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arXiv:nlin/0312062v3 [nlin.CD] 24 Feb 2004
Spectral properties and classical decays in quantum open systems
Ignacio Garc´ıa-Mata
∗
and Marcos Saraceno
†
Dto. de F´ısica, Comisi´on Nacional de Energ´ıa At´omica. Av. del Libertador 8250 (1429), Buenos Aires, Argentina.
(Dated: February 8, 2008)
We study the relationship between the spectral properties of diffusive open quantum maps and the classical
spectrum of Ruelle-Pollicott resonances. The leading resonances determine the asymptotic time regime for
several quantities of interest - the linear entropy, the Loschmidt echo and the correlations of the initial state.
A numerical method that allow an efficient calculation of the leading spectrum is developed using a truncated
basis adapted to the dynamics.
PACS numbers: 03.65.Sq, 05.45.Mt, 05.45.Pq
I. INTRODUCTION
The study of the emergence of classical features in systems
ruled by quantum mechanics is as old as quantum mechan-
ics itself. When the quantum system is isolated and the evo-
lution unitary, these features appear in the WKB semiclassi-
cal limit, which is of paramount importance in establishing
the quantum classical correspondence in integrable systems.
In its more modern form, the EBK quantization rule [1], it
shows the direct connection between tori in phase space and
quantized eigenfunctions in Hilbert space. In chaotic systems,
the relationship is more subtle and is embodied in the cele-
brated Gutzwiller trace formula [2], relating sets of unstable
periodic orbits to the density of states. The limits of appli-
cability of these semiclassical methods and the insight they
provide on the quantum dynamics of isolated chaotic systems
has inspired most of the recent research in the area of quantum
chaos.
In open quantum systems, on the contrary, the emergence
of classical features has been studied mainly in the time evo-
lution of a different set of observables, most notably the rate
of linear entropy growth (or purity decay)[3], the Loschmidt
echo or fidelity [4, 5, 6] and the decay of correlations [7].
These studies have demonstrated that in certain well defined
regimes for chaotic systems the classical Lyapunov exponent
governs these rates and that the evolution of localized quan-
tum densities in phase space becomes classical.
In this article we consider this question from the point of
view of the spectral properties of the classical and quantum
propagators. Classical densities evolve according to the Li-
ouville equation whose solution can be written in terms of
a propagator called Perron-Frobenius operator(PF)[41]. It is
unitary on L
2
. However, for chaotic systems, correlation func-
tions exhibit oscillations and exponential decay. The decay
rates are given by the poles of the resolvent of the PF opera-
tor, the so-called Ruelle-Pollicott (RP) resonances[9]. By lim-
iting the resolution of the functionalspace, one can effectively
truncate the PF to a nonunitary operator of finite size (say
N × N ) with a spectrum lying entirely inside the unit circle,
except for the simply degenerate eigenvalue 1. In the, prop-
∗
Electronic address: garciama@tandar.cnea.gov.ar
†
Electronic address: saraceno@tandar.cnea.gov.ar
erly taken, limit of infinite size and no coarse graining, the
isolated eigenvalues turn out to be the RP resonances[17, 18].
As shown in [8], the linear entropy and the Loschmidt echo,
for asymptotic times much longer than the Ehrenfest time
1
,
also show characteristic decay rates governed by the classical
RP resonances[9]. Experimental evidence of this dependence
on RP resonances was observed for the first time in [10]. Our
approach is similar in spirit to the calculations performed on
the sphere for the dissipative kicked top by Haake and col-
laborators [11, 12, 13], Fishman [14, 15] and, for the baker’s
map, by Hasegawa and Saphir [16]. We model the unitary dy-
namics by means of a quantum map and implement a diffusive
superoperator represented by a Kraus sum. Two recent works
by Blank, et al. [17] and Nonnenmacher[18] provide a rigor-
ous theoretical underpinning to our calculations for quantum
and classical maps on the torus.
The plan of the paper is as follows. Section II provides a
short account of the quantization procedurefor maps acting on
a classical surface with periodic boundary conditions in both
coordinates and momenta, i.e. a torus. In Sec.III we imple-
ment the open system dynamics with the definition of a diffu-
sion superoperator represented as a Kraus sum. The general
spectral properties of both the unitary and the noisy part, as
well as those of the combined action are studied. Sec.IV deals
with the relationship between the classical and the quantum
resonances and, utilizing recently proved theorems [17, 18],
how they coincide in specific ranges of ~ and of the noise
strength. As a consequence we show that the asymptotic time
behaviour of several quantities is classical and depends on the
Ruelle-Pollicott resonaces closer to the unit circle. A numeri-
cal method that allows the calculation of the leading spectrum
of resonances is developed. Sec.V illustrates this correspon-
dence taking the perturbed Arnold cat map as an example. We
relegate to the Appendix some notation concerning the spec-
tral decomposition of superoperatorsand the details of the nu-
merical method.
1
In our case the Ehrenfest time n
E
is related to the time it takes for an
initially localazed package to reach the borders of phase space due to ex-
ponential instability (it is sometimes called “log time”).
2
II. UNITARY DYNAMICS ON THE TORUS T
2
We picture the classical phase space as a square of unit
area with sides identified. The classical transformations will
map this square onto itself, thus providing a simple model
of Hamiltonian area preserving dynamics. The fact that the
phase space has finite area brings some well known special
features to the quantization that we briefly review. Refs.
[19, 20] provide a more extensive account.
A. The Hilbert space
As the phase space has finite area, which we normalize to
unity, the Hilbert space H
N
is finite and its dimension N sets
the value of Planck’s constant to ~ = (2πN)
−1
. The po-
sition and momentum bases are then sets of discrete states
|q i, |p i, q, p = 0, ..N − 1 which are related by the discrete
Fourier transform (DFT) of dimension N
hp|q i =
1
√
N
e
−
2πi
N
pq
. (1)
A vector |ϕ(t) i in H
N
characterizes pure states of the sys-
tem and can be represented by the amplitudes hq|ϕi, hp|ϕi in
the coordinate or momentum basis, respectively.
In the description of open systems it is imperative to rep-
resent states by a density operator ˆρ. They form a subset of
self-adjoint, positive semidefinite matrices with unit trace in
H
N
2
def
= H
N
⊗ H
∗
N
, the space of complex N × N matrices,
usually called in this context Liouville space. While Hilbert
space is the natural arena for unitary dynamics, this much
larger Liouville space sets the stage for the more general de-
scription of open quantum dynamics. It acquires the structure
of a Hilbert space with the usual introduction of the matrix
scalar product
(
ˆ
A,
ˆ
B) = Tr(
ˆ
A
†
ˆ
B). (2)
where
ˆ
A,
ˆ
B ∈ H
N
2
. Linear transformations in this space are
termed superoperators, they map operators into operators and
are represented by N
2
× N
2
matrices. In Appendix A we re-
view the various notations and properties related to this space.
B. Translations on the torus
The usual translation operator in the infinite plane R
2
is
ˆ
T
(q,p)
= e
−
i
~
(q
ˆ
P −p
ˆ
Q)
(3)
= e
−
i
~
q
ˆ
P
e
i
~
p
ˆ
Q
e
i
2~
qp
=
ˆ
U
q
ˆ
V
p
e
i
2~
qp
, (4)
where
ˆ
U and
ˆ
V generate shifts in the position and momen-
tum eigenbasis respectively. On the torus the main differ-
ence is that the infinitesimal translation operators
ˆ
P ,
ˆ
Q with
the usual commutation rules cannot be defined because posi-
tion and momentum eigenstates are discrete. However, finite
translation operators
ˆ
U and
ˆ
V that have the property
ˆ
V
p
ˆ
U
q
=
ˆ
U
q
ˆ
V
p
e
i
2π
N
qp
(5)
can be defined and they generate finite cyclic shifts in the re-
spective bases [21]. The N × N grid of coordinate and mo-
mentum states constitutes the quantum phase space for the
torus. Eq. (5) allows a definition of a translation operator
ˆ
T
(q,p)
: H
N
→ H
N
, q, p integers, analog to Eq. (3). The
action of
ˆ
T on position and momentum eigenstates is
ˆ
T
(q
1
,p
1
)
|q i = exp
i
2π
N
p
q +
q
1
2
|q + q
1
i (6)
ˆ
T
(q
1
,p
1
)
|p i = exp
−i
2π
N
q
p +
p
1
2
|p + p
1
i. (7)
These equations confirm that
ˆ
T
(q
1
,p
1
)
are indeed phase space
translations. They satisfy the Weyl group composition rule
ˆ
T
(q
1
,p
1
)
ˆ
T
(q
2
,p
2
)
=
ˆ
T
(q
1
+q
2
,p
1
+p
2
)
e
i
π
N
(p
1
q
2
−q
1
p
2
)
. (8)
The N
2
translations
ˆ
T
(q,p)
, p, q = 0, ..N − 1 satisfy the or-
thogonality relation
Tr(
ˆ
T
†
(q,p)
ˆ
T
(q
′
,p
′
)
) = N δ
qp,q
′
p
′
(9)
thus constituting an orthogonal basis for the Liouville space
H
N
2
.
The expansion of any operator
ˆ
A in this basis constitutes
the chord [19] or characteristic function representation. This
representation assigns to every
ˆ
A ∈ H
N
2
the c-number func-
tion a(q, p) = N
−1
Tr(
ˆ
A
ˆ
T
†
(q,p)
) and therefore every operator
has the expansion
ˆ
A =
X
q,p
a(q, p)
ˆ
T
(q,p)
. (10)
For representation purposes we also use a basis of “phase
point ” operators that constitute the Weyl, or center [19], rep-
resentation. In this basis the density operator is the discrete
Wigner function of the quantum state. The peculiar features
of the discrete Wigner function for Hilbert spaces of finite di-
mension have been described recently in [20].
C. Unitary Dynamics: quantum maps
A classical map is a dynamical system that usually, but not
exclusively,arises form the discretization of a continuous time
system (by means of a Poincar´e section, for example). Al-
though it is always possible, by integration of the equations
of motion, to derive the map from a Hamiltonian, this con-
nection is rather involved and in many instances it is more
useful to model specific features of Hamiltonian dynamics by
directly specifying the map equations without going through
the integration step. The same is true in quantum mechanics:
3
instead of modeling the Hamiltonian operator and integrating
it to obtain the unitary propagator, it is simpler to model di-
rectly the unitary map. Classically an area preserving map is
characterized by a finite canonical transformation and the cor-
responding quantum map is the unitary propagator that rep-
resents this canonical transformation. There are no exact and
systematic procedures to realize this correspondence. On the
2-dimensional plane R
2
relatively standard procedures (see
[22]) give an approximation of the propagator in the semi-
classical limit as
U(q
1
, q
2
) =
i
~
∂
2
S
∂q
1
∂q
2
1/2
exp
i
~
S(q
2
, q
1
)
, (11)
where S is the action along the unique classical path from q
1
to q
2
, and where for simplicity we do not consider the ex-
istence of multiple branches and Maslov indices. Only for
linear symplectic maps on R
2
this unitary propagator is ex-
act, and then S is minus the quadratic generating function of
the linear transformation. On the other hand, several ad-hoc
procedures for the quantization of specific maps have been
devised: some integrable (translations[21] and shears) and
chaotic maps, such as cat maps [23, 24], baker maps [25, 26]
and the standard map [27] . Also all maps of a “kicked” na-
ture, realized as compositions of non-commuting nonlinear
shears can be quantized, as well as periodic time dependent
Hamiltonians [28]. Once the quantum propagator has been
constructed, the advantages of using quantum maps to model
specific features of quantum dynamics become apparent. The
propagator
ˆ
U is a unitary N ×N matrix, propagation of a pure
state is achieved simply by matrix multiplication, and finally
the classical limit is obtained by letting N → ∞.
In Liouville space the evolution of the density operator ˆρ by
the map
ˆ
U is given by
ˆρ
′
=
ˆ
Uρ
ˆ
U
†
. (12)
As a linear map acting on H
N
2
Eq. (12) can be written as
ˆρ
′
=
ˆ
U ⊗
ˆ
U
†
(ˆρ) = U(ˆρ), (13)
In what follows the notation
ˆ
U ⊗
ˆ
U
†
is meant to be equivalent
to the Ad(U) notation customary in group theory. The linear
operator U
def
=
ˆ
U ⊗
ˆ
U
†
is a unitary N
2
× N
2
matrix.
III. NOISY DYNAMICS
Realistic quantum processes always involve a certain de-
gree of interaction between system and environment. In this
case the evolution of the system is not unitary and requires
a description in Liouville space. This loss of unitarity leads
to decoherence and to the emergence of classical features
[29, 30] in the evolution. When the environment is taken into
account the evolution of the system is governed by a mas-
ter equation, which takes the form of a hierarchy of integro-
differential equations. A drastic simplification follows from
the assumption that the environment reacts to the system suf-
ficiently fast, in such a manner that the system looses all prior
memory of its state, i.e. that the evolution is Markovian.
2
.
The resulting Lindblad equation [33, 34]
dˆρ
dt
= −
i
~
[
ˆ
H, ˆρ] +
+
1
~
X
j
(
ˆ
L
j
ˆρ
ˆ
L
†
j
−
1
2
ˆ
L
†
j
ˆ
L
j
ˆρ −
1
2
ˆρ
ˆ
L
†
j
ˆ
L
j
).(14)
determines the evolution of open quantum systems through a
Hamiltonian
ˆ
H that governs the unitary noiseless evolution
and the Lindblad
ˆ
L
i
operators that model the interaction with
the environment. The particular structure of the equation en-
sures that the evolutionpreservesthe total probability,the pos-
itive semi-definiteness and hermiticity of the density matrix.
The infinitesimal propagator is a linear operator in Liouville
space which can be integrated to yield a finite linear mapping
ˆρ = S(ˆρ
0
). (15)
This mapping, as a reflection of the structure of the Lindblad
equation, also has a particular form that guarantees the preser-
vation of the general properties of the density operator. The
general form, called the Kraus representation [35] is
S(ˆρ) =
X
µ
ˆ
M
µ
ˆρ
ˆ
M
†
µ
=
"
X
µ
ˆ
M
µ
⊗
ˆ
M
†
µ
#
(ˆρ). (16)
The only further restriction on the
ˆ
M
µ
operators arises from
the preservation of the trace that requires
X
µ
ˆ
M
†
µ
ˆ
M
µ
=
ˆ
I. (17)
In what follows we will select them from a certain complete
family, with a specific norm. In that case the representation
takes a more general form
S(ˆρ) =
X
µ
c
µ
ˆ
M
µ
ˆρ
ˆ
M
†
µ
(18)
where now the positivity requirements are c
µ
≥ 0 and
X
µ
c
µ
ˆ
M
†
µ
ˆ
M
µ
=
ˆ
I. (19)
Within this general framework, just as in the case of quantum
maps, we have the choice of modeling the noise through the
Lindblad operators or directly in terms of the integrated form
via the Kraus operators. In what follows we choose the latter
and thus we modelthe evolutionby specifying a quantum map
to represent the unitary evolution followed by a noisy step,
modeled by its Kraus superoperator form. An evolution of the
density matrix specified in this way is known in the literature
[36, 37, 38, 39] as a quantum operation. It includes the special
case of unitary evolution when the sum is limited to only one
term. In that case, and only then, the dynamics is reversible.
A general superoperator has no inverse.
2
Fora detailed description of quantum noise andquantum Markov processes
see [31, 32].
4
A. Quantum coarse grained dynamics
We are interested in modeling the effect of a small amount
of noise on the evolution of an otherwise unitaryquantummap
[8, 13, 18, 40]. We assume that the one step propagatorresults
from the composition of two superoperators. The first is the
unitary propagator U and the second is a quantum diffusion
superoperator D
ǫ
, defined by
D
ǫ
=
X
q,p
c
ǫ
(q, p)
ˆ
T
(q,p)
⊗
ˆ
T
†
(q,p)
, (20)
that introduces decoherence. The linear form of the full prop-
agator is
L
ǫ
= D
ǫ
◦ U (21)
and its action on a density matrix ˆρ is
L
ǫ
(ˆρ) =
X
q,p
c
ǫ
(q, p)
ˆ
T
(q,p)
ˆ
U ˆρ
ˆ
U
†
ˆ
T
†
(q,p)
. (22)
As the Kraus operators in this case are unitary the condition
(19) becomes simply
X
q,p
c
ǫ
(q, p) = 1. (23)
Subject to this condition c
ǫ
(q, p) can be an arbitrary positive
function of q and p. Its significance in terms of coarse graining
is clear: as long as c
ǫ
(q, p) is peaked around (0, 0) and of
width ǫ, the action that follows the unitary step consists in
displacing the state incoherently over a phase space region of
order ǫ. To avoid a net drift in anyparticular direction, c
ǫ
(q, p)
must be an even function of the arguments q and p. From this
imposition and the fact that
ˆ
T
†
(q,p)
=
ˆ
T
(−q,− p)
(24)
it follows, from the properties of the matrix scalar product
(2) that D
ǫ
is hermitian (see App. A for details on the scalar
product).
The spectral propertiesof the separate superoperatorsU and
D
ǫ
are simple to obtain. If the Floquet spectrum of the quan-
tum map is
ˆ
U|φ
k
i = e
iξ
k
|φ
k
i (25)
then the spectrum of U is unitary and given by
U(|φ
k
ihφ
j
|) = e
i(ξ
k
−ξ
j
)
|φ
k
ihφ
j
|. (26)
To obtain the spectrum of D
ǫ
we use the composition rule (8)
to show that
ˆ
T
(q,p)
ˆ
T
(µ,ν)
ˆ
T
†
(q,p)
= exp
i
2π
N
(νq − µp)
ˆ
T
(µ,ν)
. (27)
We then derive
D
ǫ
ˆ
T
(µ,ν)
=
X
q,p
c
ǫ
(q, p)
ˆ
T
(q,p)
ˆ
T
(µ,ν)
ˆ
T
†
(q,p)
=
X
q,p
c
ǫ
(q, p) exp
i
2π
N
(νq − µp)
ˆ
T
(µ,ν)
= ec
ǫ
(µ, ν)
ˆ
T
(µ,ν)
. (28)
Therefore, the N
2
eigenvalues of D
ǫ
are given by the 2D dis-
crete Fourier transform ec
ǫ
(µ, ν) of the coefficients c
ǫ
(q, p).
The eigenfunctions are the translation operators themselves.
Hence, using the bra-ket notation described in appendix A,
the spectral decomposition of D
ǫ
is
D
ǫ
=
X
µ,ν
T
(µ,ν)
ec
ǫ
(µ, ν)
T
(µ,ν)
, (29)
in analogy with Eq. (A10).
Physically, the action of D
ǫ
is quite simple in the chord
representation (10): if ˆρ is expanded as
ˆρ =
X
µ,ν
ρ
µ,ν
ˆ
T
(µ,ν)
(30)
then
D
ǫ
(ˆρ) =
X
µ,ν
ec
ǫ
(µ, ν)ρ
µ,ν
ˆ
T
(µ,ν)
. (31)
Thus the coefficients in the chord representation are sup-
pressed selectively according to ec
ǫ
(µ, ν).
FIG. 1: The left pane ec
ǫ
(µ, ν) shows the eigenvalues of D
ǫ
for
ǫ = 0.15 and N = 100. The DFT of this function generates the
coefficients c
ǫ
(q, p) (right pane) of the Kraus representation of D
ǫ
of
Eq. (20).
It is evident from Eqs. (20) and (29) that the diffusion su-
peroperator thus defined can be specified indistinctly either by
c
ǫ
(q, p) or by ec
ǫ
(µ, ν) . For an efficient numerical implemen-
tation of its action we have found convenient to specify the
latter as
ec
ǫ
(µ, ν) = e
−
1
2
(
ǫN
π
)
2
(
sin
2
[πµ/N]+sin
2
[πν/N]
)
, (32)
This is a smooth Gaussian-like periodic function of the integer
variables µ and ν. For large values of N it is very close to the
Gaussian
ec(µ, ν) ≃ e
−ǫ
2
(µ
2
+ν
2
)/2
(33)
This means that the action of D
ǫ
will leave essentially un-
altered the coefficients ρ
µ,ν
in a region of size ∼ 1/(ǫN )
(Fig. 1, left) around the origin while strongly suppressing
those outside. The backward DFT of ec
ǫ
(µ, ν) does not have a
simple analytic expression but from general properties of the
DFT it will also be a Gaussian like function with the comple-
mentary width ∼ ǫ/2π (Fig. 1,right).
The action of D
ǫ
progressively washes out the quantum in-
terference. This fact is clearly seen if the density matrix is
5
(a) (b)
FIG. 2: Display of the action of D
ǫ
. Panel (a) shows the Wigner
function after the step U has been applied to a position state (ρ
0
=
| q
0
ih q
0
|). Panel (b) shows the state after the full propagator L
ǫ
=
D
ǫ
◦ U has acted. The map is the perturbed cat map of Eq. (64)k =
0.02, N = 60, ǫ = 0.25.
represented by the Wigner function. On the torus the Wigner
function exhibits two different types of interference. The
stretching and folding produce quantum interference between
different parts of the extended state. Additionally, the peri-
odicity of the torus introduces interference between the state
and its images. In Fig. 2 we show the difference between the
unitary and the noisy evolution of a coordinate state by a non-
linear map. The two types of interference are clearly seen.
The long wavelength fringes on the convex side are produced
by nonlinearities. The short wavelength fringes correspond to
the images. The effect of the noise can be seen on (Fig. 2(b))
: the classical part of the state (in white) has been broadened
and the long wavelength interference has been significantly
erased. This process continues at each step of the propaga-
tion and the quantum state becomes more and more mixed
and more and more classical.
B. Spectrum of the quantum coarse gained propagator
In this section we study the general features of the spec-
trum of the combined action of the unitary map and the coarse
graining operator, given by (Eq. (22)). For finite values of ǫ
and N, L
ǫ
is a convexsum of unitary matrices and is therefore
a completely positive, contracting superoperator. Its spec-
trum has the following properties:
• It is unital i.e. it has a trivial non-degenerate unit eigen-
value corresponding to the uniform density ˆρ
∞
=
ˆ
I/N.
• The remaining spectrum is entirely contained inside the
unit circle and symmetric with respect to the real axis.
The pair of complex conjugate eigenvalues correspond
to hermitian conjugate eigenoperators.
• As
q
(L
ǫ
◦ L
†
ǫ
) = D
ǫ
the eigenvalues of D
ǫ
are also
the singular values of L
ǫ
. Therefore the spectrum is
contained exactly in the annulus
e
−
(
ǫN
π
)
2
≤ |λ
i
| ≤ e
−
1
2
(
ǫN
π
)
2
(
sin
2
[π/N]
)
(34)
In the limit of large N we can thus write
−
Nǫ
π
2
≤ ln |λ
i
| ≤ −
ǫ
2
2
. (35)
The singular values accumulate near the origin, thus
forcing most of the eigenvalues of L
ǫ
to be near zero.
On the other hand the allowed eigenvalue region ex-
tends exponentially close to the unit circle in the limit
ǫ → 0.
• The superoperator is not normal, and therefore has dis-
tinct left and right eigenoperatorscorrespondingto each
eigenvalue. The left and right eigenvalue problems are
then posed as follows for each pair of complex conju-
gate eigenvalues λ, λ
∗
L
ǫ
ˆ
R
i
= λ
i
ˆ
R
i
L
ǫ
ˆ
R
†
i
= λ
∗
i
ˆ
R
†
i
(36)
L
†
ǫ
ˆ
L
i
= λ
∗
i
ˆ
L
i
L
†
ǫ
ˆ
L
†
i
= λ
i
ˆ
L
†
i
(37)
where
ˆ
L
i
,
ˆ
R
i
conform a biorthogonal set
Tr
L
†
i
R
j
= Tr
L
i
R
j
= Tr
L
†
i
R
†
j
= δ
i,j
. (38)
and we assume the normalization Tr
L
†
i
L
i
=
Tr
R
†
i
R
i
= 1 In particular, corresponding to λ
0
= 1
we choose
ˆ
L
0
=
ˆ
R
0
=
ˆ
I/N and therefore all the re-
maining eigenoperators are traceless.
• The spectral decomposition of L
ǫ
then becomes
L
ǫ
=
X
i
|R
i
)λ
i
( L
i
|. (39)
The exact numerical calculation of the spectrum is ham-
pered by the need to diagonalize very large non-hermitianma-
trices of dimension N
2
×N
2
for values of N large enough to
extract semiclassical features from the spectrum. In Section
IVB we develop a method, specially adapted to chaotic sys-
tems, that takes account of the dynamics of the map to extract
the part of the leading spectrum relevant to asymptotic time
behavior.
IV. QUANTUM CLASSICAL CORRESPONDENCE
Chaotic evolution in phase space implies exponential
stretching and squeezing of initially localized densities. On
a timescale of the order of the Ehrenfest time t
~
, significant
quantum corrections to the classical evolution inevitably ap-
pear. However, essentially classical features emerge from
quantum chaotic dynamics when decoherence is introduced,
even in the limit of no decoherence. In this section we relate
the spectra of the propagators of densities (both classical and
quantum)with the underlying, mainly asymptotic, behaviorof
time dependent quantities.
6
Consider the classical analog for the propagation of densi-
ties in phase space. If f (x) is a classical map, and x = (q, p)
a point in phase space, then the evolution of a probability den-
sity is governed by
ρ
′
(y) =
Z
δ(y − f(x))ρ(x)dx = [Lρ] (y ) (40)
where y = (q
′
, p
′
) and L is the Perron-Frobenius (PF)
operator[41]. It is unitary on the space of square integrable
functions L
2
, and infinite dimensional. However, one is
mostly interested in the decay properties of observables much
smoother than L
2
. When the functional space on which L op-
erates is restricted by smoothness, the spectrum of PF changes
drastically, moving to the inside of the unit circle. This
smoothing can be attained by convolution with a self adjoint
compact (on L
2
) coarse graining operatorD
ǫ
[18, 42], where ǫ
is the coarse graining parameter. The coarse grained PF takes
the form
L
ǫ
= D
ǫ
◦ L, (41)
(notice the analogy with Eq. (21)). D
ǫ
damps high frequency
modes in L
2
and thus effectively truncates L to a nonunitary
operator. There is substantial difference, however, between
the spectrum of the PF for a regular map and for a chaotic
map. As the coarse graining ǫ tends to zero, parts of the spec-
trum of L
ǫ
for a regular map can be arbitrarily close to the
unit circle. On the contrary, for a chaotic map there is a finite
gap for any value of ǫ > 0. The isolated eigenvalues which
remain inside the unit circle as ǫ → 0 are the Ruelle-Pollicot
resonances. Rugh[43] and more recently Blank, et al. [17]
made formal descriptions of the spectrum of PF for Anosov
maps on the torus using tailor-made Banach spaces adapted to
the dynamics. Moreover Blank, et al. use this to analyze res-
onances of noisy propagators and prove that these resonances
are stable, i.e. independent of ǫ in the limit of small coarse-
graining . Blum and Agam[44] proposed a numerical method
to approximate the classical spectrum using similar concepts.
A formal and very thorough recent work by
Nonnenmacher[18] explores the characteristics of prop-
agators, both classical and quantum, with noise for maps on
the torus, both regular and chaotic. In that work its is proved
that, in the limit N → ∞, the spectrum of the coarse grained
quantum propagator L
ǫ
, for fixed ǫ, tends to that of the
coarse grained PF L
ǫ
([18] Theorem 1). These two theorems,
taken together, provide a solid framework for the numerical
calculation of quantum resonances of torus maps and of their
classical manifestations.
A. Asymptotic behavior
The time evolution the von Neumann entropy was used by
Zurek and Paz[3] to characterize quantum chaotic systems.
They conjectured that the rate of increase of the von Neu-
mann entropy of the decohering (chaotic) system is indepen-
dent of the strength of the coupling to the environment and is
ruled by the Lyapunov exponents. Thus classicality emerges
naturally and correspondence even for chaotic systems is re-
covered when decoherence is considered. This assertion was
extensively tested numerically [8, 40, 46, 47] mainly for the
linear entropy (closely related to the purity) which is a lower
bound of the von Neumann entropy. Other quantities, like
the Loschmidt echo[5] which also display a noise indepen-
dent Lyapunov decay. have also become of interest recently,
especially in the context of quantum information processing
and computing. Besides the linear entropy, in this section we
study the asymptotic behavior of the autocorrelation function
and the Loschmidt echo.
For purely chaotic systems, after the initial spread gov-
erned by the Lyapunov exponent, a state ˆρ evolved n times
approaches asymptotically ˆρ
∞
=
ˆ
I/N and all time depen-
dent quantities saturate to a constant value. The rate at which
these quantities saturate is given by the largest eigenvalue, in
modulus, smaller than one. Since, according to [18] the spec-
trum of L
ǫ
approachesthat of L
ǫ
, then the universalityof these
decays can also be used to characterize quantum chaos.
To display the decay towards ˆρ
∞
we subtract it from the
initial state. Thus given an arbitrary state ˆρ, we define
ˆρ
0
= ˆρ −
ˆ
I
N
, (42)
were it is clear that Tr(ˆρ
0
) = 0. Thus in all computations in-
stead of evolving an initial state, we evolve an initial traceless
pseudo-state such as the one defined in Eq. (42), orthogonal to
ˆρ
∞
. Thus, we study how the distance between the initial state
and the equilibrium state evolves. For example, for the lin-
ear entropy, after the initial Lyapunov behavior, which ends
at about the Ehrenfest time (n
E
∼ ln N), instead of satura-
tion to the equilibrium state ρ
∞
, we expect to get an unbound
growthwhich represents howthis distance decreasesexponen-
tialy, and the exponent is proportional to |λ
1
|.
Assuming for simplicity that all the eigenvalues are nonde-
generate, and that
ˆ
R
i
, i = 0, . . . , N
2
− 1, are the right eigen-
functions (see App. A) then the expansion of ˆρ
0
in terms of
ˆ
R
i
is
ˆρ
0
=
X
i6=0
r
i
ˆ
R
i
, (43)
were r
i
= Tr(
ˆ
L
†
i
ˆρ
0
) and
ˆ
L
i
is the left eigenfunction. The
pseudo state ˆρ
0
evolved n times, is given by
ˆρ
n
= L
n
ǫ
ˆρ
0
=
X
i6=0
r
i
λ
n
i
ˆ
R
i
. (44)
If the eigenvalues are ordered decreasingly, according to 1 >
|λ
1
| ≥ |λ
2
| ≥ . . . ≥ λ
N
2
−1
, then ˆρ
0
is a sum of exponentially
decaying modes. Suppose that λ
1
is real
3
, then it is clear from
Eq. (44) that
ˆρ
n
→ r
1
λ
n
1
ˆ
R
1
(45)
3
In all the numerical simulations made, this was indeed the case.
7
FIG. 3: Quantum-classical correspondence for the noisy propagator. The top row shows repeated applications of the Perron-Frobenius operator
of the perturbed Arnold cat of Eq. (64), to an initial classical (position) state. The bottom row shows the Husimi representation of ρ
0
, . . . , ρ
6
,
where ρ
0
is a position eigenstate (N = 150, ǫ = 0.2, k = 0.02).
as n → ∞. Hence the asymptotic decay to the uniform den-
sity is ruled by λ
1
. As a consequence any quantity which
depends explicitly on ˆρ
n
shows an exponential decay. Such is
the case for the autocorrelation function
C(n) = Tr(ˆρ
†
0
ˆρ
n
) (46)
From Eq. (45) we get, for large n,
C(n) → |r
1
|
2
λ
n
1
+ . . . (47)
where we used the fact that Tr(
ˆ
R
†
1
ˆ
R
1
) = 1. If λ
1
is complex
then
ˆρ ∼ λ
n
1
r
1
ˆ
R
1
+ λ
∗
n
1
r
∗
1
ˆ
R
1
,
and C(n) oscillates around λ
n
1
(oscillation also appears if, for
example, |λ
2
| ≈ |λ
1
|). Similarly, we can see that the linear
entropy
S
n
= −ln
Tr(ˆρ
2
n
)
(48)
grows linearly with 2n. Once again, using Eq. (45), the linear
entropy for large n is
S
n
∼ −ln
h
|r
1
|
2
|λ
1
|
2n
Tr
ˆ
R
†
1
ˆ
R
1
i
= −2n ln [|λ
1
|] + constants. (49)
Recently the Loschmidt echo has been extensively
studied[5] especially in the context of fidelity decay in quan-
tum algorithms[6]. The definition of the echo is
M(t) = |hψ(0) |e
i
~
(H+Σ)t
e
−
i
~
Ht
|ψ(0) i|
2
(50)
which is the return probability of a state evolved forward a
time t with a Hamiltonian H and backward with a slightly
perturbed Hamiltonian H + Σ. It can also be viewed as the
overlap between two states evolved forward with slightly dif-
ferent Hamiltonians. Then M is just a measure of howfast the
two states “separate”. Most works focus on short times where
several “universal” regimes have been identified. In particu-
lar noise independent Lyapunov decay is observed for chaotic
systems.
In terms of the density operator, and discrete time systems,
the Loschmidt echo after n steps is
M
n
= Tr[ˆρ
′
n
ˆρ
n
] = Tr
h
U’
†
n
(ˆρ
0
)U
n
(ˆρ
0
)
i
. (51)
Where the prime represents a slight difference in the map. If
the propagation occurs in a noisy environment, characterized
by D
ǫ
, it is natural to define the echo as
M
n
(ǫ) = Tr
h
L
′
†
ǫ
n
(ˆρ
0
)L
ǫ
(ˆρ
0
),
i
(52)
were Eq. (51) is recovered by making ǫ = 0.
Following the same arguments used for the autocorrelation
function and for the linear entropyit can be shownthat asymp-
totically
ln [M
n
] ∼ n [ln(|λ
′
1
|) + ln(|λ
1
|)] . (53)
Notice that Schwartz inequality implies that
Tr[ˆρ
′
n
ˆρ
n
] ≤
r
Tr
h
(ˆρ
′
n
)
2
i
Tr
h
(ˆρ
n
)
2
i
(54)
Taking the natural logarithm of the expression above we get
ln [M
n
] ≤
1
2
ln
Tr
h
(ˆρ
′
n
)
2
i
+ ln
Tr
h
(ˆρ
n
)
2
i
⇒ ln [M
n
] ≤ −
1
2
(S
′
n
+ S
n
) . (55)
So we can see that the decay of the Loschmidtecho is bounded
by the negativevalue of the averagebetween the linear entropy
of the original system and the perturbed one (see Fig. 4 in [8]).
These three examples illustrate the fact that in the regime
where the leading spectrum of L
ǫ
and L
ǫ
coincide. We then
expect all time dependent quantities to decay asymptotically
with classical decay rates.
B. Leading spectrum. Dynamics approach.
In this section we describe the method used in [8] to com-
pute the relevant eigenvalues of the coarse grained propagator.
8
This method works well for hyperbolic automorphisms of T
2
because the nontrivial spectrum of the propagator lies entirely
inside the unit circle for all values of ǫ. The existence of a gap
between 1 and λ
1
is crucial.
In any complete basis, a superoperator such as L
ǫ
acting
on H
N
2
has associated an N
2
× N
2
dimensional matrix. For
small N this represents no setback. However, in order to es-
tablish a relationship between quantum and classical we need
to consider the semiclassical limit N → ∞ and the diagonal-
ization becomes unmanageable. To overcome this problem,
we use an approach which takes advantage of the dynamics
of the map to compute an approximation of the most relevant
part of thespectrum by reducing sensibly the size of the eigen-
value equation.
Following [17, 43, 44] we construct two sets F, B ∈ H
N
2
which are explicitly adapted to the dynamics of the map
4
.
Let ˆρ
0
be an arbitrary initial density in H
N
2
, which for con-
venience we choose it to be a pure state (projected onto some
space). Then, by repeated application of L
ǫ
we generate
F = {ˆρ
0
, ˆρ
u
1
, . . . , ˆρ
u
n
, . . .}, (56)
B = {ˆρ
0
, ˆρ
s
1
, . . . , ˆρ
s
n
, . . .} (57)
where
ˆρ
u
n
= L
ǫ
(ˆρ
u
n−1
) = L
n
ǫ
(ˆρ
0
) (58)
ˆρ
s
n
= L
†
ǫ
(ˆρ
s
n−1
) = L
†
n
ǫ
(ˆρ
0
). (59)
Notice that L
†
ǫ
is the back-step propagator. Therefore, if
the dynamics is chaotic, ˆρ
u
n
and ˆρ
s
n
are increasingly smooth
along the unstable and stable (classical) directions respec-
tively. Thus they reflect the expected behavior of the left and
right eigenfunctions of L
ǫ
(see Fig. 3).
Using the bra-ket notation described in App. A, we now
construct the matrix
[L
ǫ
]
i,j
= ( ρ
s
i
|L
ǫ
|ρ
u
j
) = ( ρ
s
i
|L
ǫ
(ρ
u
j
) ) = ( ρ
s
i
|ρ
u
j+1
), (60)
where ( ρ
s
i
| = ( L
†
i
ǫ
(ˆρ
0
)| = ( ρ
0
|L
i
ǫ
. Then we build the matrix
of overlaps between elements of F and B,
O
ij
= ( ρ
s
i
|ρ
u
j
). (61)
Notice that the structure of the matrices is very simple
( ρ
s
i
|ρ
u
j
) = ( ρ
0
|L
i
ǫ
|ρ
u
j
) = ( ρ
0
|L
i
ǫ
(ρ
u
j
) )
= ( ρ
0
|ρ
u
j+i
) = ( L
†
j
ǫ
(ρ
s
i
)|ρ
0
)
= ( ρ
s
i+j
|ρ
0
)
. (62)
We remark that ˆρ
0
∈ {ˆρ
∞
}
⊥
. Because by construction L
ǫ
is
trace preserving, successive applications on an arbitrary ρ
0
re-
main in {ρ
∞
}
⊥
and therefore the eigenvalue 1 isexplicitly ex-
cluded from our calculations. Moreover, the matrix elements
4
See Florido, et al. [45] for a rigorous review on numerical methods that
can be used to find RP resonances. The method used in [44], as well as its
limitations, is analized there.
FIG. 4: Plot of the matrix elements O
ij
= L
ǫ
i(j−1)
, where j+i = n.
They are closely related to the autocorrelation C(n) = ( ρ
0
| ρ
n
),
Exponential decay is observed. The initial state is ˆρ
0
= | 0, 0 ih 0, 0 |
where | 0, 0 i is the coherent state centered at (0, 0), which is a fixed
point of the map.
in (60) and (61)decay very rapidly, providing a natural cutoff
n
max
to the sets F and B. In App. B we show that an approx-
imation of the n
max
leading eigenvalues of L
ǫ
arises from the
solution of
Det[[L
ǫ
]
i,j
− z [O]
i,j
] = 0 , (63)
i, j = 0, n
max
− 1. This method resembles the Lanczos itera-
tion method [49] that uses Krylov matrices.
The combination of small matrix computations plus a
strong dependence on the dynamics makes this method a very
efficient tool to get an approximation of the leading spectrum
of L
ǫ
for chaotic maps.
Even when some of the main advantages of this method are
evident (reduced size, leading spectrum and spectral decom-
position, etc.), some drawbacks should be pointed out. When
the classical system is nearly integrable some resonances can
remain close to the unit circle and become unitaryin the ǫ → 0
limit and therefore convergenceof the method with small ma-
trices becomes problematic. Moreover, in that case there is
a strong dependence in the initial state ˆρ
0
. If it lies in a reg-
ular island it will not explore all phase space. On the other
hand, if initialized in the chaotic region it will only explore
the chaotic sea, leaving out the regular tori. As a consequence
some part of the relevant spectrum is inevitably lost. There-
fore, the method is useful when the classical dynamics is fully
chaotic.
V. NUMERICAL RESULTS
To illustrate our approach we utilize the Arnold cat map
[23] with a small sinusoidal perturbation. The map is
p
′
= p + q − 2πk sin[2πq]
q
′
= q + p
′
+ 2πk sin[2πp
′
]
(mod 1) (64)
where k is the small perturbation parameter. The map has a
Lyapounovexponent which is almost independent of the value
of k and equal to λ = ln[(3 +
√
5)/2]. On the other hand the
9
FIG. 5: Leading spectrum of L
ǫ
for different values of ǫ and N. If λ
i
is the i-th eigenvalue, then log λ
i
= log(r
i
) + iφ
i
(where r
i
= |λ
i
|)
and the coordinates in the plots are (φ, − log(r)). The ranges of the axes are φ ∈ [−π, π] and − log(r) ∈ [0, 6]. The map is the PAC with
k = 0.02 and the matrix was truncated to dim= 12.
Ruelle resonances (computed numerically) are very sensitive
to it. Thus it is the ideal model to test the asymptotic results,
independently of the short time Lyapunov regime. The map
is a composition of two nonlinear shears and therefore it is
easily quantized as a product of two noncommuting unitary
operators. The explicit expression in the mixed representation
is
hp |
ˆ
U|q i = exp
i
2π
N
q
2
2
+ qp −
p
2
2
×
× exp
n
2πN k
cos[2πq/N] + cos[2πp/N]
o
(65)
The other advantage of using a map of this type is that the
propagation both of pure states and of density matrices can
be done by fast Fourier techniques, thus allowing relatively
large Hilbert spaces with reasonable CPU times. The minor
disadvantage is that the quantization for this particular map is
only valid for even values of N [23].
10
A. Spectrum
In Ref. [8] we have performed the classical calculation of
resonances and shown that the quantum and classical leading
spectra coincide. Here we take a slightly different approach
and just compute the quantum spectra for a range of ǫ and
N values, as shown in Fig. 5. Observe that there is an ex-
tended region where the spectrum is stable and independent
of those parameters, signifying that the eigenvalues are prop-
erties exclusive of the map, and therefore coinciding with the
classical resonances. It is clear from this figure that the limits
ǫ → 0 and N → ∞ cannot be independent. In fact, at fixed N
the limit ǫ → 0 restores unitarity and the spectrum returns to
the unit circle. Therefore, ǫ must decrease as a certain func-
FIG. 6: Top (bottom) row shows the first 4 right (left) eigenfunctions
showing the unstable (stable) manifolds for the quantum PAC with
N = 100, ǫ = 0.3, k = 0.02, and matrices truncated to dim= 12.
tion of N . An optimal relationship between N and ǫ is yet
to be established but cannot be inferred from our limited data.
However, in our range of values a dependence like ǫ ∼ 1/
√
N
seems suitable.
The method described in Sect. IV B also provides approxi-
mations to the eigenfunctions of L
ǫ
corresponding to the lead-
ing eigenvalues. Inside the safe region (see Fig. 5) of N and ǫ
we were able to reconstruct at least 8 eigenfunctions success-
fully with matrices of dimension of order 12. The accuracy of
these eigenfunctions was checked by evaluating the orthogo-
nality properties in Eq. (38) and by computing the overlaps
1
λ
j
( L
i
|L
ǫ
|R
j
)
1
λ
j
( R
i
|L
ǫ
|L
j
)
. (66)
A plot of the absolute value of the Husimi representation for
the first four eigenfunctions can be seen in Fig. 6. As was
expected, the right (left) eigenfunctions corresponding to in-
variant densities of the propagator is smooth along the classi-
cal unstable (stable) manifold of the corresponding map. The
right (left) eigenfunctions are not uniform along the unstable
(stable) manifold showingpronouncedpeaks at the positionof
short periodic points. We intend to make a systematic analysis
of this connection in a future work.
B. Asymptotic decay
In this section we study numerically the asymptotic behav-
ior of the autocorrelation function, the linear entropy and the
Loschmidt echo for the PAC. In Fig. 7 we see the growth of
C
n
= −ln[C(n)] and the growth of S
n
for the perturbed
Arnold cat defined in Eq. (64) in Sect. V. In both cases there
are two well defined regimes. Initially both grow with the
slope determined by the Lyapunov exponent of the map. For
the PAC the Lyapunov is essentially the same for a wide range
of perturbations. On the other hand, the Ruelle resonances de-
pend strongly on the perturbation
5
. Taking as initial density a
traceless pseudo-state (see Eq. (42)), time evolution of quanti-
ties show how the state approaches uniformity exponentially,
with a rate givenby the largestRP resonance. We observethat,
after the Lyapunov regime (around the Eherenfest time n
E
6
),
the slope of the growth of C
n
is given by ln |λ
1
| whereas the
slope of S
n
is given by 2 ln |λ
1
| as predicted. This factor two
arises from the square in the definition of S
n
and is clearly
seen in Fig. 7. The solid lines represent these two slopes and
were obtained by computing λ
1
using the method described
in Sec. IVB.
FIG. 7: Purity decay and Correlation decay for the PAC (Eq. (64))
with N = 450, ǫ = 0.05, k = 0.005, initial pseudo-state ˆρ =
| 0, 0 ih 0, 0 | −
ˆ
I/N, where | 0, 0 i is a coherent state centered at
(0, 0). The inset shows the evolution of S
n
for ˆρ = | 0, 0 ih 0, 0 | and
how it saturates to the constant value ln N.
In order to show the universality of the decay of the linear
entropy and the Loschmidt echo, in terms of classical quanti-
ties, in Fig. 8 we show S
n
and ln(M
n
(ǫ)) for various values
of the parameter ǫ. The linear entropy is simply the negative
logarithm of the purity Tr(ˆρ
2
n
). When ǫ ∼ 0 the purity is con-
served and equal to one, so the linear entropy does not grow.
However when ǫ 6= 0 the purity will decay at a rate propor-
tional to ǫ. At one point, as predicted in [3], the growth of
the linear entropy saturates and no longer depends on ǫ. Since
we evolved a traceless pseudo-state, with no component on
the uniform density, after the Lyapunov growth the Ruelle-
Pollicot regime appears. In the same way as for the entropy,
for small values of ǫ the asymptotic decay rate is ǫ dependent
but it saturates when rate determined by the first Ruelle res-
onance is attained. This phenomenon can clearly be seen in
5
see Fig. 3 in [8].
6
In Fig. 7 N = 450 so n
E
∼ ln N = 6.11.
11
FIG. 8: Linear entropy growth (left panel) and Loschmidt echo decay (right) for various values of ǫ, ranging from 0.001 to 0.1 and for the
PAC with N = 450, k = 0.005. Both Lyapunov and Ruelle regimes can be seen when the rates saturate at an ǫ-independent value.
Fig. 8 (left). In the right panel we display the echo and illus-
trate exactly the same feature.
VI. CONCLUSIONS
We havedevelopeda method to study numerically the spec-
tral properties of open quantum maps on the torus. The
method is particularly well suited to chaotic maps and pro-
vides reliable eigenvalues and eigenfunctions. The noise
model that we implemented utilizes phase space translations
as Kraus operators and is equivalent to coarse graining quan-
tum Markovianmaster equations. Therefore it brings out clas-
sical properties of the map and we haveshown that these prop-
erties are reflected in the asymptotic decay of several quanti-
ties. The same methods can be used to study other noise mod-
els in the context of quantum information theory, if one thinks
of the quantum map as an algorithm to be implemented and
the noise as the error source present in any implementation.
Acknowledgments
The authors profited from discussions with Eduardo
Vergini, Diego Wisniacki, Fernando Cucchietti and Stephanne
Nonnenmacher. We would like to thank the referees for use-
ful comments as well as for pointing out Ref. [45]. Financial
support was provided by CONICET and ANPCyT.
APPENDIX A: ADJOINT AND LINEAR ACTION
Let H
N
be a complex Hilbert space of dimension N. The
space of linear operators acting on H
N
is called Liouville
space L ≡ H
N
2
. Elements in L are usually represented
by N × N dimensional complex matrices. However, given
ˆ
A,
ˆ
B ∈ H
N
2
then the “canonical” inner product, which in-
duces the norm, is
(
ˆ
A,
ˆ
B) = Tr(
ˆ
A
†
ˆ
B). (A1)
Thus H
N
2
is a Hilbert space itself. Now, superoperators are
a subset of the space of linear operators acting on H
N
2
. We
introduce a bra-ket notation to simplify inner product expres-
sions but also to distinguish the two types of decompositions
we use for superopertors. Let
ˆ
A,
ˆ
B ∈ H
N
2
then the action of
a superoperator S : H
N
2
→ H
N
2
can be written as
ˆ
B = S(
ˆ
A) (A2)
or as
|B ) = S|A ) (A3)
indistinctly. The adjoint, in the bra-ket form, is defined as
usual by
( A|S(
ˆ
B) ) = ( A|S|B ) = ( S
†
(
ˆ
A)|B ), (A4)
which settles that ( A|S = ( S
†
(
ˆ
A)|. Summarizing,
ˆ
A ≡ |A ) (
ˆ
A, ·) ≡ ( A |
(
ˆ
A,
ˆ
B) ≡ ( A|B ) S(
ˆ
A) ≡ S|A )
. (A5)
One way to think about it (not absolutely necessary but help-
ful) is to think of
ˆ
A as an operator, or matrix, in an operator
space, acting on vectors, and |A ) as a vector in a vector space,
acted on by superoperators.
Now, a completely positive superoperator has a Kraus op-
erator sum representation. Suppose S is a completely positive
supeorperator then there exist a set of operators {
ˆ
M
µ
}
N
2
−1
µ=0
∈
H
N
2
, such that
S =
X
µ
ˆ
M
µ
⊗
ˆ
M
†
µ
(A6)
12
ˆ
M
µ
are the Kraus operators. Without loss of generality, if the
number of operators is smaller than N
2
we can always com-
plete the set with zeros. The adjoint action of Son an operator
ˆ
A is defined through the Kraus representation suitable for the
case of Eq. (A2)
S(
ˆ
A) =
X
µ
ˆ
M
µ
ˆ
A
ˆ
M
†
µ
. (A7)
Eq. (A7) determines how the Kronecker product symbol ⊗
should be interpreted throughout this work.
On the other hand, a superoperator S can be written as an
expansion of spectral projectors. Let
ˆ
R
i
and
ˆ
L
i
be right and
left eigenoperators of L respectively, such that
S(
ˆ
R
i
) = λ
i
ˆ
R
i
S
†
(
ˆ
L
i
) = λ
∗
i
ˆ
L
i
, i = 1, . . . , N
2
, (A8)
and assume for simplicity that λ
i
are nondegenerate. Then the
spectral projectors are
ˆ
R
i
Tr(
ˆ
L
†
i
, ·) = |R
i
)( L
i
|, (A9)
and the spectral decomposition is given by,
S =
X
i
|R
i
)λ
i
( L
i
| (A10)
S
†
=
X
i
|L
i
)λ
∗
i
( R
i
|. (A11)
Therefore, given the spectral decomposition, the two ways of
expressing the action of S on
ˆ
A are
S|A ) =
X
i
|R
i
)λ
i
( L
i
|A ) ≡
X
i
ˆ
M
µ
ˆ
A
ˆ
M
†
µ
= S(
ˆ
A)
(A12)
In more general terms Caves[39] identifies and describes the
two different ways a superoperator acts as ordinary action
(A2) and left-right action (A3). This provides two distinct
decompositions of the same superoperator.
APPENDIX B: LEADING EIGENVALUES OF A LARGE
MATRIX
In this section we describe in a general way the method
used to compute the leading eigenvalues of the superoperator
L
ǫ
in section IVB. It is based on the Lanczos power itera-
tion method[49] but was inspired by a recent work by Blum
and Agam [44]. This method is useful when only a few of the
largest (in modulus) eigenvalues is needed and also, since it
deals with large matrices, when there is an efficient subrou-
tine to implement the matrix-vector product but there is no
need to store the whole matrix in an array variable. Moreover,
convergence and accuracy depends strongly on the distance
part of the spectrum one wants to calculate and the part to be
neglected.
In this work we don’t address the question of the estimation
of errors.
Proposition 1 . Suppose A is a large, sparse matrix in C
n×n
and assume each of its eigenvalues λ
i
has multiplicity one
and that 1 ≥ |λ
0
| > |λ
1
|. . . > |λ
n−1
|. Suppose {l
i
}
n−1
i=0
and
{r
i
}
n−1
i=0
are the corresponding left and right eigenvectors
Ar
i
= λ
i
r
i
(B1)
A
†
l
j
= λ
∗
j
l
j
, (B2)
and let u
0
∈ C
n
be a vector such that
|(l
i
, u
0
)| > 0 and |(r
i
, u
0
)| > 0 ∀i < k (B3)
for some k ≤ n, where ( , ) represents as usual the inner
product. Then the first k eigenvalues can be estimated from
the reduced (k × k) eigenvalue equation
Det
h
K
T
(A
†
, u
0
, k)AK(A, u
0
, k)
−zK
T
(A
†
, u
0
, k)K(A, u
0
, k)
i
= 0 (B4)
where K(A, u
0
, k) is the Krylov matrix whose columns are
the iterates of u
0
,
K(A, u
0
, k) =
h
u
0
, Au
0
, A
2
u
0
, . . . , A
k−1
u
0
i
(B5)
and
T
as usual denotes matrix transposition.
Proof. We sketch a rather straightforward proof (though per-
haps not entirely rigorous). The sets {l
i
}
n−1
i=0
and {r
i
}
n−1
i=0
of
left and right eigenvectors of A are complete and, they can be
normalized according to
(l
i
, r
j
) = δ
ij
. (B6)
Therefore there exist two distinct expansions of u
0
u
0
=
n−1
X
i=0
α
i
r
i
(B7)
u
0
=
n−1
X
i=0
β
i
l
i
. (B8)
13
In terms of these expansions we obtain
K
T
(A
†
, u
0
, k)K(A, u
0
, k) =
P
i
β
i
l
i
P
i
β
i
λ
∗
i
l
i
.
.
.
P
i
β
i
λ
∗
k−1
i
l
i
X
j
α
j
r
j
,
X
j
α
j
λ
j
r
j
, ··· ,
X
j
α
j
λ
k−1
j
r
j
(B9)
which yields
h
K
T
(A
†
, u
0
, k)K(A, u
0
, k)
i
µν
=
X
i,j
α
j
β
i
λ
ν
j
λ
∗
µ
i
(l
i
, r
j
)
=
X
i
α
i
β
i
λ
ν
i
λ
∗
µ
i
, (B10)
and similarly
h
K
T
(A
†
, u
0
, k)AK(A, u
0
, k)
i
µν
=
X
i,j
β
i
α
j
λ
ν+1
j
λ
∗
µ
i
(l
i
, r
j
)
=
X
i
β
i
α
i
λ
ν+1
i
λ
∗
µ
i
. (B11)
Thus, Eq. (B4) can be re-written as
Det
"
n−1
X
i=0
α
i
β
i
λ
ν
i
λ
∗
µ
i
(λ
i
− z)
#
= 0. (B12)
If all the conditions of the proposition are met, this equation is
equivalent to the original full eigenvalue equation. Now, since
λ
i
are ordered by dereasing modulus and assuming that the
eigenvaluesaccumulate around zero, leaving only a few, say k
of them, with significant modulus (as is the case for the maps
studied in section IVB) then we can neglect the contribution
of the last n −k terms in the sum
7
. Thus Eq. (B12) is just the
determinant of the product of three k × k square matrices
Det[Λ
†
Ξ Λ] = Det[Λ
†
] Det[Ξ] Det[Λ] = 0 (B13)
7
Although they can be computed, in this work we don’t provide estimations
of the errors due to this truncation.
where
Λ =
1 λ
0
λ
2
0
··· λ
k−1
0
1 λ
1
λ
2
1
··· λ
k−1
1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1 λ
k−1
λ
2
k−1
··· λ
k−1
k−1
; Ξ =
α
0
β
0
(λ
0
− z) 0 ··· 0
0 α
1
β
1
(λ
1
− z) ··· 0
.
.
.
.
.
.
.
.
.
.
.
.
0 ··· 0 α
k−1
β
k−1
(λ
k−1
− z)
. (B14)
The matrix Λ is a Vandermonde matrix. The determinant of a
Vandermonde matrix Λ
k
(λ
0
, . . . , λ
k−1
) is given by
Det[Λ
k
(λ
0
, . . . , λ
k−1
)] =
Y
0≤i≤j≤k−1
(λ
j
− λ
i
). (B15)
From equation Eq. (B15) it can be readily seen that if the spec-
trum of A is non-degenerate then Λ is invertible. Moreover,
the structure of Λ determines k because in the limit of k “too
large”, Λ is singular, at least to within computing precision.
So, using properties of the determinant in the secular equation
Eq. (B13), we get
Det[Ξ] =
Y
µ
α
µ
β
µ
(λ
µ
− z) = 0. (B16)
Since, from the hypothesis, α
µ
β
µ
6= 0 then Eq. (B16) yields
the desired solution, i.e. the first k eigenvalues of A.
In practice, the usefulness of the method depends upon the
gap (1 −|λ
1
|), because it determines how fast the terms of the
sum in Eq. (B13) decay.
In section IVB the span of the sets F and B are just the
Krylovspaces[49]of L
ǫ
and L
†
ǫ
, and using the present notation
Eq. (63) is
Det
h
K
†
(L
†
ǫ
, ˆρ
0
, n
max
)L
ǫ
K(L
ǫ
, ˆρ
0
, n
max
)
−z K
†
(L
†
ǫ
, ˆρ
0
, n
max
)K(L
ǫ
, ˆρ
0
, n
max
)
i
= 0 (B17)
In analogy with Eq. (B13). The efficiency of this method
depends strongly on the spectrum configuration. The case
of the coarse-grained propagator of hyperbolic maps on the
torus[17, 18] is particularly favorable because of the signifi-
cant gap between 1 and λ
1
and because 0 is an accumulation
point, so a large number of resonances can be discarded and
14
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