Resource theory of quantum states out of thermal equilibrium.

TL;DR: It is shown that the free energy of thermodynamics emerges naturally from the resource theory of energy-preserving transformations, provided that a sublinear amount of coherent superposition over energy levels is available, a situation analogous to the sub linear amount of classical communication required for entanglement dilution.

Abstract: The ideas of thermodynamics have proved fruitful in the setting of quantum information theory, in particular the notion that when the allowed transformations of a system are restricted, certain states of the system become useful resources with which one can prepare previously inaccessible states. The theory of entanglement is perhaps the best-known and most well-understood resource theory in this sense. Here, we return to the basic questions of thermodynamics using the formalism of resource theories developed in quantum information theory and show that the free energy of thermodynamics emerges naturally from the resource theory of energy-preserving transformations. Specifically, the free energy quantifies the amount of useful work which can be extracted from asymptotically many copies of a quantum system when using only reversible energy-preserving transformations and a thermal bath at fixed temperature. The free energy also quantifies the rate at which resource states can be reversibly interconverted asymptotically, provided that a sublinear amount of coherent superposition over energy levels is available, a situation analogous to the sublinear amount of classical communication required for entanglement dilution.

Summary

Summary

• For simplicity, the authors shall first pretend that both ' T and n consist of a single type and subsequently show how to extend the argument to all strongly typical types.
• The authors begin with a single composite type, a concatenation of a type coming from the resource state and one from the reservoir state.
• Using these three conditions, the authors find that the transformation is possible for any R such that EQUATION 0.
• The formation protocol is similar to the distillation protocol and is again based on considering type transformations satisfying the three requirements of energy conservation, unitarity, and dimension conservation.
• Distillation for arbitrary resource states is related to that of stationary states, and the authors can recycle part of the previous distillation protocol.
• Now, imagine projecting the resource state onto the various energy subspaces, destroying any coherence between them.
• Thus, the authors may, instead, imagine that this projection is performed after the distillation step.
• The extra resource of the reference system is, thus, of a size sublinear in n and does not affect the rate calculations.
• This creates an interesting asymmetry between distillation and formation, akin to a similar phenomenon in the resource theory of entanglement, where Conclusions.
• The authors should emphasize that although the procedures they have described for the conversion of resource states may seem quite unnatural from a physical point of view, their use is to establish the ''in principle'' interconversion rate given in Theorem 1.

Full text

Resource Theory of Quantum States Out of Thermal Equilibrium
Fernando G. S. L. Branda
˜
o,
1,2
Michał Horodecki,
3,4
Jonathan Oppenheim,
5
Joseph M. Renes,
6,7,
*
and Robert W. Spekkens
8
1
Departamento de Fı
´
sica, Universidade Federal de Minas Gerais, Caixa Postal 702, Belo Horizonte, Minas Gerais 30123-970, Brazil
2
Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore
3
Institute for Theoretical Physics and Astrophysics, University of Gdan
´
sk, PL-80952, Gdan
´
sk, Poland
4
National Quantum Information Centre of Gdan
´
sk, 81-824 Sopot, Poland
5
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road,
Cambridge CB3 0WA, United Kingdom
6
Institut fu
¨
r Angewandte Physik, Technische Universita
¨
t Darmstadt, Hochschulstrasse 4a, 64289 Darmstadt, Germany
7
Institut fu
¨
r Theoretische Physik, ETH Zurich, CH-8093 Zu
¨
rich, Switzerland
8
Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
(Received 17 August 2012; revised manuscript received 12 March 2013; published 18 December 2013)
The ideas of thermodynamics have proved fruitful in the setting of quantum information theory, in
particular the notion that when the allowed transformations of a system are restricted, certain states of the
system become useful resources with which one can prepare previously inaccessible states. The theory of
entanglement is perhaps the best-known and most well-understood resource theory in this sense. Here, we
return to the basic questions of thermodynamics using the formalism of resource theories developed in
quantum information theory and show that the free energy of thermodynamics emerges naturally from the
resource theory of energy-preserving transformations. Specifically, the free energy quantifies the amount
of useful work which can be extracted from asymptotically many copies of a quantum system when using
only reversible energy-preserving transformations and a thermal bath at fixed temperature. The free
energy also quantifies the rate at which resource states can be reversibly interconverted asymptotically,
provided that a sublinear amount of coherent superposition over energy levels is available, a situation
analogous to the sublinear amount of classical communication required for entanglement dilution.
DOI: 10.1103/PhysRevLett.111.250404 PACS numbers: 05.30.d, 03.67.Bg, 03.67.Mn, 05.70.a
Quantum resource theories are specified by a restriction
on the quantum operations (state preparations, measure-
ments, and transformations) that can be implemented by
one or more parties. This singles out a set of states which
can be prepared under the restricted operations. If the
parties facing the restriction acquire a quantum state out-
side the restricted set of states, then they can use this state
to implement measurements and transformations that are
outside the class of allowed operations, consuming the
state in the process. Thus, such states are useful resources.
A few prominent examples serve to illustrate the idea: if
two or more parties are restricted to communicating clas-
sically and implementing local quantum operations, then
entangled states become a resource [1]; if a party is re-
stricted to quantum operations that have a particular sym-
metry, then states that break this symmetry become a
resource [2–4]; if a party is restricted to preparing states
that are completely mixed and performing unitary opera-
tions, then any state that is not completely mixed, i.e., any
state that has some purity, becomes a resource [5].
In this Letter, we develop the quantum resource theory of
states that are T athermal, i.e., not thermal at temperature T.
This provides a useful new formulation of equilibrium and
nonequilibrium thermodynamics for finite-dimensional
quantum systems, and allows us to apply new mathematical
tools to the subject. The restricted class of operations
which defines our resource theory includes only those that
can be achieved through energy-conserving unitaries and
the preparation of any ancillary system in a thermal state at
temperature T, as first studied by Janzing et al. [6] in the
context of Landauer’s principle. Here, the ancillary systems
can have an arbitrary Hilbert space and an arbitrary
Hamiltonian, and may be described as having access to a
single heat bath at temperature T. States that are not in
thermal equilibrium at temperature T are the resource in
this approach.
Quantum resource theories provide answers to questions
such as: How does one measure the quality of different
resource states? Can one particular resource state be con-
verted to another deterministically? If not, can it be done
nondeterministically, and if so with what probability?
What if one has access to a catalyst? A particularly funda-
mental problem, addressed in this Letter, is to identify the
equivalence classes of states that are reversibly intercon-
vertible in the limit of asymptotically many copies of the
resource and to determine the rates of interconversion. We
show that all T-athermal states are reversibly interconver-
tible asymptotically and that the interconversion rate is
governed by the free energies of the states involved.
The great merit of the resource theory approach is its
generality. Rather than considering the behavior of the
property of interest for some particular system with
PRL 111, 250404 (2013)
PHYSICAL REVIEW LETTERS
week ending
20 DECEMBER 2013
0031-9007=13=111(25)=250404(5) 250404-1 Ó 2013 American Physical Society

particular dynamics (as is typical in thermodynamics), one
considers instead the fundamental limits that are imposed
by the restriction defining the resource and the laws of
quantum theory. On the practical side, a better understand-
ing of a given resource helps determine how best to
implement the tasks that make use of it, and, more funda-
mentally, such an understanding may serve to clarify what
sorts of resources are even relevant for a given task. For
instance, entanglement is commonly asserted to be the
necessary resource for tasks in which the use of quantum
systems yields improved performance over the use of
classical systems. But in quantum metrology, it is asym-
metry which is relevant, not entanglement.
Finally, the resource theory approach provides a frame-
work for organizing and consolidating the results in a
given field, thermodynamics being particularly in need
of such a framework, as well as synthesizing new results.
Indeed, studying the interconvertibility of finite resources
leads to useful notions of free energy in that case, as
shown in parallel to the present work in Ref. [7], and
subsequently to a more detailed, quantitative treatment of
the Second Law in Ref. [8]. Results similar to the former
were also reported independently of [7] and the present
work by A
˚
berg [9] and later by Egloff et al. [10], who
investigated the work extractable from finite resources.
For a different approach in the same quantum setting
as ours, but more along the lines of statistical mechanics,
see Ref. [11].
Allowed operations and resource states.—We now
define the restricted class of operations and the resource
states more precisely. Given a quantum system with Hilbert
space H and Hamiltonian H, the restricted operations
are the completely positive trace-preserving maps
E: LðH Þ!LðH Þ of the form
E ðÞ¼Tr
anc
½Vð 


T
ÞV
y
; (1)
where


T
is the thermal (Gibbs) state of an arbitrary
ancillary system with Hamiltonian

H at temperature T, V
is an arbitrary unitary operation on the joint system which
commutes with the total Hamiltonian: ½V;H  I þI 

H¼0. Observe that Eð
T
Þ¼
T
, where 
T
is the Gibbs
state at temperature T associated with H. Any other state
 Þ 
T
is a resource state. While, here, we consider the
case that input and output systems and their Hamiltonians
are identical, this framework can be easily extended to the
more general case, as done by Janzing et al. [6].
The allowed operations are particularly relevant for
thermodynamics because they cannot, on their own, be
used to do work. Moreover, it is not too difficult to see
that various different kinds of T-athermal states can be
used, via the restricted class of operations, to do work. For
thermal states at a temperature T
0
distinct from T (hence T
athermal), work can be drawn using a heat engine, as we
now effectively have two heat baths at temperatures T and
T
0
; for pure states within a degenerate energy eigenspace,
work can be drawn using a Szilard engine [12]; for pure
energy eigenstates, work can be drawn directly by an
energy-conserving unitary. One is led to expect that work
can be extracted from any T-athermal state. We shall show
that asymptotically this is, indeed, the case.
It is important to note the differences between the
resource theory framework and the more usual approaches
to thermodynamics. Chiefly, all sources and sinks of en-
ergy and entropy must be explicitly accounted for: only
energy and entropy-neutral operations on the system and
thermal reservoir are allowed, rather than specific energy-
or entropy-changing operations more common in an open-
system approach. All interactions between the system and
reservoir are due to the unitary V and not an interaction
term in the total Hamiltonian. Moreover, no attempt is
made a priori to restrict the allowed operations to be
physically realistic; indeed, we assume the experimenter
has complete control over V. This ensures that the restric-
tions we find are truly fundamental, though ultimately the
operations needed to establish our main result are map-
pings between macroscopic observables and do not require
fine-grained, microscopic control. These apparent differ-
ences notwithstanding, we show in the Supplemental
Material [13] that a number of different classes of opera-
tions for thermodynamics are, in fact, equivalent.
Resource interconvertibility and free energy.—A central
question in any resource theory is that of resource inter-
conversion: Which resources can be transformed into
which others, and how easily? Generally, there exists a
quasiorder of resources: We say A  B if resource A can be
transformed into B using the allowed operations. Functions
which respect this quasiorder are known as resource
monotones. For instance, the relative entropy of entangle-
ment is a well-known resource monotone relative to local
operations and classical communication [1].
Here, we are interested in determining the optimal rate
RðA ! BÞ at which resource A can generate resource B,in
the limit of an infinite supply of A, that is, the largest R such
that A
n
 B
nR
for n !1. A simple argument, going
back to Carnot [14], implies that if the transformation
is reversible in the sense that RðB ! AÞ¼RðA ! BÞ
1
,
then the rate at which two resources can be reversibly
interconverted must achieve the optimal rate. Otherwise,
it would be possible to generate arbitrary amounts of a
resource state from a small number via cyclic transforma-
tions to and from another resource state.
That reversible interconversion is optimal (when pos-
sible) gives a simple means of characterizing the intercon-
version rate by using a ‘‘standard’’ reference resource.
Consider a transformation from A to B which proceeds
via the standard resource C: A ! C ! B. Following this
with B ! A must give a combined transformation of unit
rate, again to avoid the possibility of spontaneously
generating resources. Composing the rates, we have
RðA ! CÞRðC ! BÞRðB ! AÞ¼1,or
PRL 111, 250404 (2013)
PHYSICAL REVIEW LETTERS
week ending
20 DECEMBER 2013
250404-2

RðA ! BÞ¼
RðA ! CÞ
RðB ! CÞ
; (2)
using the fact that RðA ! BÞRðB ! AÞ¼1. With
this framework, we need only define the relative entropy
Dð k 
T
Þ¼Tr½ðln  ln
T
Þ to state the main result of
this Letter.
Theorem 1.—Using thermal operations at background
temperature T, asymptotic interconversion at nonvanishing
rate is possible between all states  and  of a system with
Hamiltonian H.For
T
the Gibbs state of temperature T
associated with H, the optimal rate is given by
Rð ! Þ¼
Dð k 
T
Þ
Dð k 
T
Þ
: (3)
Simple calculation reveals that Dð k 
T
Þ¼F

ðÞ
F

ð
T
Þ, where F

ðÞhHi

 k
B
TSðÞ is the free en-
ergy and SðÞ¼Tr½ ln the von Neumann entropy.
Thus, the free energy directly determines the optimal rate
of resource interconversion in our resource theory.
To prove the result we shall employ the connection to
free energy by constructing protocols for both distillation
of resource states into a standard state and formation of
resource states from standard states. The standard state is
chosen to have energy but no entropy, so as to represent
available work.
Before doing so, it is enlightening to note that, assuming
reversible interconversion is possible, Eq. (3) follows easily
from Ref. [15], Theorem 1, and Ref. [16], Theorem 4. This
result states that any asymptotically continuous resource
monotone f determines the interconversion rate via its
regularization f
1
ðÞ¼lim
n!1
ð1=nÞfð
n
Þ as Rð !
Þ¼f
1
ðÞ=f
1
ðÞ, provided the latter is nonzero and
finite.
Here, fðÞ¼Dð k 
T
Þ is a T-athermality monotone
(i.e., for all T-thermal operations E, D½EðÞk
T

Dð k 
T
Þ) by contractivity of the relative entropy under
quantum operations and the fact that Eð
T
Þ¼
T
. Its regu-
larization is nonzero and finite since f
1
ðÞ¼fðÞ, which
follows from the additivity of the relative entropy and the
fact the thermal state of n identical systems is just n copies
of the thermal state of one system. Finally, asymptotic
continuity follows from extensivity of energy by using
Proposition 2 of Ref. [17]; we leave the simple derivation
of this to the Supplemental Material [13].
Extensivity is crucial to the conclusion. For instance,
~
fð Þ¼Dð
T
k Þ (note the reversed order of  and 
T
)is
also a T-athermality monotone, but does not lead to the
interconversion rate; the extensivity argument fails and
~
f is
not asymptotically continuous. Nonetheless,
~
fð Þ plays an
important role in determining the resource requirements
for creating low-temperature states [6].
Distillation and formation protocols.—In order to estab-
lish Theorem 1, let us now turn to the distillation and
formation protocols. For purposes of exposition, we
specialize to the case of resources having just two non-
degenerate energy levels, call them j0i and j1i, i.e., qubits.
This, nevertheless, captures the essential aspects of the
problem. First, we consider the distillation and formation
of quasiclassical resources , meaning ½; H¼0 and take
up the case of nonstationary resources afterwards. In what
follows, we sketch the steps required to complete the proof
and leave the somewhat cumbersome mathematical details
to the Supplemental Material [13].
Both distillation and formation protocols must satisfy
three requirements, up to error terms smaller than OðnÞ:
(1) energy conservation, (2) unitarity, and (3) equality of
input and output dimensions. Without loss of generality,
we may take the total Hamiltonian to be H ¼ E
0
P
i
j1i
i
h1j
for some energy E
0
, where the sum runs over all the
qubits.
We begin the distillation protocol with ‘ copies of the
Gibbs state 
T
of H and n copies of the resource , where
 ¼ð1  pÞj0ih0jþpj1ih1j for arbitrary 0  p  1 and

T
¼ð1  qÞj0ih0jþqj1ih1j for q ¼ e
E
0
=ð1 þ e
E
0
Þ.
The aim is to effect a transformation of the form 
‘
T


n
! 
ðkÞ
j1ih1j
m
by an energy-conserving unitary,
such that m is as large as possible. The resulting exhaust
state  of k systems is arbitrary, though as an aside we
show that the optimality of the protocol implies that it has
near-Gibbs form in the Supplemental Material [13]. We
denote by R ¼ðm=nÞ the rate of distillation and  ¼ðn=lÞ
the ratio between initial resource states and Gibbs states.
The Gibbs states are free, so we allow  ! 0 as n !1.
We now use the fact that for large n, 
n
consists of
mixtures of basis states corresponding to length n binary
strings with roughly np 1s. The number t of 1s in a string is
known as its type, and more concretely we have that, up to
an error which vanishes as n !1,

n

X
t
p
t
P
t
: (4)
Here the t summation runs over strongly typical types, the
types for which t ¼ np  Oð
ffiffiffi
n
p
Þ, and P
t
denotes the pro-
jector onto the type t. See Ref. [18], Sec. 2, for more
details. An entirely similar statement holds for 
‘
T
.For
simplicity, we shall first pretend that both 
‘
T
and 
n
consist of a single type and subsequently show how to
extend the argument to all strongly typical types.
We begin with a single composite type, a concatenation
of a type coming from the resource state and one from the
reservoir state. This corresponds to a uniform mixture of
strings of length n þ ‘, each of which consists of two
substrings: the first having ‘q 1s and the second np 1s.
There are roughly e
lhðqÞ
 e
nhðpÞ
such strings, where
hðpÞ¼p lnp ð1  pÞlnð1  pÞ is the binary entropy,
expressed in nats.
Now consider a transformation which maps these
strings to new strings having at least m 1s in the rightmost
positions,
PRL 111, 250404 (2013)
PHYSICAL REVIEW LETTERS
week ending
20 DECEMBER 2013
250404-3

00...011...1
|fflfflffl{zfflfflffl}
‘q
zfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflffl{
‘
00...011...1
|fflfflffl{zfflfflffl}
np
zfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflffl{
n
! 0...001...1
|fflffl{zfflffl}
rk
|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
k
1...1
|fflffl{zfflffl}
m
where k ¼ ‘ þ n  m expresses conservation of dimen-
sion, and r and m are to be determined. Conservation of
energy requires that the number of 1s is conserved, hence,
‘q þ np ¼ rk þ m . Unitarity requires that there are at
least as many strings of length k with rk 1s as the number
of initial strings: e
khðrÞ
 e
‘hðqÞþnhðpÞ
. Roughly speaking,
this is conservation of entropy. Using these three condi-
tions, we find that the transformation is possible for any R
such that
hðqÞþhðpÞð1 þ   RÞh

q þ p  R
1 þ   R

: (5)
We now expand this with respect to  to first order and let
 ! 0. This means the heat reservoir is much larger than
the resource systems. As a result, we obtain that the
following rate can be achieved:
R ¼
hðqÞhðpÞþðp  qÞ
hðqÞþð1  qÞ
¼
Dð k 
T
Þ
Dðj1ih1jk
T
Þ
; (6)
establishing one direction necessary for Theorem 1.
In the above argument, we worked with a single com-
posite type, whereas in actuality, the initial state is a
mixture of these. Thus, we apply the protocol separately
to each composite type, assuming the number m of output
excited states to be the same for all input types, with m
fitted to the composite type containing the fewest strings
(i.e., the one consisting of strings with ‘q  Oð
ffiffiffi
‘
p
Þþ
np  Oð
ffiffiffi
n
p
Þ 1’s). To proceed as above, we need to ensure
that any variations from the above conditions are small
relative to n. Thus, we need to simultaneously fulfill
ffiffiffi
‘
p

m ¼ Rn, in order for R from (6) to be achievable, and
‘  n, in order for  ! 0. Choosing ‘ ¼ðRnÞ
3=2
, there-
fore, ensures that our estimate (6) will be accurate in the
limit n !1[19].
The formation protocol is similar to the distillation
protocol and is again based on considering type transfor-
mations satisfying the three requirements of energy con-
servation, unitarity, and dimension conservation. The
major difference is that, whereas the ideal distillation out-
put is simply the fixed-type state j1i
m
, the ideal formation
output must recreate a good approximation to the probabi-
listic mixture of type classes found in 
n
.
We construct the formation protocol in three stages. The
first two are similar to the distillation protocol. In the first, a
given type class of the Gibbs state together with the stan-
dard resource is transformed into a desired type class of the
target resource 
n
. In the second, the transformation is
extended to all the strongly typical types of the Gibbs state.
Finally, in the third step, an additional number of Gibbs
states are used to probabilistically select which type class
of the target should be output, in order to recreate the
appropriate distribution over types of the target state. In
principle, this step is irreversible, but since the number of
type classes grows only polynomially with n, the number
of extra resources required for the third step of the for-
mation protocol vanishes in the n !1limit. The similar-
ity of the first two steps with the distillation protocol
then ensures that the formation protocol achieves the
inverse rate.
Distillation for arbitrary resource states is related to
that of stationary states, and we can recycle part of the
previous distillation protocol. Suppose the resource state
has the diagonal form  ¼ pj
1
ih
1
jþð1  pÞj
2
ih
2
j,
for arbitrary orthogonal states j
k
i, implying an average
energy of hEi¼ðpjh
1
j1ij
2
þð1  pÞjh
2
j1ij
2
ÞE
0
.Inn
instances of  the total energy will overwhelmingly likely
be nhEiOð
ffiffiffi
n
p
Þ. Now, imagine projecting the resource
state onto the various energy subspaces, destroying any
coherence between them. Just as in (4), 
n
is supported
almost entirely on its typical subspace, whose size is not
larger than e
nSðÞþOð
ffiffi
n
p
Þ
. Thus, the state support in every
energy subspace is at most this large.
Now we may imagine applying the same scheme as in
the previous distillation protocol, creating as many copies
of j1i as possible. The three conditions now become k ¼
‘ þ n  m, ‘qE
0
þ nhEi¼rkE
0
þ mE
0
, and e
khðrÞ

e
‘hðqÞþnSðÞ
. An entirely similar derivation leads again to
the distillation rate found in (6). Finally, since the distil-
lation operations commute with the Hamiltonian, they
commute with the projection onto energy subspaces.
Thus, we may, instead, imagine that this projection is
performed after the distillation step. Such a projection
has no effect on the work systems, while the form of the
exhaust state is irrelevant, and therefore, we may dispense
with the projection step altogether.
The formation of arbitrary resource states is more com-
plicated than their distillation. Strictly speaking, the
desired transformation is impossible, since the inputs are
states diagonal in the energy basis and the allowed trans-
formations cannot change this fact. However, to create the
appropriate coherences between energy subspaces it suffi-
ces to use a small additional resource in the form of a
superposition over energy eigenstates.
In particular, a system in a superposition of energy levels
acts as a reference system which lifts the superselection
rule of energy conservation, as in Refs. [20,21], allowing
one to create arbitrary coherences over energy levels on the
system. However, since 
n
is almost entirely supported on
energy levels in the range nhEiOð
ffiffiffi
n
p
Þ, the formation
process requires only a reference system made from order
ffiffiffi
n
p
qubits. The extra resource of the reference system is,
thus, of a size sublinear in n and does not affect the rate
calculations. This creates an interesting asymmetry
between distillation and formation, akin to a similar phe-
nomenon in the resource theory of entanglement, where
PRL 111, 250404 (2013)
PHYSICAL REVIEW LETTERS
week ending
20 DECEMBER 2013
250404-4

distillation of entangled states does not require any com-
munication, but formation requires an amount sublinear in
the number of inputs n.
Conclusions.—We have shown that well-known results
from thermodynamics can be derived quite naturally
within the framework of the resource theory of energy-
preserving transformations and auxiliary thermal states.
We should emphasize that although the procedures we
have described for the conversion of resource states may
seem quite unnatural from a physical point of view, their
use is to establish the ‘‘in principle’’ interconversion rate
given in Theorem 1. Any more realistic reversible trans-
formation such as, for instance, the Hamiltonian method
of Ref. [22] or, for quasiclassical resources, the sequential
protocol of Ref. [23] which was developed subsequently
to the results presented here, will necessarily extract the
same amount of work.
We thank Jochen Rau and Dominik Janzing for helpful
conversations. J. M. R. acknowledges support from the
Center for Advanced Security Research Darmstadt
(CASED). R. W. S. acknowledges support from the
Government of Canada through NSERC and the Province
of Ontario through MRI. M. H. is grateful for the support of
the Foundation for Polish Science TEAM project cofi-
nanced by the EU European Regional Development
Fund. Part of this work was done at the National
Quantum Information Centre of Gdansk. The authors
acknowledge the hospitality of the Institute Mittag
Leffler within the program Quantum Information Science
(2010), where part of this work was done.
*Corresponding author.
renes@phys.ethz.ch
[1] R. Horodecki, P. Horodecki, M. Horodecki, and K.
Horodecki, Rev. Mod. Phys. 81, 865 (2009).
[2] D. Janzing and T. Beth, IEEE Trans. Inf. Theory 49, 230
(2003).
[3] G. Gour and R. W. Spekkens, New J. Phys. 10, 033023
(2008).
[4] I. Marvian and R. W. Spekkens, New J. Phys. 15, 033001
(2013).
[5] M. Horodecki, P. Horodecki, and J. Oppenheim, Phys.
Rev. A 67, 062104 (2003).
[6] D. Janzing, P. Wocjan, R. Zeier, R. Geiss, and T. Beth, Int.
J. Theor. Phys. 39, 2717 (2000).
[7] M. Horodecki and J. Oppenheim, Nat. Commun. 4, 2059
(2013).
[8] F. G. S. L. Branda
˜
o, M. Horodecki, N. H. Y. Ng, J.
Oppenheim, and S. Wehner, arXiv:1305.5278.
[9] J. A
˚
berg, Nat. Commun. 4, 1925 (2013).
[10] D. Egloff, O. C. O. Dahlsten, R. Renner, and V. Vedral,
arXiv:1207.0434.
[11] B. Schumacher and M. Westmoreland, Quantum
Processes Systems, and Information (Cambridge
University Press, Cambridge, England, 2010).
[12] J. Oppenheim, M. Horodecki, P. Horodecki, and R.
Horodecki, Phys. Rev. Lett. 89, 180402 (2002).
[13] See Supplemental Material at http://link.aps.org/
supplemental/10.1103/PhysRevLett.111.250404 for more
precise proofs of Theorem 1 and statements made in the
main text.
[14] E. Fermi, Thermodynamics (Courier Dover, New York,
1956).
[15] M. Horodecki, J. Oppenheim, and R. Horodecki, Phys.
Rev. Lett. 89, 240403 (2002).
[16] G. Gour, I. Marvian, and R. W. Spekkens, Phys. Rev. A 80,
012307 (2009).
[17] K. Horodecki, M. Horodecki, P. Horodecki, and J.
Oppenheim, Phys. Rev. Lett. 94, 200501 (2005).
[18] I. Csisza
´
r and J. Ko
¨
rner, Information Theory: Coding
Theorems for Discrete Memoryless Systems (Cambridge
University Press, Cambridge, England, 2011), 2nd ed.
[19] Additionally, the number of strings of a given type np is
given by polyðnÞe
nhðpÞ
, not simply e
nhxðpÞ
. However, in the
limit n !1, the polynomial factor is again not relevant in
our estimates.
[20] Y. Aharonov and L. Susskind, Phys. Rev. 155, 1428 (1967).
[21] S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Rev. Mod.
Phys. 79, 555 (2007).
[22] R. Alicki, M. Horodecki, P. Horodecki, and R. Horodecki,
Open Syst. Inf. Dyn. 11, 205 (2004).
[23] P. Skrzypczyk, A. J. Short, and S. Popescu,
arXiv:1302.2811.
PRL 111, 250404 (2013)
PHYSICAL REVIEW LETTERS
week ending
20 DECEMBER 2013
250404-5

Frequently asked questions

Q1. What have the authors contributed in "Resource theory of quantum states out of thermal equilibrium" ?

Brandão, Horodecki, Oppenheim, Joseph M. Renes, * and Robert W. L.