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Sandwiched R
´
enyi Convergence
for Quantum Evolutions
Alexander M
¨
uller-Hermes
1
and Daniel Stilck Franc¸a
2
1
Department of Mathematical Sciences, University of Copenhagen, 2100 Copenhagen, Denmark
2
Department of Mathematics, Technische Universit
¨
at M
¨
unchen, 85748 Garching, Germany
February 27, 2018
We study the speed of convergence of a primitive quantum time evolution
towards its fixed point in the distance of sandwiched R´enyi divergences. For
each of these distance measures the convergence is typically exponentially fast
and the best exponent is given by a constant (similar to a logarithmic Sobolev
constant) depending only on the generator of the time evolution. We establish
relations between these constants and the logarithmic Sobolev constants as
well as the spectral gap. An important consequence of these relations is the
derivation of mixing time bounds for time evolutions directly from logarithmic
Sobolev inequalities without relying on notions like l
p
-regularity. We also derive
strong converse bounds for the classical capacity of a quantum time evolution
and apply these to obtain bounds on the classical capacity of some examples,
including stabilizer Hamiltonians under thermal noise.
Alexander M
¨
uller-Hermes: muellerh@posteo.net
Daniel Stilck Franc¸a: dsfranca@mytum.de
Accepted in Quantum 2018-02-01, click title to verify 1
arXiv:1607.00041v3 [quant-ph] 26 Feb 2018
Contents
1 Introduction 3
2 Notation and Preliminaries 4
2.1 Noncommutative l
p
-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Quantum dynamical semigroups . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Logarithmic Sobolev inequalities and the spectral gap . . . . . . . . . . . . 7
3 Convergence rates for sandwiched enyi divergences 9
3.1 R´enyi-entropy production . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Sandwiched R´enyi convergence rates . . . . . . . . . . . . . . . . . . . . . . 11
3.3 Computing β
p
in simple cases . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 Comparison with similar quantities 13
4.1 Comparison with spectral gap . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2 Comparison with logarithmic Sobolev constants . . . . . . . . . . . . . . . . 15
5 Mixing times 16
5.1 Mixing in Continuous Time . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.2 Mixing in Discrete Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6 Strong converse bounds for the classical capacity 21
7 Examples of bounds for the classical capacity of Semigroups 23
7.1 Depolarizing Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
7.2 Stabilizer Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
7.3 2D Toric Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
7.4 Truncated harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . 26
8 Conclusion and open questions 28
Acknowledgments 29
A Taylor Expansion of the Dirichlet Form 29
B Interpolation Theorems and Proof of Theorem 5.3 31
References 33
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1 Introduction
Consider a quantum system affected by Markovian noise modeled by a quantum dynamical
semigroup T
t
(with time parameter t
+
) driving every initial state towards a unique
full rank state σ. Using the framework of logarithmic Sobolev inequalities as introduced
in [1, 2] the speed of the convergence towards the fixed point can be studied. Specifically,
the α
1
-logarithmic Sobolev constant (see [1, 2]) is the optimal exponent α
+
such that
the inequality
D(T
t
(ρ)kσ) e
2αt
D (ρkσ) (1)
holds for the quantum Kullback-Leibler divergence, given by D (ρkσ) = tr [ρ(ln(ρ) ln(σ))],
for all t
+
and all states ρ.
The framework of logarithmic Sobolev constants is closely linked to properties of non-
commutative l
p
-norms, and specifically to hypercontractivity [1, 2]. Noncommutative l
p
-
norms also appeared recently in the definition of generalized R´enyi divergences (so called
“sandwiched R´enyi divergences” [3, 4]). It is therefore natural to study the relationship
between logarithmic Sobolev inequalities and noncommutative l
p
-norms more closely. The
approach used here is to define constants (which we call β
p
for a parameter p [1, )),
which resemble the logarithmic Sobolev constants, but where the distance measure is a
sandwiched enyi divergence instead of the quantum Kullback-Leibler divergence. More
specifically, the constants β
p
will be the optimal exponents such that inequalities of the
form (1) hold for the sandwiched enyi divergences D
p
, given by
D
p
(ρkσ) =
1
p1
ln
tr

σ
1p
2p
ρσ
1p
2p
p

if ker (σ) ker (ρ) or p (0, 1)
+, otherwise,
(2)
instead of the quantum Kullback-Leibler divergence D.
Our main results are two-fold:
We derive inequalities between the new β
p
and other quantities such as logarithmic
Sobolev constants and the spectral gap of the generator of the time evolution. These
inequalities not only reveal basic properties of the β
p
, but can also be used as a
technical tool to strengthen results involving logarithmic Sobolev constants.
We apply our framework to derive bounds on the mixing time of quantum dynam-
ical semigroups. Using the interplay between the β
p
and the logarithmic Sobolev
constants we show how to derive a mixing time bound with the same scaling as that
of the one derived in [2] directly from a logarithmic Sobolev constant. Previously,
this was only known under the additional assumption of l
p
-regularity (see [2]) of the
generator or for the α
1
-logarithmic Sobolev constant. It is still an open question
whether l
p
-regularity holds for all primitive generators.
As an additional application of our methods we derive time-dependent strong converse
bounds on the classical capacity of a quantum dynamical semigroup. We apply these to
some examples of systems under thermal noise. These include stabilizer Hamiltonians,
such as the 2D toric code, and a truncated harmonic oscillator. To the best of our
knowledge, these are the first bounds available on the classical capacity of these channels.
We also apply our bound to depolarizing channels, whose classical capacity is known [5],
to benchmark our findings.
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2 Notation and Preliminaries
Throughout this paper M
d
will denote the space of d × d complex matrices. We will
denote by D
d
the set of d-dimensional quantum states, i.e. positive semi-definite matrices
ρ M
d
with trace 1. By M
+
d
we denote the set of positive definite matrices and by
D
+
d
= M
+
d
D
d
the set of full rank states.
In [3, 4] the following definition of sandwiched quantum enyi divergences was pro-
posed:
Definition 2.1 (Sandwiched p-R´enyi divergence). Let ρ, σ D
d
. For p (0, 1) (1, ),
the sandwiched p-R´enyi divergence is defined as:
D
p
(ρkσ) =
1
p1
ln
tr

σ
1p
2p
ρσ
1p
2p
p

if ker (σ) ker (ρ) or p (0, 1)
+, otherwise
(3)
where ker (σ) is the kernel of σ.
Note that we are using a different normalization than in [3, 4], which is more convenient
for our purposes. The logarithm in our definition is in base e, while theirs is in base 2.
When we write log in later sections we will mean the logarithm in base 2.
Taking the limit p 1 gives the usual quantum Kullback-Leibler divergence [6]
lim
p1
D
p
(ρkσ) = D (ρkσ) :=
(
tr[ρ (ln(ρ) ln(σ))] if ker (σ) ker (ρ)
+, otherwise
.
Similarly by taking the limit p we obtain the max-relative entropy [3, Theorem 5]
lim
p→∞
D
p
(ρkσ) = D
(ρkσ) = ln
kσ
1
2
ρσ
1
2
k
.
The sandwiched R´enyi divergences increase monotonically in the parameter p 1 (see [7,
Theorem 7]) and we have
D (ρkσ) = D
1
(ρkσ) D
p
(ρkσ) D
q
(ρkσ) D
(ρkσ) . (4)
for any q p 1 and all ρ, σ D
d
. Next we state two simple consequences of this
ordering, which will be useful later.
Lemma 2.1. For σ D
+
d
and p [1, +)
sup
ρ∈D
d
D
p
(ρkσ) = ln
kσ
1
k
. (5)
Proof. Using (4) for ρ D
d
we have
D
p
(ρkσ) D
(ρkσ) = ln
kσ
1
2
ρσ
1
2
k
ln
kσ
1
k
.
Here we used that any quantum state ρ D
d
fulfills ρ 1
d
. Clearly, choosing ρ =
|v
min
ihv
min
| for an eigenvector |v
min
i
d
corresponding to the eigenvalue kσ
1
k
of σ
1
achieves equality in the previous bound.
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Using (4) together with the well-known Pinsker inequality [8, Theorem 3.1] for the
quantum Kullback-Leibler divergence we have
1
2
kσ ρk
2
1
D (ρkσ) D
p
(ρkσ) (6)
for any p 1 and all ρ, σ D
d
. The constant
1
2
has been shown to be optimal in the
classical case (see [9]), i.e. restricting to ρ that commute with σ, and is therefore also
optimal here.
2.1 Noncommutative l
p
-spaces
In the following σ D
+
d
will denote a full rank reference state. For p 1 we define the
noncommutative p-norm with respect to σ as
kXk
p,σ
=
tr
σ
1
2p
Xσ
1
2p
p

1
p
(7)
for any X M
d
. The space (M
d
, k · k
p,σ
) is called a (weighted) noncommutative l
p
-
space. For a linear map Φ : M
d
M
d
and p, q 1 we define the noncommutative
p q-norm with respect to σ as
kΦk
pq
= sup
Y ∈M
d
kΦ(Y )k
q
kY k
p,σ
.
We introduce the weighting operator Γ
σ
: M
d
M
d
as
Γ
σ
(X) = σ
1
2
Xσ
1
2
.
For powers of the weighting operator we set
Γ
p
σ
(X) = σ
p
2
Xσ
p
2
for p and X M
d
. We define the so called power operator I
p,q
: M
d
M
d
as
I
p,q
(X) = Γ
1
p
σ
Γ
1
q
σ
(X)
q
p
!
(8)
for X M
d
. It can be verified that
kI
p,q
(X)k
p
p,σ
= kXk
q
q
for any X M
d
. As in the commutative theory, the noncommutative l
2
-space turns out
to be a Hilbert space, where the weighted scalar product is given by
hX, Y i
σ
= tr
h
Γ
σ
X
Y
i
(9)
for X, Y M
d
. With the above notions we can express the sandwiched p-R´enyi divergence
(3) for p > 1 in terms of a noncommutative l
p
-norm as
D
p
(ρkσ) =
1
p 1
ln
kΓ
1
σ
(ρ) k
p
p,σ
. (10)
For a state ρ D
d
the positive matrix Γ
1
σ
(ρ) M
d
is called the relative density of ρ
with respect to σ. Note that any X 0 with kXk
1
= 1 can be written as X = Γ
1
σ
(ρ)
for some state ρ D
d
. We will simply call operators X 0 that satisfy kXk
1
= 1
relative densities when the reference state is clear.
We refer to [1, 2] and references therein for proofs and more details about the concepts
introduced in this section.
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2.2 Quantum dynamical semigroups
A family of quantum channels, i.e. trace-preserving completely positive maps, {T
t
}
t
+
0
,
T
t
: M
d
M
d
, parametrized by a non-negative parameter t
+
0
is called a quantum
dynamical semigroup if T
0
= id
d
(the identity map in d dimensions), T
t+s
= T
t
T
s
for
any s, t
+
0
and T
t
depends continuously on t. Any quantum dynamical semigroup can
be written as T
t
= e
tL
(see [10, 11]) for a Liouvillian L : M
d
M
d
of the form
L(X) = S(X) κX Xκ
,
where κ M
d
and S : M
d
M
d
is completely positive such that S
(1
d
) = κ+κ
, where
S
is the adjoint of S with respect to the Hilbert-Schmidt scalar product. We will also
deal with tensor powers of semigroups. For a quantum dynamical semigroup {T
t
}
t
+
with Liouvillian L we denote by L
(n)
the Liouvillian of the quantum dynamical semigroup
{T
n
t
}
t
+
.
In the following we will consider quantum dynamical semigroups having a full rank
fixed point σ D
+
d
, i.e. the Liouvillian generating the semigroup fulfills L(σ) = 0 (implying
that e
tL
(σ) = σ for any time t
+
0
). We call a quantum dynamical semigroup (or the
Liouvillian generator) primitive if it has a unique full rank fixed point σ. In this case for
any initial state ρ D
d
we have ρ
t
= e
tL
(ρ) σ as t (see [12, Theorem 14]).
The notion of primitivity can also be defined for discrete semigroups of quantum chan-
nels. For a quantum channel T : M
d
M
d
we will sometimes consider the discrete
semigroup {T
n
}
n
. Similar to the continuous case we will call this semigroup (or the
channel T ) primitive if there is a unique full rank state σ D
+
d
with lim
n→∞
T
n
(ρ) = σ for
any ρ D
d
. We refer to [12] for other characterizations of primitive channels and sufficient
conditions for primitivity.
To study the convergence of a primitive semigroup to its fixed point σ we introduce the
time evolution of the relative density X
t
= Γ
1
σ
(ρ
t
). For any Liouvillian L : M
d
M
d
with full rank fixed point σ D
+
d
define
ˆ
L = Γ
1
σ
L Γ
σ
(11)
to be the generator of the time evolution of the relative density. Indeed it can be checked
that
X
t
= Γ
1
σ
e
tL
(ρ)
= e
t
ˆ
L
(X)
for any state ρ D
d
and relative density X = Γ
1
σ
(ρ). Note that kX
t
k
1
= kXk
1
= 1
for all t
+
0
. Clearly the semigroup generated by
ˆ
L is completely positive and unital,
but it is not trace-preserving in general. In the special case where
Γ
1
σ
L Γ
σ
= L
, (12)
the map
ˆ
L generates the adjoint of the initial semigroup, i.e. the corresponding time
evolution in Heisenberg picture. A semigroup fulfilling (12) is called reversible (or said
to fulfill detailed balance), and in this case the Liouvillian
ˆ
L is a Hermitian operator
w.r.t. the σ-weighted scalar product. We again refer to [2, 1] for more details on these
topics. For discrete semigroups we similarly set
ˆ
T = Γ
1
σ
T Γ
σ
.
One important class of semigroups are Davies generators, which describe a system
weakly coupled to a thermal bath under an appropriate approximation [13]. Describing
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them in detail goes beyond the scope of this article and here we will only review their
most basic properties. We refer to [14, 15, 16] for more details.
Suppose that we have a system of dimension d weakly coupled to a thermal bath
of dimension d
B
at inverse inverse temperature β > 0. Consider a Hamiltonian H
tot
M
d
M
d
B
of the system and the bath of the form
H
tot
= H 1
B
+ 1
S
H
B
+ H
I
,
where H M
d
is the Hamiltonian of the system, H
B
M
d
B
of the bath and
H
I
=
X
α
S
α
B
α
M
d
M
d
B
(13)
describes the interaction between the system and the bath. Here the operators S
α
and
B
α
are self-adjoint. Let {λ
k
}
k[d]
be the spectrum of the Hamiltonian H. We then
define the Bohr-frequencies ω
i,j
to be given by the differences of eigenvalues of H, that
is, ω
i,j
= λ
i
λ
j
for different values of λ. We will drop the indices on ω from now on to
avoid cumbersome notation, as is usually done. Moreover, we introduce operators S
α
(ω)
which are the Fourier components of the coupling operators S
α
and satisfy
e
iHt
S
α
e
iHt
=
X
ω
S
α
(ω)e
t
.
The canonical form of the Davies generator at inverse temperature β > 0 in the Heisenberg
picture, L
β
, is then given by
L
β
(X) = i[H, X] +
X
ω
L
ω
(X),
where
L
ω
(X) = G
α
(ω)
S
α
(ω)
XS
α
(ω)
1
2
{S
α
(ω)
S
α
(ω), X}
.
Here {X, Y } = XY + Y X is the anticommutator and G
α
: are the transition rate
functions. Their form depends on the choice of the bath model [15]. For our purposes it will
be enough to assume that these are functions that satisfy the KMS condition [17], that is,
G
α
(ω) = G
α
(ω)e
βω
. Although this presentation of the Davies generators is admittedly
very short, for our purposes it will be enough to note that under some assumptions on
the operators S
α
(ω) [18, 19] and on the transition rate functions, the semigroup generated
by L
β
converges to the thermal state
e
βH
tr
(
e
βH
)
and is reversible [17]. In the examples
considered here this will always be the case.
2.3 Logarithmic Sobolev inequalities and the spectral gap
To study hypercontractive properties and convergence times of primitive quantum dy-
namical semigroups the framework of logarithmic Sobolev inequalities has been developed
in [1, 2]. Here we will briefly introduce this theory. For more details and proofs see [1, 2]
and the references therein.
We define the operator valued relative entropy (for p > 1) of X M
+
d
as
S
p
(X) = p
d
ds
I
p+s,p
(X) |
s=0
. (14)
With this we can define the p-relative entropy:
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Definition 2.2 (p-relative entropy). For any full rank σ M
+
d
and p > 1 we define the
p-relative entropy of X M
+
d
as
Ent
p,σ
(X) = hI
q,p
(X) , S
p
(X)i
σ
kXk
p
p,σ
ln (kXk
p,σ
) , (15)
where
1
q
+
1
p
= 1. For p = 1 we can consistently define
Ent
1
(X) = tr
σ
(X) (ln
σ
(X)) ln(σ))].
by taking the limit p 1.
The p-relative entropy is not a divergence in the information-theoretic sense (e.g. it
is not contractive under quantum channels). It was originally introduced to study hyper-
contractive properties of semigroups in [1], where they also show it is positive for positive
operators. There is however a connection to the quantum relative entropy as
Ent
p,σ
I
p,1
Γ
1
σ
(ρ)

=
1
p
D (ρkσ) .
As a special case of the last equation we have
Ent
1
Γ
1
σ
(ρ)
= D(ρkσ).
We may also use it to obtain an expression for Ent
2
:
Ent
2
(X) = tr
Γ
1
2
σ
(X)
2
ln
Γ
1
2
σ
(X)
kXk
2
1
2
tr
"
Γ
1
2
σ
(X)
2
ln(σ)
#
.
We also need Dirichlet forms to define logarithmic Sobolev inequalities:
Definition 2.3 (Dirichlet form). Let L : M
d
M
d
be a Liouvillian with full rank fixed
point σ D
+
d
. For p > 1 we define the p-Dirichlet form of X M
+
d
as
E
L
p
(X) =
p
2(p 1)
D
I
q,p
(X),
ˆ
L(X)
E
σ
where
1
p
+
1
q
= 1 and
ˆ
L = Γ
1
σ
L Γ
σ
denotes the generator of the time evolution of the
relative density (cf. (11)). For p = 1 we may take the limit p 1 and consistently define
the 1-Dirichlet form by
E
L
1
(X) =
1
2
tr
h
Γ
σ
ˆ
L(X)
(ln
σ
(X)) ln(σ))
i
.
Formally, by making this choice we introduce the logarithmic Sobolev framework for
ˆ
L (i.e. the generator of the time-evolution of the relative density) instead of L
. While
this is a slightly different definition compared to [2], where the Heisenberg picture is used,
they are the same for reversible Liouvillians.
In [1] the Dirichlet forms were introduced to study hypercontractive properties of
semigroups. As we will see in Theorem 3.1, they appear naturally when we compute the
entropy production of the Sandwiched enyi divergences. From Corollary 3.1 we will be
able to infer that the Dirichlet form is positive for positive operators, a fact already proved
in [1]. Both the Ent
p,σ
and the Dirichlet form are intimately related to hypercontractive
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properties of semigroups, as we have for a relative density X, some constant α > 0 and
p(t) = 1 + e
2αt
that
d
dt
ln
kX
t
k
p(t)
=
αe
αt
(1 + e
αt
) kX
t
k
p(t)
p(t)
Ent
p(t)
(X
t
)
1
α
E
p(t)
(X
t
)
,
as shown in [1].
Notice that when working with E
L
2
we may always suppose the Liouvillian is reversible
without loss of generality. This follows from the fact that
E
L
2
(X) =
D
X,
ˆ
L(X)
E
σ
is invariant under the additive symmetrization
ˆ
L 7→
1
2
L
+ Γ
1
σ
L Γ
σ
for X 0.
We can now introduce the logarithmic Sobolev constants:
Definition 2.4 (Logarithmic Sobolev constants). For a Liouvillian L : M
d
M
d
with
full rank fixed point σ D
+
d
and p 1 the p-logarithmic Sobolev constant is defined
as
α
p
(L) = sup{α
+
: αEnt
p,σ
(X) E
L
p
(X) for all X > 0} (16)
As Ent
2
does not depend on L and, as remarked before, E
L
2
is invariant under an addi-
tive symmetrization, we may always assume without loss of generality that the Liouvillian
is reversible when working with α
2
.
For any X M
+
d
we can define its variance with respect to σ D
+
d
as
Var
σ
(X) = kXk
2
2
kXk
2
1
. (17)
This defines a distance measure to study the convergence of the semigroup. Given a
Liouvillian L : M
d
M
d
with fixed point σ D
+
d
we define its spectral gap as
λ(L) = sup
n
λ
+
: λVar
σ
(X) E
L
2
(X) for all X > 0
o
(18)
where
ˆ
L : M
d
M
d
is given by (11). We can always assume the Liouvillian to be
reversible when dealing with the spectral gap, as it again depends on E
L
2
.
The spectral gap can be used to bound the convergence in the variance (see [20]), as
for any X M
+
d
we have
d
dt
Var
σ
(X
t
) = 2
D
ˆ
L(X) , X
E
σ
(19)
and so
Var
σ
(X
t
) e
2λt
Var
σ
(X) . (20)
3 Convergence rates for sandwiched R
´
enyi divergences
In this section we consider the sandwiched R´enyi divergences of a state evolving under
a primitive quantum dynamical semigroup and the fixed point of this semigroup. It is
clear that these quantities converge to zero as the time-evolved state approaches the fixed
point. To study the speed of this convergence we introduce a differential inequality, which
can be seen as an analogue of the logarithmic Sobolev inequalities for sandwiched enyi
divergences.
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3.1 R
´
enyi-entropy production
In [18] the entropy production for the quantum Kullback-Leibler divergence of a Liouvillian
was computed. We will now derive a similar expression for the entropy production for the
p-R´enyi divergences for p > 1.
Theorem 3.1 (Derivative of the sandwiched p-R´enyi divergence). Let L : M
d
M
d
be
a Liouvillian with full rank fixed point σ D
+
d
. For any ρ D
d
and p > 1 we have
d
dt
D
p
(e
tL
(ρ)kσ)
t=0
=
p
p 1
tr
"
σ
1p
2p
ρσ
1p
2p
p1
σ
1p
2p
L(ρ) σ
1p
2p
#
tr

σ
1p
2p
ρσ
1p
2p
p
. (21)
Using the relative density X = Γ
1
σ
(ρ) and (11) this expression can be written as:
d
dt
D
p
(e
tL
(ρ)kσ)
t=0
=
p
p 1
kXk
p
p,σ
D
I
q,p
(X),
ˆ
L(X)
E
σ
(22)
with
1
p
+
1
q
= 1.
Proof. Rewriting the p-R´enyi divergence in terms of the relative density X = Γ
1
σ
(ρ) and
the corresponding generator
ˆ
L = Γ
1
σ
L Γ
σ
(see (11)) we have
D
p
(e
tL
(ρ)kσ) =
1
p 1
ln
ke
t
ˆ
L
(X) k
p
p,σ
. (23)
By the chain rule
d
dt
D
p
(e
tL
ρkσ)
t=0
=
1
p 1
kXk
p
p,σ
d
dt
ke
t
ˆ
L
(X)k
p
p,σ
t=0
.
Define the curve γ :
+
0
M
d
as γ(t) = σ
1
2p
e
t
ˆ
L
(X) σ
1
2p
and observe that
ke
t
ˆ
L
(X) k
p
p,σ
= tr[γ(t)
p
].
As the differential of the function X 7→ X
p
at A M
+
d
is given by pA
p1
, another
application of the chain rule yields
d
dt
ke
t
ˆ
L
(X)k
p
p,σ
t=0
= p
γ(0)
p1
,
dt
(0)
.
It is easy to check that
dt
(0) = σ
1
2p
ˆ
L(X) σ
1
2p
. Inserting this in the above equations and
writing it in terms of the power operator (8) we finally obtain
d
dt
D
p
(e
tL
(ρ)kσ)
t=0
=
p
p 1
kXk
p
p,σ
D
I
q,p
(X),
ˆ
L(X)
E
σ
with
1
p
+
1
q
= 1. Expanding this formula gives (21).
By recognizing the p-Dirichlet form in the previous theorem we get:
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Corollary 3.1. Let L : M
d
M
d
be a Liouvillian with full rank fixed point σ D
+
d
.
For any ρ D
d
and p > 1 we have
d
dt
D
p
(e
tL
(ρ)kσ)
t=0
= 2kXk
p
p,σ
E
L
p
(X) 0, (24)
where we used the relative density X = Γ
1
σ
(ρ).
As we remarked before, Corollary 3.1 implies that the Dirichlet form is always positive
for relative densities. To see this, recall that the divergences contract under quantum
channels [7] and therefore we have that
d
dt
D
p
(e
tL
(ρ)kσ)
t=0
0. As E
L
p
(λX) = λ
p
E
L
p
(X)
for λ > 0, this shows that it is positive for all positive operators by properly normalizing
X.
3.2 Sandwiched R
´
enyi convergence rates
For any p > 1 we introduce the functional κ
p
: M
+
d
as
κ
p
(X) =
1
p 1
kXk
p
p,σ
ln
kXk
p
p,σ
kXk
p
1
!
(25)
for X M
+
d
. For p = 1 we may again take the limit p 1 and obtain κ
1
(X) :=
lim
p1
κ
p
(X) = Ent
1
(X). Note that κ
p
is well-defined and non-negative as kXk
p,σ
kXk
1
for p 1. Strictly speaking the definition also depends on a reference state σ D
+
d
,
which we usually omit as it is always the fixed point of the primitive Liouvillian under
consideration.
Given a Liouvillian L : M
d
M
d
with full rank fixed point σ D
+
d
it is a simple
consequence of Corollary 3.1 that for ρ 6= σ
d
dt
D
p
(e
tL
(ρ)kσ)
t=0
D
p
(ρkσ)
= 2
E
L
p
(X)
κ
p
(X)
, (26)
where we used the relative density X = Γ
1
σ
(ρ), which fulfills kXk
1
= 1. This motivates
the following definition.
Definition 3.1 (Entropic convergence constant for p-R´enyi divergence). For any primitive
Liouvillian L : M
d
M
d
and p 1 we define
β
p
(L) = sup{β
+
: βκ
p
(X) E
L
p
(X) for all X > 0}. (27)
Note that as a special case we have α
1
(L) = β
1
(L). It should be also emphasized that
the supremum in the previous definition goes over any positive definite X M
+
d
and not
only over relative densities. However, it is easy to see that we can equivalently write
β
p
(L) = inf
n
E
L
p
(X)
κ
p
(X)
: X > 0
o
= inf
n
E
L
p
(X)
κ
p
(X)
: X > 0, kXk
1
= 1
o
(28)
as replacing X 7→ X/kXk
1
does not change the value of the quotient E
L
p
(X)
p
(X).
Therefore, to compute β
p
it is enough to optimize over relative densities (i.e. X > 0
fulfilling kXk
1
= 1). By inserting β
p
into (26) we have
d
dt
D
p
(e
tL
(ρ)kσ) 2β
p
(L)D
p
e
tL
(ρ)kσ
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for any ρ D
d
and Liouvillian L : M
d
M
d
with full rank fixed point σ D
+
d
. By
integrating this differential inequality we get
Theorem 3.2. Let L : M
d
M
d
be a Liouvillian with full rank fixed point σ D
+
d
. For
any p 1 and ρ D
d
we have
D
p
e
tL
(ρ)kσ
e
2β
p
(L)t
D
p
(ρkσ) (29)
where β
p
(L) is the constant defined in (27).
3.3 Computing β
p
in simple cases
In general it is not clear how to compute β
p
and it does not depend on spectral data
of L alone. This is not surprising, as the computation of the usual logarithmic Sobolev
constants α
2
or α
1
is also challenging and the exact values are only known for few Liou-
villians [21, 2, 22]. In the following we compute β
2
for the depolarizing semigroups.
Theorem 3.3 (β
2
for the depolarizing Liouvillian). Let L
σ
: M
d
M
d
denote the
depolarizing Liouvillian given by L
σ
(ρ) = tr (ρ) σ ρ with fixed point σ D
+
d
. Then
β
2
(L
σ
) =
1
1
kσ
1
k
ln (kσ
1
k
)
. (30)
Proof. Without loss of generality we can restrict to X > 0 with kXk
1
= 1 in the
minimization (28). Observe that the generator of the time evolution of the relative density
(see (11)) for the depolarizing Liouvillian is
ˆ
L
σ
(X) = tr
σ
1
2
Xσ
1
2
1 X.
An easy computation yields E
L
σ
2
(X) = kXk
2
2
1 and so
E
L
σ
2
(X)
κ
2
(X)
=
1
1
kXk
2
2
ln
kXk
2
2
.
As the function x 7→
1
1
x
ln(x)
is monotone decreasing for x 1, we have
inf
X>0
E
L
σ
2
(X)
κ
2
(X)
=
1
1
kσ
1
k
ln (kσ
1
k
)
, (31)
where we used
sup
X0,kXk
1
=1
kXk
2
2
= kσ
1
k
,
which easily follows from Lemma 2.1 by exponentiating both sides of Equation (5) and
using the correspondence between relative densities and states.
The exact value of α
2
(L
σ
) is open to the best of our knowledge, but in the case of σ =
1
d
we have α
2
L
1
d
=
2(12/d)
ln(d1)
[2, Theorem 24], which is of the same order of magnitude as
β
2
for these semigroups.
Computing β
p
for p 6= 2 seems not to be straightforward even for depolarizing channels,
but for the semigroup depolarizing to the maximally mixed state we can at least provide
upper and lower bounds.
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Theorem 3.4 (β
p
for the Liouvillian depolarizing to the maximally mixed state). Let
L(ρ) = tr(ρ)
1
d
ρ. For p 2 we have
p
2(p 1)
1
ln(d)
β
p
(L)
p
2(p 1)
d
p1
p
1
d
p1
p
ln(d)
.
Proof. The Dirichlet Form of this Liouvillian for X > 0 with kXk
1,
1
d
= 1 is given by
E
L
p
(X) =
p
2(p 1)
(kXk
p
p,
1
d
kXk
p1
p1,
1
d
).
Dividing this expression by κ
p
(X) we get
E
L
p
(X)
κ
p
(X)
=
1
kXk
p1
p1,
1
d
kXk
p
p,
1
d
2 ln
kXk
p,
1
d
. (32)
By the monotonicity of the weighted norms, we have
kXk
p1
p1,
1
d
kXk
p
p,
1
d
1
kXk
p,
1
d
and so
E
L
p
(X)
κ
p
(X)
kXk
p,
1
d
1
2kXk
p,
1
d
ln
kXk
p,
1
d
(33)
The expression on the right-hand side of (33) is monotone decreasing in kXk
p,
1
d
and so
the infimum is attained at
sup
kXk
1,
1
d
=1
kXk
p,
1
d
= d
p1
p
,
which again easily follows from Lemma 2.1. The upper bound follows from (32) as
E
L
p
(X)
κ
p
(X)
1
2 ln
kXk
p,
1
d
.
which is again monotone decreasing in kXk
p,
1
d
.
From the relations between LS constants [2, Proposition 13], it follows that for the LS
constants of the depolarizing channels we have α
p
L
1
d
α
2
L
1
d
=
2(12/d)
ln(d1)
for p 1.
The constants β
p
and α
p
are therefore of the same order in this case for small p 2.
4 Comparison with similar quantities
4.1 Comparison with spectral gap
Here we show how β
p
, see (27), compares to the spectral gap (18) of a Liouvillian. This is
motivated by similar results for logarithmic Sobolev constants, where it was shown [2, The-
orem 16] that α
1
(L) λ(L) for reversible semigroups, a result we recover and generalize
here.
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Theorem 4.1 (Upper bound spectral gap). Let L : M
d
M
d
be a primitive and
reversible Liouvillian with full rank fixed point σ D
+
d
and p 1. Then
β
p
(L) λ (L) . (34)
Proof. Let (s
i
)
d
i=1
denote the spectrum of σ
1/p
and choose a unitary U such that
σ
1/p
= Udiag (s
1
, s
2
, . . . , s
d
) U
.
As L is reversible, there is a self-adjoint eigenvector X M
d
of
ˆ
L corresponding to the
spectral gap, i.e.
ˆ
L(X) = λ(L)X. Let
0
> 0 be small enough such that Y
= 1
d
+ X
is positive for any ||
0
. For ||
0
we use Lemma A.1 of the appendix to show
β
p
(L)
E
L
p
(Y
)
κ
p
(Y
)
=
λ(L)
p
2(p1)
2
2
P
1ijd
f
p
(s
i
, s
j
)b
ij
b
ji
+ O(
3
)
2
p1
p
P
1ijd
f
p
(s
i
, s
j
)b
ij
b
ji
+ O(
3
)
(35)
where b
ij
= (U
σ
1/2p
Xσ
1/2p
U)
ij
and
f
p
(x, y) =
(
(p 1)x
p2
if x = y
x
p1
y
p1
xy
else.
(36)
Observe that f
p
(s
i
, s
j
) > 0 for s
i
, s
j
> 0. Moreover, as U
σ
1/2p
Xσ
1/2p
U is non-zero and
self-adjoint we have b
ij
b
ji
0 for all i, j and this inequality is strict for at least one choice
of i, j. Therefore, the terms of second order in in the numerator and denominator of
(35) are strictly positive, and we obtain λ(L) as the limit of the quotient as 0.
A similar argument as the one given in the previous proof shows that all real, nonzero
elements of the spectrum of
ˆ
L are upper bounds to β
p
without invoking reversibility.
Note that in the case of p = 2 (see the discussion after (16)) we may assume that the
Liouvillian is reversible without loss of generality and drop the requirement of reversibility
in the previous theorem. Alternatively, we can obtain the same statement directly from a
simple functional inequality. In this case we can also give a lower bound on β
2
in terms
of the spectral gap.
Theorem 4.2 (Upper and lower bound for β
2
). Let L : M
d
M
d
be a primitive
Liouvillian with full rank fixed point σ D
+
d
. Then
λ (L)
1
1
kσ
1
k
ln (kσ
1
k
)
β
2
(L) λ (L) . (37)
To prove Theorem 4.2 we need the following Lemma.
Lemma 4.1. For any X M
d
we have
Var
σ
(X) κ
2
(X).
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Proof. For X > 0 dividing both sides of the inequality by kXk
2
1
yields
kXk
2
2
kXk
2
1
1
kXk
2
2
kXk
2
1
ln
kXk
2
2
kXk
2
1
!
.
This follows from the elementary inequality x 1 x ln(x) for x 1, where we use the
ordering kXk
2
kXk
1
for any X M
d
.
Proof of Theorem 4.2. Using the definition of β
2
(see (27)) and Lemma 4.1 yields
β
2
Var
σ
(X) β
2
κ
2
(X) E
L
2
(X).
Now the variational definition of λ(L) (see (18)) implies the second inequality of (37).
To prove the first inequality of (37) consider the depolarizing Liouvillian
L
σ
(X) = tr(X)σ X.
By Theorem 3.3 we have
1
1
kσ
1
k
ln (kσ
1
k
)
κ
2
(X) E
L
σ
2
(X)
As E
L
σ
2
(X) = Var
σ
(X), we have E
L
σ
2
(X)
1
λ(L)
E
L
2
(X) by the variational definition of
λ(L) (see (18)). Inserting this in the above inequality finishes the proof.
4.2 Comparison with logarithmic Sobolev constants
Here we show how β
p
, see (27), compares to the logarithmic Sobolev constant α
p
.
Theorem 4.3. Let L : M
d
M
d
be a primitive Liouvillian with full rank fixed point
σ D
+
d
. Then for any p 1 we have
β
p
(L)
α
p
(L)
p
. (38)
We will need the following Lemma.
Lemma 4.2. For any full rank state σ D
+
d
, any p > 1 and X M
+
d
with kXk
1
= 1
we have
Ent
p,σ
(X)
κ
p
(X)
p
. (39)
Proof. The function p 7→ D
p
(ρkσ) is monotonically increasing [3, 7] and differentiable
(as the noncommutative l
p
-norm is differentiable in p [1, Theorem 2.7]). Thus, with
f :
+
given by f(t) = t + p we have
0 kXk
p
p,σ
d
dt
D
f(t)
(ρkσ)
t=0
=
1
(p 1)
2
kXk
p
p,σ
ln
kXk
p
p,σ
+
1
p 1
d
dt
kXk
f(t)
f(t)
t=0
.
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where we used the relative density X = Γ
1
σ
(ρ). The remaining derivative in the above
equation has been computed in [1, Theorem 2.7] and we have
d
dt
kXk
f(t)
f(t)
t=0
= hI
q,p
(X), S
p
(X)i
σ
with the operator valued entropy S
p
defined in (14) and
1
p
+
1
q
= 1. Inserting this expression
in the above equation we obtain
1
p 1
kXk
p
p,σ
ln
kXk
p
p,σ
hI
q,p
(X), S
p
(X)i
σ
. (40)
for any X M
+
d
with kXk
1
= 1, i.e. for any X = Γ
1
σ
(ρ) for some state ρ D
d
. Now
we get
pEnt
p,σ
(X) = p hI
q,p
(X), S
p
(X)i
σ
kXk
p
p,σ
ln(kXk
p
p,σ
)
p
p 1
kXk
p
p,σ
ln(kXk
p
p,σ
) kXk
p
p,σ
ln(kXk
p
p,σ
)
= κ
p
(X)
where we used (40).
Proof of Theorem 4.3. There is nothing to show for p = 1 as α
1
(L) = β
1
(L) and we can
assume p > 1. For X M
+
d
with kXk
1
= 1 we can use Lemma 4.2 and the definition of
α
p
(L) to compute
α
p
(L)
p
κ
p
(X) α
p
(L) Ent
p,σ
(X) E
L
p
(X).
By the variational definition (27) of β
p
the claim follows.
Theorem 4.3 will be applied in Section 5 to obtain bounds on the mixing time of a
Liouvillian with a positive logarithmic Sobolev constant without invoking any form of
l
p
-regularity (see [2]). As usually a logarithmic Sobolev is implied by a hypercontractive
inequality [1], we would like to remark that one can also make a similar statement as that
of Theorem 4.3 from a hypercontractive inequality. One can easily show that
||e
t
ˆ
L
||
p(t)p,σ
1 (41)
for p(t) = (p 1)e
α
p
t
+ 1 implies that β
p
(L)
α
p
p
.
5 Mixing times
In this section we will introduce the quantities of interest and prove the building blocks
to prove mixing times from the entropy production inequalities of the last sections, dis-
tinguishing between continuous and discrete time semigroups. We will mostly focus on
β
2
, as this seems to be the most relevant constant for mixing time applications. This is
justified by the fact that the underlying Dirichlet form is a quadratic form and the entropy
related to it stems from a Hilbert space norm. Moreover, as the same Dirichlet form is
also involved in computations of the spectral gap, it could be easier to adapt existing
techniques, such as the ones developed in [23, 19].
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Definition 5.1 (Mixing times). For either I =
+
or I = let {T
t
}
tI
be a primitive
semigroup of quantum channels with fixed point σ D
+
d
. We define the l
1
mixing time
for > 0 as
t
1
() = inf{t I : kT
t
(ρ) σk
1
for all ρ D
d
}.
Similarly we define the l
2
mixing time for > 0 as
t
2
() = inf{t I : Var
σ
ˆ
T
t
(X)
for all X M
+
d
with kXk
1
= 1}.
In the continuous case I =
+
we will often speak of the mixing times of the Liouvillian
generator of a quantum dynamical semigroup which we identify with the mixing times of
the semigroup according to the above definition.
5.1 Mixing in Continuous Time
It is now straightforward to get mixing times from the previous results.
Theorem 5.1 (Mixing time from entropy production). Let L : M
d
M
d
be a primitive
Liouvillian with fixed point σ D
+
d
. Then
t
1
()
1
2β
p
(L)
ln
2 ln
kσ
1
k
2
!
.
Proof. From (6) and Lemma 2.1 we have
ln
kσ
1
k
e
2β
p
(L)t
1
2
ke
tL
(ρ) σk
2
1
. (42)
for any ρ D
d
. The claim follows after rearranging the terms.
Using Theorem 4.3 we can lower bound β
p
in terms of the usual logarithmic Sobolev
constant α
p
. Combining this with Theorem 5.1 shows the following Corollary.
Corollary 5.1 (Mixing time bound from logarithmic Sobolev inequalities). Let L : M
d
M
d
be a primitive Liouvillian with fixed point σ D
+
d
. Then
t
1
()
p
2α
p
(L)
ln
2 ln
kσ
1
k
2
!
. (43)
By Corollary 5.1 a nonzero logarithmic Sobolev constant always implies a nontrivial
mixing time bound. One should say that the same bound was showed in [2] for p = 2,
however under additional assumptions (specifically l
p
-regularity [2]) on the Liouvillian in
question. While these assumptions have been shown for certain classes of Liouvillians (in-
cluding important examples like Davies generators and doubly stochastic Liouvillians [2])
they have not been shown in general. Moreover, the bound in Theorem 5.1 clearly does not
depend on p and one could in principle optimize over all β
p
. However, as the computations
in subsection 3.3 already indicate, it does not seem to be feasible to compute or bound β
p
for p 6= 2 even in simple cases and one will probably only work with β
2
in applications.
The bound from Corollary 5.1 also has the right scaling properties needed in recent
applications of rapid mixing, such as the results in [24, 25]. In particular, together with
the results in [26], the last Corollary shows that the hypothesis of Theorem 4.2 in [24] is
Accepted in Quantum 2018-02-01, click title to verify 17
always satisfied for product evolutions and not only for the special classes considered in
[2].
One may also use these techniques to get mixing times in the l
2
norms which are
stronger than the ones obtained just by considering that β
2
is a lower bound to the
spectral gap.
Theorem 5.2 (l
2
-mixing time bound). Let L : M
d
M
d
be a Liouvillian with fixed
point σ D
+
d
. Then
t
2
()
1
2β
2
(L)
ln
ln
kσ
1
k
ln(1 + )
!
. (44)
Proof. For X > 0 with kXk
1
= 1 we have Var
σ
(X) = kX 1k
2
2
= kXk
2
2
1 and thus
κ
2
(X) = (1 + Var
σ
(X)) ln (1 + Var
σ
(X)) .
In the following let X
t
= e
t
ˆ
L
(X) denote the time evolution of the relative density X.
Using (19) and the definition of β
2
(L) (see (27)) we obtain
d
dt
Var
σ
(X
t
) = 2E
L
2
(X
t
) 2β
2
(L)(1 + Var
σ
(X
t
)) ln(1 + Var
σ
(X
t
)).
Integrating this differential inequality we obtain
ln
ln(1 + Var
σ
(X))
ln(1 + )
t
2
()
Z
0
1
(1 + Var
σ
(X
t
)) ln (1 + Var
σ
(X
t
))
d
dt
Var
σ
(X
t
)
dt
2β
2
t
2
().
As 1 + Var
σ
(X) kσ
1
k
, the claim follows after rearranging the terms.
In the remaining part of the section we will discuss a converse to the previous mixing
time bounds, i.e. a lower bound on the logarithmic Sobolev constant in terms of a mixing
time. This excludes the possibility of a reversible semigroup with both small β
2
and short
mixing time with respect to the l
2
distance. For this we generalize [21, Corollary 3.11] to
the noncommutative setting.
Theorem 5.3 (LS inequality from l
2
mixing time). Let L : M
d
M
d
be a primitive,
reversible Liouvillian with fixed point σ D
+
d
. Then
1
2
α
2
(L) t
2
e
1
2β
2
(L) t
2
e
1
. (45)
Moreover, this inequality is tight.
Proof. We refer to Appendix B for a proof.
As remarked in [21], even the classical result does not hold anymore if we drop the
reversibility assumption. Therefore, this assumption is also needed in the noncommutative
setting. By considering a completely depolarizing channel it is also easy to see that no
such bound can hold in discrete time.
Theorem 5.3 implies that for reversible Liouvillians β
2
and α
2
cannot differ by a large
factor. More specifically we have the following corollary.
Accepted in Quantum 2018-02-01, click title to verify 18
Corollary 5.2. Let L : M
d
M
d
be a primitive, reversible Liouvillian with fixed point
σ D
+
d
. Then
2β
2
(L) α
2
(L) β
2
(L) ln
ln
kσ
1
k
ln(1 + e
1
)
!
. (46)
Proof. We showed the first inequality in Theorem 4.3. The second inequality follows by
combining (45) and (44).
5.2 Mixing in Discrete Time
In this section we will obtain mixing time bounds and also entropic inequalities for discrete-
time quantum channels T : M
d
M
d
. We will then use these techniques to derive mixing
times for random local channels, which we will define next. These include channels that
usually appear in quantum error correction scenarios, such as random Pauli errors on
qubits [27, Chapter 10]. They will be based on the following quantity:
Definition 5.2. For a primitive quantum channel T : M
d
M
d
with full rank fixed
point σ D
+
d
, we define
β
D
(T ) = β
2
(T
ˆ
T id
d
). (47)
Here we used
ˆ
T = Γ
1
σ
T Γ
σ
.
The definition of β
D
(T ) can be motivated by the following improved data-processing
inequality for the 2-sandwiched enyi divergence.
Theorem 5.4. Let T : M
d
M
d
be a primitive quantum channel with full rank fixed
point σ D
+
d
. Then for all ρ D
d
we have
D
2
(T (ρ)kσ) (1 β
D
(T ))D
2
(ρkσ) . (48)
Proof. Let X = σ
1
2
ρσ
1
2
denote the relative density of ρ with respect to σ. Observe that
the 2-Dirichlet form (see Definition 2.3) of the semigroup L = T
ˆ
T id
d
can be written
as
E
L
2
(X) = kXk
2
2
k
ˆ
T (X)k
2
2
.
From the definition of β
2
(L) (see (27)) it follows that
kXk
2
2
k
ˆ
T (X)k
2
2
β
2
κ
2
(X),
which is equivalent to
ln(