Spectral convergence bounds for classical and quantum Markov processes

0.39 MB·English

by  Oleg Szehr

Spectral convergence bounds for classical and quantum Markov
processes
Oleg Szehr,1,∗ David Reeb,1,† and Michael M. Wolf1,‡
1Department of Mathematics, Technische Universität München, 85748 Garching, Germany
(Dated: January 22, 2013)
3
1 We introduce a new frameworkthatyields spectralbounds on normsof functions oftran-
0
sition maps for finite, homogeneous Markov chains. The techniques employed work for
2
bounded semigroups, in particular for classical as well as for quantum Markov chains and

n they do not require additional assumptions like detailed balance, irreducibility or aperiod-
a
icity. We use the method in order to derive convergence bounds that improve significantly
J
upon known spectral bounds. The core technical observation is that power-boundedness of
1
transition maps of Markov chains enables a Wiener algebra functional calculus in order to
2
upper bound any norm of any holomorphic function of the transition map. Finally, we dis-

] cuss how generaldetailed balance conditions for quantumMarkovprocesseslead to spectral
h
convergence bounds.
p
-
h
t
a
m
Contents
[

1 I. Introduction 2
v
7
II. Preliminaries 4
2
8 A. Bounded Semigroups 4
4 B. Classical and Quantum Markov chains 4
.
1 C. Spectral properties 6
0
3
III. Limiting behavior and classical convergence estimates 7
1
: A. Limiting behavior of ( n) 7
v T n≥0
i B. Jordan and Schur convergence estimates 8
X
r IV. Main result: Spectrum and Convergence 10
a
A. Function spaces and functional calculi 11
B. Spectral bounds for the convergence of Markovian processes to stationarity 13
C. Comparison to the Schur and Jordan convergence bounds 18
D. Semigroups of Hilbert space contractions 20
E. Slow convergence for Markov chains 24
V. Convergence bounds from detailed balance 24
A. General bound 25
B. An ℓ2 bound 28
∗Electronic address: [email protected]
†Electronic address: [email protected]
‡Electronic address: [email protected]
2
Acknowledgments 30
References 31
A. An upper bound on a single Blaschke factor 32
I. INTRODUCTION
Across scientific disciplines, Markov chains are ubiquitous in algorithms as well as in
models for time evolutions. In many cases one is interested in when their limit behavior is
setting in. For algorithms this is often necessary in order to extract the right information
and for time evolutions of physical systems this is the time scale on which relaxation
or equilibration takes place. Some of the most widespread tools for bounding this time
scale are based on the spectrum of the transition map. For time-homogeneous Markov
chains with finite state space, the transition map is a stochastic matrix in the context of
classical probability distributions and a completely positive trace-preserving map in the
quantum case. Since these maps have spectral radius equal to 1, it is somehow clear that
only eigenvalues of magnitude 1 survive the limit, that the largest subdominant eigenvalue
governs thespeedofconvergence, andthattherestofthespectrumonlymattersonshorter
time scales. Let and be the transition map and its asymptotic part, respectively.
∞
T T
We seek convergence estimates of the form
n n Kµn (1)
||T −T∞|| ≤
after n time steps, where µ is the magnitude of the largest eigenvalue of inside the
T
open unit disc and K depends on the spectrum of , on n and on the dimension of the
T
underlyingspace. WedemandthatthedependenceofK onnisnotexponential, capturing
the intuition that the convergence is determined by an exponential decay as µn at larger
timescales, while for smaller n the whole spectral data is relevant. Such bounds are of
general interest for the theory of Markov chains, and they are especially important for
stochastic algorithms, which are widely used in statistics and computer science. They are
relatedtothesensitivityofthechaintoperturbations[9,10,25],areusedtostudy“cut-off”
phenomena [2] and random walks on groups [19].
Before describing our main innovation, we mention two traditional, linear algebraic,
approaches to bounding convergence times of classical Markov chains as in (1). A Jordan
decomposition of the difference yields a bound of the form Equation (1) with
∞
T − T
K = kµ−dµ+1ndµ−1, where d is the size of the largest Jordan block corresponding to any
µ
eigenvalue of magnitude µ and k is constant with respect to n but depends on as it is
T
essentially the condition number of the similarity transformation to Jordan normal form.
Unfortunately, there is no a priori bound on this condition number. An alternative way is
to use Schur’s instead of Jordan’s normal form. This leads indeed to an expression as in
Equation (1) where K can be bounded independent of , albeit not of n, and we obtain
T
roughly K µ−D+1(Dn)D, where D is the dimension of the underlying vector space. (See
∼
Section IIIB for details.) Needless to say, this “constant” seems to be far from optimal,
especially it does not capture the (correct) asymptotic n-dependence of the Jordan bound.
3
When proving bounds of the form of Equation (1), one typically employs additional
propertiesoftheMarkovchainsuchasdetailedbalance,irreducibility, aperiodicity, unique-
ness of the fixed point, Gibbs distribution of the stationary state, etc.. Clearly, these as-
sumptions are not always fulfilled—in particular in the quantum context detailed balance
seemstobealessnaturalassumptionand, furthermore, especiallyintheareaofdissipative
quantum computing [28] and dissipative state preparation [3, 6, 28], one aims at preparing
rank deficient states.
For classical Markov chains convergence estimates have been widely studied [7, 22] and
estimates basedon theJordan andSchur decompositions have beenknown formany years.
Although the latter are generally referred to as spectral convergence bounds, they do not
provide a satisfactory spectral desciption of the convergence of a Markov chain. While in
case of the Jordan bound it is not possible to compute K in terms of the eigenvalues of ,
T
the Schur bound cannot provide the correct asympotic behavior and does not reflect the
full spectral structure of . So far there is no a priori estimate as in Inequality (1) such
T
that K can simply be inferred from the localization of the eigenvalues of and such that
T
one obtains the correct asymptotic behavior of the chain. One goal of the present work is
to close this gap and to understand what information the spectrum of the transition map
of a classical or quantum Markov chain carries about the speed at which it approaches its
stationary behavior, i.e., to determine K in terms of the spectrum of .
T
Our primary interest lies in the study of classical and quantum Markovian evolutions.
However, to obtain a unified picture, in this article we will state our results more generally
forboundedsemigroupsoflinearmaps. Anyendomorphism ofavectorspace naturally
T V
generates a semigroup consisting of all n-fold concatenations n, n N. In our analysis
T ∈
we shall assume that the vector space of endomorphisms of carries a norm and that the
V
map ispower-bounded withrespecttothatnorm. Thatmeans thereisaconstant C such
T
that, for any n, n is bounded by C. This is equivalent to saying that the semigroup
||T ||
( n) generated by is bounded. The framework of bounded semigroups naturally
n≥0
T T
incorporates both classical and quantum Markov chains (see Section IIB).
We start our discussion by analyzing the asymptotic behavior of a bounded semigroup
( n) . We discuss spectrum related properties of that generate a bounded semigroup
n≥0
T T
and define the asymptotic part of the evolution in Section IIIA. In Section IIIB we
∞
T
extend the known convergence estimates based on the Jordan and Schur decompositions
to cope with bounded semigroups. Implicitly, the analysis covers quantum Markov pro-
cesses, where we state new convergence estimates. Section IV contains our main result, a
convergence estimate with the form of Equation (1), where K is fully determined by n and
the spectrum of . We start Section IV with a mathematical primer, Section IVA, con-
T
taining an introduction to an entirely new mathematical toolbox in the context of Markov
processes. We proceed by analyzing what information can be inferred from the spectrum
of about the speed at which ( n) approaches its asymptotic behavior, Section IVB.
n≥0
T T
The methods, which we employ, enable us in principle to derive spectral bounds on norms
of arbitrary functions of transition maps. When applied to power functions, we basically
obtain the sought convergence bounds. We discuss in Subsection IVC how our new bound
outperforms the convergence estimates based on the Jordan and Schur decompositions.
Nevertheless, it turns out that for many application, such as dissipative quantum com-
4
putation and state preparation, the convergence estimates obtained still are insufficient to
prove the efficiency of a possible implementation. The problem is that the convergence
time grows with D, which in turn is exponential in the number of constituent particles
(Section IVC). We discuss aspects related to the optimality of our new estimate as well as
the convergence speed of contractive Hilbert space semigroups in Section IVD. We prove
that stronger estimates, i.e. estimates such that roughly log(D) time steps bring the chain
close to stationarity cannot rely on the spectrum of the transition map alone, the latter
simply does not contain sufficient information.
As an approach to better convergence estimates in Section V we extend the detailed
balance condition for classical Markov chains and define this property in the context of
bounded semigroups, which then includes quantum evolutions. The core theorem of this
section is an extension of a convergence estimate that is frequently used to prove cut-off
behavior for classical Markov chains (Section VB).
Ourdiscussionfocusesongeneralboundedsemigroupsbutthecorrespondingstatements
about classical and quantum Markov processes are implicit, and we will frequently use
these for illustration. In what follows one can think of either as a quantum channel or
T
an ordinary stochastic matrix.
II. PRELIMINARIES
A. Bounded Semigroups
Throughout this paper will be a real or complex vector space of finite dimension D.
V
The set of linear endomorphisms of will be denoted by ( ), which shall be endowed
V L V
with a norm . For a given ( ) we consider the semigroup ( n) = n n N
n≥0
||·|| T ∈ L V T {T | ∈ }
of linear maps on generated by . Throughout, we assume that ( n) is bounded,
n≥0
V T T
i.e., there is a constant C > 0 such that sup n C.
n≥0||T || ≤
Our main approach applies for ( n) with a general norm. Nevertheless, for certain
n≥0
T
resultsconcerningconvergenceofclassicalandquantumMarkovchainsitwillbeconvenient
to endow with a scalar product . We will consider the induced Hilbert space norm
V h·|·i
(shortly, 2-norm) v = v v andtheoperatornorm(shortly, -norm)on ( )defined
|| ||2 h | i ∞ L V
by = sup ||T(v)||2. In some of our examples (e.g., classical Markov chains) it is
||T||∞ v6=0 ||v||2p
useful to fix an orthonormal basis e for . In this case we write T for the matrix
i i=1,...,D
{ } V
representation of with respectto e , i.e., T = e (e ) . We willemphasize whether
i i ij i j
T { } h |T i
or not has such additional structure in the corresponding sections.
V
B. Classical and Quantum Markov chains
We briefly review the definitions of classical and quantum Markov chains and discuss
certain related concepts.
Aclassical,finiteandtimehomogeneousMarkovprocessischaracterizedbyasemigroup
generated by a classical stochastic matrix T. More precisely, in this scenario = RD
V ∼
equipped with the canonical basis e and standard scalar product. The assertion that
i i
{ }
5
T is stochastic is equivalent to T 0 and T = 1. The latter is equivalent to saying
ij ≥ i ij
that the vector e = D e is fixed by the adjoint map, T∗(e) = e. In the context of
i=1 i ∈ V P
classicalMarkovchainsthe1-normplaysanexceptionalrole. Forv wewritev = e v
i i
and define v = PD v . The induced norm on the set of mat∈ricVes M acting onh |iis
|| ||1 i=1| i| V
called the 1-to-1 norm,
P
Mv
1
M = sup|| || .
1→1
|| || v
v6=0 1
|| ||
The 1-to-1 norm and the -norm (i.e. the 2-to-2 norm) are equivalent with
∞
D−1/2 M M D1/2 M . (2)
|| ||∞ ≤ || ||1→1 ≤ || ||∞
It is easily seen that T = 1 for any stochastic matrix T. We note that if is any
|| ||1→1 ||·||
norm such that T C holds forall stochastic matrices T, then Tn C n N; that is
|| || ≤ || || ≤ ∀ ∈
the Markov chain constitutes a bounded semigroup with constant C. Since we are working
in finite dimensions, such a semigroup is bounded with respect to any norm.
AtimehomogeneousquantumMarkovchainisalsocharacterizedbyasemigroup. Inthe
contextofquantumevolutions, however, thespace hasdifferentandadditionalstructure.
V
Inthis article we think of as the realvector space consisting of hermitian matrices acting
V
on a complex Hilbert space of dimension d, i.e., D = d2. A matrix ρ that is positive
∈ V
semidefinite (ρ 0) and has unit trace (tr[ρ] = 1) is referred to as a quantum state. An
≥
element ( ) is called positive iff X 0 implies (X) 0 for any X , and
T ∈ L V ≥ T ≥ ∈ V
trace-preserving iff tr[ (X)] = tr[X] X . is trace-preserving iff the adjoint ∗ of
T ∀ ∈ V T T
with respect to the Hilbert-Schmidt inner product X Y = tr(XY) on preserves the
T h | i V
identity matrix, ∗(1)= 1. If , with being the operator identity, acts as a positive
T T ⊗I I
map on , then is called completely positive [12, 16]. We denote by T the subset of
V⊗V T
( ) containing trace preserving and positive maps (TPPMs) and by T T the set of
+
L V ⊂
completely positive maps in T (TPCPMs). The latter describe the dynamics of a quantum
system, whenever the evolution of the system is independent of its history, and they are
called quantum channels in the realm of quantum information theory.
For X we denote by X the Schatten 1-norm of X. The induced distance
∈ V || ||1
ρ σ of two quantum states ρ and σ corresponds to the maximum probability to detect
|| − ||1
a difference between ρ and σ in an experiment, i.e.
ρ σ = sup tr(O(ρ σ)),
|| − ||1 | − |
||O|| ≤1
∞
where O stands for the largest singular value of O . For linear maps ( ) we
|| ||∞ ∈ V M ∈L V
define1 the induced 1-to-1-norm via
(X)
1
= sup ||M || .
1→1
||M|| X
X6=0 1
|| ||
1 Note that in this article we define the 1-to-1 norm with the supremum taken over Hermitian matrices.
Alternatively, the supremum could be taken over all matrices. The resulting norms are different, but
thelatter can be upperboundedin terms of 2 times theformer.
6
The diamond norm is the “stabilized version” of the 1-to-1 norm,
= ,
||M||⋄ ||M⊗I||1→1
where denotes the operator identity in ( ). It is the dual of the norm of complete
I L V
boundedness (CB-norm), i.e., we have = ∗ . The diamond norm and the
||M||⋄ ||M ||CB
1-to-1 norm are equivalent with [23]
D1/2 . (3)
||M||1→1 ≤ ||M||⋄ ≤ ||M||1→1
For any quantum channel we have = = 1 [16]. The distance of
T ||T||1→1 ||T||⋄ ||E −T||⋄
two channels , measures how well these channels can be distinguished by any quantum
E T
experiment. In the quantum context the 1-to-1 norm and the -norm (i.e., the 2-to-2
∞
norm) are equivalent with
D−1/4 D1/4 . (4)
||T||∞ ≤ ||T||1→1 ≤ ||T||∞
Note that due to the different structure of the underlying space in case of quantum
V
Markov chains, the above differs from the Inequalities (2).
If we are given a norm such that C T (or T ) the quantum Markov
+
||T|| ≤ ∀T ∈ ∀T ∈
chain generated by constitutes a semigroup bounded by C. Again, due to D < , this
T ∞
implies that the semigroup is bounded with respect to any norm.
C. Spectral properties
To each linear map ( ) we can assign a spectrum σ( ) via the usual eigenvalue
M∈ L V M
equation: we have λ σ( ) if and only if there is X = 0 with (X) = λX. We
∈ M 6 M
write m for the minimal polynomial associated with (i.e., the minimal degree, monic
M
M
polynomial that annihilates , m ( ) = 0) and m for the number of linear factors
M M
M M | |
in m . Another important object is the Blaschke product associated with m ,
M M
z λ
i
B(z) = − , (5)
1 λ¯ z
i
mYM −
wheretheproductistakenoverallisuchthatthelinearfactorz λ occursinm respect-
i M
−
ing multiplicities. Thus, the numerator of B as defined here is exactly the corresponding
minimal polynomial, m .
M
For convenience, we shall always assume that the eigenvalues in σ( ) are arranged
M
such that their magnitudes are non-decreasing. (This ordering is not unique when several
eigenvalues have the same magnitude. This ambiguity will, however, be irrelevant in the
following. Wheneverthesituationoccursthatwepickaneigenvalueofacertainmagnitude
λ we mean that we can take any eigenvalue that has this property.)
| |
For any ( ) the largest magnitude of all eigenvalues is the spectral radius,
M ∈ L V
which we denote as µ. It follows from Gelfand’s formula µ = lim k 1/k [4] that
k→∞
M
the spectral radius of any element of a bounded semigroup is at most 1. For stochastic
(cid:12)(cid:12) (cid:12)(cid:12)
matrices and TPPM it is clear that 1 is an eigenvalue of . (cid:12)(cid:12) (cid:12)(cid:12)
T
7
III. LIMITING BEHAVIOR AND CLASSICAL CONVERGENCE ESTIMATES
A. Limiting behavior of ( n)n≥0
T
In this section we begin our discussion of spectral convergence bounds for semigroups.
Based on a spectral decomposition of ( n) we introduce a map and show that
n≥0 ∞
T ∈ T T
this map reflects the behavior of ( n) for large n. In the following we extend known
n≥0
T
spectral convergence bounds for classical Markov chains to the more general semigroup
setup. We consider the classical derivations based on the Jordan and Schur decomposition
(Section IIIB). For this reason in this section we assume that carries a scalar product.
V
Our main result Theorem IV.3 will later outperform the bounds proven in this section
in terms of convergence speed even in the context of classical Markov chains. Moreover,
the techniques introduced there will allow us to consider general norms, which are not
induced by a scalar product.
Toformalizeourintuitionthatthespectrumof determinestheconvergenceproperties
T
of ( n) let us consider a Jordan decomposition of ,
n≥0
T T
= (λ + ), with (6)
i i i
T P N
i
X
= = , = δ i,j. (7)
i i i i i i j i,j i
N P P N N P P P ∀
Here, thesummation istakenoveralldistincteigenvalues of , the areprojectors whose
i
T P
rank equals the algebraic multiplicity of λ and the are the corresponding nilpotent
i i
N
blocks. All contributions to n that stem from eigenvalues of with magnitude smaller
T T
than 1 will vanish with increasing n. Hence, we expect the image of n to converge to a
T
subspace of spanned by all eigenvectors of whose eigenvalues are of magnitude one.
V T
We therefore define the linear map whose range is this subspace by
∞
T
:= λ , (8)
∞ i i
T P
|λXi|=1
where the are spectral projectors corresponding to the eigenvalues of of magnitude 1.
i
P T
Incaseswherethespectralradiusof isstrictlysmallerthan1, issimplyzero. If has
∞
T T T
only one eigenvalue of magnitude one and this eigenvalue is equals 1, thenthe sequence n
T
converges to , which is the unique rank one projection onto the stationary eigenspace of
∞
T
. In the following lemma we shall prove that mirrors the limit behavior of ( n)
∞ n≥0
T T T
also in the more general case. More precisely, as n is increasing n n approaches
||T −T∞||
0 and, for every k N, k indeed is an accumulation point of ( n) . The latter
∈ T∞ T n≥0
assertion is relevant especially in the case of classical and quantum Markov chains: the set
of stochastic matrices (or quantum channels) constitutes a closed set in the corresponding
space, which implies that is again a bona fide stochastic matrix (or quantum channel).
∞
T
Lemma III.1 (Limiting behavior of n). Let ( n) be a semigroup within ( ) such
n≥0
T T L V
that n C n N and let be as in Equation (8). Then we have that
∞
||T || ≤ ∀ ∈ T
i)alleigenvalues of withmagnitude1have trivialJordan blocks (i.e., λ = 1 = 0),
i i
T | | ⇒ N
ii) ( n n)= ( )n n N 0 ,
T −T∞ T −T∞ ∀ ∈ \{ }
8
iii) lim n n = 0,
n→∞||T −T∞||
iv) for any k N, k is contained in the closure of ( n) in ( ),
∈ T∞ T n≥0 L V
v) k C k N.
T∞ ≤ ∀ ∈
Pro(cid:12)(cid:12)of. (cid:12)i(cid:12)) Weproceed by contradiction and consider n . Since ( n) is bounded and
(cid:12)(cid:12) (cid:12)(cid:12) ||T ||∞ T n≥0
in finite dimensions all norms are equivalent there is 0 < K < with n K . On
1 ∞ ||T ||∞ ≤ 1
the other hand there is K > 0 with n K (λ + )n . If λ has a non-
2 ||T ||∞ ≥ 2 j jPj Nj ∞ i
trivialJordanblockthelattercanbelowerbounded(cid:12)(cid:12)(cid:12)b(cid:12)(cid:12)(cid:12)Py j(λjPj +N(cid:12)(cid:12)(cid:12)j(cid:12)(cid:12)(cid:12))n ∞ ≥ |λi|n−1n. It
followsthatif λ = 1andλ hasanon-trivialJordanbl(cid:12)o(cid:12)ckthen n gr(cid:12)(cid:12)owsunboundedly
| i| i (cid:12)(cid:12)P ||T ||∞ (cid:12)(cid:12)
with n.2 (cid:12)(cid:12) (cid:12)(cid:12)
ii) follows from the relations in (7) since = = 2. For n > 2 the statement
TT∞ T∞T T∞
follows by induction.
iii) By the previous assertion Tn Tn = (T T )n holds. The spectral radius µ
|| − ∞|| || − ∞ ||
of the map is strictly smaller than 1. We have from Gelfand’s formula that
∞
T − T n
lim Tn Tn 1/n = µ < 1 and hence for all n sufficiently large Tn Tn 1+µ .
n→∞|| − ∞|| || − ∞|| ≤ 2
With increasing n the right hand side goes to 0 and the claim follows. (cid:16) (cid:17)
iv) We prove that for fixed k there is a subsequence (Tnl)l that converges to T∞k, that
is liml→∞ Tnl −T∞k = 0. To achieve this we subdivide V into the invariant subspace
of corresponding to all eigenvalues of magnitude 1 and its complement. On the latter
T (cid:12)(cid:12) (cid:12)(cid:12)
subspace w(cid:12)(cid:12)e can dire(cid:12)c(cid:12)tly invoke iii). On the former subspace, it is sufficient to find, for
any ǫ > 0, a subsequence (Tnl)l with the property that |λnil −λki| ≤ ǫ simultaneously for
all i with λ = 1. The existence of such a subsequence follows from Dirichlet’s Theorem
i
| |
on simultaneous Diophantine approximation, [21] Theorem 1B.
v) By iv) for any k N and any ǫ > 0 there is n such that n k ǫ. This implies
∈ T −T∞ ≤
that k C +ǫ ǫ > 0 and hence the claimed inequality.
T∞ ≤ ∀ (cid:12)(cid:12) (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12)
B. Jordan and Schur convergence estimates
Our next aim is to understand qualitatively by how much for certain n the evolution
n differs fromits limit behavior, i.e., how smallthe quantity n n = Tn Tn
T ||T −T∞||∞ || − ∞||∞
is for any bounded semigroup. We shortly review two standard methods to obtain such
estimates. Both methods rely on the fact that n n = ( )n and perform a
T − T∞ T − T∞
transformationofT T touppertriangularform. Whilethefirstapproachistochoosethe
∞
−
Jordan normal form forT T , thesecond oneis basedon theSchur decomposition. Both
∞
−
decompositionsinvolveasimilaritytransformationAthatbringsT T touppertriangular
∞
−
form, i.e., T T = A(Λ+N)A−1 with diagonal Λ and nilpotent N. While in case of
∞
−
Jordan decomposition Λ+N has Jordan block structure, for the Schur decomposition A
is unitary.
Theorem III.2. Let ( n) be a bounded semigroup in ( ), let be the map intro-
n≥0 ∞
T L V T
duced in (8) and let µ be the spectral radius of . Then there are constants C ,C > 0
∞ 1 2
T −T
2 Seealso the derivation of thelower boundin Theorem III.2.
9
such that, for all n 1,
≥
C µn−dµ+1ndµ−1 n n C µn−dµ+1ndµ−1,
1 ≤ ||T −T∞||∞ ≤ 2
where d is the size of the largest Jordan block corresponding to any eigenvalue of
µ ∞
T −T
of magnitude µ.
Proof. We first state an upper bound on (Λ+N)n with diagonal Λ and nilpotent
|| ||∞
upper-triangular N. We note that any monomial in N and Λ vanishes if the total degree
of N is larger than or equal to D 1. Using this together with the triangle inequality in
−
the binomial expansion and exploiting the sub-multiplicativity of the -norm we find
∞
min{n,D−1}
n
(Λ+N)n N k Λ n−k. (9)
|| ||∞ ≤ k || ||∞|| ||∞
k=0 (cid:18) (cid:19)
X
Let now J(λ ) be a Jordan block with diagonal part λ 1 and nilpotent part N . We
i i i
consider the Jordan decomposition T T = A J (λ ) A−1, where the summation
− ∞ i,ν ν i
goes over i, which labels the different eigenvalues(cid:16)oLf T−T∞, a(cid:17)nd over ν, which enumerates
the Jordan blocks corresponding to an eigenvalue λ . We introduce the constant κ =
i
inf A A−1 where the infimum is taken over all A that bring T T to Jordan
|| ||∞ ∞ − ∞
form. It follows readily that
(cid:0) (cid:12)(cid:12) (cid:12)(cid:12) (cid:1)
(cid:12)(cid:12) (cid:12)(cid:12)
κ−1 Jn Tn Tn κ Jn (10)
|| ||∞ ≤ || − ∞||∞ ≤ || ||∞
with J = J (λ ). For any J (λ ) there is an n such that for all n n one has
i,ν ν i ν i 0 ≥ 0
J (λ )n J (λ )n , where J (λ ) denotes the largest Jordan block corre-
|| ν i ||∞L≤ || max max ||∞ max max
sponding to an eigenvalue λ of modulus µ. Therefore, to find an upper bound on the
max
right hand side of (10) we can subdivide J (λ ) in a nilpotent and a diagonal part
max max
and use Inequality (9). We note that for k d 1 we can bound n ndµ−1 and taking
≤ µ− k ≤
everything together we obtain for large enough n
(cid:0) (cid:1)
dµ−1
Tn Tn κ ndµ−1µn−dµ+1,
|| − ∞||∞ ≤
k=0
X
whichproves theupperboundinTheoremIII.2sinceitcanbeextendedtoanupperbound
valid forany n N by a rescaling of C . The lower bound is aconsequence of the following
1
∈
inequalities for n d 1
µ
≥ −
n
µn−dµ+1 J (λ )n Jn .
d 1 ≤ || max max ||∞ ≤ || ||∞
(cid:18) µ− (cid:19)
One problem with the above proof is that n and thus C can get large if there is a
0 2
sub-dominant eigenvalue close to the spectral radius. Another issue is that one cannot a
priori bound κ for general T. Consequently, only little is known about C and C . Most
1 2
10
awkward, C and C depend on the given channel , i.e. are not universal for all channels
1 2
T
of a given dimension. For this reason Theorem III.2 is a qualitative statement about the
asymptotic behavior of the semigroup. In contrast, the Schur decomposition allows us to
state an upper bound on the rate of convergence that only depends on n, D and µ. This
goes at the price of a rather pessimistic estimate.
Theorem III.3. Let ( n) be a bounded semigroup in ( ) such that n C n
n≥0 ∞
T L V ||T || ≤ ∀ ∈
N and let µ be the spectral radius of . For any n N it holds that
∞
T −T ∈
n n 2µn−D+1nD−1(µ+2C)D−1.
||T −T∞||∞ ≤
Proof. As already mentioned, this will be proven based on the Schur decomposition T
−
T = U(Λ+N)U†, where U is unitary. As before we can rely on the binomial expansion
∞
Inequality (9). We note that U = 1 and that for n > 1
|| ||∞
D−1 D−1
n
nk 2nD−1.
k ≤ ≤
k=0 (cid:18) (cid:19) k=0
X X
Thus, using the sub multiplicativity of the -norm it follows from (9) that
∞
Tn Tn 2nD−1µn−D+1max(1, N D−1).
|| − ∞||∞ ≤ || ||∞
In addition we have that N = T T Λ and therefore N 2C +µ.
− ∞− || ||∞ ≤
To obtain a convergence estimate for Markov chains in 1-to-1 norm we can rely on the
Inequalities(2). ThecorrespondingstatementofTheoremIII.2isimmediate. Analogously,
Theorem III.2 can be used to estimate the speed of convergence of TPPMs in 1-to-1 and
diamond norm via the Inequalities (4), (3).
Due to the lower bound in (2) the singular values of stochastic matrices are bounded
by D1/2 from which we infer that C D1/2 in this case. For positive, trace preserving
≤
maps the singular values are bounded by D1/4 ([17], or by the norm equivalence (4) and
the fact that = 1). Thus, Theorem III.3 includes a convergence bound for both
||T||1→1
classical stochastic matrices and TPPMs. For a more detailed discussion of the resulting
estimates in the quantum context see Subsection IVC.
IV. MAIN RESULT: SPECTRUM AND CONVERGENCE
Themaincontributionofthisarticleistointroduceanewformalismthatyieldsspectral
bounds on norm of functions of transition maps of Markov processes and to apply this
formalism to prove new estimates for the convergence of a such processes to stationarity.
The core technical innovation will be to employ a Wiener algebra functional calculus in
the context of bounded semigroups. To prove our estimates we will rely on the theory of
function algebras, functional calculi and model spaces. To our knowledge these concepts
have not found their way into the theory of classical or quantum Markov processes so far.
ForthisreasonatfirstwebrieflyintroducethemathematicalframeworkinSubsectionIVA.