point.rLet
usfirst
discuss
theprop=
t
t
t
+
gs
(5)
1 2 3
S
lytically
for all values of the paramete
r those refrigerators which
are operating
|)
BIAS
TRACELESS
MATRIX
gh010
is efficiency
a .dimensionless
parameter
depending
takes
the form
arnot
–(2)
which
from
hereon
we
l parameters
of the
theCarnot
modelpoint.
(namely
igerators
around
From pi , g, Ei ,
miltonian
couples
mperatures
Tclear
and
CARNOT
POINT
r S = t 1t 2t 3 + gs
Eq. (5), it is
that
fors gis=a traceless
0, r S is matrix
C,R,H ),
space
of asthe
single
off-diagonal
termother
(see than
[18] the
for details).
ble state,
it isfree
nothing
portant
property
of each
the
solution
it can
g
g can
where
g isisthat
a
dimensionless
t ofconstant
thermal
state
for
qubit.CARNOT
Hence,
ALSO
TRUE
AROUND
(BALL
OF SEP STATES) param
wn
that
the refrigerator
cools
qubit
1 whenever of the model (n
ent
is present
at the Carnot
point.
e still
producing
upon
allMore
parameters
n this
case,
onefinds
qubit
is in a stationthis
remainsthat
true
for
a1small
ur
ofstatement
the
refrigerand
temperatures
TC,R,H ), and s is a
thatset
is diagonal,
corresponding
of all r S inwith
thewith
vicinity
of the temper,eSthe
the
eigenvalues
a single
off-diagonal term (see
< TCall. Moreover,
the
efficiency
of the refrigThus
refrigerators
which
are highly
y H0. The above
The important property of the soluti
ted to heat baths ati
the refrigerator cools qubit
1 whenever
r˙S =
0 i.e.
case, onefindsthat
qubit 1 isincan
aCARNOT
stationhown
in [18], this AROUND
equation
be solvedPOINT
anaitemper[ H0The
+ Hsolution
diagonal,
withofcorresponding
int , r S] = Â pi ( t i ⌦Tr i (
ysfor
all values
the parameters.
Moreover, the efficiency of the refrigi
Ce. form
STEADY
STATE
ction
between
o the Carnot
limitthe
in the limit g ! 0.
m
As
shown
in
[18],
this
equation
can
Carnot point.rLet
usfirst
discuss
theprop=
t
t
t
+
gs
(5)
1 2 3
S
lytically
for all values of the paramete
r those refrigerators which
are operating
|)
BIAS
TRACELESS
MATRIX
gh010
is efficiency
a .dimensionless
parameter
depending
takes
the form
arnot
–(2)
which
from
hereon
we
l parameters
of the
theCarnot
modelpoint.
(namely
igerators
around
From pi , g, Ei ,
miltonian
couples
mperatures
Tclear
and
CARNOT
POINT
r S = t 1t 2t 3 + gs
Eq. (5), it is
that
fors gis=a traceless
0, r S is matrix
C,R,H ),
space
of asthe
single
off-diagonal
termother
(see than
[18] the
for details).
ble state,
it isfree
nothing
portant
property
of each
the
solution
it can
g
g can
where
g isisthat
a
dimensionless
t ofconstant
thermal
state
for
qubit.CARNOT
Hence,
ALSO
TRUE
AROUND
(BALL
OF SEP STATES) param
wn
that
the refrigerator
cools
qubit
1 whenever of the model (n
ent
is present
at the Carnot
point.
e still
producing
upon
allMore
parameters
n this
case,
onefinds
qubit
is in a stationthis
remainsthat
true
for
a1small
ur
ofstatement
the
refrigerand
temperatures
TC,R,H ), and s is a
ENTANGLEMENT
ISof
DETRIMENTAL
FOR EFFICIENCY
e
that
is
diagonal,
with
corresponding
temperthe
set
of
all
r
in
the
vicinity
the
S
,S the eigenvalues
with a single off-diagonal term (see
< TCall. Moreover,
the
efficiency
of the refrigThus
refrigerators
which
are highly
y H0. The above
The important property of the soluti
ENTANGLEMENT?
efficient functionefficient
withoutfunction
entanglement.
To see this Tocally,
that
without entanglement.
see this
STEADY
STATE
let us first
rewrite
therewrite
following
letr us
first
r S inform
the following form is present.
S in
be found.
r S = w| GHZ i hGHZ
(1 − iw
) sdiag| + ( 1 − (6)
r S = |w+| GHZ
hGHZ
w) sdiag
(6)
ment (i)
al
p
p
on all t
| GHZ
= i (|
101i ) / 2 is (ii)
tripartite
where | GHZWHERE
i =where
(|010
i + ii |101
) /0102i +is i |tripartite
genuine tr
entangled
state (of the Greenberger-Horne-Zeilinger
entangled state (of
the Greenberger-Horne-Zeilinger
multiparti
and sdiagdensity
is a diagonal
hence
form), and sdiag form),
is a diagonal
matrix,density
hencematrix,
the specifi
to a fully
state. While
there
corresponding tocorresponding
a fully separable
state.separable
While there
forms of en
is noofunique
notion of in
entanglement
in multipartite
is no unique notion
entanglement
multipartite
turns
out that theof
entanglement
of
Entangle
systems, it turnssystems,
out that itthe
entanglement
states of of states
the form (6) can be conveniently characterized.
the paper,
the form (6) can be conveniently characterized.
We first note that in the vicinity of anyglement
Carnot th
We first note that in the vicinity of any Carnot
point, the state r S has full rank and off-diagonal
terms
We
will se
point, the state r Swhich
has full
rank
and
off-diagonal
terms
are small compared to diagonal ones.performan
Hence,
which are small in
compared
to diagonal
ones.
Hence,
this regime,
the state can
be decomposed
as task
r S = of
the
in this regime, the
r Sa=diagonal sepa( 1 −state
e) s 0can+be
er decomposed
( p) , where s 0as is
entanglement, our main tool will be a class of entanthe constra
At this point it is useful to recall that entanglement
and pj g ⌧
glement
developed
in [29,
which
allow
commenta cf
can witnesses
appear under
several forms
in a30]
state
of 3 qubits.
choosing
one toIndeed,
fully characterize
entanglement
of states
ENTANGLEMENT?
Weconclu
first
there can
bethe
bipartite
entanglement
alongofa
of our
the form
r Sbipartition
. Moreover,
also provide
given
(e.g.these
qubitwitnesses
1 versus qubits
2 and 3),
the
precise
c
tional
cons
function
without
entanglement.
To to
see
this To
cally,
that
or genuine
tripartite
entanglement.
In
detect
aefficient
meaningful
entropy
based
measure
oforder
multipartite
efficient
function
without
entanglement.
see
this
the
strength
we
look
fo
STEADY
STATE
our
main
tool
will
a
class
ofcondientanlet usentanglement,
first rewrite
r us
therewrite
following
form
is present.
constrain
first
r Sbe
insufficient
the
following
form the
S in
entanglement
[33]letand
necessary
and
smallest
v
glement
witnesses developed
in [29, [31].
30] which
allow
comment
fu
be found.
tions for
biseparability
for
our
system
Formally,
tion,
but
n
r Sfully
= wcharacterize
| GHZ i hGHZ
+| GHZ
(1 − iw
) sdiag
r S the
= |wentanglement
hGHZ
| +states
( 1 − (6)
w) sdiag We first
(6) p
one to
of
of
ment (i) al
these the
witnesses
are
given
by
inequalities
of
the
form
glement
is
p
form r S. Moreover, these witnesses
p also provide
tional
constr
(ii)
on all t
!multipartite
| GHZ
= i (|
i +is
i |tripartite
101i ) / 2 iswe
tripartite
where
| GHZWHERE
i =where
(|010
ibased
+ ii |101
) /0102of
find
theforop
a meaningful
entropy
measure
look
genuine tr
p (of and
entangled
state
the sufficient
Greenberger-Horne-Zeilinger
entangled
state
(of
the
Greenberger-Horne-Zeilinger
entanglement
[33]
and
necessary
condithat
theval
st
smallest
WS ( r ) = 2 |r 3,6| − Â r k,kr 9− k,9− k
0 (7)
multiparti
form),
and
s
is
a
diagonal
density
matrix,
hence
tions
forsbiseparabi
lity
for
our
system
[31].
Formally,
diag
form),
and
is
a
diagonal
density
matrix,
hence
tion,
but
no
ment
witn
diag
k2 S
the specifi
ENTANGLEMENT
WITNESSES
a fully
state. While
there
these
witnesses
given
by to
inequalities
of
the form
glement
is hp
corresponding
tocorresponding
aare
fully
separable
state.separable
While
there
metries),
!
where
ofnotion
the density
matrix
andin multipartite
forms
en
iselements
noofunique
of in
entanglement
find
theof
opti
i,j denotes
is no runique
notion
entanglement
multipartite
partition. T
p
itthe
out
that
the entanglement
states
the
set S depends
on
and
of entanthat
the of
stat
Entangle
W
( r ) =systems,
2 that
|the
r 3,6|partition
−turns
r k,k
r 9−type
0 (7)
Âentanglement
k,9− kof states
systems,
it Sturns
out
of of
servewitnes
that
ment
the form (6)
be conveniently
characterized.
kcan
2 SWhen
glement
one
is
interested
in.
inequality
(7)
is
the
paper,
the form (6) can be conveniently characterized.
peratures
metries),
We
first
note
that
in
the
vicinity
of
any
Carnothen
violated,
its
LHS
gives
the
concurrence
[32]
of
C
|
RH
glement
th
where
r
denotes
elements
of
the
density
matrix
and
i,j
We first note
that in the vicinity of any Carnot partition.
to the
case
Th
point,
the
state
r
has
full
rank
and
off-diagonal
terms
S
MEASURE
OF
ENTANGLEMENT
We
will
se
(Spoint,
= the
{ 2the
}set
),state
R
(
S
=
{
1
}
),
CR
|
H
(
S
=
{
3
}
)
or
the
S|CH
depends
on
the
partition
and
type
of
entanr Swhich
has full
rank
and
off-diagonal
termsones.
serve
thatInth
arable.
are
small
compared
to
diagonal
Hence,
glement
one
is
interested
in.
When
inequality
(7)
is
performan
genuine
multipartite
concurrence
(see Refs.
[33,
34])
which are
small in
compared
to
diagonal
ones.
Hence,
peratures
wa
provides
this
regime,
the
state
can
be
decomposed
as
r
=
violated,
its
LHS
gives
the
concurrence
[32]
of
C
|
RH
S of
the
task
for
S
=
{
1,
2,
3
}
.
When
inequality
(7)
holds,
no
entanto
the
case
0
0
in this regime, the
r Sa=
GUHNE
SEEVINCK NJP
HUBER et
PRL
2010 w
isal.quantifie
( 1 −state
e) s can+be
er decomposed
( p) ,&where
s as2010,
is
diagonal
sepa-
ENTANGLEMENT ZOO
1. ENTANGLEMENT BETWEEN ANY BIPARTITION
2. GENUINE TRIPARTITE ENTANGLEMENT
ENTANGLEMENT ZOO
1. ENTANGLEMENT BETWEEN ANY BIPARTITION
2. GENUINE TRIPARTITE ENTANGLEMENT
DOES THIS ENTANGLEMENT PLAY ANY ROLE?
COOLING
CONSIDER A GIVEN OBJECT (QUBIT) TO BE COOLED
FIX: ENERGY, BATH (TEMPERATURE TC, COUPLING)
COOLING
CONSIDER A GIVEN OBJECT (QUBIT) TO BE COOLED
FIX: ENERGY, BATH (TEMPERATURE TC, COUPLING p1)
CONSIDER GIVEN RESSOURCES: HOT BATH (TH) COLD BATH (TR)
COOLING
CONSIDER A GIVEN OBJECT (QUBIT) TO BE COOLED
FIX: ENERGY, BATH (TEMPERATURE TC, COUPLING p1)
CONSIDER GIVEN RESSOURCES: HOT BATH (TH) COLD BATH (TR)
1. OPTIMIZE COOLING  TS (LOWEST T FOR QUBIT)
FREE PARAMETERS: E2 and g, p2, p3 << Ei
COOLING
CONSIDER A GIVEN OBJECT (QUBIT) TO BE COOLED
FIX: ENERGY, BATH (TEMPERATURE TC, COUPLING p1)
CONSIDER GIVEN RESSOURCES: HOT BATH (TH) COLD BATH (TR)
1. OPTIMIZE COOLING  TS (LOWEST T FOR QUBIT)
FREE PARAMETERS: E2 and g, p2, p3 << Ei
2. OPTIMIZE COOLING IMPOSING SEPARABILITY  TS*
to the case w hen the system is constrained to
{ 1} ), CR| H (S = { 3} ) or the
arable. In the regime w here TR ⌧ TH , entang
oncurrence (see Refs. [33, 34])
provides
a significant enhancement in cooling
COOLING
ENHANCEMENT
inequality (7) holds, no
entanis quantified by the ratio
he given bipartition.
T − TS
the Carnot point w e find,
RELATIVE COOLING ENHANCEMENT z = C
.
he parameter space numeriTC − TS⇤
to the case w hen the system is constrained to
{ 1} ), CR| H (S = { 3} ) or the
arable. In the regime w here TR ⌧ TH , entang
oncurrence (see Refs. [33, 34])
provides
a significant enhancement in cooling
COOLING
ENHANCEMENT
inequality (7) holds, no
entanis quantified by the ratio
he given bipartition.
T − TS
the Carnot point w e find,
RELATIVE COOLING ENHANCEMENT z = C
.
he parameter space numeriTC − TS⇤
1.0006
1.0004
1.0002
1.0000
1.0006
1000
1.0004
900
1.0002
800
1.0004
700
TH
1.0002
600
1.0000
500
1.0002
400
1.0001
300
(a)
1
3
2
1.0000
4
TR
NO ENHANCEMENT
1
2
3
4
(b)
COOLING ENHANCEMENT
TH = 300K
1.0008
TH = 600K
TH = 900K
1.0006
1.0004
1.0002
1.0000
0.010
0.020
0.030
MONOTONOUS RELATION BTW COOLING ENHANCEMENT
AND ENTANGLEMENT (CONCURRENCE)
COOLING ENHANCEMENT
TH = 300K
1.0008
TH = 600K
TH = 900K
1.0006
1.0004
1.0002
1.0000
0.010
0.020
0.030
MONOTONOUS RELATION BTW COOLING ENHANCEMENT
AND ENTANGLEMENT (CONCURRENCE)
FUNCTIONAL RELATIONSHIP?
Entanglement
vsTRANSPORT
Energy Transport
ENERGY
Optimized
entangleme
across R||C
correspond
In vs Energ
ENTANGLEMENT:
ENERGY IN / ENERGY OUT
OPEN QUESTIONS
• BEYOND WEAK COUPLING REGIME
• OTHER MODELS
• MACROSCOPIC FRIDGES
• HEAT ENGINES
• QUANTUM EFFECTS IN BATHS