# Tight Hardness of the Non-commutative Grothendieck Problem

Jop Briët Courant Institute of Mathematical Sciences, New York University.
Supported by a Rubicon grant from the Netherlands Organisation for Scientific
Research (NWO). E-mail<EMAIL_ADDRESS>Oded Regev Courant Institute of
Mathematical Sciences, New York University. Supported by the Simons
Collaboration on Algorithms and Geometry and by the National Science
Foundation (NSF) under Grant No. CCF-1320188. Any opinions, findings, and
conclusions or recommendations expressed in this material are those of the
authors and do not necessarily reflect the views of the NSF. Rishi Saket
IBM Research, Bangalore, India. Email<EMAIL_ADDRESS>
###### Abstract

We prove that for any $\varepsilon>0$ it is NP-hard to approximate the non-
commutative Grothendieck problem to within a factor $1/2+\varepsilon$, which
matches the approximation ratio of the algorithm of Naor, Regev, and Vidick
(STOC’13). Our proof uses an embedding of $\ell_{2}$ into the space of
matrices endowed with the trace norm with the property that the image of
standard basis vectors is longer than that of unit vectors with no large
coordinates.

##  1 Introduction

The subject of this paper, the non-commutative Grothendieck problem, has its
roots in celebrated work of Grothendieck [Gro53], sometimes (jokingly?)
referred to as “ _Grothendieck’s résumé_.” His paper laid the foundation for
the study of the geometry of tensor products of Banach spaces, though its
significance only became widely recognized after it was revamped by
Lindenstrauss and Pełczyński [LP68]. The main result of the paper, now known
as Grothendieck’s inequality, shows a close relationship between the following
two quantities. For a complex $d\\!\times\\!d$ matrix $M$ let

$\operatorname{\mathsf{OPT}}(M)=\sup_{\alpha_{i},\beta_{j}}\Big{|}\sum_{i,j=1}^{d}M_{ij}\alpha_{i}\overline{\beta_{j}}\Big{|},$
(1)

where the supremum goes over scalars on the complex unit circle, and let

$\operatorname{\mathsf{SDP}}(M)=\sup_{a_{i},b_{j}}\Big{|}\sum_{i,j=1}^{d}M_{ij}\langle
a_{i},b_{j}\rangle\Big{|},$ (2)

where the supremum goes over vectors on a complex Euclidean unit sphere of any
dimension. Since the circle is the sphere in dimension one, we clearly have
$\operatorname{\mathsf{SDP}}(M)\geq\operatorname{\mathsf{OPT}}(M)$.
Grothendieck’s inequality states that there exists a universal constant
$K_{G}^{\mathbb{C}}<\infty$ such that for each positive integer $d$ and any
$d\\!\times\\!d$ matrix $M$, we also have $\operatorname{\mathsf{SDP}}(M)\leq
K_{G}^{\mathbb{C}}\,\operatorname{\mathsf{OPT}}(M).$ This result found an
enormous number of applications both within and far beyond its original scope
and we give some examples below (see [KN12, Pis12] for extensive surveys).
Despite this, finding the optimal value of $K_{G}^{\mathbb{C}}$ is the only
one of six problems posed in [Gro53] that remains unsolved today; the current
best upper and lower bounds are $1.4049$ [Haa87] and $1.338$ [Dav84],
respectively. The situation is similar for the real variant of the problem,
where all objects involved are over the real numbers and the constant is
denoted $K_{G}$; see [BMMN13] for recent progress on the problem of
determining $K_{G}$.

The non-commutative Grothendieck problem, which we will refer to simply as the
_NCG_ , is the optimization problem in which we are asked to maximize a given
bilinear form over two unitary matrices. More explicitly, we are given a four-
dimensional array of complex numbers $(T_{ijkl})_{i,j,k,l=1}^{d}$ and are
asked to find or approximate the value

$\operatorname{\mathsf{OPT}}(T)=\sup_{A,B}\Big{|}\sum_{i,j,k,l=1}^{d}T_{ijkl}A_{ij}\overline{B_{kl}}\Big{|},$
(3)

where the supremum is over pairs of $d\\!\times\\!d$ unitary matrices. (The
word “non-commutative” simply refers to the fact that optimization is over
matrices.) It is not difficult to see that the (commutative) Grothendieck
problem of computing $\operatorname{\mathsf{OPT}}(M)$ as in (1) is the special
case where $T$ has $T_{iijj}=M_{ij}$ and zeros elsewhere. Seen at first, the
problem might seem overly abstract, but in fact, as we will illustrate below,
it captures many natural questions as special cases. Grothendieck conjectured
that his namesake inequality has an extension that relates (3) and the
quantity

$\operatorname{\mathsf{SDP}}(T)=\sup_{\vec{A},\vec{B}}\Big{|}\sum_{i,j,k,l=1}^{d}T_{ijkl}\big{\langle}\vec{A}_{ij},\vec{B}_{kl}\big{\rangle}\Big{|},$
(4)

where $\vec{A},\vec{B}$ range over all $d\\!\times\\!d$ matrices whose entries
are complex _vectors_ of arbitrary dimension satisfying a certain “unitarity”
constraint.111Namely, we require that $\vec{A}^{*}\vec{A}=\mathds{1}$ and
$\vec{A}\vec{A}^{*}=\mathds{1}$ and similarly for $\vec{B}$, where the
multiplication of two vector-entried matrices is a scalar-valued matrix
computed just like a normal matrix multiplication except the scalar
multiplication is replaced by an inner product, e.g., the $(i,j)$-coordinate
of $\vec{A}\vec{A}^{*}$ is given by
$\sum_{k}\langle\vec{A}_{ik},\vec{A}_{jk}\rangle$. Namely, he conjectured that
there exists a universal constant $K<\infty$ such that for every positive
integer $d$ and array $T$ as above, we have
$\operatorname{\mathsf{OPT}}(T)\leq\operatorname{\mathsf{SDP}}(T)\leq
K\operatorname{\mathsf{OPT}}(T)$, where the first inequality follows
immediately from the definition. Over twenty-five years after being posed, the
non-trivial content of Grothendieck’s conjecture,
$\operatorname{\mathsf{SDP}}(T)\leq K\operatorname{\mathsf{OPT}}(T)$, was
finally settled in the positive by Pisier [Pis78]. This result is now known as
the non-commutative Grothendieck inequality. In contrast with the commutative
case, and somewhat surprisingly, the optimal value of $K$ is known: Haagerup
[Haa85] lowered Pisier’s original estimate to $K\leq 2$ and this was later
shown to be sharp by Haagerup and Itoh [HI95].

#### Algorithmic applications.

The importance of Grothendieck’s inequality to computer science was pointed
out by Alon and Naor [AN06], who placed it in the context of approximation
algorithms for combinatorial optimization problems. They observed that
computing $\operatorname{\mathsf{SDP}}(M)$ is a semidefinite programming (SDP)
problem that can be solved efficiently (to within arbitrary precision), and
they translated an upper bound of about $1.78$ on $K_{G}$ due to Krivine
[Kri79] to an efficient rounding scheme that turns SDP vectors into a feasible
solution for the real Grothendieck problem (1) achieving value at least
$\operatorname{\mathsf{SDP}}(M)/1.78$. A generic argument following from
[BdOFV14] shows that whatever the value of $K_{G}$ is, there exists a brute-
force-based algorithm achieving value at least
$(1/K_{G}-\varepsilon)\operatorname{\mathsf{SDP}}(M)$ for any $\varepsilon>0$.
The Grothendieck problem shows up in a number of different areas such as graph
partition problems and computing the Cut-Norm of a matrix [AN06], in
statistical physical where it gives ground state energies in the spin model
[KNS10], and in quantum physics where it is related to Bell inequalities
[Tsi87].

In the same spirit, Naor, Regev, and Vidick [NRV14] recently translated the
_non-commutative_ Grothendieck inequality into an efficient SDP-based
approximation algorithm for the NCG problem (3) that achieves value at least
$\operatorname{\mathsf{SDP}}(T)/2$. They also considered the real variant and
a Hermitian variant, for which they gave analogous algorithms achieving value
at least $\operatorname{\mathsf{SDP}}(T)/2\sqrt{2}$. This in turn implies
efficient constant-factor approximation algorithms for a variety of problems,
including the Procrustes problem and two robust versions of Principle
Component Analysis [NRV14] and quantum XOR games [RV14]. Bandeira et al.
[BKS13] prove better approximation guarantees for a special case of the Little
NCG (considered below) and show that it captures the Procrustes problem and
another natural problem called the Global Registration Problem.

#### Hardness of approximation.

For simplicity we momentarily turn to the real setting, but similar results
hold over the complex numbers. Since the Grothendieck problem contains MaxCut
as a special case (by taking $M$ to be the positive semidefinite Laplacian
matrix of a graph), Håstad’s inapproximability result [Hås01] implies that it
is NP-hard to approximate the value (1) to any factor larger than
$16/17\approx.941$. Based on the current best-known lower bound of about
$1.676$ on $K_{G}$, Khot and O’Donnell [KO09] proved that (1) is Unique-Games-
hard to approximate to within a factor larger than $1/1.676\approx.597$.
Moreover, despite the fact that the exact value of $K_{G}$ is still unknown,
Raghavendra and Steurer [RS09] were able to improve this Unique Games hardness
to $1/K_{G}$. (See for instance [Kho10, Tre12] for background on the Unique
Games conjecture.)

#### Our result.

Whereas the hardness situation for the commutative version of Grothendieck’s
problem is reasonably well understood (apart from the yet-unknown exact value
of $K_{G}$), no tight hardness result was previously known for the non-
commutative version. In fact, we are not even aware of any hardness result
that is better than what follows from the commutative case. Here we settle
this question.

######  Theorem 1.1.

For any constant $\varepsilon>0$ it is NP-hard to approximate the optimum (3)
of the non-commutative Grothendieck problem to within a factor greater than
$1/2+\varepsilon$.

#### Little Grothendieck.

In fact, we prove a stronger result than Theorem 1.1 that concerns a special
case of the NCG called the _Little NCG_. Let us start by describing the (real
case of the) _commutative_ Little Grothendieck problem (a.k.a. the positive-
semidefinite Grothendieck problem). A convenient way to phrase it is as asking
for the operator norm of a linear map
$\mathcal{F}:\mathbb{R}^{n}\to\ell_{1}^{d}$, defined as
$\lVert\mathcal{F}\rVert=\sup_{a}\lVert{\mathcal{F}}(a)\rVert_{\ell_{1}}$
where the vector $a$ ranges over the $n$-dimensional Euclidean unit ball. It
turns out that that this is a special case of (the real version of) Eq. (1):
for any ${\mathcal{F}}$ there exists a positive semidefinite $d\\!\times\\!d$
matrix $M$ such that
$\operatorname{\mathsf{OPT}}(M)=\lVert\mathcal{F}\rVert^{2}$; and vice versa,
one can also map any such $M$ into a corresponding operator $\mathcal{F}$
(see, e.g., [Pis12] or Section 6). We wish to highlight that for such
instances, the constant $K_{G}$ may be replaced by the smaller value $\pi/2$
[Rie74] and that this value is known to be optimal [Gro53]. Moreover, Nesterov
made this algorithmic, namely, he showed an algorithm that approximates
$\lVert\mathcal{F}\rVert$ as above to within $\sqrt{2/\pi}$ [Nes98]. Finally,
Khot and Naor [KN09], as part of a more general result, showed that this is
tight: the Unique-Games-hardness threshold for the Little Grothendieck problem
is exactly $\sqrt{2/\pi}$.

The Little _non-commutative_ Grothendieck problem is formulated in terms of
the (normalized) trace norm, also known as the Schatten-1 norm, which for a
$d\\!\times\\!d$ matrix $A$ is given by $\lVert
A\rVert_{S_{1}}=d^{-1}\operatorname{{\mathsf{Tr}}}\sqrt{A^{*}A}$. In other
words, $\lVert A\rVert_{S_{1}}$ is the average of the singular values of $A$.
The space of matrices endowed with this norm is denoted by $S_{1}$ and by
$S_{1}^{d}$ if we restrict to $d\\!\times\\!d$ matrices. The problem then asks
for the operator norm of a linear map $\mathcal{F}:\mathbb{C}^{n}\to
S_{1}^{d}$. This problem is a special case of the NCG where
$\operatorname{\mathsf{OPT}}(T)=\lVert\mathcal{F}\rVert^{2}$ (see Section 6).
In particular, it follows from [Haa85, NRV14] that there is an efficient SDP-
based $1/\sqrt{2}$-approximation algorithm for the Little NCG. Our stronger
result alluded to above shows tight hardness for the Little NCG, which
directly implies Theorem 1.1.

######  Theorem 1.2.

For any constant $\varepsilon>0$ it is NP-hard to approximate the Little non-
commutative Grothendieck problem to within a factor greater than
$1/\sqrt{2}+\varepsilon$.

While this result applies to the complex case, an easy transformation shows
that it directly implies the same result for the real and Hermitian cases
introduced in [NRV14] (see Section 5.1). Finally, as we show in the “warm-up”
section of this paper (Section 4), we also get a tight NP-hardness result for
the _commutative_ Little Grothendieck problem, strengthening the unique-games-
based result of [KN09].

######  Theorem 1.3.

For any constant $\varepsilon>0$ it is NP-hard to approximate the real Little
commutative Grothendieck problem to within a factor greater than
$\sqrt{2/\pi}+\varepsilon$. Similarly, the complex case is NP-hard to
approximate to within a factor greater than $\sqrt{\pi/4}+\varepsilon$.

#### Techniques.

Nearly all recent work on hardness of approximation, including for commutative
Grothendieck problems [RS09, KN09], uses the machinery of Fourier analysis
over the hypercube, influences, or the majority is stablest theorem. Our
attempts to apply these techniques here failed. Instead, we use a more direct
approach similar to that taken in [GRSW12] and avoid the use of the hypercube
altogether. The role of dictator functions is played in our proof simply by
the standard basis vectors of $\mathbb{C}^{n}$. The dictatorship test, which
is our main technical contribution, comes in the form of a linear operator
${\mathcal{F}}:\mathbb{C}^{n}\to S_{1}^{d}$ with the following notable
property: it maps the $n$ standard basis vectors to matrices with trace norm
1, and it maps any unit vector with no large coordinate to a matrix with trace
norm close to $1/\sqrt{2}$. Roughly speaking, one can think of $\mathcal{F}$
as identifying an interesting subspace of $S_{1}^{d}$ in which the unit ball
looks somewhat like the intersection of the Euclidean ball with a (scaled)
$\ell_{\infty}$ ball.

A first attempt to construct an operator $\mathcal{F}$ as above might be to
map each standard basis vector to a random unitary matrix. This, however,
leads to a very poor map – while standard basis vectors are mapped to matrices
of trace norm 1, vectors with no large coordinates are mapped to matrices of
trace norm close to $8/(3\pi)\approx 0.848$ by Wigner’s semicircle law.
Another natural approach is to look at the construction by Haagerup and Itoh
[HI95] (see also [Pis12, Section 11] for a self-contained description) which
shows the factor-2 lower bound in the non-commutative Grothendieck inequality,
i.e., the tight integrality gap of the SDP (4). Their construction relies on
the so-called CAR algebra (after canonical anticommutation relations) and
provides an _isometric_ mapping from $\mathbb{C}^{n}$ to $S_{1}$, i.e., all
unit vectors are mapped to matrices of trace norm $1$. Directly modifying this
construction (akin to how, e.g., Khot et al. [KKMO07] obtained tight hardness
of MaxCut by restricting the tight integrality gap instances by Feige and
Schechtman [FS02] from the sphere to the hypercube) does not seem to work.
Instead, our construction of $\mathcal{F}$ relies on a different (yet related)
algebra known as the Clifford algebra. The Clifford algebra was used before in
a celebrated result by Tsirelson [Tsi87] (to show that Grothendieck’s
inequality can be interpreted as a statement about XOR games with
entanglement). His result crucially relies on the fact that the Clifford
algebra gives an _isometric_ mapping from $\mathbb{R}^{n}$ to $S_{1}$. Notice
that this is again an isometric embedding, but now only over the reals. Our
main observation here (Lemma 5.2) is that the same mapping, when extended to
$\mathbb{C}^{n}$, exhibits intriguing cancellations when the phases in the
input vectors are not aligned, and this leads to the construction of
$\mathcal{F}$ (Lemma 5.1). Even though the proof of this fact is simple, we
find it surprising; we are not aware of any previous application of those
complex Clifford algebras in computer science (or elsewhere for that matter).

#### Open questions.

For the real and Hermitian cases there is a gap of $\sqrt{2}$ between the
guarantee of the [NRV14] algorithms and our hardness result. It also would be
interesting to explore whether hardness of approximation results can be
derived to some of the applications of the NCG, including the Procrustes
problem and robust Principle Component Analysis. We believe that our embedding
would be useful there too.

#### Outline.

The rest of the paper is organized as follows. In Section 2, we set some
notational conventions, gather basic preliminary facts about relevant Banach
spaces, and give a detailed formulation of the Smooth Label Cover problem. In
Section 3, we prove hardness of approximation for the problem of computing the
norm of a general class of Banach-space-valued functions, closely following
[GRSW12]. In Section 4, as a “warm up,” we prove Theorem 1.3 using the generic
result of Section 3 and straightforward applications of real and complex
versions of the Berry-Esséen Theorem. Section 5 contains our main technical
contribution, which we use there to finish the proof of our main result
(Theorem 1.2).

#### Acknowledgements.

We thank Steve Heilman and Thomas Vidick for early discussions.

##  2 Preliminaries

#### Notation and relevant Banach spaces.

For a positive integer $n$ we denote $[n]=\\{1,\dots,n\\}$. For a graph $G$
and vertices $v,w\in V(G)$ we write $v\sim w$ to denote that $v$ and $w$ are
adjacent. We write $\mbox{\rm Pr}_{e\sim v}[\cdot]$ for the expectation with
respect to a uniformly distributed random edge with $v$ as an endpoint. For a
finite set $U$ we denote by $\mathbf{E}_{u\in U}[\cdot]$ the expectation with
respect to the uniform distribution over $U$. All Banach spaces are assumed to
be finite-dimensional. Recall that for Banach spaces $X,Y$ the operator norm
of a linear operator $\mathcal{F}:X\to Y$ is given by

$\lVert\mathcal{F}\rVert=\sup_{x\in X:\,\lVert x\rVert_{X}\leq
1}\lVert\mathcal{F}(x)\rVert_{Y}.$

For a real number $p\geq 1$, the $p$-norm of a vector $a\in\mathbb{C}^{n}$ is
given by

$\lVert a\rVert_{\ell_{p}}=\left(\sum_{i=1}^{n}|a_{i}|^{p}\right)^{1/p}.$

As usual we implicitly endow $\mathbb{C}^{n}$ with the Euclidean norm $\lVert
a\rVert_{\ell_{2}}$. For a finite set $U$ endowed with the uniform probability
measure we denote by $L_{p}(U)$ the space of functions $f:U\to\mathbb{C}$ with
the norm

$\lVert f\rVert_{L_{p}(U)}=\left(\mathbf{E}_{u\in
U}\big{[}|f(u)|^{p}\big{]}\right)^{1/p}.$

More generally, for a Banach space $X$ we denote by $L_{p}(U,X)$ the space of
functions $f:U\to X$ with the norm

$\lVert f\rVert_{L_{p}(U,X)}=\left(\mathbf{E}_{u\in U}\Big{[}\lVert
f(u)\rVert_{X}^{p}\Big{]}\right)^{1/p}.$

We will write $L_{p}(X)$ if $U$ is not explicitly given and $\lVert
f\rVert_{L_{p}}$ instead of $\lVert f\rVert_{L_{p}(U,X)}$ when there is no
danger of ambiguity. Note that $L_{2}(U,\mathbb{C}^{n})$ is a Hilbert space.

#### Smooth Label Cover.

An instance of Smooth Label Cover is given by a quadruple $(G,[n],[k],\Sigma)$
that consists of a regular connected (undirected) graph $G=(V,E)$, a label set
$[n]$ for some positive integer $n$, and a collection
$\Sigma=\big{(}(\pi_{ev},\pi_{ew}):\,e=(v,w)\in E\big{)}$ of pairs of maps
both from $[n]$ to $[k]$ associated with the endpoints of the edges in $E$.
Given an _assignment_ $A:V\to[n]$, we say that an edge $e=(v,w)\in E$ is
_satisfied_ if $\pi_{ev}\big{(}A(v)\big{)}=\pi_{ew}\big{(}A(w)\big{)}$.

The following hardness result for Smooth Label Cover, given in [GRSW12],222For
convenience, we make implicit some of the parameters in the statement of the
theorem. is a slight variant of the original construction due to Khot [Kho02].
The theorem also describes the various structural properties, including
smoothness, that are satisfied by the hard instances.

######  Theorem 2.1.

For any positive real numbers $\zeta,\gamma$ there exist positive integers
${n=n(\zeta,\gamma)}$, $k=k(\zeta,\gamma)$, and $t=t(\zeta)$, and a Smooth
Label Cover instance $(G,[n],[k],\Sigma)$ as above such that:

  * •

(Hardness): It is NP-hard to distinguish between the following two cases:

    * –

(YES Case): There is an assignment that satisfies all edges.

    * –

(NO Case): Every assignment satisfies less than a $\zeta$-fraction of the
edges.

  * •

(Structural properties):

    * –

(Smoothness): For every vertex $v\in V$ and distinct $i,j\in[n]$, we have

$\mbox{\rm Pr}_{e\sim v}\left[\pi_{ev}(i)=\pi_{ev}(j)\right]\leq\gamma.$ (5)

    * –

For every vertex $v\in V$, edge $e\in E$ incident on $v$, and $i\in[k]$, we
have $|\pi_{ev}^{-1}(i)|\leq t$; that is, at most $t$ elements in $[n]$ are
mapped to the same element in $[k]$.

    * –

(Weak Expansion): For any $\delta>0$ and vertex subset $V^{\prime}\subseteq V$
such that $|V^{\prime}|=\delta\cdot|V|$, the number of edges among the
vertices in $|V^{\prime}|$ is at least $(\delta^{2}/2)|E|$.

##  3 Hardness for general Banach-space valued operators

The following proposition shows hardness of approximation for the problem of
computing the norm of a linear map from $\mathbb{C}^{n}$ to any Banach space
that allows for a “dictatorship test,” namely, a linear function that maps the
standard basis vectors to long vectors, and maps “spread” unit vectors to
short vectors. As stated, the proposition assumes the underlying field to be
$\mathbb{C}$; we note that the proposition holds with exactly the same proof
also in the case of the real field $\mathbb{R}$.

######  Theorem 3.1.

Let $(X_{n})_{n\in\mathbb{N}}$ be a family of finite-dimensional Banach
spaces, and $\eta$ and $\tau$ be positive numbers such that $\eta>\tau$.
Suppose that for each positive integer $n$ there exists a linear operator
${f:\mathbb{C}^{n}\to X_{n}}$ with the following properties:

  * •

For any vector $a\in\mathbb{C}^{n}$, we have
$\|f(a)\|_{X_{n}}\leq\|a\|_{\ell_{2}}$.

  * •

For each standard basis vector $e_{i}$, we have
$\|f(e_{i})\|_{X_{n}}\geq\eta$.

  * •

For any $\varepsilon>0$, there is a $\delta=\delta(\varepsilon)>0$ such that
$\|f(a)\|_{X_{n}}>(\tau+\varepsilon)\|a\|_{\ell_{2}}$ implies
$\|a\|_{\ell_{4}}>\delta\|a\|_{\ell_{2}}$.

Then, for any $\varepsilon^{\prime}>0$ there exists a positive integer $n$
such that it is NP-hard to approximate the norm of an explicitly given linear
operator $\mathcal{F}:L_{2}\to L_{1}(X_{n})$ to within a factor greater than
$(\tau/\eta)+\varepsilon^{\prime}$.

###  3.1 The hardness reduction

To set up the reduction, we begin by defining a linear operator
$\mathcal{F}=\mathcal{F}_{\zeta,\gamma}$ for any choice of positive real
numbers $\zeta,\gamma$. Afterwards we show that there is a choice of these
parameters giving the desired result. For positive real numbers
$\zeta,\gamma$, let $n$, $k$, and $t$ be positive integers (depending on
$\zeta,\gamma)$ and $(G,[n],[k],\Sigma)$ a Smooth Label Cover instance as in
Theorem 2.1, where $G=(V,E)$ is a regular graph. Note that $\zeta$ controls
the “satisfiability” of the instance in the NO case, that $\gamma$ controls
the “smoothness,” and that $t$ depends on $\zeta$ only. Endow the vertex set
$V$ with the uniform probability measure. To define $\mathcal{F}$ we consider
a special linear subspace $\mathcal{H}$ of the Hilbert space
$L_{2}(V,\mathbb{C}^{n})$. It will be helpful to view a vector ${\bf a}\in
L_{2}(V,\mathbb{C}^{n})$ as an assignment ${\bf a}=(a_{v})_{v\in V}$ of
vectors $a_{v}\in\mathbb{C}^{n}$ to $V$. Let $\mathcal{H}\subseteq
L_{2}(V,\mathbb{C}^{n})$ be the subspace of vectors ${\bf a}=(a_{v})_{v\in V}$
that satisfy for every $e=(v,w)\in E$ and $j\in[k]$ the homogeneous linear
constraint

$\sum_{i\in\pi_{ev}^{-1}(j)}a_{v}(i)=\sum_{i\in\pi_{ew}^{-1}(j)}a_{w}(i),$ (6)

where $a_{v}(i)$ denotes the $i$th coordinate of the vector $a_{v}$. Notice
that if an assignment $A:V\to[n]$ satisfies the edge $e=(v,w)$, then the
standard basis vectors $a_{v}=e_{A(v)}$ and $a_{w}=e_{A(w)}$ satisfy (6);
indeed, if $\pi_{ev}\big{(}A(v)\big{)}=\pi_{ew}\big{(}A(w)\big{)}=j^{\prime}$
then both sides of (6) equal 1 if $j=j^{\prime}$ and equal zero otherwise.

Now let $\eta$, $\tau$ and $f$ be as in Theorem 3.1. We associate with the
Smooth Label Cover instance $(G,[n],[k],\Sigma)$ from above the linear
operator $\mathcal{F}:\mathcal{H}\to L_{1}(V,X_{n})$ given by,

$\left(\mathcal{F}(\mathbf{a})\right)(v)=f(a_{v}).$ (7)

The operator $\mathcal{F}$ thus maps a $\mathbb{C}^{n}$-valued assignment
${\bf a}=(a_{v})_{v\in V}$ satisfying (6) to an $X_{n}$-valued assignment
given by $f(a_{v})$ for each $v\in V$. Theorem 3.1 follows from the following
two lemmas, which we prove in Sections 3.2 and 3.3, respectively.

######  Lemma 3.2 (Completeness).

Suppose that there exists an assignment $A:V\to[n]$ that satisfies all the
edges in $E$. Then, $\lVert\mathcal{F}\rVert\geq\eta$.

######  Lemma 3.3 (Soundness).

For any $\varepsilon>0$ there exists a choice of $\zeta,\gamma>0$ such that if
${\lVert\mathcal{F}\rVert>\tau+4\varepsilon}$ then there exists an assignment
that satisfies at least a $\zeta$-fraction of the edges of $G$.

###### Proof of Theorem 3.1.

Let $\varepsilon>0$ be arbitrary, let $\zeta,\gamma$ be as in Lemma 3.3, and
let $n=n(\zeta,\gamma)$ and $k=k(\zeta,\gamma)$ be as in Theorem 2.1. We use
the reduction described above, which maps a Smooth Label Cover instance
$(G,[n],[k],\Sigma)$ to the linear operator $\mathcal{F}:L_{2}\to
L_{1}(X_{n})$ specified in (7). By Lemma 3.2, YES instances are mapped to
${\mathcal{F}}$ satisfying $\lVert\mathcal{F}\rVert\geq\eta$, whereas by Lemma
3.3, NO instances are mapped to $\mathcal{F}$ satisfying
$\lVert\mathcal{F}\rVert<\tau+4\varepsilon$. We therefore obtain hardness of
approximation to within a factor $(\tau+4\varepsilon)/\eta$. Since
$\varepsilon$ is arbitrary, we are done. ∎

###  3.2 Completeness

Here we prove Lemma 3.2.

###### Proof of Lemma 3.2.

Let $A:V\to[n]$ be an assignment that satisfies all the edges. Consider the
vector $\mathbf{a}\in L_{2}(V,\mathbb{C}^{n})$ where $a_{v}=e_{A(v)}$ and
notice that ${\|\mathbf{a}\|_{L_{2}}=1}$. Since $A$ satisfies all edges,
$\mathbf{a}$ satisfies the constraint (6) for every $e\in E$ and $j\in[n]$,
and thus $\mathbf{a}$ lies in the domain $\mathcal{H}$ of $\mathcal{F}$.
Moreover, by the second property of $f$ given in Theorem 3.1,

$\|\mathcal{F}(\mathbf{a})\|_{L_{1}(V,X_{n})}=\mathbf{E}_{v\in V}\big{[}\lVert
f(a_{v})\rVert_{X_{n}}\big{]}\geq\eta.$

Hence, $\|\mathcal{F}\|\geq\eta$. ∎

###  3.3 Soundness

Here we prove Lemma 3.3 and show that among the family of operators
$\mathcal{F}=\mathcal{F}_{\zeta,\gamma}$ as in (7), for any $\varepsilon>0$
there is a choice of $\zeta,\gamma>0$ such that if
$\lVert\mathcal{F}\rVert>\tau+4\varepsilon$, then there exists an assignment
satisfying a $\zeta$-fraction of the edges in the Smooth Label Cover instance
associated with $\mathcal{F}$. To begin, assume that
$\|\mathcal{F}\|>\tau+4\varepsilon$ for some ${\varepsilon>0}$. Let
$\mathbf{b}\in\mathcal{H}$ be a vector such that $\|\mathbf{b}\|_{L_{2}}=1$
and

$\mathbf{E}_{v\in V}\big{[}\lVert f(b_{v})\rVert\big{]}=\lVert\mathcal{F}({\bf
b})\rVert_{L_{1}}\geq\tau+4\varepsilon.$ (8)

The weak expansion property in Theorem 2.1 implies that it suffices to find a
“good” assignment for a large subset of the vertices, as any large set of
vertices will induce a large set of edges. For $\delta=\delta(\varepsilon)$ as
in Theorem 3.1, we will consider set of vertices

$V_{0}=\\{v\in V\,\mid\,\|b_{v}\|_{\ell_{4}}>\delta\varepsilon\>\>\textnormal{
and }\>\>\|b_{v}\|_{\ell_{2}}\leq 1/\varepsilon\\}\;.$ (9)

The following lemma shows that $V_{0}$ contains a significant fraction of
vertices.

######  Lemma 3.4.

For $V_{0}\subseteq V$ defined as in (9), we have
$|V_{0}|\geq\varepsilon^{2}|V|$.

###### Proof.

Define the sets

$\displaystyle V_{1}$ $\displaystyle=\\{v\in
V\,\mid\,\|b_{v}\|_{\ell_{4}}\leq\delta\varepsilon\>\>\textnormal{ and
}\>\>\|b_{v}\|_{\ell_{2}}<\varepsilon\\},$ $\displaystyle V_{2}$
$\displaystyle=\\{v\in
V\,\mid\,\|b_{v}\|_{\ell_{4}}\leq\delta\varepsilon\>\>\textnormal{ and
}\>\>\|b_{v}\|_{\ell_{2}}\geq\varepsilon\\},$ $\displaystyle V_{3}$
$\displaystyle=\\{v\in V\,\mid\,\|b_{v}\|_{\ell_{2}}>1/\varepsilon\\}\;.$

From (8), we have

$\sum_{v\in V_{0}}\|f(b_{v})\|_{X_{n}}+\sum_{v\in
V_{1}}\|f(b_{v})\|_{X_{n}}+\sum_{v\in V_{2}}\|f(b_{v})\|_{X_{n}}+\sum_{v\in
V_{3}}\|f(b_{v})\|_{X_{n}}\geq(\tau+4\varepsilon)|V|.$ (10)

We bound the four sums on the left-hand side of (10) individually. Since (by
the first item in Theorem 3.1) we have
$\|f(b_{v})\|_{X_{n}}\leq\|b_{v}\|_{\ell_{2}}$, and since
$\|b_{v}\|_{\ell_{2}}\leq 1/\varepsilon$ for every $v\in V_{0}$, the first sum
in (10) can be bounded by

$\sum_{v\in V_{0}}\|f(b_{v})\|_{X_{n}}\leq|V_{0}|/\varepsilon.$ (11)

Similarly, using the definition of $V_{1}$ the second sum in (10) is at most
$\varepsilon|V|$. Next, from the third property of $f$ in Theorem 3.1, for
each $v\in V_{2}$, we have
$\|f(b_{v})\|_{X_{n}}\leq(\tau+\varepsilon)\|b_{v}\|_{\ell_{2}}$. Therefore,
the third sum in (10) is bounded as

$\displaystyle\sum_{v\in V_{2}}\|f(b_{v})\|_{X_{n}}$
$\displaystyle\leq(\tau+\varepsilon)\sum_{v\in V_{2}}\|b_{v}\|_{\ell_{2}}$
$\displaystyle\leq(\tau+\varepsilon){|V_{2}|}^{\frac{1}{2}}\Big{(}\sum_{v\in
V_{2}}\|b_{v}\|_{\ell_{2}}^{2}\Big{)}^{\frac{1}{2}}\ \ \ \ \textnormal{(By
Cauchy-Schwartz)}$
$\displaystyle\leq(\tau+\varepsilon){|V|}^{\frac{1}{2}}\Big{(}\sum_{v\in
V}\|b_{v}\|_{\ell_{2}}^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle=(\tau+\varepsilon)|V|,$ (12)

where the last inequality uses $\|\mathbf{b}\|_{L_{2}}=1$. Finally, the fourth
sum in (10) is bounded by

$\displaystyle\sum_{v\in V_{3}}\|f(b_{v})\|_{X_{n}}$
$\displaystyle\leq\sum_{v\in V_{3}}\|b_{v}\|_{\ell_{2}}\ \ \ \ \textnormal{(By
the property of }f\textnormal{)}$ $\displaystyle<\sum_{v\in
V_{3}}\varepsilon\|b_{v}\|^{2}_{\ell_{2}}$
$\displaystyle\leq\varepsilon\sum_{v\in V}\|b_{v}\|_{\ell_{2}}^{2}$
$\displaystyle=\varepsilon|V|\|\mathbf{b}\|_{L_{2}}^{2}=\varepsilon|V|.$ (13)

Combining the above with Equation (10) yields,
$|V_{0}|/\varepsilon\geq\varepsilon|V|$, which proves Lemma 3.4. ∎

Lemma 3.4 and the weak expansion property implies that the set $E(V_{0})$ of
edges induced by $V_{0}$ has cardinality

$|E(V_{0})|\geq(\varepsilon^{4}/2)|E|.$ (14)

We set out to show that there exists an assignment to the vertices in $V_{0}$
that satisfies a significant fraction of edges in $E(V_{0})$. Roughly
speaking, we do this by randomly assigning each ${v\in V_{0}}$ one of the
coordinates of the vector $b_{v}$ at which it has large magnitude. (Assigning
the largest coordinate may not work.) The following simple proposition shows
that those vectors indeed have large coordinates.

######  Proposition 3.5.

Let $\beta=\beta(\varepsilon)$ be given by $\beta=\delta^{2}\varepsilon^{3}$.
Then, for each $v\in V_{0}$, we have $\lVert
b_{v}\rVert_{\ell_{\infty}}\geq\beta$.

###### Proof.

For every $v\in V_{0}$, we have

$\delta^{4}\varepsilon^{4}\leq\|b_{v}\|_{\ell_{4}}^{4}\leq\lVert
b_{v}\rVert_{\ell_{\infty}}^{2}\|b_{v}\|_{\ell_{2}}^{2}\leq\lVert
b_{v}\rVert_{\ell_{\infty}}^{2}/\varepsilon^{2},$ (15)

giving the claim. ∎

For the to-be-determined value of $\zeta$ let $t=t(\zeta)$ be as in Theorem
2.1 and for each $v\in V_{0}$ let

$A^{v}_{1}=\Big{\\{}i\in[n]\,\Big{|}\,|b_{v}(i)|\geq\frac{\beta}{4}\Big{\\}}\quad\quad\text{and}\quad\quad
A^{v}_{2}=\Big{\\{}i\in[n]\,\Big{|}\,|b_{v}(i)|\geq\frac{\beta}{4t}\Big{\\}}.$

By Proposition 3.5 these sets are nonempty and clearly $A^{v}_{1}\subseteq
A^{v}_{2}$. Moreover, since $\|b_{v}\|_{\ell_{2}}\leq 1/\varepsilon$, we have,

$|A^{v}_{1}|\leq\frac{16}{\varepsilon^{2}\beta^{2}}\quad\quad\textnormal{and}\quad\quad|A^{v}_{2}|\leq\frac{16t^{2}}{\varepsilon^{2}\beta^{2}}.$
(16)

Now consider a random assignment $A:V_{0}\to[n]$ that independently assigns
each vertex $v\in V_{0}$ a uniformly random label from $A^{v}_{1}$ and assigns
the remaining vertices in $V$ some fixed arbitrary label. The following lemma
shows that on average, this assignment satisfies a significant fraction of
edges.

######  Lemma 3.6.

There exists a $\gamma>0$ depending only on $\varepsilon$ and $\zeta$ such
that for some absolute constant $c>0$ the expected fraction of edges in $E$
satisfied by the random assignment $A$ given above is at least
$c\varepsilon^{8}\beta^{4}$.

Setting $\gamma$ appropriately as in the above lemma and
$\zeta=c\varepsilon^{8}\delta^{4}$ then gives Lemma 3.3; indeed, notice that
then $\zeta$, and therefore also $\gamma$, depend on $\varepsilon$ alone.

The remainder of this section is devoted to the proof of Lemma 3.6. Let
$E^{\prime}\subseteq E(V_{0})$ be the subset of edges $e=(v,w)$ whose
projections $\pi_{ev}$ and $\pi_{ew}$ are injective on the subsets $A^{v}_{2}$
and $A^{w}_{2}$ respectively. Formally,

$E^{\prime}=\big{\\{}e=(v,w)\in
E(V_{0})\,\big{|}\,|\pi_{ev}(A^{v}_{2})|=|A^{v}_{2}|,\textnormal{ and
}|\pi_{ew}(A^{w}_{2})|=|A^{w}_{2}|\big{\\}}.$ (17)

We set the parameter $\gamma$ according to the following proposition which
shows a lower bound on $|E^{\prime}|$ using the smoothness property. Recall
that $t$ is a function of $\zeta$ only.

######  Proposition 3.7.

There exists an absolute constant $c^{\prime}>0$ such that for any $\gamma\leq
c^{\prime}\varepsilon^{8}\beta^{4}/t^{4}$, the set $E^{\prime}$ has
cardinality $|E^{\prime}|\geq(\varepsilon^{4}/4)|E|$.

###### Proof.

Consider any vertex $v\in V_{0}$. By the smoothness property of Theorem 2.1
and a union bound over all distinct pairs $i,j\in A_{v}^{2}$, the fraction of
edges $e\in E$ incident on $v$ that _do not_ satisfy

$|\pi_{ev}(A^{v}_{2})|=|A^{v}_{2}|,$ (18)

is at most

$\frac{\gamma\,|A^{v}_{2}|^{2}}{2}\leq\frac{1}{2}\left(\frac{\varepsilon^{8}\beta^{4}}{2^{10}\cdot
t^{4}}\right)\left(\frac{16^{2}\cdot
t^{4}}{\varepsilon^{4}\beta^{4}}\right)=\frac{\varepsilon^{4}}{8},$

via an appropriate setting of $c^{\prime}$. Therefore, the number of edges in
$E$ that are incident on some $v\in V_{0}$ and do not satisfy (18) is at most

$\sum_{v\in
V_{0}}\frac{\varepsilon^{4}}{8}\deg(v)\leq\frac{\varepsilon^{4}}{8}\sum_{v\in
V}\deg(v)\\\ \leq\frac{\varepsilon^{4}}{4}|E|.$

Thus,

$|E^{\prime}|\geq|E(V_{0})|-(\varepsilon^{4}/4)|E|\geq(\varepsilon^{4}/4)|E|,$

by Equation (14). ∎

The following proposition shows that for an edge $e=(v,w)\in E^{\prime}$, the
label sets $A^{v}_{1}$ and $A^{w}_{1}$ intersect under projections given by
$e$.

######  Proposition 3.8.

For every edge $e=(v,w)\in E^{\prime}$, we have
$\pi_{ev}(A^{v}_{1})\cap\pi_{ew}(A^{w}_{1})\neq\emptyset$.

###### Proof.

From Proposition 3.5, let $i^{*}\in[n]$ be such that
$|b_{v}(i^{*})|\geq\beta$. Note that $i^{*}\in A_{1}^{v}$. Let
${j^{*}=\pi_{ev}(i^{*})}$. Clearly it suffices to show that there exists an
$i^{\prime}\in A_{1}^{w}$ such that ${\pi_{ew}(i^{\prime})=j^{*}}$, as this
implies that $j^{*}\in\pi_{ev}(A^{v}_{1})\cap\pi_{ew}(A^{w}_{1})$.

Recall that since $\mathbf{b}\in\mathcal{H}$, the vector $\mathbf{b}$
satisfies the constraint (6), in particular,

$\bigg{|}\sum_{i\in\pi_{ev}^{-1}(j^{*})}b_{v}(i)\bigg{|}=\bigg{|}\sum_{i\in\pi_{ew}^{-1}(j^{*})}b_{w}(i)\bigg{|}\;.$
(19)

We show that because $i^{*}\in A_{1}^{v}$, the left-hand side must be large.
Therefore the right hand side is also large, from which we conclude that there
must exist a coordinate $i^{\prime}\in\pi_{ew}^{-1}(j^{*})$ such that
$|b_{w}(i^{\prime})|$ is large, and so $i^{\prime}\in A_{1}^{w}$.

Recall from the second structural property in Theorem 2.1 that
$|\pi_{ev}^{-1}(j^{*})|\leq t$. Moreover, since $\pi_{ev}$ acts injectively on
the set $A_{2}^{v}$ and since $i^{*}\in A_{2}^{v}$, no index $i\neq i^{*}$
such that $\pi_{ev}(i)=\pi_{ev}(i^{*})$ can belong to $A_{2}^{v}$. Hence, by
the triangle inequality, the left-hand side of (19) is at least

$\displaystyle|b_{v}(i^{*})|-\sum_{\begin{subarray}{c}i\in\pi_{ev}^{-1}(j^{*})\\\
i\neq
i^{*}\end{subarray}}|b_{v}(i)|\geq\beta-t\cdot\left(\frac{\beta}{4t}\right)=\frac{3\beta}{4}.$
(20)

Combining (19), (20), and the triangle inequality lets us bound the right-hand
side of (19) by

$\displaystyle\frac{3\beta}{4}$
$\displaystyle\leq\left|\sum_{i\in\pi_{ew}^{-1}(j^{*})}b_{w}(i)\right|$
$\displaystyle\leq\sum_{i\in\pi_{ew}^{-1}(j^{*})\cap
A_{2}^{w}}|b_{w}(i)|+\sum_{i\in\pi_{ew}^{-1}(j^{*})\smallsetminus
A_{2}^{w}}|b_{w}(i)|$ $\displaystyle\leq\sum_{i\in\pi_{ew}^{-1}(j^{*})\cap
A_{2}^{w}}|b_{w}(i)|+t\,\frac{\beta}{4t},$ (21)

where the last inequality uses the same facts as above. Since $\pi_{ew}$ acts
injectively on $A_{2}^{w}$, there is at most one index
$i\in\pi_{ew}^{-1}(j^{*})$ that also belongs to $A_{2}^{w}$, meaning that the
sum in (21) consists of at most one term. We see that sum must is at least
$\beta/2$ and in particular, there is an $i^{\prime}\in\pi_{ew}^{-1}(j^{*})$
such that $|b_{w}(i^{\prime})|\geq\beta/2$. We conclude that $i^{\prime}\in
A_{1}^{w}$ and $\pi_{ew}(i^{\prime})=j^{*}=\pi_{ev}(i^{*})$, proving the
claim. ∎

###### Proof of Lemma 3.6.

By Proposition 3.8 and Equation (16) any edge $e=(v,w)\in E^{\prime}$ is
satisfied by the assignment $A$ with probability at least
$1/(|A^{v}_{1}||A^{w}_{1}|)\geq(\varepsilon^{4}\beta^{4})/256$. Since by
Proposition 3.7, we have $|E^{\prime}|\geq(\varepsilon^{4}/4)|E|$, the
expected fraction of satisfied edges is at least
$\varepsilon^{8}\beta^{4}/1024$. ∎

##  4 The commutative case

Recall that the _commutative_ Little Grothendieck problem asks for the norm of
a linear operator $\mathcal{F}:L_{2}\to L_{1}$. In this section we use Theorem
3.1 to prove Theorem 1.3, the tight hardness result for this problem. We first
consider the real case of Theorem 1.3, and then the complex case in Section
4.2.

###  4.1 The real case

The real case of Theorem 1.3 follows easily by combining Theorem 3.1 with the
following simple lemma.

######  Lemma 4.1.

For every positive integer $n$ there exists a map $f:\mathbb{R}^{n}\to L_{1}$
with the following properties:

  * •

For any vector $a\in\mathbb{R}^{n}$, we have $\lVert
f(a)\rVert_{L_{1}}\leq\lVert a\rVert_{\ell_{2}}$.

  * •

For each standard basis vector $e_{i}$, we have $\lVert
f(e_{i})\rVert_{L_{1}}=1$. If $\lVert
f(a)\rVert_{L_{1}}>(\sqrt{2/\pi}+\varepsilon)\lVert a\rVert_{\ell_{2}}$ then
$\lVert a\rVert_{\ell_{4}}>(\varepsilon/K)\lVert a\rVert_{\ell_{2}}$, where
$K<\infty$ is a universal constant.

This shows that there is an $L_{1}$-valued function $f$ that satisfies the
conditions of the real variant of Theorem 3.1 for $\tau=\sqrt{2/\pi}$,
$\eta=1$ and $\delta(\varepsilon)=(\varepsilon/K)$. Hence, it is NP-hard to
approximate the norm of a linear operator ${\mathcal{F}}:L_{2}\to
L_{1}(L_{1})$ over $\mathbb{R}$ to a factor $\sqrt{2/\pi}+\varepsilon$ for any
$\varepsilon>0$. The real case of Theorem 1.3 then follows from the fact that
$L_{1}(L_{1})$ is isometrically isomorphic to $L_{1}$.

The proof of Lemma 4.1 uses the following version of the Berry–Esséen Theorem
(see for example [O’D14, Chapter 5.2, Theorem 5.16]).

######  Theorem 4.2 (Berry–Esséen Theorem).

There exists a universal constant $K<\infty$ such that the following holds.
Let $n$ be a positive integer and let $Z_{1},\dots,Z_{n}$ be independent
centered $\\{-1,1\\}$-valued random variables. Then, for any vector
$a\in\mathbb{R}^{n}$ such that $\lVert
a\rVert_{\ell_{\infty}}\leq\varepsilon\lVert a\rVert_{\ell_{2}}$, we have

$\mathbf{E}\Big{[}\Big{|}\sum_{i=1}^{n}a_{i}Z_{i}\Big{|}\Big{]}\leq\Big{(}\sqrt{\frac{2}{\pi}}+K\varepsilon\Big{)}\lVert
a\rVert_{\ell_{2}}.$

###### Proof of Lemma 4.1.

Endow $\\{-1,1\\}^{n}$ with the uniform probability measure and define the
function $f:\mathbb{R}^{n}\to L_{1}(\\{-1,1\\}^{n})$ by

$\big{(}f(a)\big{)}(Z_{1},\dots,Z_{n})=\sum_{i=1}^{n}a_{i}Z_{i}.$

The first property follows since

$\lVert f(a)\rVert_{L_{1}}\leq\lVert
f(a)\rVert_{L_{2}}=\left(\mathbf{E}\Big{[}\Big{|}\sum_{i=1}^{n}a_{i}Z_{i}\Big{|}^{2}\Big{]}\right)^{1/2}=\lVert
a\rVert_{\ell_{2}}.$

The second property is trivial. The third property follows from Theorem 4.2.
Indeed, the theorem implies that if for some $\varepsilon>0$, we have

$\lVert
f(a)\rVert_{L_{1}}=\mathbf{E}\Big{[}\Big{|}\sum_{i=1}^{n}a_{i}Z_{i}\Big{|}\Big{]}>\Big{(}\sqrt{\frac{2}{\pi}}+\varepsilon\Big{)}\lVert
a\rVert_{\ell_{2}},$

then $\lVert a\rVert_{\ell_{\infty}}>(\varepsilon/K)\lVert
a\rVert_{\ell_{2}}$. Since $\lVert a\rVert_{\ell_{4}}\geq\lVert
a\rVert_{\ell_{\infty}}$ the last property follows. ∎

###  4.2 The complex case

A similar argument to the one above shows the complex case of Theorem 1.3.
This follows from the following complex analogue of Lemma 4.1.

######  Lemma 4.3.

For every positive integer $n$ there exists a map $f:\mathbb{C}^{n}\to L_{1}$
with the following properties

  * •

For any vector $a\in\mathbb{C}^{n}$, we have $\lVert
f(a)\rVert_{L_{1}}\leq\lVert a\rVert_{\ell_{2}}$.

  * •

For each standard basis vector $e_{i}$, we have $\lVert
f(e_{i})\rVert_{L_{1}}=1$. If $\lVert
f(a)\rVert_{L_{1}}>(\sqrt{\pi/4}+\varepsilon)\lVert a\rVert_{\ell_{2}}$ then
$\lVert a\rVert_{\ell_{4}}>(\varepsilon^{2}/K)\lVert a\rVert_{\ell_{2}}$,
where $K<\infty$ is a universal constant.

This shows that there is an $L_{1}$-valued function $f$ that satisfies the
conditions of Theorem 3.1 for $\tau=\sqrt{\pi/4}$, $\eta=1$ and
$\delta(\varepsilon)=(\varepsilon^{2}/K)$. Hence, it is NP-hard to approximate
the norm of a linear operator ${\mathcal{F}}:L_{2}\to L_{1}(L_{1})$ over
$\mathbb{C}$ to a factor $\sqrt{\pi/4}+\varepsilon$ for any $\varepsilon>0$.

The proof of Lemma 4.3 is based on the following complex analogue the
Berry–Esséen Theorem. Since we could not find this precise formulation in the
literature we include a proof below for completeness.

######  Lemma 4.4 (Complex Berry–Esséen Theorem).

There exists a universal constant $K<\infty$ such that the following holds.
Let $Z_{1},\dots,Z_{n}$ be independent uniformly distributed random variables
over $\\{1,i,-1,-i\\}$. Then, for any vector $a\in\mathbb{C}^{n}$ such that
$\lVert a\rVert_{\ell_{\infty}}\leq\varepsilon\lVert a\rVert_{\ell_{2}}$, we
have

$\mathbf{E}\Big{[}\Big{|}\sum_{j=1}^{n}Z_{j}a_{j}\Big{|}\Big{]}\leq\Big{(}\sqrt{\frac{\pi}{4}}+K\sqrt{\varepsilon}\Big{)}\lVert
a\rVert_{\ell_{2}}.$

The proof is based on the following multi-dimensional version of the
Berry–Esséen theorem due to Bentkus [Ben05, Theorem 1.1].

######  Theorem 4.5 (Bentkus).

Let $X_{1},\dots,X_{n}$ be independent $\mathbb{R}^{d}$-valued random
variables such that $\mathbf{E}[X_{j}]=0$ for each $j\in[n]$. Let
$S=X_{1}+\cdots+X_{n}$ and assume that the covariance matrix of $S$ equals
$\mathds{1}_{d}$. Let $g\sim\mathcal{N}(0,\mathds{1}_{d})$ be standard
Gaussian vector in $\mathbb{R}^{d}$ with the same covariance matrix as $S$.
Then, for any measurable convex set $A\subseteq\mathbb{R}^{d}$, we have

$\big{|}\mbox{\rm Pr}[S\in A]-\mbox{\rm Pr}[g\in A]\big{|}\leq
c(d)\sum_{j=1}^{n}\mathbf{E}\big{[}\lVert X_{j}\rVert_{\ell_{2}}^{3}\big{]},$

where $c(d)=O(d^{1/4})$.

We also use the following standard tail bound.

######  Lemma 4.6 (Hoeffding’s inequality [Hoe62]).

Let $X_{1},\dots,X_{n}$ be independent real-valued random variables such that
for each $i\in[n]$, $X_{i}\in[a_{i},b_{i}]$ for some $a_{i}<b_{i}$. Let
$S=X_{1}+\cdots+X_{n}$. Then, for any $t>0$,

$\mbox{\rm Pr}\big{[}|S-\mathbf{E}[S]|>t\big{]}\leq
2e^{-2t^{2}/\sum_{i=1}^{n}(b_{i}-a_{i})^{2}}.$

###### Proof of Lemma 4.4.

Let $a\in\mathbb{C}^{n}$ be some vector. By homogeneity we may assume that
$\lVert a\rVert_{\ell_{2}}=1$. Set $\varepsilon=\lVert
a\rVert_{\ell_{\infty}}$. For each $j\in[n]$ define the random vector
$X_{j}\in\mathbb{R}^{2}$ by
$X_{j}=\sqrt{2}[\Re(Z_{j}a_{j}),\Im(Z_{j}a_{j})]^{\mathsf{T}}$. Let
$S=X_{1}+\cdots+X_{n}$, and let $T\geq 1$ be some number to be set later. We
have

$\mathbf{E}[\lVert S\rVert_{\ell_{2}}]=\int_{0}^{\infty}\mbox{\rm Pr}[\lVert
S\rVert_{\ell_{2}}>t]dt=\int_{0}^{T}\mbox{\rm Pr}[\lVert
S\rVert_{\ell_{2}}>t]dt+\int_{T}^{\infty}\mbox{\rm Pr}[\lVert
S\rVert_{\ell_{2}}>t]dt\;.$ (22)

We now bound each integral separately. Notice that $\mathbf{E}[X_{j}]=0$, and
$\mathbf{E}[X_{j}X_{j}^{\mathsf{T}}]=|a_{j}|^{2}\mathds{1}_{2}$. It follows
that the covariance matrix of $S$ equals $\mathds{1}_{2}$. If we let
$g\sim\mathcal{N}(0,\mathds{1}_{2})$ be a standard Gaussian vector in
$\mathbb{R}^{2}$, then it follows from Theorem 4.5 (for $d=2$) that for any
$t>0$, we have

$\displaystyle\big{|}\mbox{\rm Pr}\big{[}\lVert S\rVert_{\ell_{2}}\leq
t\big{]}-\mbox{\rm Pr}\big{[}\lVert g\rVert_{\ell_{2}}\leq t\big{]}\big{|}$
$\displaystyle\leq c\sum_{j=1}^{n}\mathbf{E}\big{[}\lVert
X_{j}\rVert_{\ell_{2}}^{3}\big{]}$ $\displaystyle\leq
c\max_{j\in[n]}\mathbf{E}[\lVert
X_{j}\rVert_{\ell_{2}}]\sum_{k=1}^{n}\mathbf{E}\big{[}\lVert
X_{k}\rVert_{\ell_{2}}^{2}\big{]}$ $\displaystyle\leq 2\sqrt{2}c\varepsilon.$
(23)

By (23), the first integral in (22) is at most

$2\sqrt{2}c\varepsilon T+\mathbf{E}[\lVert
g\rVert_{\ell_{2}}]=2\sqrt{2}c\varepsilon T+\sqrt{\pi/2}\;,$ (24)

where we used that $\lVert g\rVert_{\ell_{2}}$ is distributed according to a
$\chi_{2}$ distribution.

We now bound the second integral in (22). The first coordinate $S_{1}$ is a
sum of independent random variables, $\sqrt{2}\Re(Z_{j}a_{j})$, which are
centered and have magnitude at most $\sqrt{2}|a_{j}|$. Similarly, the same
holds for $S_{2}$. Lemma 4.6 therefore gives,

$\displaystyle\int_{T}^{\infty}\mbox{\rm Pr}[\lVert S\rVert_{\ell_{2}}>t]dt$
$\displaystyle\leq\int_{T}^{\infty}\mbox{\rm
Pr}[|S_{1}|>t/\sqrt{2}]dt+\int_{T}^{\infty}\mbox{\rm
Pr}[|S_{2}|>t/\sqrt{2}]dt$ $\displaystyle\leq
4\int_{T}^{\infty}e^{-t^{2}/4}dt$ $\displaystyle\leq
4\int_{T}^{\infty}te^{-t^{2}/4}dt$ $\displaystyle=8e^{-T^{2}/4},$ (25)

where in the last inequality we used the assumption $T\geq 1$.

Now set $T=\sqrt{4/\varepsilon}$. Combining (22), (24) and (25), we get

$\mathbf{E}\Big{[}\Big{|}\sum_{j=1}^{n}Z_{j}a_{j}\Big{|}\Big{]}=\frac{1}{\sqrt{2}}\mathbf{E}\big{[}\lVert
S\rVert_{\ell_{2}}\big{]}\leq\frac{1}{\sqrt{2}}\left(\sqrt{\frac{\pi}{2}}+2\sqrt{2}c\varepsilon
T+8e^{-T^{2}/4}\right)\leq\sqrt{\frac{\pi}{4}}+K\sqrt{\varepsilon}.\qed$

The proof of Lemma 4.3 is nearly identical to that of Lemma 4.1, now based on
Lemma 4.4 and the function $f:\mathbb{C}^{n}\to L_{1}(\\{1,i,-1,-i\\}^{n})$
given by $\big{(}f(a)\big{)}(Z_{1},\dots,Z_{n})=a_{1}Z_{1}+\cdots+a_{n}Z_{n},$
where $\\{1,i,-1,-i\\}^{n}$ is endowed with the uniform probability measure.

##  5 The non-commutative case

In this section we complete the proof of our main theorem (Theorem 1.2). The
following lemma gives the linear matrix-valued map $f$ mentioned in the
introduction that will allow us to conclude.

######  Lemma 5.1.

Let $n$ be a positive integer and let $d=2^{2n+\lceil n/2\rceil}$. Then, there
exists a linear operator $f~{}:~{}\mathbb{C}^{n}~{}\to~{}\mathbb{C}^{d\times
d}$ such that for any vector $a\in\mathbb{C}^{n}$, we have

$\lVert f(a)\rVert_{S_{1}}\leq\sqrt{\frac{\lVert a\rVert_{\ell_{2}}^{2}+\lVert
a\rVert_{\ell_{4}}^{2}}{2}}\;.$

In particular, $\lVert f(a)\rVert_{S_{1}}\leq(\lVert a\rVert_{\ell_{2}}+\lVert
a\rVert_{\ell_{4}})/\sqrt{2}$. Moreover, for each basis vector $e_{i}$ we have
$\lVert f(e_{i})\rVert_{S_{1}}=1$.

Theorem 1.2 now follows easily by combining the above lemma with Theorem 3.1.
Indeed, Lemma 5.1 shows that the conditions of Theorem 3.1 hold for
$\tau=2^{-1/2}$, $\eta=1$ and $\delta(\varepsilon)=\sqrt{2}\varepsilon$. It is
therefore NP-hard to approximate the norm of a linear operator
$\mathcal{F}:L_{2}\to L_{1}(S_{1})$ to within a factor
$1/\sqrt{2}+\varepsilon$ for any $\varepsilon>0$. This implies the theorem
because $L_{1}(S_{1})$ is isometrically isomorphic to $S_{1}$. To see the last
fact, we use the map that takes a matrix-valued function $g$ on a finite
measure space $U$ to a block diagonal matrix with blocks proportional to
$g(u)$ for $u\in U$ and use the fact that the trace norm of a block diagonal
matrix is the average trace norm of the blocks.

The rest of this section is devoted to the proof of Lemma 5.1. For a complex
vector $a\in\mathbb{C}^{n}$ let $\Re(a),\Im(a)\in\mathbb{R}^{n}$ denote its
real and imaginary parts, respectively, and define

$\Lambda(a)=\sqrt{\lVert\Re(a)\rVert_{\ell_{2}}^{2}\lVert\Im(a)\rVert_{\ell_{2}}^{2}-\big{\langle}\Re(a),\Im(a)\big{\rangle}^{2}}.$
(26)

Note that this value is the area of the parallelogram in $\mathbb{R}^{n}$
generated by the vectors $\Re(a)$ and $\Im(a)$.

######  Lemma 5.2.

Let $n$ be a positive integer and let $d^{\prime}=2^{\lceil n/2\rceil}$. Then,
there exists a operator $C:\mathbb{C}^{n}\to\mathbb{C}^{d^{\prime}\times
d^{\prime}}$ such that for any vector $a\in\mathbb{C}^{n}$, we have

$\lVert C(a)\rVert_{S_{1}}=\frac{1}{2}\sqrt{\lVert
a\rVert_{\ell_{2}}^{2}+2\Lambda(a)}+\frac{1}{2}\sqrt{\lVert
a\rVert_{\ell_{2}}^{2}-2\Lambda(a)}.$ (27)

Though we will not use it here, let us point out that the map $C$ becomes an
isometric embedding if we restrict it to $\mathbb{R}^{n}$, since
$\Lambda(a)=0$ for real vectors.

###### Proof.

We begin by defining a set of pairwise anti-commuting matrices as follows. The
Pauli matrices are the four Hermitian matrices

$I=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix},\quad\quad X=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\quad\quad Y=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},\quad\quad Z=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$

Using these we define $2\lceil n/2\rceil$ matrices in $\mathbb{C}^{d\times d}$
by

$\displaystyle C_{2j-1}$ $\displaystyle=\underbrace{Z\otimes\cdots\otimes
Z}_{\text{$j-1$ times}}\otimes X\otimes\underbrace{I\otimes\cdots\otimes
I}_{\text{$\lceil n/2\rceil-j$ times}}$ $\displaystyle C_{2j}$
$\displaystyle=\underbrace{Z\otimes\cdots\otimes Z}_{\text{$j-1$
times}}\otimes Y\otimes\underbrace{I\otimes\cdots\otimes I}_{\text{$\lceil
n/2\rceil-j$ times}},$

for each $j\in[\lceil n/2\rceil]$. It is easy to verify that these matrices
have trace zero, that they are Hermitian, unitary, and that they pairwise
anti-commute. In particular, they satisfy $C_{j}^{2}=I$. For a vector
$a\in\mathbb{C}^{n}$ we define the map $C$ by
$C(a)=a_{1}C_{1}+\cdots+a_{n}C_{n}$. Note that for a real vector
$x\in\mathbb{R}^{n}$, the matrix $C(x)$ is Hermitian and that it satisfies
$C(x)^{2}=\|x\|^{2}_{\ell_{2}}I$. If a real vector $z\in\mathbb{R}^{n}$ is
orthogonal to $x$ then by expanding the definitions of the matrices $C(x)$ and
$C(z)$ and using the above properties we find that they anti-commute:

$\displaystyle C(x)C(z)$ $\displaystyle=\langle x,z\rangle I+\sum_{j\neq
k}x_{j}z_{k}C_{j}C_{k}$ $\displaystyle=0-\sum_{j\neq k}x_{j}z_{k}C_{k}C_{j}$
$\displaystyle=-C(z)C(x)\;.$

This shows that the matrix $C(x)C(z)$ is skew-Hermitian, which implies that it
has purely imaginary eigenvalues. Since this matrix has trace zero and
satisfies
$C(x)C(z)\big{(}C(x)C(z)\big{)}^{*}=\|x\|^{2}_{\ell_{2}}\|z\|^{2}_{\ell_{2}}I$,
half the eigenvalues equal $i\|x\|_{\ell_{2}}\|z\|_{\ell_{2}}$ and the other
half equal $-i\|x\|_{\ell_{2}}\|z\|_{\ell_{2}}$.

We show that $C$ satisfies (27). Let $x=\Re(a)$ and $y=\Im(a)$ so that
$C(a)=C(x)+iC(y)$. Write $y=y^{\parallel}+y^{\perp}$ where $y^{\parallel}$ is
parallel to $x$ and $y^{\perp}$ is orthogonal to $x$. Then,

$\displaystyle C(a)C(a)^{*}$ $\displaystyle=$
$\displaystyle\big{(}C(x)+iC(y)\big{)}\big{(}C(x)-iC(y)\big{)}$
$\displaystyle=$ $\displaystyle\lVert
a\rVert_{\ell_{2}}^{2}I-i\big{(}C(x)C(y)-C(y)C(x)\big{)}$ $\displaystyle=$
$\displaystyle\lVert a\rVert_{\ell_{2}}^{2}I-2iC(x)C(y^{\perp}),$

where in the last line we used the fact that $C(y^{\parallel})$ commutes with
$C(x)$ while $C(y^{\perp})$ anti-commutes with $C(x)$. Using what we deduced
above for the matrix $C(x)C(y^{\perp})$ we see that half of the eigenvalues of
$C(a)C(a)^{*}$ equal $\lVert a\rVert_{\ell_{2}}^{2}+2\lVert
x\rVert_{\ell_{2}}\lVert y^{\perp}\rVert_{\ell_{2}}$ and the other half equal
$\lVert a\rVert_{\ell_{2}}^{2}-2\lVert x\rVert_{\ell_{2}}\lVert
y^{\perp}\rVert_{\ell_{2}}$. Hence,

$\lVert C(a)\rVert_{S_{1}}=\frac{1}{2}\sqrt{\lVert
a\rVert_{\ell_{2}}^{2}+2\lVert x\rVert_{\ell_{2}}\lVert
y^{\perp}\rVert_{\ell_{2}}}+\frac{1}{2}\sqrt{\lVert
a\rVert_{\ell_{2}}^{2}-2\lVert x\rVert_{\ell_{2}}\lVert
y^{\perp}\rVert_{\ell_{2}}}$

The claim now follows because $\lVert x\rVert_{\ell_{2}}\lVert
y^{\perp}\rVert_{\ell_{2}}$ is precisely the area of the parallelogram
generated by the vectors $x$ and $y$. ∎

We denote the entry-wise product of two vectors $a,b\in\mathbb{C}^{n}$ by
$a\circ b=(a_{1}b_{1},\dots,a_{n}b_{n})$.

######  Proposition 5.3.

Let $\omega$ be a vector chosen uniformly from $\\{1,i,-1,-i\\}^{n}$. Then,
for any $a\in\mathbb{C}^{n}$, we have

$4\mathbf{E}_{\omega}\big{[}\Lambda(a\circ\omega)^{2}\big{]}=\lVert
a\rVert_{\ell_{2}}^{4}-\lVert a\rVert_{\ell_{4}}^{4}.$

###### Proof.

Fix a vector $a\in\mathbb{C}^{n}$. Define the random vectors
$x_{\omega}=\Re(a\circ\omega)$ and $y_{\omega}=\Im(a\circ\omega)$. Then
$\Lambda(a\circ\omega)^{2}=\lVert x_{\omega}\rVert_{\ell_{2}}^{2}\lVert
y_{\omega}\rVert_{\ell_{2}}^{2}-\langle x_{\omega},y_{\omega}\rangle^{2}$. For
each $j\in[n]$ we factor $a_{j}=\alpha_{j}e^{i\phi_{j}}$ and
$\omega_{j}=e^{i\psi_{j}}$, where $\alpha_{j}\in\mathbb{R}_{+}$ and
$\phi_{j},\psi_{j}\in[0,2\pi]$. Note that $\psi_{1},\dots,\psi_{n}$ are
independent uniformly distributed random phases in $\\{0,\pi/2,\pi,3\pi/2\\}$.
Then,

$\displaystyle\lVert x_{\omega}\rVert_{\ell_{2}}^{2}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\alpha_{j}^{2}\cos^{2}(\phi_{j}+\psi_{j})$
$\displaystyle\lVert y_{\omega}\rVert_{\ell_{2}}^{2}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\alpha_{j}^{2}\sin^{2}(\phi_{j}+\psi_{j})$
$\displaystyle\langle x_{\omega},y_{\omega}\rangle$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\alpha_{j}^{2}\cos(\phi_{j}+\psi_{j})\sin(\phi_{j}+\psi_{j}).$

With this it is easy to verify that

$\displaystyle\lVert x_{\omega}\rVert_{\ell_{2}}^{2}\lVert
y_{\omega}\rVert_{\ell_{2}}^{2}-\langle x_{\omega},y_{\omega}\rangle^{2}=$
$\displaystyle\sum_{j\neq
k}\alpha_{j}^{2}\alpha_{k}^{2}\cos^{2}(\phi_{j}+\psi_{j})\sin^{2}(\phi_{k}+\psi_{k})-$
$\displaystyle\sum_{j\neq
k}\alpha_{j}^{2}\alpha_{k}^{2}\cos(\phi_{j}+\psi_{j})\sin(\phi_{j}+\psi_{j})\cos(\phi_{k}+\psi_{k})\sin(\phi_{k}+\psi_{k}).$
(28)

By independence of $\psi_{j}$ and $\psi_{k}$ when $j\neq k$ and the elementary
identities $\mathbf{E}[\cos^{2}(\phi_{j}+\psi_{k})]=1/2$,
$\mathbf{E}[\sin^{2}(\phi_{j}+\psi_{k})]=1/2$ and
$\mathbf{E}[\cos(\phi_{j}+\psi_{j})\sin(\phi_{j}+\psi_{j})]=0$, the
expectation of (28) equals

$\mathbf{E}_{\omega}\big{[}\Lambda(a\circ\omega)^{2}\big{]}=\frac{1}{4}\sum_{j\neq
k}a_{j}^{2}a_{k}^{2}=\frac{1}{4}\big{(}\lVert a\rVert_{\ell_{2}}^{4}-\lVert
a\rVert_{\ell_{4}}^{4}\big{)}.\qed$

We remark that in the above proof, it suffices if
$\omega\in\\{1,i,-1,-i\\}^{n}$ is chosen from a pairwise independent family.
Using this in the proof below, allows one to prove Lemma 5.1 with a smaller
parameter $d$.

###### Proof of Lemma 5.1.

Let $C$ be the map given by Lemma 5.2. Define the map

$f(a)=\bigoplus_{\omega}C(a\circ w)$

where $\omega$ ranges over over $\\{1,i,-1,-i\\}^{n}$. By convexity of the
square function, Jensen’s inequality, and the fact that $\lVert
a\circ\omega\rVert_{\ell_{2}}=\lVert a\rVert_{\ell_{2}}$, we have

$\displaystyle\lVert f(a)\rVert_{S_{1}}^{2}$
$\displaystyle=\Big{(}\mathbf{E}_{\omega}\big{[}\lVert
C(a\circ\omega)\rVert_{S_{1}}\big{]}\Big{)}^{2}$
$\displaystyle\stackrel{{\scriptstyle\text{Lemma~{}\ref{lem:Tmap}}}}{{=}}\left(\mathbf{E}_{\omega}\left[\frac{1}{2}\sqrt{\lVert
a\circ\omega\rVert_{\ell_{2}}^{2}+2\Lambda(a\circ\omega)}+\frac{1}{2}\sqrt{\lVert
a\circ\omega\rVert_{\ell_{2}}^{2}-2\Lambda(a\circ\omega)}\right]\right)^{2}$
$\displaystyle\leq\mathbf{E}_{\omega}\left[\left(\frac{1}{2}\sqrt{\lVert
a\rVert_{\ell_{2}}^{2}+2\Lambda(a\circ\omega)}+\frac{1}{2}\sqrt{\lVert
a\rVert_{\ell_{2}}^{2}-2\Lambda(a\circ\omega)}\right)^{2}\right]$
$\displaystyle=\frac{\lVert
a\rVert_{\ell_{2}}^{2}}{2}+\frac{1}{2}\mathbf{E}_{\omega}\left[\sqrt{\lVert
a\rVert_{\ell_{2}}^{4}-4\Lambda(a\circ\omega)^{2}}\right].$ (29)

Concavity of the square-root function, Jensen’s inequality and Proposition 5.3
gives that the expectation in (29) is at most

$\left(\lVert
a\rVert_{\ell_{2}}^{4}-4\mathbf{E}\big{[}\Lambda(a\circ\omega)^{2}\big{]}\right)^{1/2}=\left(\lVert
a\rVert_{\ell_{2}}^{4}-\lVert a\rVert_{\ell_{2}}^{4}+\lVert
a\rVert_{\ell_{4}}^{4}\big{]}\right)^{1/2}=\lVert a\rVert_{\ell_{4}}^{2}\;.$

Hence,

$\lVert f(a)\rVert_{S_{1}}\leq\sqrt{\frac{\lVert a\rVert_{\ell_{2}}^{2}+\lVert
a\rVert_{\ell_{4}}^{2}}{2}}\;.$

For the second claim observe that for any standard basis vector $e_{j}$ and
$\omega\in\\{1,i,-1,-i\\}^{n}$, the vector $e_{j}\circ\omega$ is either purely
real or purely imaginary. This implies $\Lambda(e_{j}\circ\omega)=0$. Hence,
by Lemma 5.2,

$\lVert
f(e_{i})\rVert_{S_{1}}=\frac{1}{2}\mathbf{E}_{\omega}\left[\sqrt{\lVert
e_{j}\circ\omega\rVert_{\ell_{2}}^{2}+2\Lambda(e_{j}\circ\omega)}+\sqrt{\lVert
e_{j}\circ\omega\rVert_{\ell_{2}}^{2}-2\Lambda(e_{j}\circ\omega)}\right]=1.\qed$

###  5.1 The real and Hermitian variants

We end this section by showing that our hardness result of Theorem 1.2 also
holds for two variants of the Little NCG, the real variant and the Hermitian
variant. Both variants were introduced (in the context of the “big” NCG) in
[NRV14], partly for the purpose of using them in applications. The real
variant asks for the operator norm of a linear map $\mathcal{F}$ from
$\mathbb{R}^{n}$ to a space $\mathbb{R}^{d\times d}$ endowed with the
Schatten-1 norm; in the Hermitian variant, the linear map is from
$\mathbb{R}^{n}$ to the space $H_{d}\subseteq\mathbb{C}^{d\times d}$ of
Hermitian matrices, again endowed with the Schatten-1 norm. In both cases the
operator norm is given by $\lVert\mathcal{F}\rVert=\sup_{a}\lVert
F(a)\rVert_{S_{1}}$ with the supremum over real unit vectors $a$. Both the
real and Hermitian variants follow directly by combining the lemma shown below
and the real version of Theorem 3.1. Let us denote by $\mathbb{S}^{d\times
d}\subseteq\mathbb{R}^{d\times d}$ the space of real symmetric matrices.

######  Lemma 5.4.

Let $n$ be a positive integer and let $d$ be as in Lemma 5.1. Then, there
exists a linear operator $f:\mathbb{R}^{n}~{}\to~{}\mathbb{S}^{4d\times 4d}$
satisfying the conditions stated in Lemma 5.1 (with $a\in\mathbb{R}^{n}$).

The lemma follows by applying the map $\rho$ of the elementary claim below to
the restriction of the operator $f$ of Lemma 5.1 to $\mathbb{R}^{n}$.

######  Claim 5.5.

For every positive integer $d$ there exists a map $\rho:\mathbb{C}^{d\times
d}\to\mathbb{S}^{4d\times 4d}$ such that for any matrix
$A\in\mathbb{C}^{d\times d}$, we have $\lVert\rho(A)\rVert_{S_{1}}=\lVert
A\rVert_{S_{1}}$. Moreover, $\rho$ is linear over the real numbers, that is,
for any $\alpha\in\mathbb{R}$ and $A,B\in\mathbb{C}^{d\times d}$, we have
$\rho(\alpha A)=\alpha\rho(A)$ and $\rho(A+B)=\rho(A)+\rho(B)$.

###### Proof.

The proof follows by combining two standard transformations taking complex
matrices to Hermitian matrices and real matrices, respectively. Let
$A\in\mathbb{C}^{d\times d}$ be a matrix with singular values
$\sigma_{1}\geq\cdots\geq\sigma_{d}$. The first transformation is given by
$A\mapsto\big{[}\begin{smallmatrix}0&A\\\ A^{*}&0\end{smallmatrix}\big{]}$. By
[HJ13, Theorem 7.3.3], the last matrix has eigenvalues
$\sigma_{1}\geq\cdots\geq\sigma_{d}\geq-\sigma_{d}\geq\cdots\geq-\sigma_{1}$.
Notice that this transformation is linear over the reals since the adjoint is
such. Let $B\in\mathbb{C}^{d\times d}$ be a Hermitian matrix with eignevalues
$\lambda_{1}\geq\cdots\geq\lambda_{d}$. The second transformation is given by
$B\mapsto\big{[}\begin{smallmatrix}\Re(B)&\Im(B)\\\
-\Im(B)&\Re(B)\end{smallmatrix}\big{]}$. Then the last matrix is symmetric and
by [HJ13, 1.30.P20 (g), p. 71], that matrix has the same eigenvalues as $B$
but with doubled multiplicities, that is, the matrix has eigenvalues
$\lambda_{1}\geq\lambda_{1}\geq\cdots\geq\lambda_{d}\geq\lambda_{d}$. Notice
that this transformation is also linear over the reals. Let $\rho$ be the
composition of these maps. Then the matrix $\rho(A)$ has the same singular
values as $A$ but with quadrupled multiplicities, which implies that
$\lVert\rho(A)\rVert_{S_{1}}=\lVert A\rVert_{S_{1}}$, and $\rho$ is linear
over the reals. ∎

##  6 Little versus big Grothendieck theorem

For completeness, we include here the well-known relation between the little
and big Grothendieck problems. We focus on the non-commutative case; the
commutative case is similar and can be found in, e.g., [Pis12, Section 5].
This discussion clarifies how to derive Theorem 1.1 from Theorem 1.2.

Consider a linear map $\mathcal{F}:\mathbb{C}^{n}\to S_{1}^{d}$. A standard
and easy-to-prove fact is that for two finite-dimensional Banach spaces $X,Y$,
the operator norm of a linear map $\mathcal{G}:X\to Y$ equals the norm of its
adjoint $\mathcal{G}^{*}:Y^{*}\to X^{*}$. As a result,
$\lVert\mathcal{F}\rVert=\lVert\mathcal{F}^{*}\rVert$. Notice that since
Hilbert space is self-dual and the dual of $S_{1}$ is the space $S_{\infty}$
of matrices endowed with the Schatten-$\infty$ norm (i.e., the maximum
singular value), we have that
$\mathcal{F}^{*}:S_{\infty}^{d}\to\mathbb{C}^{n}$. In particular,

$\lVert\mathcal{F}^{*}\rVert=\sup\lVert\mathcal{F}^{*}(A)\rVert_{2}\;,$

where the supremum is taken over all $A$ of Schatten-$\infty$ norm at most
$1$. Equivalently, since any matrix with Schatten-$\infty$ norm at most 1 lies
in the convex hull of the set of unitary matrices, we could take the supremum
over all unitary matrices $A$. Next, recall that in the NCG problem we are
given a bilinear form $T:\mathbb{C}^{d\times d}\times\mathbb{C}^{d\times
d}\to\mathbb{C}$, and asked to compute
$\operatorname{\mathsf{OPT}}(T)=\sup_{A,B}\big{|}T(A,B)\big{|}$, where the
supremum ranges over unitary matrices. Define the bilinear form
$T(A,B)=\langle\mathcal{F}^{*}(A),\mathcal{F}^{*}(B)\rangle$. By Cauchy-
Schwarz,

$\operatorname{\mathsf{OPT}}(T)=\sup_{A}\lVert\mathcal{F}^{*}(A)\rVert_{2}^{2}=\lVert\mathcal{F}^{*}\rVert^{2}=\lVert\mathcal{F}\rVert^{2}\;,$

where the supremum is over all unitary $A$, showing that the Little NCG is a
special case of the “big” NCG.

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