PHYSICAL REVIEW A 71, 052327 共2005兲
Entanglement and majorization in „1 + 1…-dimensional quantum systems
Román Orús
Departament d’Estructura i Constituents de la Matèria, Universitat Barcelona, 08028, Barcelona, Spain
共Received 27 January 2005; published 25 May 2005兲
Motivated by the idea of entanglement loss along renormalization group flows, analytical majorization
relations are proven for the ground state of 共1 + 1兲-dimensional conformal field theories. For any of these
theories, majorization is proven to hold in the spectrum of the reduced density matrices in a bipartite system
when changing the size L of one of the subsystems. Continuous majorization along uniparametric flows is also
proven as long as part of the conformal structure is preserved under the deformation and some monotonicity
conditions hold as well. As particular examples of our derivations, we study the cases of the XX, Heisenberg,
and XY quantum spin chains. Our results provide in a rigorous way explicit proofs for all the majorization
conjectures raised by Latorre, Lütken, Rico, Vidal, and Kitaev in previous papers on quantum spin chains.
DOI: 10.1103/PhysRevA.71.052327
PACS number共s兲: 03.67.Mn, 03.65.Ud, 03.67.Hk
I. INTRODUCTION
In the last few years, the emerging field of quantuminformation science 关1兴 has developed tools and techniques
for the analysis of quantum systems which have been proved
to be useful in other fields of physics. The studies of the
many-body Hamiltonians, quantum phase transitions, and
quantum correlations—or entanglement—that these systems
develop are examples of this interdisciplinary research. In
fact, understanding entanglement has been realized as one of
the most challenging and interesting problems in physics 关2兴.
Another interesting application of the tools of quantuminformation science has been the use of majorization theory
关4兴 in order to analyze the structure present in the ground
state—also called the vacuum—of some models along renormalization group 共RG兲 flows 共for a recent review on the RG,
see 关3兴兲. Following this idea, Latorre et al. 关5兴 proposed that
irreversibility along RG flows may be rooted in properties of
the vacuum only, without the necessity of accessing the
whole Hamiltonian of the system and its excited states. The
vacuum of a theory may already have enough information in
order to envisage irreversibility along RG trajectories. Such
an irreversibility was cast into the idea of an entanglement
loss along RG flows, which proceeded in three constructive
steps for 共1 + 1兲-dimensional quantum systems. First, due to
the fact that the central charge of a 共1 + 1兲-dimensional conformal field theory is in fact a genuine measure of the bipartite entanglement present in the ground state of the system
关6,7兴, there is a global loss of entanglement due to the c
theorem of Zamolodchikov 关8兴, which assures that the value
of the central charge at the ultraviolet fixed point is bigger
than or equal to its value at the infrared fixed point 共cuv
艌 cir兲. Second, given the splitting of the system into two
contiguous pieces, there is a monotonic loss of entanglement
due to the monotonicity numerically observed for the entanglement entropy between the two subsystems along the
flow, decreasing when going away from the critical fixed
共ultraviolet兲 point. Third, this loss of entanglement is seen to
be fine grained, since it follows from a very strict set of
majorization ordering relations, which the eigenvalues of the
reduced density matrix of the subsystems are numerically
1050-2947/2005/71共5兲/052327共7兲/$23.00
seen to perfectly obey. This last step motivated the authors of
关5兴 to affirm that there was a fine-grained entanglement loss
along RG flows rooted in properties of the vacuum, at least
for 共1 + 1兲-dimensional quantum systems. In fact, a similar
fine-grained entanglement loss had already been numerically
observed by Vidal et al. in 关6兴, for changes in the size of the
bipartition described by the corresponding ground-state density operators, at conformally invariant critical points.
In this work, we analytically prove the links between conformal field theory 共CFT兲, the RG, and entanglement that
were conjectured in the recent papers 关5,6兴 for quantum spin
chains. We develop, in the bipartite scenario, a detailed and
analytical study of the majorization properties of the eigenvalue spectrum obtained from the reduced density matrices
of the ground state for a variety of 共1 + 1兲-dimensional quantum models. Our approach is based on infinitesimal variations of the parameters defining the model—magnetic fields,
anisotropies—or deformations in the size of the block L for
one of the subsystems. We prove in these situations that there
are strict majorization relations underlying the structure of
the eigenvalues of the considered reduced density matrices
or, as defined in 关5兴, there is a fine-grained entanglement
loss. The result of our study is presented in terms of two
theorems. On the one hand, we prove exact continuous majorization relations in terms of deformations of the size of the
block L that is considered. On the other hand, we are also
able to prove continuous majorization relations as a function
of the parameters defining the model. The validity of the two
theorems is also exemplified by explicit analytical examples
based on previous work of Peschel, Kaulke, and Legeza 关9兴.
This paper is structured as follows. In Sec. II we remember the concepts of global, monotonic, and fine-grained entanglement loss, as defined in 关5兴. In Sec. III we analytically
prove continuous majorization relations for any
共1 + 1兲-dimensional CFT when the size of the subsystem L is
changed, and exemplify this result with the case of the XX
model. In Sec. IV we prove continuous majorization relations with respect to the flows in parameter space for
共1 + 1兲-dimensional quantum systems under perturbations
which preserve part of the conformal structure of the partition function. This result is analytically exemplified by the
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©2005 The American Physical Society
PHYSICAL REVIEW A 71, 052327 共2005兲
ROMÁN ORÚS
Heisenberg and XY quantum spin chains. Finally, Sec. V
collects the conclusions of our study. We also review in Appendix A the definition of majorization and produce two different lemmas which will be used in our calculations.
II. GLOBAL, MONOTONIC, AND FINE-GRAINED
ENTANGLEMENT LOSS
Consider the pure ground state 共or vacuum兲 兩⍀典 of a given
physical system which depends on a particular set of parameters, and let us perform a bipartition of the system into two
pieces A and B. The density matrix for A, describing all the
physical observables accessible to A, is given by ␳A
= trB共兩⍀典具⍀兩兲, and analogously for B. In this section we will
focus our discussion on the density matrix for the subsystem
A, so we will drop the subindex A from our notation. Let us
consider a change in one—for simplicity—of the parameters
on which the resultant density matrix depends, say, the parameter t, which can be either an original parameter of the
system or the size of the region A. In other words, we make
the change ␳共t1兲 → ␳共t2兲, where t1 ⫽ t2. In order to simplify
even more our discussion let us assume that t2 ⬎ t1. We wish
to understand how this variation of the parameter alters the
inner structure of the ground state and, in particular, how it
modifies the entanglement between the two parties A and B.
Because we are considering entanglement at two different
points t2 and t1, we assume for simplicity that the entanglement between A and B is bigger at the point t1 than at the
point t2, so we have an entanglement loss when going from t1
to t2.
Our characterization of this entanglement loss will
progress through three stages, as in 关5兴, refining at every step
the underlying ordering of quantum correlations. These three
stages will be respectively called global, monotonic, and
fine-grained entanglement loss.
共a兲 Global entanglement loss. The simplest way to
quantify the loss of entanglement between A and B when
going from t1 to t2 is by means of the entanglement entropy
S(␳共t兲) = −tr关␳共t兲ln ␳共t兲兴. Since at t2 the two parties are less
entangled than at t1, we have that
S„␳共t1兲… ⬎ S„␳共t2兲…,
共2.1兲
which is a global assessment between points t2 and t1. This is
what we shall call global entanglement loss.
共b兲 Monotonic entanglement loss. A more refined
quantification of entanglement loss can be obtained by imposing the monotonicity of the derivative of the entanglement entropy when varying parameter t. That is, the condition
dS
⬍0
dt
共2.2兲
the spectra of the underlying reduced density matrix that
becomes more and more ordered as we change the value of
the parameter. It is then natural to ask if it is possible to
characterize the reordering of the density matrix eigenvalues
along the flow beyond the simple entropic inequality discussed before and thereby unveil some richer structure. The
finest notion of reordering when changing the parameter is
then given by the monotonic majorization 共see Appendix A兲
of the eigenvalue distribution along the flow. If we call ␳ជ 共t兲
the vector corresponding to the probability distribution of the
spectra arising from the density operator ␳共t兲, then the condition
␳ជ 共t1兲 Ɱ ␳ជ 共t2兲
whenever t2 ⬎ t1 will reflect the strongest possible ordering
of the ground state along the flow. This is what we call
fine-grained entanglement loss, and it is fine grained since
this condition involves a whole tower of inequalities to be
simultaneously satisfied 共see Appendix A兲. In what follows
we will see that this precise majorization condition will
appear in different circumstances when studying
共1 + 1兲-dimensional quantum systems.
III. FINE-GRAINED ENTANGLEMENT LOSS
WITH THE SIZE OF THE BLOCK IN
„1 + 1…-DIMENSIONAL CFT
A complete analytical study of majorization relations for
any 共1 + 1兲-dimensional conformal field theory is presented
in the bipartite scenario when the size of the considered subsystems changes, i.e., under deformations in the interval of
the accessible region for one of the two parties. This size will
be represented by the length L of the space interval for which
we consider the reduced density matrix ␳L after tracing out
all the degrees of freedom corresponding to the rest of the
universe. Our main result in this section can be cast into the
following theorem.
Theorem. ␳L Ɱ ␳L⬘ if L 艌 L⬘ for all possible
共1 + 1兲-dimensional CFTs, whenever L 艌 L⬘ Ⰷ ⑀ , ⑀ being the
ultraviolet cutoff of the theory.
¯
Proof. Let Z共␶ ,¯␶兲 = q−btr共qL0+L0兲 be the partition function
of a subsystem of size L on a torus 关10兴, where q = e2␲i␶ ,
␶ = i␬ / ln共L / ⑀兲 with ␬ a positive constant, ⑀ being an ultraviolet cutoff and b ⬅ 共c + c̄兲 / 24 a combination of the holomorphic and antiholomorphic central charges that define the universality class of the model. The unnormalized density
¯
matrix ␳L can then be written as ␳L = q−bqL0+L0, since ␳L can
be understood as a propagator and 共L0 + L̄0兲 is the generator
of translations in time 共dilatations in the conformal plane兲
关10兴. Furthermore, we have that
¯
implies a stronger condition in the structure of the ground
state under deformations of the parameter. This monotonic
behavior of the entanglement entropy is what we shall call
monotonic entanglement loss.
共c兲 Fine-grained entanglement loss. When monotonic
entanglement loss holds, we can wonder whether, in fact, it is
共2.3兲
tr共qL0+L0兲 = 1 + n1q␣1 + n2q␣2 + ¯ ,
共3.1兲
due to the fact that L0 + L̄0 is diagonal in terms of highestweight states 兩h , h̄典 : 共L0 + L̄0兲兩h , h̄典 = 共h + h̄兲兩h , h̄典, with h 艌 0
and h̄ 艌 0; the coefficients ␣1 , ␣2 , … ⬎ 0 , ␣i ⫽ ␣ j ∀ i ⫽ j, are
related to the scaling dimensions of the descendant operators,
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ENTANGLEMENT AND MAJORIZATION IN 共1 + 1兲-…
PHYSICAL REVIEW A 71, 052327 共2005兲
⬁
and n1 , n2,… are degeneracies. The normalized distinct eigenvalues of ␳L are then given by
H=
x
y
+ ␴ny␴n+1
兲.
兺 共␴xn␴n+1
共3.7兲
n=1
␭1 =
1
,
共1 + n1q␣1 + n2q␣2 + ¯ 兲
␭2 =
q ␣1
,
共1 + n1q␣1 + n2q␣2 + ¯ 兲
The system described by the XX model is critical since it has
no mass gap, and therefore there is an underlying conformal
symmetry in the structure of the model. Taking the ground
state and tracing out all but a block of L contiguous spins, the
density matrix ␳L describing this block can be written, in the
large-L limit, as a thermal state of free fermions 共see 关9兴兲:
⯗
q␣共l−1兲
␭l =
.
␣1
共1 + n1q + n2q␣2 + ¯ 兲
共3.2兲
¯
Let us define Z̃共q兲 ⬅ tr共q共L0+L0兲兲 = 共1 + n1q␣1 + n2q␣2 + ¯ 兲.
The behavior of the eigenvalues in terms of deformations
with respect to the parameter L follows from
dZ̃共q兲 Z̃共q兲 − 1 dq d ln共L/⑀兲
=
艌 0,
d ln共L/⑀兲 dL
dL
q
and therefore
冉 冊
共3.4兲
Because ␭1 is always the biggest eigenvalue ∀ L, the first
cumulant automatically satisfies continuous majorization
when decreasing the size of the interval L. The variation of
the rest of the eigenvalues ␭l with respect to L reads as
follows:
冉
冊
共3.9兲
冉兺 冊
1
␳L共n0,n1,…,nL−1兲 = exp −
nk⑀k = ␳L共n0兲 ¯ ␳L共nL−1兲,
ZL
k=0
共3.10兲
with ␳共n␣兲 = 共1 / ZL␣兲e−n␣⑀␣, where ZL␣ = 共1 + e−⑀␣兲 is the partition
function for the mode ␣, and n␣ = 0,1 ∀ ␣. It is worth noticing that the partition function of the whole block ZL can then
be written as a product over the L modes:
ZL =
Imposing the condition that d␭ / dL 艌 0 ∀ l ⫽ 1 is equivalent
to the condition
∀ l ⫽ 1,
k = 0,1,…,L − 1.
L−1
共3.5兲
Z̃共q兲 − 1
␲2
共2k + 1兲,
2 ln L
The eigenvalues of the density matrix ␳L can then be written
in terms of noninteractive fermionic modes
Z̃共q兲 − 1 dq
d␭l d q␣共l−1兲
q␣共l−1兲−1
=
.
=
␣共l−1兲 −
dL dL Z̃共q兲
dL
Z̃共q兲
Z̃共q兲
␣共l−1兲 艌
共3.8兲
where ZL is the partition function for a given L, and H⬘
L−1
= 兺k=0
⑀kd†k dk, with fermionic operators d†k , dk and the dispersion relation
⑀k =
共3.3兲
e−H⬘
,
ZL
L−1
1
d␭1 d
=
艋 0.
dL dL Z̃共q兲
冉 冊
␳L =
共3.6兲
Z̃共q兲
which is always satisfied for L Ⰷ ⑀, since the right-hand
side of this expression decreases exponentially fast. Therefore, if L is much larger than the ultraviolet cutoff of the
theory ⑀, then 共d␭1 / dL兲 艋 0 and 共d␭l / dL兲 艌 0 ∀ l ⫽ 1. Under
these conditions the monotonicity lemma on majorization
from Appendix A applies, from which we immediately obtain that ␳L Ɱ ␳L⬘ if L 艌 L⬘ Ⰷ ⑀. This proof is valid for all
possible 共1 + 1兲-dimensional conformal field theories since it
is based only on completely general assumptions. 䊐
A. Analytical finite-L majorization
for the critical quantum XX model
Let us exemplify the previous theorem with the particular
case of the quantum XX model for which the exact spectrum
of ␳L can be explicitly computed. The Hamiltonian of the
model without magnetic field is given by the expression
共1 + e−⑀ 兲.
兿
k=0
k
共3.11兲
Once the density matrix of the subsystem is well characterized with respect to its size L, it is not difficult to prove
that ␳L Ɱ ␳L⬘ if L 艌 L⬘. In order to see this, we will fix our
attention on the majorization within each mode and then we
will apply the direct product lemma from Appendix A for the
whole subsystem. We initially have to observe the behavior
in L of the biggest probability defined by each individual
distribution for each one of the modes, that is, PL␣ = 1 / ZL␣
= 共1 + e−⑀␣兲−1, for ␣ = 0,1, … , L − 1. It is straightforward to see
that
dPL␣
e −⑀␣ d ⑀ ␣
=
⬍ 0,
dL 共1 + e−⑀␣兲2 dL
共3.12兲
which implies that PL␣ decreases if L increases ∀ ␣. This
involves majorization within each mode ␣ = 0,1, … , L − 2
when decreasing L by one unit. In addition, we need to see
what happens with the last mode ␣ = L − 1 when the size of
the system is reduced from L to L − 1. Because this mode
disappears for the system of size L − 1, its probability distribution turns out to be represented by the probability vector
共1,0兲, which majorizes any probability distribution of two
components. Combining these results with the direct product
lemma from Appendix A, we conclude that this example for
the quantum XX model matches our previous theorem.
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ROMÁN ORÚS
IV. FINE-GRAINED ENTANGLEMENT LOSS
ALONG UNIPARAMETRIC FLOWS IN
„1 + 1…-DIMENSIONAL QUANTUM SYSTEMS
We study in this section strict continuous majorization
relations along uniparametric flows, under the conditions of
integrable deformations and monotonicity of the eigenvalues
in parameter space. The main result of this section can be
cast into the next theorem.
Theorem. Consider a 共1 + 1兲-dimensional physical theory
which depends on a set of real parameters gជ = 共g1 , g2 , …兲,
such that
共1兲 there is a nontrivial conformal point gជ *, for which the
model is conformally invariant;
共2兲 the deformations from gជ * in parameter space in the
positive direction of a given unitary vector ê preserve part
of the conformal structure of the model, i.e., the eigenvalues
of the reduced density matrices of the vacuum ␳共gជ 2兲 are still
of the form given in Eq. 共3.2兲 for values of the parameters
gជ 1 = gជ * + aê;
ជ ជ q共gជ 兲兴兩 ជ 艋 0, where q共gជ 兲 are the corresponding
共3兲 ê · 关ⵜ
g
g1
parameter-dependent conformal q factors.
Then, far enough away from the conformal point there is
continuous majorization of the eigenvalues of the reduced
density matrices of the ground state along the flow in the
parameters gជ in the positive direction of ê, i.e.,
␳共gជ 1兲 Ɱ ␳共gជ 2兲,
gជ 1 = gជ * + aê,
gជ 2 = gជ * + a⬘ê,
a⬘ 艌 a Ⰷ 0.
共4.1兲
Proof. If the eigenvalues are assumed to be of the form
given by Eq. 共3.2兲, then it is straightforward to see that
ជ ជ ␭ 共gជ 兲兴兩 ជ 艌 0, which assures that the first cumulant satê · 关ⵜ
g 1
g1
isfies majorization. Now, if we impose the rest of the eigenvalues to be monotonically decreasing functions along this
direction in parameter space, we obtain again the condition
given in Eq. 共3.6兲, which is satisfied far enough from the
conformal point in the positive direction of ê. Applying the
monotonicity lemma from Appendix A, the theorem is
proved. 䊐
The applicability of this theorem is based on the conditions we had to assume as hypothesis. Indeed, these conditions are naturally satisfied by many interesting models. We
now wish to illustrate this point with the analytical examples
of the Heisenberg and XY quantum spin chains.
Consider the Hamiltonian of the Heisenberg quantum spin
chain
⬁
x
y
z
+ ␴ny␴n+1
+ ⌬␴zn␴n+1
兲,
兺 共␴xn␴n+1
冉兺 冊
⬁
1
␳⌬共n0,n1,…,n⬁兲 = exp −
n k⑀ k ,
Z⌬
k=0
共4.2兲
n=1
where ⌬ is the anisotropy parameter. This model is noncritical for ⌬ ⬎ 1 and critical at ⌬ = 1. From the pure ground state
of the system, half of it is traced out, getting an infinitedimensional density matrix which describes half of the sys-
共4.3兲
with dispersion relation
⑀k = 2k arccosh共⌬兲,
共4.4兲
and nk = 0,1 for k = 0,1, … , ⬁. The physical branch of the
function arccosh 共⌬兲 is defined for ⌬ 艌 1 and is a monotonically increasing function as ⌬ increases. Additionally, the
whole partition function Z⌬ can be decomposed as an infinite
direct product of the different free fermionic modes.
From the last equations, it is not difficult to see that
␳⌬ Ɱ ␳⌬⬘ if ⌬ 艋 ⌬⬘. Fixing the attention on a particular mode
␣, we evaluate the derivative of the biggest probability for
this mode, P⌬␣ = 共1 + e−⑀␣兲−1. This derivative is seen to be
dP⌬␣
2␣
=
⬎0
−
⑀
d⌬ 共1 + e k兲2冑⌬2 − 1
共4.5兲
for ␣ = 1,2, … , ⬁ and 0 for ␣ = 0. It follows from this fact that
all the modes independently majorize their respective probability distributions as ⌬ increases, with the peculiarity that
the 0th mode remains unchanged along the flow, its probability distribution being always 共 21 , 21 兲. The particular behavior
of this mode is responsible for the appearance of the “cat”
state that is the ground state for large values of ⌬ 共in that
limit, the model corresponds to the quantum Ising model
without magnetic field兲. These results, together with the direct product lemma from Appendix A, make this example
match our previous theorem.
B. Analytical majorization along uniparametric flows
for the quantum XY model
Similar results to the one obtained for the Heisenberg
model can be obtained as well for a more generic quantum
spin chain. Let us consider the quantum XY model, as described by the Hamiltonian
⬁
H=−
A. Analytical majorization along the anisotropy
flow for the Heisenberg quantum spin chain
H=
tem 共N / 2 contiguous spins in the limit N → ⬁兲. The resulting
reduced density matrix ␳⌬ can be written as a thermal density
matrix of free fermions 关9兴, in such a way that its eigenvalues
are given by
x
y
+ 共1 − ␥兲␴ny␴n+1
+ 2␭␴zn兴,
兺 关共1 + ␥兲␴xn␴n+1
n=1
共4.6兲
where ␥ can be regarded as the anisotropy parameter and ␭
as the magnetic field. The phase diagram of this model is
shown in Fig. 1, where it is seen that there exist different
critical regions depending on the values of the parameters.
Consider the ground state of this Hamiltonian of infinite
number of spins, and trace out half of the system 共if the size
of the system is N, we trace out N / 2 contiguous spins, and
take the limit N → ⬁兲, for given values of ␭ and ␥. The resulting density matrix ␳共␭,␥兲 can be written as a thermal state
of free fermions, and its eigenvalues are given by 共see 关9兴兲
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PHYSICAL REVIEW A 71, 052327 共2005兲
⑀␣ = 共2␣ + 1兲␲
I共冑1 − x2兲
= 共2␣ + 1兲⑀ ,
I共x兲
共4.12兲
where x = ␥ / 共冑␭2 + ␥2 − 1兲. The variation of the biggest eigenvalue with respect to ␭ is
dP␭␣ 共2␣ + 1兲e−共2␣+1兲⑀ d⑀
=
.
d␭
共1 + e−共2␣+1兲⑀兲2 d␭
It is easy to see that
冋
冉
共4.13兲
冊 册
I共冑1 − x2兲 dI共x兲 dx
␲ dI共冑1 − x2兲
d⑀ d⑀ dx
=
=
−
d␭ dx d␭ I共x兲
dx
I共x兲
dx d␭
⬎ 0,
FIG. 1. Phase diagram of the quantum XY model.
␳共␭,␥兲共n0,n1,…,n⬁兲 =
冉兺 冊
⬁
1
Z共␭,␥兲
n k⑀ k ,
exp −
共4.7兲
k=0
where nk = 0,1, and the single-mode energies ⑀k are given by
⑀k =
再
2k⑀
if ␭ ⬍ 1,
共2k + 1兲⑀
if ␭ ⬎ 1,
冎
共4.8兲
with k = 0,1, … , ⬁. The parameter ⑀ is defined by the relation
⑀=␲
I共冑1 − x2兲
,
I共x兲
共4.9兲
since both 共d⑀ / dx兲 ⬍ 0 and 共dx / d␭兲 ⬍ 0. Therefore, dP␭␣ / d␭
⬎ 0 for ␣ = 0,1, … , ⬁. This derivation shows mode-bymode majorization when ␭ increases. Combining this result with the direct product lemma from Appendix A, we
see that this example matches our theorem.
共2兲 +冑1 − ␥2 ⬍ ␭ ⬍ 1. For this case, we show that ␳␭
Ɱ ␳␭⬘ if ␭ 艌 ␭⬘. In particular, the probability distribution for
the 0th fermionic mode remains constant and equal to 共 21 , 21 兲,
which brings a “cat” state for low values of ␭. Similar to the
latter case, the biggest probability for mode ␣ is P␭␣ = 共1
+ e−⑀␣兲−1, with
⑀␣ = 2␣␲
I共x兲 being the complete elliptic integral of the first kind
I共x兲 =
冕
␲/2
and x being given by
x=
共冑␭2 + ␥2 − 1兲/␥
␥/共冑␭ + ␥ − 1兲
2
共4.10兲
冑1 − x2sin2共␪兲
0
再
d␪
2
if ␭ ⬍ 1,
if ␭ ⬎ 1,
冎
共4.14兲
共4.11兲
with the condition ␭2 + ␥2 ⬎ 1 共external region of the
Baruoch-McCoy circle 关11兴兲.
We note that the probability distribution defined by the
eigenvalues of ␳共␭,␥兲 is the direct product of distributions for
each one of the separate modes. Therefore, in order to study
majorization we can focus separately on each one of these
modes, in the same way as we already did in the previous
examples. We wish now to consider our analysis in terms of
the flows with respect to the magnetic field ␭ and with respect to the anisotropy ␥ in a separate way.
1. Flow along the magnetic field ␭
We consider in this subsection a fixed value of ␥ while the
value of ␭ changes, always satisfying the condition ␭2 + ␥2
⬎ 1. Therefore, at this point we can drop ␥ from our notation. We separate the analysis of majorization for the regions
1 ⬍ ␭ ⬍ ⬁ and +冑1 − ␥2 ⬍ ␭ ⬍ 1 for reasons that will become
clearer during the example but that already can be realized
just by looking at the phase space structure in Fig. 1.
共1兲 1 ⬍ ␭ ⬍ ⬁. We show that ␳␭ Ɱ ␳␭⬘ if ␭ 艋 ␭⬘. In this
region of parameter space, the biggest probability for the
mode ␣ is P␭␣ = 共1 + e−⑀␣兲−1, with
I共冑1 − x2兲
= 2␣⑀ ,
I共x兲
共4.15兲
and x = 共冑␭2 + ␥2 − 1兲 / ␥. Its derivative with respect to ␭ is
dP␭␣
2␣e−2␣⑀ d⑀
=
.
d␭ 共1 + e−2␣⑀兲2 d␭
共4.16兲
It is easy to see that this time 共d⑀ / d␭兲 ⬍ 0, and therefore
dP␭␣ / d␭ ⬍ 0 for ␣ = 1,2, … , ⬁, which brings majorization individually for each one of these modes when ␭ decreases.
The mode ␣ = 0 needs special attention; from Eq. 共4.16兲 it is
seen that dP␭␣=0 / d␭ = 0, therefore the probability distribution
for this mode remains constant and equal to 共 21 , 21 兲 all along
the flow. This is a marginal mode that brings the system to a
“cat” state that appears as the ground state of the system for
low values of ␭. Notice that this peculiarity is rooted on the
particular form of the dispersion relation given in Eq. 共4.8兲,
which is proportional to 2k instead of 2k + 1 for this region in
parameter space. These results, together with the direct product lemma from Appendix A, prove that this example
matches our theorem as well.
2. Flow along the anisotropy ␥
In this subsection, the magnetic field ␭ is fixed and the
anisotropy ␥ is the only free parameter of the model, always
satisfying ␭2 + ␥2 ⬎ 1. Thus, at this point we can drop ␭ from
our notation. We will see that ␳␥ Ɱ ␳␥⬘ if ␥ 艌 ␥⬘, in the two
regions 1 ⬍ ␭ ⬍ ⬁ and +冑1 − ␥2 ⬍ ␭ ⬍ 1. In particular, in the
region +冑1 − ␥2 ⬍ ␭ ⬍ 1, the probability distribution for the
0th fermionic mode remains constant and equal to 共 21 , 21 兲.
Let us consider the biggest probability for the mode ␣ , P␥␣
= 共1 + e−⑀␣兲−1, with ⑀␣ = ␻⑀, where
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ROMÁN ORÚS
␻=
再
2␣
if ␭ ⬍ 1,
共2␣ + 1兲
if ␭ ⬎ 1,
冎
共4.17兲
and ⑀ is as defined in previous sections. It is easy to verify
that
dP␥␣
␻e−␻⑀␣ d⑀ dx
=
⬍0
d␥ 共1 + e−␻⑀␣兲2 dx d␥
共4.18兲
for ␣ = 0,1, … , ⬁ if ␭ ⬎ 1 and for ␣ = 1,2, … , ⬁ if ␭ ⬍ 1. The
mode ␣ = 0 for ␭ ⬍ 1 needs special attention: it is seen that
dP␭␣=0 / d␭ = 0, and therefore the probability distribution for
this mode remains constant and equal to 共 21 , 21 兲 all along the
flow. These results, together with the direct product lemma
from Appendix A, show that this case can also be accommodated in our theorem.
other approaches are also possible 关15兴, majorization may be
a unique tool in order to envisage irreversibility of RG flows
in terms of properties of the vacuum only, and some numerical results in this direction have already been observed in
systems of different dimensionality along uniparametric
flows 关16兴. New strict mathematical results could probably
be achieved in these situations following the ideas that we
have presented throughout this work.
ACKNOWLEDGMENTS
The author is grateful for very fruitful and enlightening
discussions with J. I. Latorre, C. A. Lütken, E. Rico, and G.
Vidal about the content of this paper, and for financial support from Projects No. FPA2001-3598, GC2001SGR-00065,
and IST-199-11053.
V. CONCLUSIONS
APPENDIX A: LEMMAS ON MAJORIZATION
In this paper we have provided in a rigorous way explicit
proofs for all the majorization conjectures raised by Latorre,
Lütken, Rico, Vidal, and Kitaev in previous papers on quantum spin chains 关5,6兴. In particular, we have developed a
completely general proof of majorization relations underlying the structure of the vacuum with respect to the size of the
block L for all possible 共1 + 1兲-dimensional conformal field
theories, as long as L is much larger than the ultraviolet
cutoff of the theory. This has been exemplified with the particular case of the XX model, for which the explicit calculation of the eigenvalues of the reduced density matrix can be
performed. We have proven as well the existence of a finegrained entanglement loss for 共1 + 1兲-dimensional quantum
systems along uniparametric flows, assuming that perturbations in parameter space preserve part of the conformal structure of the partition function, and some monotonicity conditions hold as well. This has been exemplified by the
Heisenberg and XY models. Our results provide solid mathematical grounds for the existence of majorization relations
along RG flows underlying the structure of the vacuum of
共1 + 1兲-dimensional quantum spin chains.
Understanding the entanglement structure of the vacuum
of 共1 + 1兲-dimensional models is a major task in quantuminformation science. For instance, spin chains like the ones
described in the particular examples of this paper can be used
as possible approximations to the complicated interactions
that take place in the register of a quantum computer 关12兴.
Entanglement across a quantum phase transition has also an
important role in quantum algorithm design, and in particular
in quantum algorithms by adiabatic evolution 关13兴. Additionally, the properties of quantum state transmission through
spin chains are also intimately related to the entanglement
properties present in the chain 关14兴. Consequently, our precise characterization of entanglement in terms of majorization relations should be helpful for the design of more powerful quantum algorithms and quantum state transmission
protocols.
It would also be of interest trying to relate the results
presented in this paper to possible extensions of the c theorem 关8兴 to systems with more than 共1 + 1兲 dimensions. While
This Appendix includes the formal definitions of majorization 关4兴 as well as the two lemmas that are used in the
main part of this work.
1. Definitions
N
N
Let xជ , yជ 苸 R be two vectors such that 兺i=1
xi = 兺i=1
y i = 1,
which represent two different probability distributions. We
say that distribution yជ majorizes distribution xជ , written xជ
Ɱ yជ , if and only if there exist a set of permutation matrices
兵P j其 and probabilities p j 艌 0 , 兺p j = 1, such that
N
xជ =
兺j p jP jyជ .
共A1兲
Since, from the previous definition, xជ can be obtained by
means of a probabilistic combination of permutations of yជ ,
we get the intuitive notion that distribution xជ is more disordered than yជ .
Notice that in Eq. 共A1兲, 兺 j p j P j = D defines a doubly stochastic matrix, i.e., D has nonnegative entries and each row
and column sums to unity. Then, xជ Ɱ yជ if and only if xជ
= Dyជ , D being a doubly stochastic matrix.
Another equivalent definition of majorization can be
stated in terms of a set of inequalities between the two distributions. Consider the components of the two vectors
sorted in decreasing order, written as 共z1 , … , zN兲 ⬅ zជ↓. Then,
xជ ↓ Ɱ yជ ↓ if and only if
k
k
x i 艋 兺 y i,
兺
i=1
i=1
k = 1,…,N.
共A2兲
Throughout this work, these probability sums are called cumulants.
A powerful relation between majorization and any convex
function f over the set of probability vectors states that
xជ Ɱ yជ ⇒ f共xជ 兲 艋 f共yជ 兲. From this relation it follows that the
N
xilog xi of a probabilcommon Shannon entropy H共xជ 兲 ⬅ −兺i=1
ity distribution satisfies H共xជ 兲 艌 H共yជ 兲 whenever xជ Ɱ yជ . In what
follows we present the two lemmas that are used all along
our work.
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ENTANGLEMENT AND MAJORIZATION IN 共1 + 1兲-…
PHYSICAL REVIEW A 71, 052327 共2005兲
Let ␭ 共gជ 兲 ⬅ 兵␭␣共gជ 兲其␣N=1 be a sorted probability distribution
depending on a set of continuous real parameters gជ
= 共g1 , g2 , …兲, and let ê be a unitary vector in a given direction of the parameter space. Given a point gជ 1 in the space of
parameters and the points gជ 2 along the positive direction of
ê , gជ 2 = gជ 1 + aê, with a 艌 0, if the following conditions are
ជ ជ ␭ 共gជ 兲兴兩 ជ 艌 0;
satisfied ∀ gជ 2 : ␭1共gជ 2兲 ⬎ ␭␣共gជ 2兲 , ␣ ⫽ 1; ê · 关ⵜ
g 1
g2
ជ ជ ⌰ 共gជ 兲兴兩 ជ 艋 0, where ⌰ 共gជ 兲 ⬅ 兺N ␭ 共gជ 兲 , k
and ê · 关ⵜ
g k
g2
k
i=N−k i
= 0,1, … , N − 3, is the partial sum of the 共k + 1兲 smallest probabilities for a given value of gជ , then there exists continuous
majorization along the flow in the positive direction of ê, that
is, ␭ជ ↓共gជ 1兲 Ɱ ␭ជ ↓共gជ 2兲 ∀ gជ 2 = gជ 1 + aê , a 艌 0.
ជ ជ ␭ 共gជ 兲兴兩 ជ 艌 0, and ␭ 共gជ 兲 is the
Proof. By assumption ê · 关ⵜ
g 1
g2
1 2
biggest probability ∀ gជ 2 so the first cumulant is always
monotonically increasing along the positive direction of ê. In
order to check the rest of the cumulants, we realize that for
the mth cumulant, which is the sum of the m biggest probជ ជ 关␭ 共gជ 兲 + ¯
abilities with m ⬎ 1, we have that ê · ⵜ
g 1
ជ
ជ
+ ␭m共g兲兴兩gជ 2 is in fact equal to −ê · ⵜgជ 关兺i⫽1,…,m␭i共gជ 兲兴兩gជ 2
ជ ជ⌰
ជ 兲兴兩gជ 2, which is 艌0. As a consequence of
= −ê · 关ⵜ
g N−m−1共g
the former equation, all the cumulants are, indeed, monotonically increasing functions for any m ⬎ 1 along the positive
direction of the unitary vector ê. Therefore, we conclude that
there exists continuous majorization along the flow in the
parameters gជ in the positive direction of ê, that is, ␭ជ ↓共gជ 1兲
Ɱ ␭ជ ↓共gជ 2兲 if gជ 2 = gជ 1 + aê , a 艌 0, which proves the desired
lemma. 䊐
Notice that this lemma is suitable for being applied to the
particular case in which all the probabilities except one are
decreasing functions of one parameter only, and there are no
crossings between them along the flow in this parameter.
This turns out to be the typical situation that we face in all
the studied examples throughout this article. As can be seen,
this lemma is still more general, and can be applied to different situations than the ones that we have analyzed in our
work.
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2. Direct product lemma [5]
If pជ 1 Ɱ pជ 2 , qជ 1 Ɱ qជ 2 then 共pជ 1 丢 qជ 1兲 Ɱ 共pជ 2 丢 qជ 2兲. This means
that majorization is preserved under the direct product operation.
Proof. If pជ 1 Ɱ pជ 2 and qជ 1 Ɱ qជ 2 then pជ 1 = D p pជ 2 and
qជ 1 = Dqqជ 2 where D p , Dq are both doubly stochastic matrices.
Therefore 共pជ 1 丢 qជ 1兲 = 共D p 丢 Dq兲共pជ 2 丢 qជ 2兲, where 共D p 丢 Dq兲 is a
doubly stochastic matrix in the direct product space, and so
共pជ 1 丢 qជ 1兲 Ɱ 共pជ 2 丢 qជ 2兲. 䊐
3. Monotonicity lemma
ជ↓
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