1. Introduction and Preliminaries
In the work of N. J. Kalton [
1,
2,
3], we can find novel ideas and methods for the stability of functional equations that depart from the classical methods of Hyers, Ulam and Rassias [
4]. In Ref. [
3] (see Theorem 2.2), Kalton provides a sharp bound on the stability of the additive map in
for the so-called singular case. His proof makes use of probabilistic and geometric methods in Banach space theory. This paper ends with a sketch on how the theory of twisted sums in Banach space theory could be used to obtain the same result. In this note, we develop this last idea and use it to obtain an improvement in the linear stability of Wigner’s symmetry theorem (see Theorem 3).
Wigner’s celebrated symmetry theorem [
5] is not only central for physics, but it also finds an important role in many preservers’ problems. A
preserver problem deals with the characterization of maps, primarily on matrix spaces and operator algebras that preserve certain functional, subset, or an invariant. In particular, in the field of Quantum Information Theory (QIT) it has been shown [
6] that the only mapping
T that preserves the
divergences (this includes the von Neumann and relative entropy) is a Wigner symmetry transformation, i.e., of the form
where
U is either a unitary or antiunitary transformation on
. It turns out that most of the proofs of different preservers problems can be reduced to Wigner’s theorem. Therefore, it is natural to expect that sharp bounds on the stability of Wigner’s theorem could provide good approximations for a wide range of almost-preserving problems. It is worth pointing out that there exists a close relation between geometric functional analysis and many questions in QIT [
7]. This is the point of view that we want to motivate here.
It has been recently shown [
8] that an arbitrary
almost-symmetry in quantum theory, i.e., a transformation on the set of pure states
in a separable complex Hilbert space
that almost preserves the transition probabilities up to an error
, can be approximated by a linear map
H if and only if
is a finite-dimensional Hilbert space. For an infinite-dimensional Hilbert space, the approximation is in a weak sense (see Theorem 2-(i) in [
8]). The quality of the approximation for a
d-dimensional Hilbert space
was obtained to be
where
is the Hilbert–Schmidt norm. The main idea for the upper bound in Equation (
1) was to consider an almost-linear extension of
f with some particular domain and codomain, followed by an application of the geometric Hahn–Banach theorem. In this work, we explore how the quality of the approximation depends on the consideration of various classes of almost-linear extensions. These extensions now have arbitrary finite-dimensional Banach spaces as domain and codomain.
Throughout this note, we will be entirely concerned with finite-dimensional Banach spaces and the twisted sums generated by almost-linear maps. A map between Banach spaces will be called almost-linear if it satifies the following two conditions:
- (i)
for all and ,
- (ii)
there exists a
such that for any finite sequence
,
and
,
We will show that, for every almost-linear map
F, there exists a linear map
H whose distance to
F depends additively on
and on some geometric invariants of the domain and target space of
F (see Theorem 1). The Banach space numbers used to express the results are the type and cotype constants which we introduce now. Let
be a sequence of independent real Gaussian random variables, i.e., for each Borel subset
, each random variable has a distribution
Let
X be a Banach space with norm
and let
,
. For every positive interger
n, we define
to be the smallest constants such that for arbitrary sequences
, we have
The space X is said to be of Gaussian type p (resp. Gaussian cotype q) if (resp. ). One can analogously define the Rademacher type and cotype by exchanging the Gaussian sequence by a Rademacher sequence. The results shown in this note are valid for both notions of type and cotype.
For
we denote by
the Hermitian part of the
d-dimensional
r-Schatten class and by
the classical space of
r-summable sequences in
; the space
is a real Banach space with norm
.
Table 1 summarizes the behaviour of the type and cotype constants for the
r-Schatten classes that we use (see Ref. [
9] for details).
We now introduce some notation. The set of rank-one projections in is denoted by . The unit ball of a space Z is written as . The convex hull of a set S is the set of convex combinations of elements of S, which we denote by . The set of linear maps between X and Y is . A linear projection is a linear map such that . Finally, we denote by the Hilbert–Schmidt inner product in the real vector space of Hermitian matrices .
In the next section, we introduce a special space which will generate the linear approximation to the almost-linear map
. This space is an extension of
X and
Y and is called a twisted sum (basically because it “twists” the unit ball of
X and
Y according to
F). Twisted sums were extensively studied by Kalton [
1] in the context of the three-space problem. In particular, Kalton showed that twisted sums are in correspondence with quasi-linear maps; this is a weaker condition than almost-linearity, but, for our purposes, it suffices to say that any almost-linear map is a quasi-linear map. See Ref. [
10] for a detailed exposition of this topic.
2. Finite-Dimensional Twisted Sums
Let
be two Banach spaces with dimension
, respectively. The twisted sum of
Y and
X is a
-dimensional space
Z that contains a subspace
that is isomorphic to
Y and such that
is isomorphic to
X. The twisted sums that interest us are constructed with an almost-linear function
F. Consider
and the Cartesian product
(the order is important) endowed with the quasi-norm:
Then,
is
-isometric to
Y and
-isometric to
X. Note that, since
F is homogeneous,
and
implies that
. Although
we can still endow
Z with a norm. The twisted sum
Z can be made into a Banach space with the norm
The fact that the above expression defines a norm will be shown below. The completion of a quasi-Banach space
Z whose dual is non-trivial with respect to this norm is known as the Banach envelope of
Z [
11]. In order to avoid charged notation, we also denote the Banach envelope by
Z.
Lemma 1. Let be a quasi-norm on Z, then the following equivalent expressions define a norm on Z. For , Moreover, for the quasi-norm defined by Equation (3), we have the following equivalence: Proof. We show first that the first expression indeed defines a norm. Since
is a quasi-norm, the only property that we need to check is the triangle inequality. This can be verified by
as those are valid decompositions of
. We show now that Equations (
5) and (
6) are the same. Let
be the infimum of Equation (
6). Then, there exist
, positive real numbers
,
and
with quasi-norm one such
. This is a valid decomposition of
z and
. On the other hand, let
be the decomposition that achieves the infimum in Equation (
5) so that
. Then,
The norm of
can be computed as
as the supremum over a convex function is achieved at the extremal points. Thus, the dual of the quasi-Banach space
Z and its Banach envelope coincide. Thus, Equation (
7) is just the usual expression in terms of the dual. We now compare the quasi-norm in Equation (
3) with the norm of its envelope.
Since
is defined by the infimum of
over all the decompositions of
z, Equation (
5), we immediately have the first inequality in Equation (
8). For the second inequality, let
, then, using Equation (
2),
□
Additionally, we can understand the resulting twisted sum
Z with a norm as in Equation (
5) as the space with unit ball [
12]
We write for the (Banach envelope of) twisted sum of Y and X generated by the almost-linear map .
4. Applications
The following result gives an improvement on Theorem 2-(ii) in [
8].
Theorem 3 (Linear Stability of Wigner’s theorem)
. Let be a function that satisfiesThen, there exists a universal constant C and a linear map such that, for all where . We call a map
that satisfies Equation (
10) an
almost-symmetry. In order to prove Theorem 3, we make use of the following lemmas (c.f. Theorem 1 in [
12]). First, we need the type constant of a twisted sum (cf. Lemma 16.6-7 in [
16]).
Lemma 2. Let Z be the twisted sum of Y and X; then, The type 2 constant of a Banach space of dimension
d can be obtained from the type constant restricted to families of size
as stated by the following lemma. This result follows from a cone version of Caratheodory’s theorem (see Lemma 6.1 in [
17]).
Lemma 3. Let X be a d-dimensional Banach space. Then, and for any .
Proof. The first step of the proof consists of extending the function f to such that . We take x in the unit sphere of and identify it with its antipodal point . We choose a fixed spectral decomposition for both elements, say , and define . Then, we can extend F homogeneously from the unit sphere to any by multiplying x or with so that or . We call again this extension F. By construction, F is a real homogeneous map. Note that this extension is not unique, but we do not need this here.
As proven in Lemma 2 of Ref. [
8],
F is an almost-linear map
with
. If we use Theorem 1 with the twisted sum
, we will see that we cannot obtain anything better than a linear dependence on
d. However, we will be able to obtain a better dimension dependence if we consider a dual construction, namely with
. For that matter, we use Lemmas 2 and 3 in order to estimate the type 2 constant of
. From Equation (
11) and
, we obtain
for all
. It is known that, for a general Banach space
E,
(Proposition 12.3 in [
9]). Thus, for all two-dimensional subspaces of
Z, the type is less than
and
(this can be alternatively derived from a classical result of John and the relation between the Banach–Mazur distance and type 2 constants). It follows from induction that
which, in turn, implies
The dimension of the real vector space of Hermitian matrices
is
. Therefore, we obtain from Lemma 3 with
It follows from Theorem 1 and
that there exists a linear map
such that
□
The following proposition is essentially due to Kalton. It can be shown using Theorem 2.2 in [
3] as
and
are isomorphic Hilbert-spaces. We present here a proof using the notions of (co)type and Theorem 1.
Proposition 1 (Stability of Global Symmetries)
. Let be a continuous function that satisfiesThen, there exists a linear map and an absolute constant C such that, for all Proof of Proposition 1. The first step consists of showing that the function f can be extended to a continuous homogeneous function on the whole space without paying much.
Lemma 4. Let be a continuous function that satisfies Then, there exists a continuous and homogeneous function such thatand Proof of Lemma 4. Let us extend
f to
where
This function is homogeneous, i.e.,
for all
, and continuous as
f and
are also continuous. Using Equation (
12) and the triangle inequality, we obtain the new almost-symmetry condition
Hence, for any
Therefore, from the linearity of the inner product, we obtain Equation (
13). Finally, we show that
f and
F are
-close. From Equation (
12),
Thus, with Equation (
15), we have
which is less than
for all
. □
We consider now the twisted sum
generated by the almost-linear map
F. Before applying Theorem 1, we estimate the type 2 constant of
Z. Since
is a Hilbert space, it has a type 2 constant equal to one and we obtain from Lemma 2 that
As in the proof of Theorem 3, all two-dimensional subspaces of
Z have a type less than
and
. It follows from induction that, for
,
Hence, from Lemma 3 with
Accordingly, from
and Theorem 1, there exists a linear map
such that, for all
Finally, from Equation (
14) and the triangle inequality, we obtain
□