Quasirandom quantum channels
Tom Bannink
1
, Jop Bri
¨
et
2
, Farrokh Labib
1
, and Hans Maassen
3
1
CWI, QuSoft, Science Park 123, 1098 XG Amsterdam, Netherlands. Supported by the Gravitation-grant NETWORKS-
024.002.003 from the Dutch Research Council (NWO).
2
CWI, QuSoft, Science Park 123, 1098 XG Amsterdam, Netherlands. Supported by the Gravitation-grant NETWORKS-
024.002.003 from the Dutch Research Council (NWO). Additionally supported by an NWO VENI grant
3
QuSoft, Korteweg-de Vries Institute for Mathematics, Radboud University.
Mixing (or quasirandom) properties of the natural transition matrix associated to
a graph can be quantified by its distance to the complete graph. Different mixing
properties correspond to different norms to measure this distance. For dense graphs,
two such properties known as spectral expansion and uniformity were shown to be
equivalent in seminal 1989 work of Chung, Graham and Wilson. Recently, Conlon and
Zhao extended this equivalence to the case of sparse vertex transitive graphs using the
famous Grothendieck inequality.
Here we generalize these results to the non-commutative, or ‘quantum’, case, where
a transition matrix becomes a quantum channel. In particular, we show that for ir-
reducibly covariant quantum channels, expansion is equivalent to a natural analog of
uniformity for graphs, generalizing the result of Conlon and Zhao. Moreover, we show
that in these results, the non-commutative and commutative (resp.) Grothendieck
inequalities yield the best-possible constants.
1 Introduction
In a seminal work [8], Chung, Graham and Wilson — building on work of Thomason [33, 34] —
proved that several seemingly distinct notions of quasirandomness for graphs are equivalent. In
particular, they identified seven properties found in random graphs with high probability, that
always coexist simultaneously in any large dense graph. Two of these properties are spectral
expansion and uniformity (defined below). A question of Chung and Graham [7] on the equivalence
of these two properties in sparse graphs resulted in a line of research culminating in recent work
of Conlon and Zhao [9], which introduced a surprising new item to the armory of combinatorics:
the famous Grothendieck inequality [13]. In this paper, we draw a parallel line in the context of
quantum information theory, where quantum channels take the place of graphs. In addition, we
give a streamlined proof of the main result of [9] and show that the use of Grothendieck’s inequality
yields an optimal constant. Similarly, we show that the non-commutative Grothendieck inequality
gives an optimal constant in the quantum setting.
Spectral expansion and uniformity. Spectral expansion is a linear-algebraic property given
in terms of the transition matrix of a graph. This transition matrix is the normalized adjacency
matrix, which for a d-regular graph G = (V, E) is given by A
uv
= e({u}, {v})/d, where e(S, T )
denotes the number of edges connecting subsets S, T ⊆ V . We say that the graph G is an (n, d, λ)
graph if |V | = n, it is d-regular and all but the largest eigenvalue of A, which is always 1, have
modulus at most λ. The smallest value of λ for which this holds is denoted by λ(G). Spectral
expansion then refers to the property that λ(G) is much smaller than 1, in which case G is referred
Tom Bannink: tombannink@gmail.com
Jop Bri
¨
et: j.briet@cwi.nl
Farrokh Labib: labib@cwi.nl
Hans Maassen: H.Maassen@math.ru.nl
Accepted in Quantum 2020-06-17, click title to verify. Published under CC-BY 4.0. 1
arXiv:1908.06310v3 [quant-ph] 8 Jul 2020
to as a (spectral) expander. Expanders have many important applications in mathematics and
computer science (we refer to [23] for an extensive survey). One such application is in randomized
algorithms, which can exploit the fact that a random walk on an expander rapidly mixes (i.e. quickly
converges to its limit distribution) to significantly reduce the amount of randomness needed.
Uniformity is a combinatorial property of the configuration of the edges. An n-vertex d-regular
graph G = (V, E) is -uniform if for all S, T ⊆ V ,
e(S, T ) −
d
n
|S||T|
≤ dn (1)
and (G) denotes the smallest value of for which this holds. Uniformity then refers to the property
that this parameter is much smaller than 1; trivially any graph is 1-uniform. Intuitively, this says
that for any two vertex subsets, the number of edges between those sets is close to the expected
number of edges in a random graph with the same edge density.
A basic result known as the Expander Mixing Lemma [23] shows that for any regular graph G
we have (G) ≤ λ(G), which is to say that spectral expansion implies uniformity. A sequence G
n
of d
n
-regular graphs is called dense if d
n
≥ Ω(n), and sparse if d
n
/n −→ 0. It was shown in [8]
that in the dense case, a converse to the Expander Mixing Lemma (G
n
) ≤ o(1) ⇒ λ(G
n
) ≤ o(1)
also holds. In contrast, Krivelevich and Sudakov [25] showed that this is false for sparse graphs,
thereby answering the question posed in [7]. Their counterexample is not regular, however (and
a later one from [4] is not connected). But in [9] it was shown that even regular sparse graphs
(where d
n
≤ o(n)) can simultaneously satisfy (G
n
) ≤ o(1) and λ(G
n
) ≥ Ω(1). Surprisingly,
Kohayakawa, R¨odl, and Schacht [24] showed that Cayley graphs over abelian groups, including
sparse ones, do again admit such a converse. Cayley graphs are an important class of regular
graphs that include for instance the famous Ramanujan graphs of Margulis [27] and Lubotzky,
Phillips and Sarnak [26]. Conlon and Zhao [9] generalized this to all Cayley graphs and showed
that this implies the same for all vertex-transitive graphs in general, for which they showed that
λ(G) ≤ 4K
G
(G), where 1.6769 . . . ≤ K
G
< 1.7822 . . . is the famous Grothendieck constant, whose
exact value is currently unknown; the bounds shown here are the best known and were shown by
Davie and Reeds (independently) in [11, 30] and Braverman et al. in [5], respectively. Spectral
expansion and uniformity are thus equivalent notions of quasirandomness for dense graphs and
vertex-transitive graphs.
Quasirandomness in quantum information theory. A transition matrix, such as the nor-
malized adjacency matrix of a graph, maps probability vectors
1
to probability vectors. A natural
non-commutative generalization of a transition matrix is a quantum channel, a completely positive
trace preserving linear map Φ : M
n
(C) → M
n
(C); see Section 2 for formal definitions. Quantum
channels are the most general operations on quantum systems that are physically realizable. They
encapsulate the “classical” transition matrices by restricting them to diagonal matrices whose
diagonals form probability vectors; we discuss this in more detail in Section 3. In quantum infor-
mation theory, general linear maps from M
n
(C) to itself are referred to as superoperators. Since
superoperators are in one-to-one correspondence with bilinear forms on M
n
(C) ×M
n
(C), they also
appear in the context of (generalizations of) Bell inequalities from physics in the form of quantum
XOR games [10, 31], as well as in combinatorial optimization [28]. The graph-theoretic concepts
mentioned above have natural analogues for superoperators, which we discuss next.
In independent work, Hastings [18] and Ben-Aroya, Schwartz and Ta-Schma [3] introduced
quantum expanders as a special class of quantum channels defined analogously to spectral ex-
panders. For a superoperator Φ, the expansion parameter is given by
λ(Φ) = kΦ −Πk
S
2
→S
2
= sup
k(Φ −Π)(X)k
S
2
: kXk
S
2
≤ 1
, (2)
where Π : X 7→
1
n
Tr(X)Id is the projection onto the identity, kXk
S
2
=
p
hX, Xi is the Frobenius
(or Schatten-2) norm and hX, Y i =
1
n
Tr(Y
∗
X) is the normalized trace inner product. A quantum
channel is an expander if λ(Φ) is much smaller than 1. Also quantum expanders found many
applications, one of which is again randomness reduction, where randomness takes on the form
1
We use the convention of writing probability vectors as column vectors intead of row vectors.
Accepted in Quantum 2020-06-17, click title to verify. Published under CC-BY 4.0. 2
of random unitary matrices. Since a k-qubit unitary requires 4
k
real parameters, sampling one
from the uniform distribution (Haar probability measure) is very expensive. A 1-design is a fixed
collection of unitaries U
1
, . . . , U
m
such that the superoperator Φ : X 7→
1
m
P
m
i=1
U
i
XU
∗
i
exactly
effects the projection Π, thus mimicking in a finite way the Haar measure on U(n). Quantum
expanders can be used to construct approximate 1-designs, meaning that Φ(X) and Π(X) are
close in trace distance
2
instead of precisely equal. Another application is in cryptography where
Ambainis and Smith [1] used quantum expanders to construct short quantum one-time pads. It
was shown in [18] that truly random quantum channels (given by independent Haar-uniform U
i
as
described above) are quantum expanders with high probability, supporting the idea that this is a
notion of quasirandomness.
In this work we introduce a natural notion of uniformity for superoperators, informally given
by how well they mimic the action of Π on projectors on subspaces, which may be thought of as
generalizations of vertex subsets in graphs. This is similar to Hasting’s notion of edge expansion for
quantum channels [18]. In particular, we say that Φ is -uniform if for any two subspaces V, W ⊆ C
n
with associated projections P
V
, P
W
, it holds that
|hP
V
, (Φ −Π)(P
W
)i| ≤ . (3)
Let (Φ) denote the smallest for which this holds. As we show in Section 3.3, the parameters λ(Φ)
and (Φ) reduce to their graphical analogs under a suitable embedding of graphs into quantum
channels.
Finally, also symmetry, which in the graph-theoretic context takes the form of vertex transitiv-
ity, is an important property of quantum channels. In particular, irreducibly covariant quantum
channels, which turn out to generalize vertex-transitive graphs (see Section 3), play an important
role in questions about the capacity of quantum channels as noisy transmitters of quantum infor-
mation [22]. A now famous result of Hastings [19] shows that the minimum output capacity in
general does not have the intuitively natural property of being sub-additive under tensor products.
However, it was shown earlier by Holevo [21], that the capacity is additive for the subclass of
irreducibly covariant quantum channels.
Summary of our results. In this work we make a first step in the study of the equivalence of
quasirandom properties for quantum channels, or superoperators in general, and show optimality
in the case of vertex-transitive graphs and covariant quantum channels.
• (Section 3.2) Our main result shows that under irreducible covariance, expansion and uni-
formity are equivalent for superoperators. In particular, while a simple analogue of the
classical Expander Mixing Lemma implies that (Φ) ≤ λ(Φ) in general, we show using a
non-commutative version of Grothendieck’s inequality due to Haagerup [14], that for this
class of superoperators, also λ(Φ) ≤ 2π
2
(Φ) always holds. This implies the same result for
vertex-transitive graphs with C-weighted edges, essentially proved in [9] with the factor 2
replaced by the complex Grothendieck constant 1.3380 . . . ≤ K
C
G
≤ 1.4049 . . . .
• (Section 3.3) We show that a construction of sparse regular graphs from [9] can be embedded
to give a sequence of quantum channels Φ
n
that are not irreducibly covariant and for which
it holds that (Φ
n
) ≤ o(1) and λ(Φ
n
) ≥ Ω(1).
• (Section 3.4) We show that for randomizing channels, a notion introduced in [2], the two
notions of quasirandomness are also equivalent. This can be interpreted as a generalization
of the same statement for dense graphs proved in [8].
• (Section 4.1) We show that the result of [9] cannot be improved in the sense that the factors
4K
G
and π
2
K
C
G
are optimal in the case of vertex-transitive graphs with R-weighted and
C-weighted edges, respectively.
• (Section 4.2) Our work leaves open whether the factor 2π
2
in our main result is optimal.
However, our proof consists of two steps, the first of which gives a factor 2 and the sec-
ond a factor π
2
, and we show these steps are individually optimal. We prove that the
2
The trace distance is the distance induced by the Schatten-1 norm, defined in Section 2.
Accepted in Quantum 2020-06-17, click title to verify. Published under CC-BY 4.0. 3
first step is optimal by showing that an example of Haagerup and Ito [16] for the non-
commutative Grothendieck inequality is irreducibly covariant, which uses some representa-
tion theory of SO(n). The optimality of the second step follows directly from a result of [9].
Acknowledgements We would like to thank M¯aris Ozols, Michael Walter and Freek Witteveen
for fruitful discussions.
2 Preliminaries
Write [n] = {1, . . . , n}. For a finite set S, write E
s∈S
for
1
|S|
P
s∈S
. For a compact set S, write C(S)
for the set of continuous functions from S to C. For a compact group Γ, write E
g∈Γ
for the the
integral with respect to the (unique) Haar probability measure on Γ.
Write M
n
(C) for the set of complex n×n matrices and let U(n) = {X ∈ M
n
(C) : X
∗
X = Id} be
the set of unitary matrices. Here, all maps of the form Φ : M
n
(C) → M
n
(C) are linear, and we refer
to these as superoperators. A superoperator Φ is unital if Φ(Id) = Id and it is completely positive if
for all k ∈ N the superoperator Id ⊗Φ : M
k
⊗M
n
→ M
k
⊗M
n
maps positive semidefinite matrices
to positive semidefinite matrices. Completely positive superoperators that are trace preserving are
called quantum channels.
We normalize inner products so that for x, y ∈ C
n
we define hy, xi = E
i∈[n]
y
i
x
i
and for matrices
X, Y ∈ M
n
(C) we have hY, Xi =
1
n
Tr[Y
∗
X].
Norms. For p ∈ [1, ∞), x ∈ C
n
and X ∈ M
n
(C), the L
p
norm and (normalized) Schatten-p
norm are defined by
kxk
L
p
=
E
i∈[n]
|x
i
|
p
1/p
and kXk
S
p
=
1
n
Tr
(X
∗
X)
p/2
1/p
and kxk
L
∞
= max
i
|x
i
| and kXk
S
∞
= sup{|hXx, yi| : kxk
L
2
, kyk
L
2
≤ 1}. Note that for the
identity matrix Id ∈ M
n
we have kIdk
S
p
= 1 for all p ∈ [1, ∞].
Proposition 2.1. Let p ≥ 1 and let X ∈ M
n
(C). Then kXk
S
p
≥ k(X
11
, . . . , X
nn
)k
L
p
.
Proof. For a vector x ∈ C
n
, denote by Diag(x) the n × n matrix with x on the diagonal and for
a matrix X denote by diag(X) the matrix where we set the off-diagonal elements to 0. A small
computation shows that
E
s∈{±1}
n
Diag(s) X Diag(s) = diag(X).
Since the Schatten-p norms are invariant under conjugation with a unitary matrix, applying the
above with the triangle inequality gives
k(X
11
, . . . , X
nn
)k
L
p
= kdiag(X)k
S
p
≤ E
s∈{±1}
n
kDiag(s) X Diag(s)k
S
p
= kXk
S
p
.
For q ∈ [1, ∞], define q
0
∈ [1, ∞] to be its dual given by
1
q
+
1
q
0
= 1. For p, q ∈ [1, ∞], a matrix
A ∈ M
n
(C) and a superoperator Φ : M
n
(C) → M
n
(C), define
kAk
L
p
→L
q
= sup{|hy, Axi| : kxk
L
p
≤ 1, kyk
L
q
0
≤ 1}
kΦk
S
p
→S
q
= sup{|hY, Φ(X)i| : kXk
S
p
≤ 1, kY k
S
q
0
≤ 1}.
If G is a d-regular graph on n vertices with normalized adjacency matrix A, then λ(G) = kA −
1
n
Jk
L
2
→L
2
, where J is the all-ones matrix. Also recall from (2) that for a superoperator Φ the
expansion parameter is λ(Φ) = kΦ − Πk
S
2
→S
2
.
Also define the cut norms by
kAk
cut
= max{|hy, Axi| : x, y ∈ {0, 1}
n
}
kΦk
cut
= sup{|hY, Φ(X)i| : X, Y projectors}.
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It is then not hard to see that if G is a d-regular graph on n vertices with normalized adjacency
matrix A, then (G) = kA −
1
n
Jk
cut
. Similarly, we have (Φ) = kΦ − Πk
cut
.
We have the following relation between these norms, the proof of which is a simple generalization
of the same result from [9] for matrices.
Lemma 2.2. For any superoperator Φ, we have kΦk
cut
≤ kΦk
S
∞
→S
1
≤ π
2
kΦk
cut
and π
2
is the
best possible constant.
Proof. First note that the cut norm as defined above can also be written as
kΦk
cut
= sup{|hY, Φ(X)i| : X, Y 0 , kXk
S
∞
, kY k
S
∞
≤ 1}, (4)
because the set {X : X 0, kXk
S
∞
≤ 1} is the convex hull of the set of projectors. Hence, by
linearity the supremum in (4) will always be attained by projectors.
The first inequality of the lemma follows by dropping the positive semidefinite constraint. For
the second inequality, let z be a complex number of norm 1, and w a uniform random complex
number of norm 1. Then
z = π E
w
[w 1
{<(z ¯w)≥0}
].
Note that E
w
[f(w)] =
1
2π
R
2π
0
f(e
iθ
)dθ, hence the equality follows by using
R
π/2
−π/2
cos(θ)dθ = 2.
We have kΦk
S
∞
→S
1
= sup{|hY, Φ(X)i| : kXk
S
∞
, kY k
S
∞
≤ 1}. The set of matrices X such that
kXk
S
∞
≤ 1 is the convex hull of the set of unitary matrices, so by linearity we can assume that the
supremum in kΦk
S
∞
→S
1
is obtained by unitary X, Y . Unitary matrices are diagonalizable, so write
X = UAU
∗
and Y = V BV
∗
with U, V unitary and A, B diagonal. Let u, w ∈ C, |u| = |w| = 1 be
uniform random complex numbers and define diagonal matrices A
0
, B
0
as A
0
ii
(w) = 1
{<(A
ii
¯w)≥0}
and B
0
ii
(u) = 1
{<(B
ii
¯u)≥0}
. By the above we have A = π E
w
[wA
0
(w)] and similar for B, so we
have X = π E
w
[wUA
0
(w)U
∗
] and Y = π E
u
[uV B
0
(u)V
∗
]. Now, UA
0
(w)U
∗
and V B
0
(u)V
∗
are
projections for all values of w and u, as required in the definition of the cut norm. Therefore
kΦk
S
∞
→S
1
= |hY, Φ(X)i| = π
2
|E
u,w
¯uwhV B
0
(u)V
∗
, Φ(UA
0
(w)U
∗
)i|
≤ π
2
E
u,w
|hV B
0
(u)V
∗
, Φ(UA
0
(w)U
∗
)i|
≤ π
2
E
u,w
kΦk
cut
= π
2
kΦk
cut
,
completing the first part of the proof. Conlon and Zhao show that π
2
is the best possible constant
in the commutative case, using the matrix A ∈ M
n
(C) given by A
st
= e
2πi(s−t)/n
. This matrix
satisfies kAk
L
∞
→L
1
= n and one can show kAk
cut
= (π
−2
+ o(1))n. By Proposition 3.7 in Sec-
tion 3.3, their example can be embedded into a superoperator with the same norms so π
2
is also
the best possible constant here.
Define the Grothendieck norm of of a matrix A ∈ M
n
(C) by
kAk
G
:= sup
n
1
n
n
X
i,j=1
A
ij
hx
i
, y
j
i
: d ∈ N, x
i
, y
j
∈ C
d
, kx
i
k
L
2
≤ 1, ky
j
k
L
2
≤ 1
o
.
Then, the complex Grothendieck constant is given by
K
C
G
:= sup
n
kAk
G
kAk
L
∞
→L
1
: n ∈ N, A ∈ M
n
(C)
o
.
The current best upper and lower bounds on K
C
G
are 1.4049 [15] and 1.338 [11], respectively. The
real version of the Grothendieck constant, denoted by K
G
and mentioned in the introduction, is
obtained by replacing the underlying field in the above quantities by the reals.
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Some basic group theory. Given a graph G = (V, E), a permutation π : V → V is an
automorphism of G if for all u, v ∈ V , we have {π(u), π(v)} ∈ E ⇔ {u, v} ∈ E. The automorphisms
of G form a group under composition, which we call Aut(G). Then, G is said to be vertex transitive
if for every u, v ∈ V , there is a π ∈ Aut(G) such that π(u) = v. For superoperators, we have the
following analogous definitions. A unitary representation of a group Γ on C
n
is a homomorphism
from Γ to U(n) and it is irreducible if the only subspaces of C
n
that are left invariant by the group
action are the zero-dimensional subspace and C
n
itself.
Definition 2.3 (Irreducible covariance). A superoperator Φ : M
n
(C) → M
n
(C) is irreducibly
covariant if there exist a compact group Γ and continuous irreducible unitary representations
U, V : Γ → U(n) such that for all g ∈ Γ and X ∈ M
n
(C), we have
Φ(U(g)XU
∗
(g)) = V (g)Φ(X)V
∗
(g).
3 Converse expander mixing lemmas
In this section, we prove the “converse expander mixing lemmas” announced in the first and third
bullet in the introduction as well as the examples announced in the second bullet. As a warm-up,
we start with a proof of the commutative case due to Conlon and Zhao, which we reprove in a
slightly different manner analogous to how we will prove the non-commutative case.
3.1 Commutative case
In the following, let S be a compact set and Γ be a compact group acting continuously and
transitively on S. The Haar probability measure on Γ induces a measure on S (by pullback)
according to which the L
p
-norm (for p ∈ [1, ∞)) and inner product of f, g ∈ C(S) are given by
kfk
L
p
=
E
π∈Γ
f
π(s
0
)
p
1
p
and hf, gi = E
π∈Γ
f
π(s
0
)
g
π(s
0
)
, (5)
where (by transitivity) s
0
can be taken to be some arbitrary but fixed element of S. We lift the
action of Γ on S to an action on C(S) by precomposition, that is, for any function f ∈ C(S)
and element π ∈ Γ, define the function f
π
by f
π
(s) := f(π(s)). Furthermore, for a linear map
A: C(S) → C(S) define A
π
by A
π
f := (Af
π
)
π
−1
and say that A is transitive covariant with
respect to Γ if for any π ∈ Γ we have A
π
= A.
3
We sometimes omit the group and simply say A
is transitive covariant if such a group Γ exists.
In [9], the following result is proved (over the real numbers) for the case S = [n], in which case
transitive covariant linear maps A are simply n ×n matrices which commute with the permutation
matrices of a transitive subgroup Γ of S
n
. However, their proof easily implies the more general
version below.
Theorem 3.1 (Conlon–Zhao). Let S be as above and let A : C(S) → C(S) be a linear map that
is transitive covariant with respect to Γ. Then,
kAk
L
2
→L
2
≤ K
C
G
kAk
L
∞
→L
1
.
Here we give a somewhat more streamlined proof of this result based on a well-known factor-
ization version of Grothendieck’s inequality [13] (see also [29]), which will serve as a stepping stone
to the proof of the non-commutative case.
4
In our setting the inequality asserts the following
Theorem 3.2 (Commutative Grothendieck inequality (factorization)). Let S be as above and let
A : C(S) → C(S) be a linear map. Then, there exist probability measures λ, ν on S such that for
all f, g ∈ C(S), we have
|hg, Afi| ≤ K
C
G
kAk
L
∞
→L
1
Z
S
|f(s)|
2
dλ(s)
1/2
Z
S
|g(s)|
2
dν(s)
1/2
.
3
In general one says A is covariant with respect to Γ, but we say transitive to emphasize that we require Γ to
act transitively on S.
4
The main difference is that in [9], the result is first proved for weighted Cayley graphs, after which it is shown
that this implies the result for transitive covariant matrices.
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Proof of Theorem 3.1. It follows from the triangle inequality and transitivity that
|hg, Afi| ≤ E
π∈Γ
|hg, A
π
fi| = E
π∈Γ
|hg
π
, Af
π
i|.
By Theorem 3.2 and the AM-GM inequality there are probability measures λ, ν on S such that
the above right-hand side is at most
K
C
G
kAk
L
∞
→L
1
2
E
π∈Γ
Z
S
|f
π
(s)|
2
dλ(s) +
Z
S
|g
π
(s)|
2
dν(s)
=
K
C
G
kAk
L
∞
→L
1
2
(kfk
2
L
2
+ kgk
2
L
2
),
where we switched the order of the integrals (using Tonelli’s theorem) and the expression (5) for
the L
2
norm. For kfk
L
2
= kgk
L
2
= 1 this shows kAk
L
2
→L
2
≤ K
C
G
kAk
L
∞
→L
1
.
3.2 Non-commutative case
Our main technical result is as follows.
Theorem 3.3. Let Φ : M
n
(C) → M
n
(C) be an irreducibly covariant superoperator. Then,
kΦk
S
∞
→S
1
≤ kΦk
S
2
→S
2
≤ 2kΦk
S
∞
→S
1
.
Since the supremum in kΦk
S
∞
→S
1
is taken over X, Y with S
∞
-norm equal to 1, the first
inequality of the theorem follows from the fact that kXk
S
2
≤ kXk
S
∞
. As projectors have Schatten-
∞ norm 1, the first inequality also easily implies the analogue of the Expander Mixing Lemma,
that is, (Φ) ≤ λ(Φ), where λ(Φ) and (Φ) are as in (2) and (3), respectively; note that when Φ
is irreducibly covariant, so is Φ − Π. The second inequality is proved at the end of this section
and in Section 4.2 we show that the factor 2 in the theorem is optimal. With Lemma 2.2, which
relates the uniformity parameter (Φ) to kΦ −Πk
S
∞
→S
1
, Theorem 3.3 then immediately gives the
following result stated in the introduction.
Corollary 3.4 (Converse Quantum Expander Mixing Lemma). Let Φ : M
n
(C) → M
n
(C) be an
irreducibly covariant superoperator. Then, λ(Φ) ≤ 2π
2
(Φ).
In this non-commutative setting we use the following analog of Theorem 3.2 (a factorization
version of the non-commutative Grothendieck inequality), proved by Haagerup in [14]; see also [29].
A density matrix is a positive semidefinite matrix with trace equal to 1.
Theorem 3.5 (Haagerup). Let Φ : M
n
(C) → M
n
(C) be a superoperator. Then, there exist density
matrices ρ
1
, ρ
2
, σ
1
, σ
2
such that for any X, Y ∈ M
n
(C), we have
|hY, Φ(X)i| ≤ kΦk
S
∞
→S
1
(Tr[ρ
1
X
∗
X] + Tr[ρ
2
XX
∗
])
1/2
(Tr[σ
1
Y
∗
Y ] + Tr[σ
2
Y Y
∗
])
1/2
. (6)
We also use the following lemma.
Lemma 3.6. Let Γ be a compact group. Then, a unitary representation U : Γ → U (n) is irreducible
if and only if for any X ∈ M
n
(C), we have
E
g∈Γ
U(g)XU(g)
∗
= Tr(X)
1
n
Id.
Proof. By Schur’s lemma, if U is an irreducible representation, then for T ∈ M
n
(C)
h
∀g ∈ Γ U(g)T U(g)
∗
= T
i
⇐⇒
h
∃λ ∈ C T = λ Id
i
.
Let T
X
= E
g∈Γ
U(g)XU(g)
∗
, then by the group structure we have U(g)T
X
U(g)
∗
= T
X
for all
g ∈ Γ. Therefore, if U is irreducible then T
X
= λ
X
Id. By taking the trace, it follows that
λ
X
= Tr(X)/n. In the other direction, if U is reducible then there exists a projector P onto an
irreducible subspace that is left invariant, i.e. U(g)P U(g)
∗
= P for all g ∈ Γ, so T
P
6= λId.
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Proof of Theorem 3.3. Denote by Γ and U, V : Γ → U(n) the group and irreducible representations
such that Φ is irreducibly covariant with respect to Γ (see Definition 2.3). For any X, Y ∈ M
n
(C)
write X
g
= U(g)XU
∗
(g) and Y
g
= V (g)Y V
∗
(g), then we have
|hY, Φ(X)i| = E
g∈Γ
|hY
g
, Φ(X
g
)i|.
By Theorem 3.5 and the AM-GM inequality, there exist density matrices ρ
1
, ρ
2
, σ
1
, σ
2
such that
the right hand side is bounded from above by
1
2
kΦk
S
∞
→S
1
E
g∈Γ
Tr[ρ
1
X
∗
g
X
g
] + Tr[ρ
2
X
g
X
∗
g
] + Tr[σ
1
Y
∗
g
Y
g
] + Tr[σ
2
Y
g
Y
∗
g
]
.
By Lemma 3.6 we have E
g∈Γ
X
∗
g
X
g
= E
g∈Γ
U(g)X
∗
XU
∗
(g) =
1
n
Tr[X
∗
X]Id = kXk
2
S
2
Id. Let ρ be
a density matrix, then E
g∈Γ
Tr[ρX
∗
g
X
g
] = kXk
2
S
2
. The same holds for E
g∈Γ
Tr[ρX
g
X
∗
g
] but with
U, and for Y with V , so we see that the above quantity is equal to
kΦk
S
∞
→S
1
kXk
2
S
2
+ kY k
2
S
2
.
If kXk
S
2
= kY k
S
2
= 1 we obtain kΦk
S
2
→S
2
≤ 2kΦk
S
∞
→S
1
.
3.3 Embedding graphs into quantum channels
In this subsection, we elucidate the claim that quantum channels generalize graphs and prove
the result stated in the second bullet in the introduction, namely that there are non-irreducible
quantum channels for which a converse expander mixing lemma does not hold.
We consider the following embeddings. For A ∈ M
n
(C), define Φ
A
: M
n
(C) → M
n
(C) as
Φ
A
(X) =
X
i,j
A
ij
X
jj
E
ii
, (7)
where E
ij
is the matrix with a single 1 at position (i, j). When A is a transition matrix, i.e., its
column sums are 1, then it is not hard to see that Φ
A
is completely positive and trace preserving
and that Φ
1
n
J
= Π. Several other ways exist to create quantum expanders from expander graphs,
see for example [20] and [17], but as we show below, our embedding given above carries over all
relevant properties of the graph we consider here.
Conlon and Zhao [9] give an infinite sequence of d-regular graphs G
n
that are o(1)-uniform
but for which λ(G
n
) ≥ 1/2. Combined with the following proposition, this immediately gives the
result stated in the second bullet in the introduction.
Proposition 3.7. Let A ∈ M
n
(C) and p, q ∈ [1, ∞]. Then, for Φ
A
as in (7), we have
kΦ
A
− Πk
S
p
→S
q
= kA −
1
n
Jk
L
p
→L
q
and kΦ
A
− Πk
cut
= kA −
1
n
Jk
cut
.
Proof. Let B = A −
1
n
J, then Φ
A
− Π = Φ
B
. By compactness and definition of k · k
S
p
→S
q
we can assume there is an X ∈ M
n
(C) such that kΦ
B
k
S
p
→S
q
= kΦ
B
(X)k
S
q
/kXk
S
p
. Write
X = diag(x) + X
other
where x ∈ C
n
is the diagonal of X, and X
other
are the off-diagonal entries.
Note that by definition of Φ
B
we have Φ
B
(X) = Φ
B
(diag(x)) = diag(Bx). By definition of
Schatten norms, kdiag(x)k
S
p
= kxk
L
p
and by Proposition 2.1 we have kXk
S
p
≥ kxk
L
p
. We have
kBk
L
p
→L
q
≥
kBxk
L
q
kxk
L
p
≥
kdiag(Bx)k
S
q
kXk
S
p
=
kΦ
B
(X)k
S
q
kXk
S
p
= kΦ
B
k
S
p
→S
q
Now let y ∈ C
n
be such that kBk
L
p
→L
q
= kByk
L
q
/kyk
L
p
. Then
kΦ
B
k
S
p
→S
q
≥
kΦ
B
(diag(y))k
S
q
kdiag(y)k
S
p
=
kdiag(By)k
S
q
kyk
L
p
=
kByk
L
q
kyk
L
p
= kBk
L
p
→L
q
.
This proves the first part.
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The cut norm of a matrix takes the supremum over x, y ∈ {0, 1}
n
. Instead we can relax this
to x, y ∈ [0, 1]
n
, since by linearity the supremum will always be attained by the extreme points.
Similarly, for the superoperator case, we use Equation (4). Then, there exist x, y ∈ [0, 1]
n
such that
kBk
cut
= |hBx, yi|. We have diag(x), diag(y) 0 and kdiag(x)k
S
∞
, kdiag(y)k
S
∞
≤ 1. Therefore
kΦ
B
k
cut
≥ |hdiag(y), Φ
B
(diag(x))i| = |hdiag(y), diag(Bx)i| = |hy, Bxi| = kBk
cut
.
In the other direction, let X, Y ∈ M
n
(C) such that X, Y 0 and kXk
S
∞
, kY k
S
∞
≤ 1. Define x, y to
be the diagonals of X, Y , i.e. x
i
= X
ii
and y
i
= Y
ii
. By Proposition 2.1 we have kxk
L
∞
, kyk
L
∞
≤ 1.
Since X, Y 0 we know all diagonal entries of X and Y are real and non-negative, so we have
x, y ∈ [0, 1]
n
. We conclude
kBk
cut
≥ |hy, Bxi| = |hdiag(y), diag(Bx)i| = |hY, Φ
B
(X)i| = kΦ
B
k
cut
,
completing the proof.
Note that kA −
1
n
Jk
L
2
→L
2
is the second largest eigenvalue in absolute value of the matrix A,
so spectral expansion is preserved under the embedding of graphs into quantum channels. Also,
uniformity is preserved since the cut-norm does not change.
The following proposition shows that the embedding (7) preserves transitivity. This shows that
our Theorem 3.3 generalizes the main result of [9], albeit with a slightly worse constant.
Proposition 3.8. For any A ∈ M
n
(C), A is vertex transitive if and only if Φ
A
is irreducibly
covariant.
Proof. Suppose A is vertex transitive. Let π ∈ Aut(A) be a permutation and P
π
∈ M
n
(C) be the
associated permutation matrix, so that P
π
AP
∗
π
= A. Then,
Φ
A
(P
π
XP
∗
π
) =
X
i,j
A
ij
(P
π
XP
∗
π
)
jj
E
ii
=
X
i,j
A
ij
X
π
−1
(j)π
−1
(j)
E
ii
=
X
i,j
A
iπ(j)
X
jj
E
ii
=
X
i,j
A
π(i)π(j)
X
jj
E
π(i)π(i)
=
X
i,j
A
π(i)π(j)
X
jj
(P
π
E
ii
P
∗
π
) = P
π
Φ
A
(X)P
∗
π
.
This shows that for all π ∈ Aut(A) we have Φ
A
(P
π
XP
∗
π
) = P
π
Φ
A
(X)P
∗
π
.
Let T = {c ∈ C : |c| = 1} be the complex unit circle. For α ∈ T
n
, define U
α
:= diag(α). We
have U
α
E
ii
U
∗
α
= |α
i
|
2
E
ii
= E
ii
and (U
α
XU
∗
α
)
ii
= |α
i
|
2
X
ii
= X
ii
. Therefore
Φ
A
(U
α
XU
∗
α
) =
X
i,j
A
ij
(U
α
XU
∗
α
)
jj
E
ii
=
X
i,j
A
ij
X
jj
U
α
E
ii
U
∗
α
= U
α
Φ
A
(X)U
∗
α
.
We combine these two observations as follows. First we have that
E
α∈T
n
U
α
XU
∗
α
ij
= E
α∈T
n
α
i
X
ij
α
j
=
Z
2π
0
Z
2π
0
e
iθ
i
X
ij
e
−iθ
j
dθ
i
dθ
j
= X
ii
δ
ij
If A is vertex transitive then for all x ∈ C
n
we have E
π∈Aut(A)
P
π
diag(x) P
∗
π
= (E
i
x
i
) Id. Therefore
E
π∈Aut(A)
α∈T
n
(P
π
U
α
)X(P
π
U
α
)
∗
= E
π∈Aut(A)
P
π
E
α∈T
n
U
α
XU
∗
α
P
∗
π
=
Tr(X)
n
Id.
Letting G ⊂ M
n
(C) be the subgroup generated by the U
α
and P
π
for π ∈ Aut(A), we see that for
any g ∈ G
Φ
A
(gXg
∗
) = gΦ
A
(X)g
∗
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and by the previous equation and Lemma 3.6, G acts irreducibly on C
n
(and it is unitary). This
proves Φ is irreducibly covariant with respect to the group G with equal representations.
For the other direction, let U : G → U(n) be the irreducible representation such that Φ
A
is
irreducibly covariant, i.e. Φ
A
(U(g)XU
∗
(g)) = U(g)Φ
A
(X)U
∗
(g) for all g ∈ G. Define P
g
∈ M
n
(C)
as (P
g
)
ij
= |U(g)
ij
|
2
so that (U(g)E
jj
U(g)
∗
)
ii
= (P
g
)
ij
. Then
A
kl
= Tr[E
kk
Φ
A
(E
ll
)] = Tr[U (g)E
kk
U(g)
∗
Φ
A
(U(g)E
ll
U(g)
∗
)]
=
X
ij
A
ij
(P
g
)
jl
(P
g
)
ik
= (P
T
g
AP
g
)
kl
,
showing P
T
g
AP
g
= A. Since U(g) is unitary, P
g
is doubly stochastic so by Birkhoff’s Theorem P
g
is
a convex combination of permutation matrices, i.e., P
g
= E
i
Π
i
for some (not necessarily uniform)
probability distribution and where Π
i
is a permutation matrix. We have
A
kl
= (P
T
g
AP
g
)
kl
= E
i
E
j
(Π
T
i
AΠ
j
)
kl
= E
i
E
j
A
π
i
(k) π
j
(l)
.
Since A is {0, 1}-valued, it follows that if A
kl
= 1 then all elements of the convex combination on
the right-hand side must be 1, and if A
kl
= 0 then all elements of the right hand side must be 0.
Therefore, for all i we have Π
T
i
AΠ
i
= A. By irreducibility, we have for all k, l that
1
n
=
Tr[E
kk
]
n
Id
ll
=
E
g∈G
U(g)E
kk
U
∗
(g)
ll
= E
g∈G
|U(g)
lk
|
2
,
showing E
g∈G
(P
g
)
lk
= 1/n. It follows that there is a g ∈ G such that (P
g
)
lk
> 0. Decomposing P
g
into permutation matrices shows there is a Π ∈ Aut(A) such that Π
lk
= 1. This holds for all k, l,
proving the lemma.
3.4 Randomizing superoperators
We prove the following analogue of one of the results from [8] showing that for any d-regular
graph G, it holds that λ(G) ≤
2(G)/δ
2
1/4
, where δ = d/n is the edge density. This in particular
establishes a tight relation between spectral expansion and uniformity for sequences of graphs with
δ
n
≥ Ω(1). For A ∈ M
n
(C), we have kAk
L
1
→L
∞
= n sup
ij
|A
ij
|, and for an n-vertex d-regular
graph with normalized adjacency matrix A we have sup
ij
|A
ij
| =
1
d
so kA − J/nk
L
1
→L
∞
=
1
δ
− 1
with J being the all-ones matrix. Therefore, a sequence of graphs with normalized adjacency
matrices A
n
is dense exactly when kA
n
− J
n
/nk
L
1
→L
∞
≤ O(1), where J
n
is the all-ones n by n
matrix .
Let Π be the projector onto the identity matrix. A superoperator Φ is said to be η-randomizing
if kΦ − Πk
S
1
→S
∞
≤ η, which when η ≤ O(1), may thus be seen as an analogue of density. Note
that by Proposition 3.7 the embedding of any dense graph is O(1)-randomizing.
Proposition 3.9. Let Φ : M
n
(C) → M
n
(C) be a superoperator that is O(1)-randomizing. Then,
λ(Φ) ≤ O((Φ)
1/4
).
To prove Proposition 3.9, we require the following lemma.
Lemma 3.10. Let Φ : M
n
(C) → M
n
(C) be a superoperator and let C = kΦk
S
1
→S
∞
. Then we
have kΦk
S
2
→S
2
≤
C
3
kΦk
S
∞
→S
1
1/4
.
Proof. Note that by definition of C we have |hQ, Φ(P )i| ≤ CkQk
S
1
kP k
S
1
. Let X, Y ∈ M
n
(C) be
such that hY, Φ(X)i = kΦk
S
2
→S
2
with kXk
S
2
= kY k
S
2
= 1. Write X =
1
n
P
n
i=1
λ
i
P
i
and Y =
1
n
P
n
i=1
µ
i
Q
i
with P
i
, Q
i
rank-1 matrices with kQ
i
k
S
1
= kP
i
k
S
1
= 1. We have kλk
L
2
= kµk
L
2
= 1
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and by applying Cauchy-Schwarz twice,
|hY, Φ(X)i|
4
=
E
ij
λ
i
µ
j
hQ
j
, Φ(P
i
)i
4
≤
E
i
λ
2
i
2
E
i
E
j
µ
j
hQ
j
, Φ(P
i
)i
2
2
=
E
i,j,j
0
µ
j
µ
j
0
hQ
j
, Φ(P
i
)ihP
i
, Φ
∗
(Q
j
0
)i
2
≤
E
j,j
0
µ
2
j
µ
2
j
0
E
j,j
0
E
i
hQ
j
, Φ(P
i
)ihP
i
, Φ
∗
(Q
j
0
)i
2
= E
i,i
0
,j,j
0
hQ
j
, Φ(P
i
)ihP
i
, Φ
∗
(Q
j
0
)ihQ
j
0
, Φ(P
i
0
)ihP
i
0
, Φ
∗
(Q
j
)i,
where all indices are averaged from 1 to n. Now we see
|hY, Φ(X)i|
4
≤ E
i,j
hQ
j
, Φ(P
i
)i
D
E
j
0
hQ
j
0
, Φ(P
i
)iQ
j
0
, Φ
E
i
0
hP
i
0
, Φ
∗
(Q
j
)iP
i
0
E
≤ E
i,j
|hQ
j
, Φ(P
i
)i| kΦk
S
∞
→S
1
kE
j
0
hQ
j
0
, Φ(P
i
)iQ
j
0
k
S
∞
kE
i
0
hP
i
0
, Φ
∗
(Q
j
)iP
i
0
k
S
∞
≤ E
i,j
|hQ
j
, Φ(P
i
)i| kΦk
S
∞
→S
1
max
j
0
|hQ
j
0
, Φ(P
i
)i| max
i
0
|hQ
j
, Φ(P
i
0
)i|
≤ C
3
kΦk
S
∞
→S
1
.
Proof of Proposition 3.9. Let Π(X) =
1
n
Tr[X]Id be the projector on to the identity. By assump-
tion, we have kΦ − Πk
cut
= (Φ). Define C = kΦ − Πk
S
1
→S
∞
. Using Lemma 2.2 and Lemma 3.10
applied to Φ −Π we find kΦ −Πk
S
2
→S
2
≤ (C
3
π
2
(Φ))
1/4
.
4 Optimality of constants
4.1 Commutative case
In this section we prove the fourth bullet point in our introduction. Theorem 3.1 shows that K
C
G
bounds the ratio of the L
2
→ L
2
and L
∞
→ L
1
norms, and Lemma 2.2 (the matrix version) shows
that π
2
bounds the ratio of the L
∞
→ L
1
norm and the cut norm. We now prove the optimality
of the combined inequality.
Let S
m−1
= {x ∈ C
m
: kxk
L
2
= 1} denote the (m − 1)-dimensional unit sphere endowed with
its Haar probability measure µ.
Theorem 4.1. For any > 0 there exist positive integers m, k and a transitive covariant linear
map M : C(S
m−1
× [k]) → C(S
m−1
× [k]) such that kMk
L
2
→L
2
≥ (π
2
K
C
G
− )kMk
cut
.
The optimality of π
2
between the L
∞
→ L
1
norm and the cut norm is already covered in
Lemma 2.2. We show that K
C
G
is optimal in the sense that Theorem 3.1 cannot be improved
(despite the fact that the exact value of the Grothendieck constant K
C
G
is unknown). We do this in
Lemma 4.2 below. Then in Lemma 4.3 we show that any map can be lifted to one on a bigger space
with appropriately bounded cut norm. The combination of these lemmas proves our theorem.
In the introduction we also mentioned the optimal constant 4K
G
in the case where the field is
R instead of C. The proofs below still apply in this case, with only small modifications.
Lemma 4.2. For any > 0 there exists a positive integer m and a transitive covariant linear map
B : C(S
m−1
) → C(S
m−1
) such that kBk
L
2
→L
2
≥ (K
C
G
− )kBk
L
∞
→L
1
.
Proof. By definition of the Grothendieck constant, for any > 0 there exists an n ∈ N and a linear
map A ∈ M
n
(C) such that kAk
G
≥ (K
C
G
− )kAk
L
∞
→L
1
. This map A might not be transitive
covariant, so from it we will now construct a transitive covariant linear map B : C(S
2n−1
) →
C(S
2n−1
) such that kBk
L
∞
→L
1
≤ kAk
L
∞
→L
1
and kBk
L
2
→L
2
≥ kAk
G
. This idea is based on a
lemma found in [6].
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Let x
i
, y
j
∈ S
2n−1
be the vectors that attain the Grothendieck norm for A, which can always
be assumed to be 2n-dimensional since there are only 2n of them, so
kAk
G
=
1
n
X
i,j
A
ij
hx
i
, y
j
i
.
Define the map B by
hf, B(g)i =
1
n
X
i,j
A
ij
Z
U(2n)
f(Ux
i
)g(Uy
j
)dU.
To bound kBk
L
∞
→L
1
we have to bound |hf, B(g)i| for f, g : S
2n−1
→ [−1, 1]. By the triangle
inequality,
|hf, B(g)i| ≤
Z
U(2n)
1
n
X
i,j
A
ij
f(Ux
i
)g(Uy
j
)
dU ≤
Z
U(2n)
kAk
L
∞
→L
1
dU ≤ kAk
L
∞
→L
1
.
Now for each i ∈ [2n] let f
i
∈ C(S
2n−1
) be given by f
i
(x) = x
i
(i.e. the i-th coordinate). Then,
1
2n
2n
X
i=1
hf
i
, B(f
i
)i ≤
1
2n
2n
X
i=1
kBk
L
2
→L
2
kf
i
k
2
L
2
= kBk
L
2
→L
2
Z
S
2n−1
1
2n
2n
X
i=1
x
2
i
dµ(x)
= kBk
L
2
→L
2
.
On the other hand,
1
2n
2n
X
i=1
hf
i
, B(f
i
)i =
1
n
X
i,j
A
ij
Z
U(2n)
hUx
i
, Uy
j
idU =
1
n
X
i,j
A
ij
hx
i
, y
j
i = kAk
G
,
so we conclude kBk
L
2
→L
2
≥ kAk
G
. We will show B is transitive covariant with respect to Γ =
U(2n). To show B is invariant, we have to prove that for all V ∈ U(2n) we have hf
V
, B(g
V
)i =
hf, B(g)i. Indeed,
hf
V
, B(g
V
)i =
1
n
X
i,j
A
ij
Z
U(2n)
f(V U x
i
)g(V Uy
j
)dU
=
1
n
X
i,j
A
ij
Z
U(2n)
f(U
0
x
i
)g(U
0
y
j
)dU
0
= hf, B(g)i,
which completes the proof.
Lemma 4.3. Let S be any compact set and let B : C(S) → C(S) be a linear map. For any > 0
there exists a k ∈ N and a linear map M : C(S × [k]) → C(S × [k]) such that
kMk
cut
kMk
L
2
→L
2
≤
1
π
2
+
kBk
L
∞
→L
1
kBk
L
2
→L
2
and if B is transitive covariant then so is M.
Proof. We will choose k large enough, to be determined later. For any f, g ∈ C(S × [k]) define
f
i
∈ C(S) as f
i
(s) := f (s, i), and similar for g
i
. Define ω = e
2πi/k
. Define a linear map M :
C(S × [k]) → C(S × [k]) as
M(f)
(t, j) :=
1
k
k
X
i=1
ω
i−j
B(f
i
)(t), for t ∈ S and j ∈ [k].
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We then have
hg, M(f)i
S×[k]
=
1
k
2
D
X
i
ω
i
g
i
, B
X
j
ω
j
f
j
E
S
where one factor of
1
k
comes from our normalization of the inner product. This implies
hg, M(f)i
S×[k]
≤ kBk
L
∞
→L
1
1
k
k
X
i=1
ω
i
g
i
L
∞
1
k
k
X
j=1
ω
j
f
j
L
∞
. (8)
If f, g ∈ C(S × [k]) are the [0, 1]-valued functions that attain the cut norm of M, then by (8)
kMk
cut
≤
1
π
2
+
kBk
L
∞
→L
1
,
where we used Lemma 4.4 to bound
1
k
P
k
i=1
ω
i
g
i
L
∞
.
Let u, v ∈ C(S) with kuk
L
2
= kvk
L
2
= 1 be such that kBk
L
2
→L
2
= hv, B(u)i
S
. Now define
f
(u)
, g
(v)
∈ C(S × [k]) as f
(u)
(s, i) := ω
−i
u(s) and g
(v)
(s, i) := ω
−i
v(s), which also have L
2
-norm
equal to 1. We then see
kMk
L
2
→L
2
≥
g
(v)
, M(f
(u)
)
S×[k]
= hv, B(u)i
S
= kBk
L
2
→L
2
.
The combination of these observations completes the first part of the proof. Now assume B is
transitive covariant with respect to Γ, so B(f
π
)(π
−1
(s)) = B(f)(s) for all s ∈ S and π ∈ Γ.
Define a new group Γ
0
as the cartesian product Γ
0
= Γ × Z
k
. For (π, m) ∈ Γ
0
define the action
(π, m) : S × [k] → S × [k] as (π, m)(s, i) = (π(s), i + m). By entering f
(π,m)
into the definition
of M it follows that M
(π,m)
= M, so M is transitive covariant with respect to Γ
0
, completing the
proof.
Lemma 4.4. Let > 0, then there exists a k
0
∈ N such that for all k ≥ k
0
and x ∈ [0, 1]
k
we have
1
k
k
X
j=1
e
2πi j/k
x
j
≤
1
π
+ .
Proof. First let k
0
be arbitrary, to be determined later and k ≥ k
0
. Define y ∈ [−1, 1]
k
as
y
i
= 2x
i
− 1, then
1
k
k
X
j=1
e
2πi j/k
x
j
=
1
2
1
k
k
X
j=1
e
2πi j/k
y
j
=
1
2
e
2πiφ
1
k
k
X
j=1
e
2πi j/k
y
j
.
In the first equality we used that
P
k
j=1
e
2πi j/k
= 0. In the second equality we used that there
exists a φ such that the full expression becomes real and positive. Since e
iθ
= cos(θ) + i sin(θ) and
the full expression is real, we know the sin component vanishes and therefore
1
2
1
k
k
X
j=1
e
2πi(φ+j/k)
y
j
=
1
2
1
k
k
X
j=1
cos(2π(φ + j/k))y
j
.
Now note that cos(2π(φ + j/k))y
j
≤
cos(2π(φ + j/k))
and hence
1
2
1
k
k
X
j=1
cos(2π(φ + j/k))
k→∞
−→
1
2
Z
1
0
cos
2π(φ + x)
dx =
1
π
.
This completes the proof.
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4.2 Non-commutative case
In the non-commutative case we show optimality of Theorem 3.3. By Lemma 2.2, the factor π
2
between the cut-norm and S
∞
→ S
1
-norm is also optimal. In contrast with the commutative case,
our work leaves the optimality of the combined inequality in Corollary 3.4 as an open problem.
Straightforward analogues of the techniques employed in Lemma 4.3 did not follow through in the
non-commutative case.
Proposition 4.5. For any > 0, there exists a positive integer n and an irreducibly covariant
superoperator Φ : M
n
(C) → M
n
(C) such that kΦk
S
2
→S
2
≥ (2 −)kΦk
S
∞
→S
1
.
One of the forms of the non-commutative Grothendieck inequality, equivalent to Theorem 3.5,
is the following [29]. Let Φ : M
n
(C) → M
n
(C) be a linear map and x
i
, y
j
∈ M
n
(C) finite sets of
matrices. Then,
X
i
hx
i
, Φ(y
i
)i
≤ K
0
G
kΦk
S
∞
→S
1
k
P
i
x
∗
i
x
i
k + k
P
i
x
i
x
∗
i
k
2
·
k
P
i
y
∗
i
y
i
k + k
P
i
y
i
y
∗
i
k
2
1/2
(9)
where K
0
G
≤ 2 and the norms on the right hand side are operator norms k·k
S
∞
. To show tightness,
i.e. K
0
G
≥ 2, Haagerup and Itoh [16] (see [29] for a survey) gave an explicit family of operators
for which (9) gives a lower bound of K
0
G
approaching 2. We will show that slight modifications of
these operators are irreducibly covariant, which proves Proposition 4.5. It is instructive to repeat
their construction. The proof uses techniques familiar in the context of the antisymmetric Fock
space, but our proof is self contained.
Lemma 4.6 ([16]). For each n ∈ N there exists a d ∈ N and a linear map Φ : M
d
(C) → M
d
(C)
with sets of matrices {x
i
}, {y
i
} such that (9) yields K
0
G
≥ (2n + 1)/(n + 1).
Proof. Let H = C
2n+1
and consider the antisymmetric k-fold tensor product H
∧k
which is a linear
subspace of the k-fold tensor product H
⊗k
. A basis of H
∧k
is formed by vectors e
i
1
∧e
i
2
∧···∧e
i
k
with i
1
< ··· < i
k
where the e
i
are standard basis vectors of H. Here ∧ is the wedge product or
exterior product, which has the property x ∧y = −y ∧ x and is given by x ∧ y = x ⊗y −y ⊗x, for
x, y ∈ H. We will consider k = n and k = n + 1 so that the dimension of H
∧k
is d =
2n+1
n
for
both k = n and k = n + 1.
For 1 ≤ i ≤ (2n + 1), define c
i
: H
∧n
→ H
∧(n+1)
as c
i
(x) := e
i
∧ x, which physicists call the
fermionic creation operator. Its adjoint c
∗
i
: H
∧(n+1)
→ H
∧n
is known as the annihilation operator.
By the antisymmetric property, c
i
(x) = 0 whenever e
i
was present in x, i.e., when x = e
i
∧ x
0
.
The operator c
i
c
∗
i
, also known as the number operator, is a projector onto the space spanned by
basis vectors in which e
i
is present. The operator c
∗
i
c
i
is a projector onto the space where e
i
is not
present. Since there are always (n + 1) vectors present in H
∧(n+1)
and (n + 1) vectors not present
in H
∧n
, we have
2n+1
X
i=1
c
i
c
∗
i
= (n + 1)Id
H
∧(n+1)
and
2n+1
X
i=1
c
∗
i
c
i
= (n + 1)Id
H
∧n
.
We will now argue that
hc
i
, c
j
i :=
1
d
Tr(c
∗
i
c
j
) = δ
i,j
n + 1
2n + 1
, (10)
k
2n+1
X
i=1
α
i
c
i
k
S
1
= kαk
L
2
n + 1
√
2n + 1
for α ∈ C
2n+1
. (11)
The δ
i,j
in (10) follows because hx, c
∗
i
c
j
xi = 0 for any x = e
k
1
∧ ··· ∧ e
k
n
when i 6= j. The factor
n+1
2n+1
follows by taking the trace of one of the sums above and noting that by symmetry in i,
every term of the sum must have the same trace. To prove (11), first note that for any unitary
U ∈ U(2n + 1) we have
U
⊗(n+1)
· c
i
· (U
⊗n
)
−1
=
X
j
U
ji
c
j
, (12)
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which can be shown by proving it for all basis states:
U
⊗(n+1)
c
i
(U
⊗n
)
−1
(e
k
1
∧ ... ∧ e
k
n
) = U
⊗(n+1)
c
i
(U
−1
e
k
1
∧ ... ∧ U
−1
e
k
n
)
= U
⊗(n+1)
(e
i
∧ U
−1
e
k
1
∧ ... ∧ U
−1
e
k
n
)
= (Ue
i
∧ e
k
1
∧ ... ∧ e
k
n
)
= (
X
j
U
ji
e
j
∧ e
k
1
∧ ... ∧ e
k
n
)
=
X
j
U
ji
c
j
(e
k
1
∧ ... ∧ e
k
n
).
The trace-norm is unitarily invariant, so (12) implies kc
i
k
S
1
= k
P
j
U
ji
c
j
k
S
1
. Since c
∗
i
c
i
is a
projector, we have
p
c
∗
i
c
i
= c
∗
i
c
i
and hence kc
i
k
S
1
=
1
d
Tr(c
∗
i
c
i
). Now let α ∈ C
2n+1
with
P
i
|α
i
|
2
=
1, then there is a unitary U ∈ U(2n + 1) such that the i-th row of U is α. Note that kαk
L
2
=
1/
√
2n + 1 since we use normalized L
2
-norms, which implies (11).
Since the dimensions of H
∧n
and H
∧(n+1)
are equal, we can identify the space of linear maps
L(H
∧n
, H
∧(n+1)
) with M
d
(C) (by choosing bases for H
∧n
and H
∧(n+1)
), and define the following
operator Φ : M
d
(C) → M
d
(C),
Φ(x) =
2n+1
X
i=1
hc
i
, xic
i
.
Consider (9) for Φ with x
i
= y
i
= c
i
. For the left hand side, note that by (10) we have
2n+1
X
j=1
hc
j
, Φ(c
j
)i
=
2n+1
X
i,j=1
hc
i
, c
j
ihc
j
, c
i
i
=
(n + 1)
2
2n + 1
.
For the right-hand side of (9), we require kΦk
S
∞
→S
1
= sup
kxk
S
∞
=1
kΦ(x)k
S
1
. For any x ∈ M
d
(C),
define v
(x)
∈ C
2n+1
as v
(x)
i
= hc
i
, xi. Note that kvk
L
2
= sup
kαk
L
2
=1
|hv, αi|. First apply (11) to
obtain
kΦ(x)k
S
1
= k
2n+1
X
i=1
hc
i
, xic
i
k
S
1
= kv
(x)
k
L
2
n + 1
√
2n + 1
= sup
kαk
L
2
=1
|hv
(x)
, αi|
n + 1
√
2n + 1
.
Using (11) again, we compute sup
kxk
S
∞
=1
|hv
(x)
, αi| for arbitrary α with kαk
L
2
= 1,
sup
kxk
S
∞
=1
|hv
(x)
, αi| = sup
kxk
S
∞
=1
1
2n + 1
hx,
X
i
α
i
c
i
i
=
1
2n + 1
k
X
i
α
i
c
i
k
S
1
=
n + 1
(2n + 1)
√
2n + 1
.
We obtain kΦk
S
∞
→S
1
= (n + 1)
2
/(2n + 1)
2
. Now (9) yields
(n+1)
2
2n+1
≤ K
0
G
(n+1)
2
(2n+1)
2
· (n + 1) and
therefore
2n+1
n+1
≤ K
0
G
.
We use the following fact from [12, Theorem 19.14], about the representations of the odd
dimensional complex special orthogonal groups on wedge products of complex vector spaces.
Lemma 4.7. Let n, k ∈ N, N := 2n + 1 and let R
k
: SO(N, C) → GL((C
N
)
∧k
) be given by
A 7→ A
⊗k
. This representation is irreducible.
Below, we actually need that the real special orthogonal group SO(N, R) acts irreducibly on
the same anti-symmetric space. Fortunately, this is implied by Lemma 4.7; see [12, pp. 439]. We
will also use the fact that R
k
and R
N−k
are unitarily equivalent to each other. This is the content
of the following proposition [32, Proposition IX.10.4].
Proposition 4.8. For positive integer n and N = 2n + 1 and k ∈ {1, . . . , N }, let R
k
be the
representation as in lemma 4.7. Then, there exists an isometry V
k
: (C
N
)
∧k
→ (C
N
)
∧(N−k)
such
that
V
k
R
k
(A) = R
N−k
(A)V
k
, ∀A ∈ SO(N, R).
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Proof of Proposition 4.5. Let d be the dimension of (C
N
)
∧n
and let Φ : M
d
(C) → M
d
(C) be as in
the proof of Lemma 4.6. For each k ∈ N, let R
k
: SO(N, R) → GL(H
∧k
) be the representation
A 7→ A
⊗k
, which is irreducible by Lemma 4.7. Define, for notational convenience, π := R
n+1
and
ρ := R
n
. We first show that for all A ∈ SO(N, R), we have
Φ(π(A)xρ
∗
(A)) = π(A) Φ(x) ρ
∗
(A). (13)
For the left-hand side, note that
Φ(π(A)xρ
∗
(A)) =
X
i
c
i
, π(A)xρ
∗
(A)
c
i
=
X
i
π(A)
∗
c
i
ρ(A), xic
i
=
X
i
D
X
j
A
ij
c
j
, x
E
c
i
=
X
ij
A
ij
hc
j
, xic
i
,
where we used (12) from the proof of Lemma 4.6 and noting that SO(N, R) ⊂ U(N) is a subgroup.
Using (12) again for the right-hand side, we have
π(A) Φ(x) ρ
∗
(A) =
X
i
hc
i
, xiπ(A)c
i
ρ
∗
(A)
=
X
i
hc
i
, xi
X
j
A
ji
c
j
=
X
ij
A
ij
hc
j
, xic
i
.
which proves (13).
Define a new superoperator Φ
0
: M
d
(C) → M
d
(C) by
Φ
0
(x) = Φ(xV
∗
)V,
where V := V
n+1
is the isometry as in Proposition 4.8 (we view V as a matrix in M
d
(C) by
choosing basis). We first note that this Φ
0
might also be used in Lemma 4.6 to show that the non-
commutative Grothendieck constant is 2, since Schatten-norms are unitarily invariant. Hence, if
we show that Φ
0
is irreducibly covariant, we are done. This follows from the following computation,
where we use (13) and the fact that V π(A) = ρ(A)V for all A ∈ SO(N, R):
Φ
0
π(A)xπ(A)
∗
= Φ
π(A)xπ(A)
∗
V
∗
V
= Φ
π(A)xV
∗
ρ(A)
∗
V
(13)
= π(A) Φ(xV
∗
) ρ(A)
∗
V
= π(A) Φ(xV
∗
) V π(A)
∗
= π(A) Φ
0
(x) π
∗
(A),
where the second-last line follows since ρ(A)
∗
= V π(A)
∗
V
∗
. Hence, Φ
0
is irreducibly covariant
with respect to the irreducible representation π of SO(N, R).
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