Robust Bell inequalities from communication com-
plexity
Sophie Laplante
1
, Mathieu Lauri
`
ere
2
, Alexandre Nolin
1
, J
´
er
´
emie Roland
3
, and Gabriel
Senno
4
1
IRIF, Universit
´
e Paris-Diderot, Paris, France
2
ORFE, Princeton University, Princeton, NJ 08544, USA
3
Universit
´
e Libre de Bruxelles, Brussels, Belgium
4
ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels
(Barcelona), Spain
June 5, 2018
Sophie Laplante: laplante@irif.fr
Mathieu Lauri
`
ere: lauriere@princeton.edu
Alexandre Nolin: nolin@irif.fr
J
´
er
´
emie Roland: jroland@ulb.ac.be
Gabriel Senno: gabriel.senno@icfo.eu,
A first version of this work appeared in [32].
arXiv:1606.09514v3 [quant-ph] 4 Jun 2018
Abstract
The question of how large Bell inequality violations can be, for quantum distributions, has
been the object of much work in the past several years. We say that a Bell inequality is
normalized if its absolute value does not exceed 1 for any classical (i.e. local) distribution.
Upper and (almost) tight lower bounds have been given for the quantum violation of these
Bell inequalities in terms of number of outputs of the distribution, number of inputs, and
the dimension of the shared quantum states. In this work, we revisit normalized Bell
inequalities together with another family: inefficiency-resistant Bell inequalities. To be
inefficiency-resistant, the Bell value must not exceed 1 for any local distribution, including
those that can abort. This makes the Bell inequality resistant to the detection loophole,
while a normalized Bell inequality is resistant to general local noise. Both these families
of Bell inequalities are closely related to communication complexity lower bounds. We
show how to derive large violations from any gap between classical and quantum commu-
nication complexity, provided the lower bound on classical communication is proven using
these lower bound techniques. This leads to inefficiency-resistant violations that can be
exponential in the size of the inputs. Finally, we study resistance to noise and inefficiency
for these Bell inequalities.
1 Introduction
The question of achieving large Bell violations has been studied since Bell’s seminal paper
in 1964 [6]. In one line of investigation, proposals have been made to exhibit families
of distributions which admit unbounded violations [33, 39, 40, 43]. In another, various
measures of nonlocality have been studied, such as the amount of communication necessary
and sufficient to simulate quantum distributions classically [7–9, 14, 37, 38, 44, 49, 50],
or the resistance to detection inefficiencies and noise. More recently, focus has turned to
giving upper and lower bounds on violations achievable, in terms of various parameters:
number of players, number of inputs, number of outputs, dimension of the quantum state,
and amount of entanglement [14, 22, 23].
Up until quite recently, violations were studied in the case of specific distributions
(measuring Bell states), or families of distributions. Buhrman et al. [12] gave a construc-
tion that could be applied to several problems which had efficient quantum protocols (in
terms of communication) and for which one could show a trade-off between communication
and error in the classical setting. This still required an ad hoc analysis of communication
problems. Recently Buhrman et al. [13] proposed the first general construction of quantum
states along with Bell inequalities from any communication problem. The quantum states
violate the Bell inequalities when there is a sufficiently large gap between quantum and
classical communication complexity (a super-quadratic gap is necessary, unless a quantum
protocol without local memory exists).
Table 1 summarizes the best known upper and lower bounds on quantum violations
achievable with normalized Bell inequalities.
Parameter Upper bound
Ad hoc lower
bounds
Best possible
lower bound
from [13]
Number of inputs N 2
c
≤ N [14, 23, 34]
√
N
log(N)
[22]
√
c
q
≤ log(N)
Number of outputs K O(K) [22] Ω
K
(log(K))
2
[12] ≤ log(K)
Dimension d O(d) [23] Ω
d
(log(d))
2
[12] ≤ log log(d)
Table 1: Bounds on quantum violations of bipartite normalized Bell inequalities, in terms of the dimen-
sion d of the local Hilbert space, the number of settings (or inputs) N and the number of outcomes
(or outputs) K per party. In the fourth column, we compare ad hoc results to the recent construc-
tions of [13] (Theorem 2) which gives a lower bound of
√
c
q
, where c (resp. q) stands for the classical
(resp. quantum) communication complexity of simulating a distribution. We give upper bounds on
their construction in terms of the parameters d, N, K.
1.1 Our results
We revisit the question of achieving large Bell violations by exploiting known connec-
tions with communication complexity. Strong lower bounds in communication complexity,
equivalent to the partition bound, amount to finding inefficiency-resistant Bell inequali-
ties [31]. These are Bell functionals that are bounded above by 1 on all local distributions
that can abort, i.e. local distributions with an additional abort outcome ⊥ for each party.
In an experimental Bell test, such a constraint would provide resistance to the detection
Accepted in Quantum 2018-05-12, click title to verify 1
loophole, by associating an event where one of the detectors does not click to an abort
outcome.
First, we study the resistance of normalized Bell inequalities to inefficiency. We show
that, up to a constant factor in the value of the violation, any normalized Bell inequal-
ity can be made resistant to inefficiency while maintaining the normalization property
(Theorem 1).
Second, we show how to derive large Bell violations from any communication problem
for which the partition bound is bounded below and the quantum communication com-
plexity is bounded above. The problems studied in communication complexity are far
beyond the quantum set, but we show how to easily derive a quantum distribution from a
quantum protocol. The Bell value we obtain is 2
c−2q
, where c is the partition lower bound
on the classical communication complexity of the problem considered, and q is an upper
bound on its quantum communication complexity (Theorem 3 and Corollary 1). The
quantum distribution has one extra output per player compared to the original distribu-
tion and uses the same amount of entanglement as the quantum protocol plus as many
EPR pairs as needed to teleport the quantum communication in the protocol. Next, we
show that the magnitude of the violations is unaltered in the presence of white noise, i.e.
when the ideal quantum state giving rise to the correlations gets added a maximally mixed
stated with some probability δ (Theorem 4).
Finally, we provide tools to build Bell inequalities from communication lower bounds
in the literature. Lower bounds used in practice to separate classical from quantum com-
munication complexity are usually achieved using corruption bounds and its variants. In
Theorem 6, we give an explicit construction which translates these bounds into a suitable
Bell functional. Table 2 summarizes the new results or the improvements that we obtain
in this work.
Problem Normalized Bell violations [13]
Inefficiency-resistant Bell
violations (this work)
VSP [28, 45]
Ω
6
√
n/
√
log n
d = 2
Θ(n log n)
, K = 2
Θ(n)
2
Ω(
3
√
n)−O(log n)
d = 2
O(log n)
, K = 3
DISJ [1, 46, 47] N/A
2
Ω(n)−O(
√
n)
d = 2
O(
√
n)
, K = 3
TRIBES [11, 21] N/A
2
Ω(n)−O(
√
n log
2
n)
d = 2
O(
√
n log
2
n)
, K = 3
ORT [11, 48] N/A
2
Ω(n)−O(
√
n log n)
d = 2
O(
√
n log n)
, K = 3
Table 2: Comparison of the Bell violations obtained by the general construction of Buhrman et al. [13]
for normalized Bell violations (second column) and this work, for inefficiency-resistant Bell violations
(see Propositions 4, 5, 6, and 7). The parameter n is the size of the input (typically, N = 2
n
.) See
Section 6.2 for the definitions of the communication problems appearing in this table. The construction
of Buhrman et al. only yields a violation when the gap between classical and quantum complexities is
more than quadratic. In the case where the gap is too small to prove a violation, we indicate this with
“N/A”.
Accepted in Quantum 2018-05-12, click title to verify 2
1.2 Related work
The study of the maximum violation of Bell inequalities began with Tsirelson [51], who
showed that for two-outcome correlation Bell inequalities, the maximum violation is bounded
above by Grothendieck’s constant. Tsirelson also raised the question of whether one can
have unbounded violations of Bell inequalities. More precisely, he asked whether there
exist families of Bell inequalities for which the amount of the violation grows arbitrarily
large.
The first answer to this question came from Mermin [39], who gave a family of Bell
inequalities for which a violation exponential in the number of parties is achieved. In the
years that followed, several new constructions appeared for number of parties and number
of inputs [3, 33, 35, 40, 43].
The study of upper bounds on violations of normalized Bell inequalities resumed in [14],
where an upper bound of O(K
2
) (with K the number of outputs per player) and of
2
c
≤ N (with c the communication complexity and N the number of inputs per player)
were proven. In [23] the authors proved a bound of O(d) in terms of the dimension d of
the local Hilbert space, and in [22], the bound in terms of the number of outputs was
improved to O(K). In [22], Bell inequalities are constructed for which a near optimal,
but probabilistic, violation of order Ω(
√
m/ log m), with N = K = d = m, is proven.
In [12], the same violation, although requiring N = 2
m
inputs, is achieved for a family of
Bell inequalities and quantum distributions built using the quantum advantage in one-way
communication complexity for the Hidden Matching problem (with K = d = m). In the
same paper, a violation of order Ω(m/(log m)
2
), with K = d = m and N = 2
m
/m is
achieved with the Khot-Vishnoi game; later, Junge et al. proved in [25] that N can be
reduced to O(m
8
) . Recently, an asymmetric version of this same game was introduced to
allow one of the parties to only make dichotomic measurements, with a smaller (although
almost optimal for this scenario) violation Ω(
√
m/(log m)
2
) [42].
For inefficiency-resistant Bell inequalities, the bounds in [22] do not apply. In fact,
Laplante et al. proved in [31] a violation exponential in the dimension and the number
of outputs for this type of Bell functionals, achieved by a quantum distribution built, as
in [12], from the Hidden Matching communication complexity problem.
The connection exhibited in [12] between Bell violations and communication complex-
ity is generalized by Buhrman et al. in [13] where a fully general construction is given to
go from a quantum communication protocol for a function f to a Bell inequality and a
quantum distribution violating it when the gap between classical and quantum commu-
nication complexity is larger than quadratic. The downside to this construction is that
the quantum distribution has a double exponential (in the communication) number of
outputs and the protocol to implement it uses an additional double exponential amount
of entanglement. Also, this result does not apply for quantum advantages in a zero-error
setting.
2 Preliminaries
2.1 Quantum nonlocality
Local, quantum, and nonsignaling distributions have been widely studied in quantum
information theory since the seminal paper of Bell [6]. In an experimental setting, two
players share an entangled state and each player is given a measurement to perform.
The outcomes of the measurements are predicted by quantum mechanics and follow some
probability distribution p(a, b|x, y), where a is the outcome of Alice’s measurement x, and
Accepted in Quantum 2018-05-12, click title to verify 3
b is the outcome of Bob’s measurement y.
We consider bipartite distribution families of the form p = (p(·, ·|x, y))
(x,y)∈X×Y
with
inputs (x, y) ∈ X × Y determining a probability distribution p(·, ·|x, y) over the outcomes
(a, b) ∈ A × B, with the usual positivity and normalization constraints. The set of proba-
bility distribution families is denoted by P. For simplicity, we call simply “distributions”
such probability distribution families. The expression “Alice’s marginal” refers to her
marginal output distribution, that is
P
b
p(·, b|x, y) (and similarly for Bob).
The local deterministic distributions, denoted L
det
, are the ones where Alice outputs
according to a deterministic strategy, i.e., a (deterministic) function of x, and Bob inde-
pendently outputs as a function of y, without communicating. The local distributions L are
obtained by taking distributions over the local deterministic strategies. Operationally, this
corresponds to protocols with shared randomness and no communication. Geometrically,
L is the convex hull of L
det
.
A Bell test [6] consists of estimating all the probabilities p(a, b|x, y) and computing
a Bell functional, or linear function, on these values. The Bell functional B is chosen
together with a threshold τ so that any local classical distribution ` satisfies the Bell
inequality B(`) ≤ τ, but the chosen distribution p exhibits a Bell violation: B(p) > τ .
By normalizing B, we can assume without loss of generality that ` satisfies B(`) ≤ 1 for
any ` ∈ L, and B(p) > 1.
In this paper, we will also consider strategies that are allowed to abort the protocol
with some probability. When they abort, they output the symbol ⊥ (⊥ denotes a new
symbol which is not in A ∪ B). We will use the notation L
⊥
det
and L
⊥
to denote local
strategies that can abort, where ⊥ is added to the possible outputs for both players.
When ` ∈ L
⊥
det
or L
⊥
, `(a, b|x, y) is not conditioned on a, b 6= ⊥ since ⊥ is a valid output
for such distributions.
The quantum distributions, denoted Q, are the ones that result from applying mea-
surements x, y to their part of a shared bipartite quantum state. Each player outputs his
or her measurement outcome (a for Alice and b for Bob). In communication complexity
terms, these are zero-communication protocols with shared entanglement. If the players
are allowed to abort, then the corresponding set of distributions is denoted Q
⊥
.
Boolean (and other) functions can be cast as sampling problems. Consider a boolean
function f : X ×Y → {0, 1} (nonboolean functions and relations can be handled similarly).
First, we split the output so that if f(x, y) = 0, Alice and Bob are required to output
the same bit, and if f(x, y) = 1, they output different bits. Let us further require Alice’s
marginal distribution to be uniform, likewise for Bob, so that the distribution is well
defined. Call the resulting distribution p
f
, that is, for any a, b ∈ {0, 1} and (x, y) ∈ X ×Y,
we have p
f
(a, b|x, y) = 1/2 if a ⊕ b = f(x, y), and p
f
(a, b|x, y) = 0 otherwise, ⊕ being the
1-bit XOR.
If p
f
were local, f could be computed with one bit of communication using shared
randomness: Alice sends her output to Bob, and Bob XORs it with his output. If p
f
were
quantum, there would be a 1-bit protocol with shared entanglement for f. Interesting
distributions have nontrivial communication complexity and, therefore, they usually lie
well beyond these sets.
Finally, a distribution is nonsignaling if for each player, its marginal output distri-
butions, given by p
A
(a|x, y) =
P
b
p(a, b|x, y), for Alice, and p
B
(b|x, y) =
P
a
p(a, b|x, y),
for Bob, do not depend on the other player’s input. When this is the case, we write the
marginals as p
A
(a|x) and p
B
(b|y). Operationally, this means that each player cannot in-
fluence the statistics of what the other player observes with his own choice of input. We
note with C the set of nonsignaling distributions, also referred to as the causal set, and we
Accepted in Quantum 2018-05-12, click title to verify 4
note C
⊥
when we allow aborting. The well-known inclusion relations between these sets
are L ⊂ Q ⊂ C ⊂ P.
For any Boolean function f, the distribution p
f
is nonsignaling since the marginals
are uniform. A fundamental question of quantum mechanics has been to establish ex-
perimentally whether nature is truly nonlocal, as predicted by quantum mechanics, or
whether there is a purely classical (i.e., local) explanation to the phenomena that have
been predicted by quantum theory and observed in the lab.
2.2 Measures of nonlocality
We have described nonlocality as a yes/no property, but some distributions are some-
how more nonlocal than others. To have a robust measure of nonlocality, it should fulfill
some common sense properties: for a fixed distribution, the measure should be bounded; it
should also be convex, since sampling from the convex combination of two distributions can
be done by first picking randomly one of the two distributions using shared randomness,
and then sampling from that distribution. We also expect such a measure of nonlocality
to have various equivalent formulations. Several measures have been proposed and stud-
ied: resistance to noise [2, 24, 26, 43], resistance to inefficiency [31, 35, 36], amount of
communication necessary to reproduce them [9, 14, 37, 44, 49, 50], information-theoretic
measures [10, 16, 17], etc.
In the form studied in this paper, normalized Bell inequalities were first studied in [14],
where they appeared as the dual of the linear program for a well-studied lower bound on
communication complexity, known as the nuclear norm ν [34] (the definition is given in
Section 2.3). There are many equivalent formulations of this bound. For distributions
arising from boolean functions, it has the mathematical properties of a norm, and it is
related to winning probabilities of XOR games. It can also be viewed as a gauge, that is,
a quantity measuring by how much the local set must be expanded in order to contain
the distribution considered. For more general nonsignaling distributions, besides having
a geometrical interpretation in terms of affine combinations of local distributions, it has
also been shown to be equivalent to the amount of local noise that can be tolerated before
the distribution becomes local [23].
A subsequent paper [31] studied equivalent formulations of the partition bound, one of
the strongest lower bounds in communication complexity [20]. This bound also also has
several formulations: the primal formulation can be viewed as resistance to detector inef-
ficiency, and the dual formulation is given in terms of inefficiency-resistant Bell inequality
violations.
In this paper, we show how to deduce large violations on quantum distributions from
large violations on nonsignaling distributions, provided there are efficient quantum com-
munication protocols for the latter.
2.3 Communication complexity and lower bounds
In classical communication complexity (introduced by [52]), two players each have a share
of the input, and wish to compute a function on the full input. Communication complexity
measures the number of bits they need to exchange to solve this problem in the worst case,
over all inputs of a given size n. In this paper we consider a generalization of this model,
where instead of computing a function, they each produce an output, say a and b, which
should follow, for each (x, y), some prescribed distribution p(a, b|x, y) (which depends on
their inputs x, y). We assume that the order in which the players speak does not depend on
the inputs. This is without loss of generality at a cost of a factor of 2 in the communication.
Accepted in Quantum 2018-05-12, click title to verify 5
We use the following notation for communication complexity of distributions. R
(p)
is the minimum number of bits exchanged in the worst case between players having access
to shared randomness in order to output with the distribution p up to in total variation
distance for all x, y. We call total variation distance between distributions the distance
denoted by |.|
1
, and defined as |p − p
0
|
1
= max
x,y
P
a,b
|p(a, b|x, y) − p
0
(a, b|x, y)|. We use
Q
to denote quantum communication complexity (see [54]) and Q
∗
when, in addition to
the communication of quantum states, the players are allowed to share entanglement.
To give upper bounds on communication complexity it suffices to give a protocol and
analyze its complexity. Proving lower bounds is often a more difficult task, and many tech-
niques have been developed to achieve this. The methods we describe here are complexity
measures which can be applied to any function. To prove a lower bound on communi-
cation, it suffices to give a lower bound on one of these complexity measures, which are
bounded above by communication complexity for any function. We describe here most of
the complexity measures relevant to this work.
The nuclear norm ν, given here in its dual formulation and extended to nonsignaling
distributions, is expressed by the following linear program [14, 34]. (There is a quantum
analogue, γ
2
, which is not needed in this work. We refer the interested reader to the
definition for distributions in [14]).
Definition 1 ([14, 34]). The nuclear norm ν of a nonsignaling distribution p ∈ C is given
by
ν(p) = max
B
B(p)
subject to |B(`)|≤ 1 ∀` ∈ L
det
.
With error , ν
(p) = min
p
0
∈C:|p
0
−p|
1
≤
ν(p
0
). We call any Bell functional that satisfies
the constraint in the above linear program normalized Bell functional.
In this definition and in the rest of the paper, unless otherwise specified (in partic-
ular in Lemma 1), B ranges over vectors of real coefficients B
a,b,x,y
and B(p) denotes
P
a,b,x,y
B
a,b,x,y
p(a, b|x, y), where a, b ranges over the nonaborting outputs and x, y ranges
over the inputs. So even when B and p have coefficients on the abort events, we do not
count them. Table 1 summarizes the known upper and lower bounds on ν for various
parameters. The (log of the) nuclear norm is a lower bound on classical communication
complexity.
Proposition 1 ([14, 34]). For any nonsignaling distribution p ∈ C, R
(p)+1 ≥ log(ν
(p)),
and for any Boolean function f, R
(f) ≥ log(ν
(p
f
)).
As lower bounds on communication complexity of Boolean functions go, ν is one of
the weaker bounds, equivalent to the smooth discrepancy [20], and no larger than the
approximate nonnegative rank and the smooth rectangle bounds [29]. More significantly
for this work, up to small multiplicative constants, for Boolean functions, (the log of) ν is a
lower bound on quantum communication, so it is useless to establish gaps between classical
and quantum communication complexity. (This limitation, with the upper bound in terms
of the number of outputs on normalized Bell violations, is a consequence of Grothendieck’s
theorem [18].)
The classical and quantum efficiency bounds, given here in their dual formulations,
are expressed by the following two convex optimization programs. The classical bound is
a generalization to distributions of the partition bound of communication complexity [20,
31]. This bound is one of the strongest lower bounds known, and can be exponentially
Accepted in Quantum 2018-05-12, click title to verify 6
larger than ν (an example is the Vector in Subspace problem [28]). It is always at least
as large as the relaxed partition bound which is in turn always at least as large as the
smooth rectangle bound [20, 27]. Its weaker variants have been used to show exponential
gaps between classical and quantum communication complexity. The definition we give
here is a stronger formulation than the one given in [31]. We show they are equivalent in
Appendix E.
Definition 2 ([31]). The -error efficiency bound of a distribution p ∈ P is given by
eff
(p) = max
B,β
β
subject to B(p
0
) ≥ β ∀p
0
∈ P s.t. |p
0
− p|
1
≤ ,
B(`) ≤ 1 ∀` ∈ L
⊥
det
.
We call any Bell functional that satisfies the second constraint in the above program
inefficiency-resistant Bell functional. The -error quantum efficiency bound of a p ∈ P is
eff
∗
(p) = max
B,β
β
subject to B(p
0
) ≥ β ∀p
0
∈ P s.t. |p
0
− p|
1
≤ ,
B(q) ≤ 1 ∀q ∈ Q
⊥
.
We denote eff = eff
0
and eff
∗
= eff
∗
0
the 0-error bounds.
Although it may not be straightforward from the above definition due to the presence
of absolute values, the program for the classical efficiency bound is linear, as a consequence
of L
⊥
det
being a polytope (See Appendix E and Ref. [31]).
For any given distribution p, its classical communication complexity is bounded from
below by the (log of the) efficiency bound. For randomized communication complexity with
error , the bound is log(eff
) and for quantum communication complexity, the bound is
log(eff
∗
). Note that for any p ∈ Q, the quantum communication complexity is 0 and the
eff
∗
bound is 1. For any function f, the efficiency bound eff
(p
f
) is equivalent to the
partition bound [20, 31].
Proposition 2 ([31]). For any p ∈ P and any 0 ≤ < 1/2, R
(p) ≥ log(eff
(p)) and
Q
∗
(p) ≥
1
2
log(eff
∗
(p)). For any p ∈ C and any 0 ≤ ≤ 1, ν
(p) ≤ 2eff
(p).
Theorem 3 in Section 4 below involves upper bounds on the quantum efficiency bound.
To give an upper bound on the quantum efficiency bound of a distribution p, it is more
convenient to use the primal formulation, and upper bounds can be given by exhibiting
a local (or quantum) distribution with abort which satisfies the following two properties:
the probability of aborting should be the same on all inputs x, y, and conditioned on not
aborting, the outputs of the protocol should reproduce the distribution p. The efficiency
bound is inverse proportional to the probability of not aborting, so the goal is to abort as
little as possible.
Proposition 3 ([31]). For any distribution p ∈ P, eff
∗
(p) = 1/η
∗
, with η
∗
the optimal
value of the following optimization problem (nonlinear, because Q
⊥
is not a polytope).
max
q∈Q
⊥
,ζ
ζ
subject to q(a, b|x, y) = ζp(a, b|x, y) ∀x, y, a, b ∈ X×Y×A×B
Moreover, for any 0 ≤ ≤ 1, eff
∗
(p) = min
p
0
∈P:|p
0
−p|
1
≤
eff
∗
(p
0
).
Accepted in Quantum 2018-05-12, click title to verify 7
3 Properties of Bell inequalities
Syntactically, there are two differences between the normalized Bell functionals (Defini-
tion 1) and the inefficiency-resistant ones (Definition 2). The first difference is that the
normalization constraint is relaxed: for inefficiency-resistant functionals, the Bell value
for local distributions is, unlike the case for normalized Bell functionals, not bounded in
absolute value; it is only bounded from above. Since this is a maximization problem, this
relaxation allows for larger violations. This difference alone would not lead to a satis-
factory measure of nonlocality, since one could obtain unbounded violations by shifting
and dilating the Bell functional. The second difference prevents this. The upper bound is
required to hold for a larger set of local distributions, those that can abort. This is a much
stronger condition. Notice that a local distribution can selectively abort on configurations
that would otherwise tend to keep the Bell value small, making it harder to satisfy the
constraint.
In this section, we show that normalized Bell violations can be modified to be resis-
tant to local distributions that abort, while preserving the violation on any nonsignaling
distribution, up to a multiplicative factor of 3 (plus a term independent of the input and
output sizes). This means that we can add the stronger constraint of resistance to local
distributions that abort to Definition 1, incurring a loss of essentially a factor of 3, and
the only remaining difference between the resulting linear programs is the relaxation of
the lower bound (dropping the absolute value) for local distributions that abort.
Theorem 1. Let B be a normalized Bell functional on A × B × X × Y and p ∈ C
a nonsignaling distribution such that B(p) ≥ 1. Then there exists a normalized Bell
functional B
∗
on (A∪{⊥}) ×(B∪{⊥}) ×X ×Y with 0 coefficients on the ⊥ outputs such
that : ∀p ∈ C, B
∗
(p) ≥
1
3
B(p) −
2
3
, and ∀` ∈ L
⊥
det
, |B
∗
(`)| ≤ 1.
The formal proof of Theorem 1 is deferred to Appendix A, and we will only give its
high-level structure in this part of the paper. First, we show (see Observation 1) how
to rescale a normalized Bell functional so that it saturates its normalization constraint.
Then, Definition 3 adds weights to abort events to make the Bell functional resistant to
inefficiency. Finally, Lemma 1 removes the weights on the abort events of a Bell functional
while keeping it bounded on the local set with abort, without dramatically changing the
values it takes on the nonsignaling set. Our techniques are similar to the ones used in [36].
4 Exponential violations from communication bounds
Recently, Buhrman et al. gave a general construction to derive normalized Bell inequalities
from any sufficiently large gap between classical and quantum communication complexity.
Theorem 2 ([13]). For any function f for which there is a quantum protocol using q qubits
of communication but no prior shared entanglement, there exists a quantum distribution
q ∈ Q and a normalized Bell functional B such that B(q) ≥
√
R
1/3
(f)
6
√
30q
(1 − 2
−q
)
2q
.
Their construction is quite involved, requiring protocols to be memoryless, which they
show how to achieve in general, and uses multiport teleportation to construct a quantum
distribution. The Bell inequality they construct expresses a correctness constraint.
In this section, we show how to obtain large inefficiency-resistant Bell violations for
quantum distributions from gaps between quantum communication complexity and the
efficiency bound for classical communication complexity. We first prove the stronger of
Accepted in Quantum 2018-05-12, click title to verify 8
two statements, which gives violations of
eff
(p)
eff
?
0
(p)
. For any problem for which a classical
lower bound c is given using the efficiency or partition bounds or any weaker method
(including the rectangle bound and its variants), and any upper bound q on quantum
communication complexity, it implies a violation of 2
c−2q
.
Theorem 3. For any distribution p ∈ P and any 0 ≤
0
≤ ≤ 1, if (B, β) is a feasible
solution to the dual of eff
(p) and (ζ, q) is a feasible solution to the primal for eff
?
0
(p),
then there is a quantum distribution q ∈ Q such that B(q) ≥ ζβ and B(`) ≤ 1, ∀` ∈ L
⊥
det
,
and in particular, if both are optimal solutions, then B(q) ≥
eff
(p)
eff
?
0
(p)
. The distribution q
has one additional output per player compared to the distribution p.
Proof. Let (B, β) be a feasible solution to the dual of eff
(p), p
0
be such that eff
?
0
(p) =
eff
?
(p
0
) with |p
0
− p|
1
≤
0
, and (ζ, q) be a feasible solution to the primal for eff
?
(p
0
).
From the constraints, we have q ∈ Q
⊥
, q(a, b|x, y) = ζp
0
(a, b|x, y) for all (a, b, x, y) ∈
A × B × X × Y, B(`) ≤ 1 for all ` ∈ L
⊥
det
, and B(p
00
) ≥ β for all p
00
s.t. |p
00
− p|
1
≤ .
Then B(q) = ζB(p
0
) ≥ ζβ. However, q ∈ Q
⊥
but technically we want a distribution
in Q (not one that aborts). So we add a new (valid) output ‘A’ to the set of outputs
of each player, and they should output ‘A’ instead of aborting whenever q aborts. The
resulting distribution, say q ∈ Q (with additional outcomes ‘A’ on both sides), is such that
B(q) = B(q) (since the Bell functional B does not have any weight on ⊥ or on ‘A’).
Theorem 2 and Theorem 3 are both general constructions, but there are a few signifi-
cant differences. Firstly, Theorem 3 requires a lower bound on the partition bound in the
numerator, whereas Theorem 2 only requires a lower bound on communication complexity
(which could be exponentially larger). Secondly, Theorem 2 requires a quantum commu-
nication protocol in the denominator, whereas our theorem only requires an upper bound
on the quantum efficiency bound (which could be exponentially smaller). Thirdly, our
bound is exponentially larger than Buhrman et al.’s for most problems considered here,
and applies to subquadratic gaps, but their bounds are of the more restricted class of
normalized Bell inequalities. These differences imply, in principle, different applicability
regimes for the two constructions, which we depict in Figure 1.
Theorem 3 gives an explicit Bell functional provided an explicit solution to the effi-
ciency (partition) bound is given and the quantum distribution is obtained from a solution
to the primal of eff
?
(Proposition 3). Recall that a solution to the primal of eff
?
is pro-
vided by a quantum zero-communication protocol that can abort, which conditioned on
not aborting, outputs following p. We can also start from a quantum protocol, as we
show below. From the quantum protocol, we derive a quantum distribution using stan-
dard techniques.
Corollary 1. For any distribution p ∈ P and any 0 ≤
0
≤ ≤ 1 such that R
(p) ≥
log(eff
(p)) ≥ c and Q
∗
0
(p) ≤ q, there exists an explicit inefficiency-resistant B derived
from the efficiency lower bound, and an explicit quantum distribution q ∈ Q derived from
the quantum protocol such that B(q) ≥ 2
c−2q
.
Proof. Let (B, β) be an optimal solution to eff
(p) and let c be such that eff
(p) = β ≥ 2
c
.
By optimality of B, we have B(p
0
) ≥ 2
c
for any p
0
such that |p
0
−p|
1
≤ . Since Q
∗
0
(p) ≤
q, there exists a q-qubit quantum protocol (possibly using preshared entaglement) for
some distribution p
0
with |p
0
− p|
1
≤
0
≤ . Then, we can use teleportation to obtain
a 2q classical bit, entanglement-assisted protocol for p
0
. We can simulate it without
communication by picking a shared 2q-bit random string and running the protocol but
without sending any messages. If the measurements do not match the string, output a
Accepted in Quantum 2018-05-12, click title to verify 9
R Q
2
eff Q
R Q
eff eff
?
DISJORT
TRIBES
VSP
?
Figure 1: An illustration of the different applicability regions of Theorem 2 and Theorem 3. There
are problems in the literature for which both constructions apply, e.g. VSP, and problems for which
only our construction applies, e.g. DISJ, ORT, and TRIBES (see Table 2). Whether there exists
any problem in the region in which R Q
2
but eff ≈ eff
∗
(indicated by a question mark in the
picture), for which Theorem 2 applies but our Theorem 3 does not, is, to our current knowledge, an
open question.
new symbol ‘A’ (not in the output set of the quantum protocol and different from ⊥). We
obtain a quantum distribution q such that B(q) = B(p
0
)/2
2q
≥ 2
c−2q
.
Most often, communication lower bounds are not given as efficiency or partition bounds,
but rather using variants of the corruption bound. We show in Section 6.1 how to map a
corruption bound to explicit Bell coefficients.
5 Noise-resistant violations from communication bounds
Normalized Bell inequalities are naturally resistant to any constant level of local noise
in the sense that the value of the Bell functional can only decrease by constant additive
and multiplicative terms. Indeed, if the observed distribution is
˜
p = (1 − ε)p + ε` for
some ` ∈ L, then B(
˜
p) ≥ (1 − ε)B(p) − ε since | B(`) |≤ 1. In inefficiency-resistant
Bell inequalities, relaxing the absolute value leads to the possibility that B(`) has a large
negative value for some local `. (Indeed, such large negative values are inherent to large
gaps between ν and eff .) If this distribution were used as adversarial noise, the observed
distribution, (1 − ε)p + ε`, could have a Bell value much smaller than B(p). This makes
inefficiency-resistant Bell inequalities susceptible to adversarial local noise.
In Theorem 4 below, whose proof we defer to Appendix B, we show that the quantum
violations arising from our construction in Corollary 1 are resistant to the addition of
white noise. More specifically, we consider the standard noise model (see e.g., [15, 26, 36])
in which the (ideal) quantum state |ψi gets added a maximally mixed state with some
probability δ,
ρ := (1 − δ)|ψihψ| + δ
I
d
.
We show that even in this noisy scenario, the inefficiency-resistant Bell functional B
and the quantum distribution q resulting from the construction in Corollary 1 are such
that B(q) ≥ 2
c−2q
. Intuitively, resistance to this kind of noise follows from: 1) B arising
from a lower bound on bounded-error communication complexity and 2) the teleportation
measurements needed to construct q still giving uniform outcomes when we replace a
maximally entangled with a maximally mixed state.
Accepted in Quantum 2018-05-12, click title to verify 10
Theorem 4. For any p ∈ P and any 0 ≤
0
≤ ≤ 1 such that log(eff
(p)) ≥ c and
Q
∗
0
(p) ≤ q, there exists an explicit inefficiency-resistant B, POVMs {E
a|x
}
a∈A∪{S}
over
a Hilbert space H
A
and {E
b|y
}
b∈B∪{S}
over a Hilbert space H
B
and a quantum state |ψi ∈
H
A
⊗ H
B
, such that for any 0 ≤ δ ≤ −
0
, we have that
B(q
δ
) ≥ 2
c−2q
,
with
q
δ
(a, b|x, y) = tr
E
a|x
⊗ E
b|y
(1 − δ)|ψihψ|+ δ
I
dim(H
A
⊗ H
B
)
.
Notice that q has one additional output per player (labelled S) compared to p.
6 Explicit constructions
6.1 From corruption bound to Bell inequality violation
We now explain how to construct an explicit Bell inequality violation from the corruption
bound. The corruption bound, introduced by Yao in [53], is a very useful lower bound
technique. It has been used for instance in [46] to get a tight Ω(n) lower bound on the
randomized communication complexity of Disjointness (whereas the approximate rank,
for example, can only show a lower bound of Θ(
√
n)). Let us recall that a rectangle R of
X × Y is a subset of that set of the form R
A
× R
B
, where R
A
⊆ X and R
B
⊆ Y.
Theorem 5 (Corruption bound [4, 30, 53]). Let f be a (possibly partial) Boolean function
on X × Y. Given γ, δ ∈ (0, 1), suppose that there is a distribution µ on X × Y such that
for every rectangle R ⊆ X ×Y
µ(R ∩ f
−1
(1)) > γµ(R ∩ f
−1
(0)) − δ
Then, for every ∈ (0, 1), 2
R
(f)
≥
1
δ
µ(f
−1
(0)) −
γ
.
See, e.g., Lemma 3.5 in [5] for a rigorous treatment. For several problems, such a µ is
already known. In Theorem 6 below, whose proof we defer to Appendix C, we show how
to construct a Bell inequality violation from this type of bound.
Theorem 6. Let f be a (possibly partial) Boolean function on X × Y, where X, Y ⊆
{0, 1}
n
. Fix z ∈ {0, 1}. Let µ be an input distribution, and (U
i
)
i∈I
(resp. (V
j
)
j∈J
) be a
family of pairwise nonoverlapping subsets of f
−1
(¯z) (resp. of f
−1
(z)). Assume that there
exists g : → (0, +∞) such that, for any rectangle R ⊆ X ×Y
X
i∈I
u
i
µ(R ∩ U
i
) ≥
X
j∈J
v
j
µ(R ∩ V
j
) − g(n). (1)
Then, the Bell functional B given by the following coefficients: for all a, b, x, y ∈ {0, 1} ×
{0, 1} × X ×Y,
B
a,b,x,y
=
1/2(−u
i
g(n)
−1
µ(x, y)) if (x, y) ∈ U
i
and a ⊕ b = z,
1/2(v
j
g(n)
−1
µ(x, y)) if (x, y) ∈ V
j
and a ⊕ b = z,
0 otherwise.
(2)
satisfies
B(`) ≤ 1, ∀` ∈ L
⊥
det
, (3)
B(p
f
) =
1
2 · g(n)
X
j
v
j
µ(V
j
) (4)
Accepted in Quantum 2018-05-12, click title to verify 11
and for any p
0
∈ P such that |p
0
− p
f
|
1
≤ :
B(p
0
) ≥
1
2 · g(n)
X
j
v
j
µ(V
j
) −
X
j
|v
j
|µ(V
j
) +
X
i
|u
i
|µ(U
i
)
. (5)
For many other problems in the literature, such as Vector in Subspace and Tribes,
stronger variants of the corruption bound are needed to obtain good lower bounds. These
stronger variants have been shown to be no stronger than the partition bound (more
specifically, the relaxed partition bound) [27]. The generalization in Theorem 6 of the
hypothesis of Theorem 5, which the reader might have noticed, allow us to construct
explicit Bell functionals also for these problems.
6.2 Some specific examples
Using Corollary 1 and the construction to go from a corruption bound (or its variants)
to a Bell inequality (Theorem 6), we give explicit Bell inequalities and violations for
several problems studied in the literature. Since our techniques also apply to small gaps,
we include problems for which the gap between classical and quantum communication
complexity is polynomial.
Vector in Subspace In the Vector in Subspace Problem VSP
0,n
, Alice is given an n/2
dimensional subspace of an n dimensional space over R, and Bob is given a vector. This
is a partial function, and the promise is that either Bob’s vector lies in the subspace, in
which case the function evaluates to 1, or it lies in the orthogonal subspace, in which case
the function evaluates to 0. Note that the input set of VSP
0,n
is continuous, but it can be
discretized by rounding, which leads to the problem
g
VSP
θ,n
(see [28] for details). Klartag
and Regev [28] show that the VSP can be solved with an O(log n) quantum protocol, but
the randomized communication complexity of this problem is Ω(n
1/3
). As shown in [27],
this is also a lower bound on the relaxed partition bound. Hence Corollary 1 yields the
following.
Proposition 4. There exists a Bell inequality B and a quantum distribution q
V SP
∈ Q
such that B (q
V SP
) ∈ 2
Ω(n
1/3
)−O(log n)
and for all ` ∈ L
⊥
det
, B(`) ≤ 1.
Note that the result of [28] (Lemma 4.3) is not of the form needed to apply Theorem 6.
It is yet possible to obtain an explicit Bell functional following the proof of Lemma 5.1
in [27].
Disjointness In the Disjointness problem, the players receive two sets and have to de-
termine whether they are disjoint or not. More formally, the Disjointness predicate is
defined over X = Y = P([n]) by DISJ
n
(x, y) = 1 iff x and y are disjoint. It is also
convenient to see this predicate as defined over length n inputs, where DISJ
n
(x, y) = 1
for x, y ∈ {0, 1}
n
if and only if |{i : x
i
= 1 = y
i
}| = 0. The communication complexity
for DISJ
n
is Ω(n) using a corruption bound [46] and there is a quantum protocol using
O(
√
n) communication [1]. Combining these results with ours, we obtain the following.
Proposition 5. There is a quantum distribution q
DISJ
∈ Q and an explicit Bell inequal-
ity B satisfying: B(q
DISJ
) = 2
Ω(n)−O(
√
n)
, and for all ` ∈ L
⊥
det
, B(`) ≤ 1.
The proof is deferred to the Appendix (see Section D.1).
Accepted in Quantum 2018-05-12, click title to verify 12
Tribes. Let r ≥ 2, n = (2r + 1)
2
. Let TRIBES
n
: {0, 1}
n
× {0, 1}
n
→ {0, 1} be defined
as: TRIBES
n
(x, y) :=
√
n
V
i=1
√
n
W
j=1
(x
(i−1)
√
n+j
∧ y
(i−1)
√
n+j
)
!
. The Tribes function has an
Ω(n) classical lower bound [19] using the smooth rectangle bound and a O(
√
n(log n)
2
)
quantum protocol [11]. Combining these results with ours, we obtain the following.
Proposition 6. There is a quantum distribution q
TRIBES
∈ Q and an explicit Bell in-
equality B satisfying: B(q
TRIBES
) = 2
Ω(n)−O(
√
n(log n)
2
)
, and for all ` ∈ L
⊥
det
, B(`) ≤ 1.
The proof is deferred to the Appendix (see Section D.2).
Gap Orthogonality. The Gap Orthogonality (ORT) problem was introduced by Sher-
stov as an intermediate step to prove a lower bound for the Gap Hamming Distance (GHD)
problem [48]. We derive an explicit Bell inequality for ORT from Sherstov’s lower bound
of Ω(n), shown in [27] to be a relaxed partition bound. (Applying Corollary 1 also gives
a (nonexplicit) violation for GHD.) The quantum upper bound is O(
√
n log n) by the
general result of [11]. In the ORT problem, the players receive vectors and need to tell
whether they are nearly orthogonal or far from orthogonal. More formally, we consider the
input space {−1, +1}
n
(to stick to the usual notations for this problem), and we denote
h·, ·i the scalar product on {−1, +1}
n
. Let ORT
n
: {−1, +1}
n
× {−1, +1}
n
→ {−1, +1}
be the partial function defined as in [48] by: ORT
n
(x, y) = −1 if |hx, yi| ≤
√
n, and
ORT
n
(x, y) = +1 if |hx, yi| ≥ 2
√
n. Combining the results mentioned above with ours, we
obtain the following.
Proposition 7. There is a quantum distribution q
ORT
∈ Q and an explicit Bell inequality
B satisfying: B(q
ORT
) = 2
Ω(n)−O(
√
n log n)
, and for all ` ∈ L
⊥
det
, B(`) ≤ 1.
The proof is deferred to the Appendix (see Section D.3).
7 Discussion
We have given three main results. First, we showed that normalized Bell inequalities
can be modified to be bounded in absolute value on the larger set of local distributions
that can abort without significantly changing the value of the violations achievable with
nonsignaling distributions. Then, we showed how to derive large inefficiency-resistant Bell
violations from any gap between the partition bound and the quantum communication
complexity of some given distribution p. The distributions q achieving the large violations
are relatively simple (only 3 outputs for boolean distributions p) and are resistant to the
addition of white noise to the ideal entangled state giving rise to them. Finally, we
showed how to construct explicit Bell inequalities when the separation between classical
and quantum communication complexity is proven via the corruption bound.
From a practical standpoint, the specific Bell violations we have studied are prob-
ably not feasible to implement, because the parameters needed are still impractical or
the quantum states are infeasible to implement. However, our results suggest that we
could consider functions with small gaps in communication complexity, in order to find
practical Bell inequalities that are robust against uniform noise and detector inefficiency.
As an example, let us show how we could design a Bell test from a gap in communica-
tion complexity using our techniques. Consider a boolean function f with a communi-
cation complexity lower bound R
(p
f
) ≥ log(eff
(p
f
)) ≥ c(n), and accepting a quan-
tum protocol using at most q(n) qubits of communication with error at most
0
< ,
Accepted in Quantum 2018-05-12, click title to verify 13
where n is the size of the inputs. The construction from Theorem 4 then leads to a
Bell inequality B, a quantum state |ψi ∈ H
A
⊗ H
B
and POVMs {E
a|x
} and {E
b|y
}
over H
A
and H
B
respectively, such that, for any δ ≤ −
0
, B(q
δ
) ≥ 2
c(n)−2q(n)
for
q
δ
(a, b|x, y) = tr
h
E
a|x
⊗ E
b|y
(1 − δ)|ψihψ|+ δ
I
dim(H
A
⊗H
B
)
i
. More precisely, |ψi is ob-
tained by preparing q(n) EPR pairs shared between Alice and Bob, and the POVMs {E
a|x
}
and {E
b|y
} arise from the players performing the local unitaries described by the quantum
protocol together with a total of q(n) local Bell measurements (each qubit communicated
by Alice to Bob in the quantum communication protocol for p
f
will turn into a Bell mea-
surement on Alice’s side, and vice versa on Bob’s side) and a final measurement on both
sides to obtain outcomes (a, b) in p
f
. Practically, these measurements will not always be
successful due to the imperfect efficiency of the detectors. Suppose that the efficiency for
each Bell measurement is η
0
, and that the overall efficiency for the final measurements is at
least η
2q(n)
0
(these final measurements involve 2q(n) qubits, which justifies such an expected
scaling). Therefore, factoring in this efficiency and assuming that the noise affecting the
experiment is δ < −
0
, the observed Bell value will be
˜
B(q
0
) ≥ 2
c(n)−2q(n)
η
2q(n)
0
, hence
we still observe a Bell violation (a value
˜
B(q
0
) > 1) as long as n is chosen high enough (or
the detector efficiencies are large enough) to have η
0
> 2
−(
c(n)
2q(n)
−1)
.
The preceding analysis, although accounting for detectors whose efficiency decreases
exponentially with the number of EPR measurements q(n), assumes that the (uniform)
δ-noise affects the quantum state both globally and independently of n. This is a quite
standard noise model, used in various studies of Bell inequalities such as [15, 26, 36]. In
a practical scenario, one might nevertheless want to consider an arguably more realistic
noise model where the error increases with the dimension of the global quantum state (this
would happen for example in the case of noisy teleportation, where each individual EPR
pair gets added a maximally mixed state). As is, our construction is not resistant to such
a noise model as it is based on a classical lower bound for bounded-error communication
complexity, which can therefore only tolerate bounded noise. One way around this issue
would be to turn the bounded-error classical lower bound into a bound valid for low
success probabilities using standard amplification techniques (see, e.g. [30]). Another
possible solution would be to reduce the noise in the quantum protocol by using quantum
error correction (see, e.g. [41]). We leave these ideas for future work.
Lastly, we comment on upper bounds for the violation of inefficiency-resistant Bell
inequalities. First, since (the log of) the efficiency bound is a lower bound on commu-
nication complexity, these violations are bounded above by the number of inputs per
side. Next, for dimension d and number of outcomes K, we obtain the upper bound
eff
(q) ≤ 2
O((
Kd
)
2
log
2
(K))
for quantum distributions, by combining known bounds. In-
deed, we know that R
(p) ≤ O((
Kν(p)
)
2
log
2
(K)) for any p ∈ C (see [14]). Combining
this with the bounds eff
(p) ≤ 2
R
(p)
(Proposition 2), and ν(q) ≤ O(d) for any q ∈ Q
(see [23]), gives the desired upper bound. Hence unbounded violations are possible for
K = 3 outputs per side.
Acknowledgments
We would like to acknowledge the following sources of funding for this work: the Euro-
pean Union Seventh Framework Programme (FP7/2007-2013) under grant agreement no.
600700 (QALGO), the Argentinian ANPCyT (PICT-2014-3711), the Laboratoire Inter-
national Associ´ee INFINIS, the Belgian ARC project COPHYMA and the Belgian Fonds
de la Recherche Scientifique - FNRS under grant no. F.4515.16 (QUICTIME), and the
Accepted in Quantum 2018-05-12, click title to verify 14
French ANR Blanc grant RDAM ANR-12-BS02-005.
References
[1] S. Aaronson and A. Ambainis. Quantum search of spatial regions. Theory of Com-
puting, 1:47–79, 2005. DOI: 10.4086/toc.2005.v001a004.
[2] A. Ac´ın, T. Durt, N. Gisin, and J. I. Latorre. Quantum nonlocality in two three-level
systems. Physical Review A, 65:052325, 2002. DOI: 10.1103/PhysRevA.65.052325.
[3] M. Ardehali. Bell inequalities with a magnitude of violation that grows exponen-
tially with the number of particles. Physical Review A, 46:5375–5378, 1992. DOI:
10.1103/PhysRevA.46.5375.
[4] L. Babai, P. Frankl, and J. Simon. Complexity classes in communication complexity
theory. In Proc. 27th FOCS, pages 337–347. IEEE, 1986. DOI: 10.1109/SFCS.1986.15.
[5] P. Beame, T. Pitassi, N. Segerlind, and A. Wigderson. A strong direct product
theorem for corruption and the multiparty communication complexity of disjointness.
Computational Complexity, 15(4):391–432, 2006. DOI: 10.1007/s00037-007-0220-2.
[6] J. S. Bell. On the Einstein Podolsky Rosen paradox. Physics, 1:195, 1964. DOI:
10.1103/PhysicsPhysiqueFizika.1.195.
[7] C. Branciard and N. Gisin. Quantifying the nonlocality of Greenberger-Horne-
Zeilinger quantum correlations by a bounded communication simulation pro-
tocol. Physical Review Letters, 107(2):020401, 2011. DOI: 10.1103/Phys-
RevLett.107.020401.
[8] J. B. Brask and R. Chaves. Bell scenarios with communication. Journal of Physics A:
Mathematical and Theoretical, 50(9):094001, 2017. DOI: 10.1088/1751-8121/aa5840.
[9] G. Brassard, R. Cleve, and A. Tapp. Cost of exactly simulating quantum entangle-
ment with classical communication. Physical Review Letters, 83(9):1874, 1999. DOI:
10.1103/PhysRevLett.83.1874.
[10] N. Brunner, D. Cavalcanti, A. Salles, and P. Skrzypczyk. Bound nonlocality
and activation. Physical Review Letters, 106:020402, 2011. DOI: 10.1103/Phys-
RevLett.106.020402.
[11] H. Buhrman, R. Cleve, and A. Wigderson. Quantum vs classical communication and
computation. In Proc. 30th STOC, pages 63–68, 1998. DOI: 10.1145/276698.276713.
[12] H. Buhrman, O. Regev, G. Scarpa, and R. de Wolf. Near-optimal and ex-
plicit Bell inequality violations. Theory of Computing, 8(1):623–645, 2012. DOI:
10.4086/toc.2012.v008a027.
[13] H. Buhrman, L. Czekaj, A. Grudka, Mi. Horodecki, P. Horodecki, M. Markiewicz,
F. Speelman, and S. Strelchuk. Quantum communication complexity advantage im-
plies violation of a Bell inequality. Proceedings of the National Academy of Sciences,
113(12):3191–3196, 2016. DOI: 10.1073/pnas.1507647113.
[14] J. Degorre, M. Kaplan, S. Laplante, and J. Roland. The communication complexity of
non-signaling distributions. Quantum Information & Computation, 11(7-8):649–676,
2011. URL http://dl.acm.org/citation.cfm?id=2230916.2230924.
[15] T. Durt, D. Kaszlikowski, and M.
˙
Zukowski. Violations of local realism with quantum
systems described by N-dimensional Hilbert spaces up to N = 16. Physical Review
A, 64(2):024101, 2001. DOI: 10.1103/PhysRevA.64.024101.
[16] M. Forster, S. Winkler, and S. Wolf. Distilling nonlocality. Physical Review Letters,
102:120401, 2009. DOI: 10.1103/PhysRevLett.102.120401.
Accepted in Quantum 2018-05-12, click title to verify 15
[17] R. Gallego, L. E. W¨urflinger, A. Ac´ın, and M. Navascu´es. Operational framework
for nonlocality. Physical Review Letters, 109:070401, 2012. DOI: 10.1103/Phys-
RevLett.109.070401.
[18] A. Grothendieck. R´esum´e de la th´eorie m´etrique des produits tensoriels topologiques.
Boletim da Sociedade de Matem´atica de S˜ao Paulo, 8:1–79, 1953. URL https://
www.revistas.usp.br/resenhasimeusp/article/view/74836.
[19] P. Harsha and R. Jain. A strong direct product theorem for the tribes function via the
smooth-rectangle bound. In Proc. 33rd FSTTCS, volume 24, pages 141–152, 2013.
ISBN 978-3-939897-64-4. DOI: 10.4230/LIPIcs.FSTTCS.2013.141.
[20] R. Jain and H. Klauck. The partition bound for classical communication com-
plexity and query complexity. In Proc. 25th CCC, pages 247–258, 2010. DOI:
10.1109/CCC.2010.31.
[21] T. S. Jayram, R. Kumar, and D. Sivakumar. Two applications of information com-
plexity. In Proc. 35th STOC, pages 673–682, 2003. ISBN 1-58113-674-9. DOI:
10.1145/780542.780640.
[22] M. Junge and C. Palazuelos. Large violation of Bell inequalities with low entan-
glement. Communications in Mathematical Physics, 306(3):695–746, 2011. DOI:
10.1007/s00220-011-1296-8.
[23] M. Junge, C. Palazuelos, D. P´erez-Garc´ıa, I. Villanueva, and M. M. Wolf. Unbounded
violations of bipartite Bell inequalities via operator space theory. Communications
in Mathematical Physics, 300(3):715–739, 2010. DOI: 10.1007/s00220-010-1125-5.
[24] M. Junge, C. Palazuelos, D. P´erez-Garc´ıa, I. Villanueva, and M. M. Wolf. Operator
space theory: A natural framework for Bell inequalities. Physical Review Letters, 104:
170405, 2010. DOI: 10.1103/PhysRevLett.104.170405.
[25] M. Junge, T. Oikhberg, and C. Palazuelos. Reducing the number of questions
in nonlocal games. Journal of Mathematical Physics, 57(10):102203, 2016. DOI:
10.1063/1.4965831.
[26] D. Kaszlikowski, P. Gnaci´nski, M.
˙
Zukowski, W. Miklaszewski, and A. Zeilinger. Vi-
olations of local realism by two entangled N -dimensional systems are stronger than
for two qubits. Physical Review Letters, 85:4418–4421, 2000. DOI: 10.1103/Phys-
RevLett.85.4418.
[27] I. Kerenidis, S. Laplante, V. Lerays, J. Roland, and D. Xiao. Lower bounds on
information complexity via zero-communication protocols and applications. SIAM
Journal on Computing, 44(5):1550–1572, 2015. DOI: 10.1137/130928273.
[28] B. Klartag and O. Regev. Quantum one-way communication can be exponentially
stronger than classical communication. In Proc. 43rd STOC, pages 31–40, 2011. ISBN
978-1-4503-0691-1. DOI: 10.1145/1993636.1993642.
[29] G. Kol, S. Moran, A. Shpilka, and A. Yehudayoff. Approximate nonnegative rank is
equivalent to the smooth rectangle bound. In Automata, Languages, and Program-
ming, pages 701–712. Springer, 2014. DOI: 10.1007/978-3-662-43948-7˙58.
[30] E. Kushilevitz and N. Nisan. Communication complexity. Cambridge University
Press, 1997. ISBN 978-0-521-56067-2. DOI: 10.1017/CBO9780511574948.
[31] S. Laplante, V. Lerays, and J. Roland. Classical and quantum partition bound and
detector inefficiency. In Proc. 39th ICALP, pages 617–628, 2012. DOI: 10.1007/978-
3-642-31594-7˙52.
[32] S. Laplante, M. Lauri`ere, A. Nolin, J. Roland, and G. Senno. Robust Bell Inequalities
from Communication Complexity. In Anne Broadbent, editor, 11th Conference on the
Theory of Quantum Computation, Communication and Cryptography (TQC 2016),
volume 61 of Leibniz International Proceedings in Informatics (LIPIcs), pages 5:1–
Accepted in Quantum 2018-05-12, click title to verify 16
5:24, Dagstuhl, Germany, 2016. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik.
ISBN 978-3-95977-019-4. DOI: 10.4230/LIPIcs.TQC.2016.5. URL http://drops.
dagstuhl.de/opus/volltexte/2016/6686.
[33] W. Laskowski, T. Paterek, M.
˙
Zukowski, and
ˇ
C Brukner. Tight multipartite Bell’s
inequalities involving many measurement settings. Physical Review Letters, 93(20):
200401, 2004. DOI: 10.1103/PhysRevLett.93.200401.
[34] N. Linial and A. Shraibman. Lower bounds in communication complexity based on
factorization norms. Random Structures & Algorithms, 34(3):368–394, 2009. ISSN
1098-2418. DOI: 10.1002/rsa.20232.
[35] S. Massar. Nonlocality, closing the detection loophole, and communication complex-
ity. Physical Review A, 65:032121, 2002. DOI: 10.1103/PhysRevA.65.032121.
[36] S. Massar, S. Pironio, J. Roland, and B. Gisin. Bell inequalities resistant to detector
inefficiency. Physical Review A, 66:052112, 2002. DOI: 10.1103/PhysRevA.66.052112.
[37] T. Maudlin. Bell’s inequality, information transmission, and prism models. In PSA:
Proceedings of the Biennial Meeting of the Philosophy of Science Association, pages
404–417. JSTOR, 1992. DOI: 10.1086/psaprocbienmeetp.1992.1.192771.
[38] K. Maxwell and E. Chitambar. Bell inequalities with communication assistance.
Physical Review A, 89(4):042108, 2014. DOI: 10.1103/PhysRevA.89.042108.
[39] D. N. Mermin. Extreme quantum entanglement in a superposition of macroscopically
distinct states. Physical Review Letters, 65(15):1838, 1990. DOI: 10.1103/Phys-
RevLett.65.1838.
[40] K. Nagata, W. Laskowski, and T. Paterek. Bell inequality with an arbitrary num-
ber of settings and its applications. Physical Review A, 74(6):062109, 2006. DOI:
10.1103/PhysRevA.74.062109.
[41] M. A Nielsen and I. Chuang. Quantum computation and quantum information.
AAPT, 2002. DOI: 10.1017/CBO9780511976667.
[42] C. Palazuelos and Z. Yin. Large bipartite Bell violations with dichotomic measure-
ments. Physical Review A, 92:052313, 2015. DOI: 10.1103/PhysRevA.92.052313.
[43] D. P´erez-Garc´ıa, M. M. Wolf, C. Palazuelos, I. Villanueva, and M. Junge. Unbounded
violation of tripartite Bell inequalities. Communications in Mathematical Physics, 279
(2):455–486, 2008. DOI: 10.1007/s00220-008-0418-4.
[44] S. Pironio. Violations of Bell inequalities as lower bounds on the communication cost
of nonlocal correlations. Physical Review A, 68(6):062102, 2003. DOI: 10.1103/Phys-
RevA.68.062102.
[45] R. Raz. Exponential separation of quantum and classical communication com-
plexity. In Proc. 31st STOC, pages 358–367, 1999. ISBN 1-58113-067-8. DOI:
10.1145/301250.301343.
[46] A. A. Razborov. On the distributional complexity of disjointness. Theoretical
Computer Science, 106(2):385 – 390, 1992. ISSN 0304-3975. DOI: 10.1016/0304-
3975(92)90260-M.
[47] A. A. Razborov. Quantum communication complexity of symmetric predicates.
Izvestiya: Mathematics, 67(1):145, 2003. DOI: 10.1070/IM2003v067n01ABEH000422.
[48] A. A. Sherstov. The communication complexity of gap Hamming distance. Theory
of Computing, 8(1):197–208, 2012. DOI: 10.4086/toc.2012.v008a008.
[49] M. Steiner. Towards quantifying non-local information transfer: finite-bit
non-locality. Physics Letters A, 270(5):239–244, 2000. DOI: 10.1016/S0375-
9601(00)00315-7.
[50] B. F. Toner and D. Bacon. Communication cost of simulating Bell correlations.
Physical Review Letters, 91(18):187904, 2003. DOI: 10.1103/PhysRevLett.91.187904.
Accepted in Quantum 2018-05-12, click title to verify 17
[51] B. S. Tsirel’son. Quantum analogues of the Bell inequalities. The case of two spa-
tially separated domains. Journal of Soviet Mathematics, 36(4):557–570, 1987. DOI:
10.1007/BF01663472.
[52] A. C. C. Yao. Some complexity questions related to distributed computing. In Proc.
11th STOC, pages 209–213, 1979. DOI: 10.1145/800135.804414.
[53] A. C. C. Yao. Lower bounds by probabilistic arguments. In Proc. 24th FOCS, pages
420–428. IEEE, 1983. DOI: 10.1109/SFCS.1983.30.
[54] A. C. C. Yao. Quantum circuit complexity. In Proc. 34th FOCS, pages 352–361.
IEEE, 1993. DOI: 10.1109/SFCS.1993.366852.
A Proof of Theorem 1
Observation 1. Let B be a nonconstant normalized Bell functional and p ∈ C such that
B(p) ≥ 1. Consider `
−
∈ L
⊥
det
such that B(`
−
) = m = min{B(`)|` ∈ L
⊥
det
} and `
+
∈ L
⊥
det
such that B(`
+
) = M = max{B(`)|` ∈ L
⊥
det
}. We have m < M because B is nonconstant.
The Bell functional
˜
B defined by
˜
B(·) =
1
M−m
(2B(·) − M − m), is such that
˜
B(`
+
) = 1,
˜
B(`
−
) = −1, |
˜
B(`)| ≤ 1 for all ` ∈ L
⊥
det
, and
˜
B(p) ≥ B(p) since B is normalized.
Definition 3 below is the first step of the construction. It takes two marginal distribu-
tions m
A
and m
B
, and a normalized Bell functional B, and constructs a Bell functional
B
⊥
m
A
,m
B
whose value over every distribution p ∈ C
⊥
coincides with the value of B over
the distribution p
0
∈ C obtained from p by replacing the abort events with samples from
m
A
and m
B
.
Definition 3. For all two families of distributions, m
A
= (m
A
(·|x))
x∈X
over outcomes in
A for Alice and m
B
= (m
B
(·|y))
y∈Y
over outcomes in B for Bob, and any normalized Bell
functional B with coefficients only on nonaborting events, we define the Bell functional
B
⊥
m
A
,m
B
on (A ∪ {⊥}) × (B ∪ {⊥}) × X ×Y by
(B
⊥
m
A
,m
B
)
a,b,x,y
= B
a,b,x,y
+ χ
{⊥}
(a)
X
a
0
6=⊥
m
A
(a
0
|x)B
a
0
,b,x,y
+ χ
{⊥}
(b)
X
b
0
6=⊥
m
B
(b
0
|y)B
a,b
0
,x,y
+ χ
{⊥}
(a)χ
{⊥}
(b)
X
a
0
,b
0
6=⊥
m
A
(a
0
|x)m
B
(b
0
|y)B
a
0
,b
0
,x,y
where χ
S
is the indicator function for set S taking value 1 on S and 0 everywhere else.
Observation 2. Let f
m
A
,m
B
: C
⊥
→ C be the function that replaces abort events on Alice’s
(resp. Bob’s) side by a sample from m
A
(resp. m
B
) (note that f
m
A
,m
B
preserves locality).
Then, for every m
A
, m
B
and B as in Definition 3, the Bell functional B
⊥
m
A
,m
B
satisfies
that B
⊥
m
A
,m
B
(p) = B(f
m
A
,m
B
(p)), ∀p ∈ C
⊥
, so B
⊥
m
A
,m
B
(p) = B(p), for all p ∈ C, and
|B
⊥
m
A
,m
B
(`)| ≤ 1, for all ` ∈ L
⊥
.
Next, in Lemma 1 below, we do without the abort coefficients in the Bell functionals
B
⊥
m
A
,m
B
.
Lemma 1. Let B
0
be a normalized Bell functional on A
⊥
× B
⊥
× X × Y (possibly with
nonzero weights on ⊥). Then the Bell functional B
00
on the same set defined by
B
00
a,b,x,y
= B
0
a,b,x,y
− B
0
a,⊥,x,y
− B
0
⊥,b,x,y
+ B
0
⊥,⊥,x,y
, (6)
for all (a, b, x, y) ∈ (A ∪ {⊥}) × (B ∪ {⊥}) × X × Y
satisfies :
Accepted in Quantum 2018-05-12, click title to verify 18
1. If a = ⊥ or b = ⊥ then B
00
a,b,x,y
= 0
2. for all p ∈ C,
B
00
(p) = B
0
(p) − B
0
(p
A,⊥
) − B
0
(p
⊥,B
) + B
0
(p
⊥,⊥
), (7)
where p
A,⊥
∈ L
⊥
(resp. p
⊥,B
∈ L
⊥
) is the local distribution obtained from p if Bob (resp.
Alice) replaces all of his (resp. her) outputs by ⊥, and p
⊥,⊥
∈ L
⊥
is the local distribution
where both Alice and Bob always output ⊥. In Item 2 above, for all p
0
,
B
0
(p
0
) =
X
(a,b)∈A
⊥
×B
⊥
X
(x,y)∈X×Y
B
0
a,b,x,y
p
0
(a, b|x, y)
where the first sum is also over the abort events.
Proof. Item 1 follows from (6). We prove Item 2. For p ∈ C
⊥
with marginals p
A
and
p
B
, we have: for all y ∈ Y , p
A
(a|x) =
P
b∈B
⊥
p(a, b|x, y), and for all x ∈ X, p
B
(b|y) =
P
a∈A
⊥
p(a, b|x, y). For the remainder of this proof, summations involving a (resp. b) are
over a ∈ A
⊥
(resp. b ∈ B
⊥
).
By definition, p
A,⊥
(a, b|x, y) = p
A
(a|x)χ
{⊥}
(b), p
⊥,B
(a, b|x, y) = χ
{⊥}
(a)p
B
(b|y), and
p
⊥,⊥
(a, b|x, y) = χ
{⊥}
(a)χ
{⊥}
(b). We have:
B
00
(p) =
X
a,b,x,y
h
B
0
a,b,x,y
− B
0
a,⊥,x,y
− B
0
⊥,b,x,y
+ B
0
⊥,⊥,x,y
i
p(a, b|x, y)
=
X
a,b,x,y
B
0
a,b,x,y
p(a, b|x, y) −
X
a,x,y
B
0
a,⊥,x,y
X
b
p(a, b|x, y)
−
X
b,x,y
B
0
⊥,b,x,y
X
a
p(a, b|x, y) +
X
x,y
B
0
⊥,⊥,x,y
X
a,b
p(a, b|x, y)
= B
0
(p) −
X
a,x,y
B
0
a,⊥,x,y
p
A
(a|x) −
X
b,x,y
B
0
⊥,b,x,y
p
B
(b|y) +
X
x,y
B
0
⊥,⊥,x,y
= B
0
(p) − B
0
(p
A,⊥
) − B
0
(p
⊥,B
) + B
0
(p
⊥,⊥
).
We are now ready to prove Theorem 1.
Proof of Theorem 1. If B is constant, since it is normalized by assumption, we have B ≡ 1.
Thus, we can simply take B
∗
defined by: for all (x, y) ∈ X × Y, B
∗
a,b,x,y
= B
a,b,x,y
if
(a, b) ∈ A × B, and B
∗
a,b,x,y
= 0 otherwise.
Now, let us assume that B is not constant and let `
−
, `
+
∈ L
det
, and
˜
B constructed
from B as in Observation 1 satisfying
˜
B(`
−
) = −1 and
˜
B(`
+
) = 1. Since `
−
and
`
+
are deterministic distributions, we have: `
−
= `
−
A
⊗ `
−
B
and `
+
= `
+
A
⊗ `
+
B
, for
some marginals `
−
A
, `
−
B
, `
+
A
, and `
+
B
. We consider the replacing Bell functional B
⊥
`
−
A
,`
−
B
(resp. B
⊥
`
+
A
,`
+
B
) from Definition 3 constructed from (
˜
B, `
−
A
, `
−
B
) (resp. from (
˜
B, `
+
A
, `
+
B
)).
Taking B
0
=
1
2
(B
⊥
`
−
A
,`
−
B
+ B
⊥
`
+
A
,`
+
B
), we have |B
0
(`)| ≤ 1, for all ` ∈ L
⊥
, and there-
fore we can apply Lemma 1 to get B
00
from B
0
. Since B
0
(p
⊥,⊥
) =
1
2
(B
⊥
`
−
A
,`
−
B
(p
⊥,⊥
) +
B
⊥
`
+
A
,`
+
B
(p
⊥,⊥
)) =
1
2
(
˜
B(`
−
) +
˜
B(`
+
)) = 0, by (7) we have for all p ∈ C
⊥
, B
00
(p) =
B
0
(p) − B
0
(p
A,⊥
) − B
0
(p
⊥,B
). Hence, denoting B
∗
=
1
3
B
00
, B
∗
satisfies all the required
properties since |B
0
(`)| ≤ 1 for all ` ∈ L
⊥
and therefore we have for all p ∈ C, B
∗
(p) ≥
1
3
B
0
(p)−
1
3
|B
0
(p
A,⊥
)|−
1
3
|B
0
(p
⊥,B
)| ≥
1
3
B
0
(p)−
2
3
, and for all ` ∈ L
⊥
, |B
∗
(`)| ≤
1
3
|B
0
(`)|+
1
3
|B
0
(`
A,⊥
)| +
1
3
|B
0
(`
⊥,B
)| ≤ 1.
Accepted in Quantum 2018-05-12, click title to verify 19
B Proof of Theorem 4
Proof. Let (B, β) be an optimal solution to eff
(p) and let c be such that eff
(p) = β ≥ 2
c
.
Recall that, by optimality of B, we have
B(p
0
) ≥ 2
c
for any p
0
such that |p
0
− p|
1
≤ . (8)
As in the proof of Corollary 1, we go from a q-qubit quantum protocol for a distribution p
0
with |p
0
−p|
1
≤
0
to a 2q-bit entanglement-assisted protocol for p
0
by replacing quantum
communication with teleportation. Let H
A
0
(resp. H
B
0
) be Alice’s (resp. Bob’s) local
Hilbert spaces in this protocol and, without loss of generality, let the initial state of the
protocol be
|ψi := |β
00
i
⊗q
|φi
with |β
00
i := 1/
√
2(|00i + |11i) and some fixed |φi ∈ H
A
0
⊗ H
B
0
. Next, we go to a zero-
communication protocol Π by instructing the players not to communicate the teleportation
measurements’ outcomes (and hence not to perform the correcting unitaries) and to output
a new symbol S /∈ A∪B∪{⊥} whenever they get a teleportation measurement’s outcome
different from 00 (the outcome corresponding to the case in which no correcting unitary
is required); notice that this happens with probability 1 − 2
−2q
. The local unitaries and
local measurements performed by the players during the execution of Π effectively induce
POVMs {E
a|x
}
a∈A∪{S}
and {E
b|y
}
b∈B∪{S}
over H
A
:= H
A
0
⊗ C
2
q
and H
B
:= C
2
q
⊗ H
B
0
respectively and a corresponding quantum distribution q such that the probability of the
players outputting (a, b) on inputs (x, y) is given by q(a, b|x, y) = tr
h
(E
a|x
⊗ E
b|y
)|ψihψ|
i
.
Notice that
q(a 6= S, b 6= S|x, y) = tr
h
(I − E
S|x
) ⊗ (I − E
S|y
)|ψihψ|
i
= 2
−2q
, (9)
which together with the fact that we started from a quantum protocol for p
0
, implies that
q is of the form
q(a, b|x, y) =
1
2
2q
p
0
(a, b|x, y) + (1 −
1
2
2q
)r
1
(a, b|x, y),
with r
1
supported on events in which at least one of the players output S. Therefore,
letting B be the Bell functional such that B
a,b,x,y
= B
a,b,x,y
if a 6= S and b 6= S, and
B
a,b,x,y
= 0 otherwise, we have that
B(q) =
1
2
2q
B(p
0
) ≥ 2
c−2q
.
Next, suppose that we run protocol Π with the initial state
ρ := (1 − δ)|ψihψ| + δ
I
2
2q
⊗
I
dim(H
0
A
) dim(H
0
B
)
!
,
and let q
δ
be the resulting quantum distribution. Notice that the 2q teleportation mea-
surements will still give uniform outcomes when performed with I/4 instead of |β
00
i, i.e.
(9) still holds when we replace |ψi with ρ, and hence we have that
q
δ
(a, b|x, y) = tr
h
(E
a|x
⊗ E
b|y
)ρ
i
= (1 − δ)
1
2
2q
p
0
(a, b|x, y) + (1 −
1
2
2q
)r
1
(a, b|x, y)
+
δ
1
2
2q
n(a, b|x, y) + (1 −
1
2
2q
)r
2
(a, b|x, y)
=
1
2
2q
(1 − δ)p
0
(a, b|x, y) + δn(a, b|x, y)
+ (1 −
1
2
2q
) [(1 − δ)r
1
(a, b|x, y) + δr
2
(a, b|x, y)]
Accepted in Quantum 2018-05-12, click title to verify 20
for distributions r
2
supported on events with at least one of the players outputting S and
n on events in which none of the players output S.
Finally, leting p
00
:= (1 − δ)p
0
+ δn, and noticing that
|p − p
00
|
1
= |(1 − δ)p
0
+ δn|
1
≤
0
+ δ|p
0
− n|
1
≤
0
+ δ ≤ ,
with the last inequality following from δ ≤ −
0
, we get from (8) that,
B(q
δ
) =
1
2
2q
B(p
00
) ≥ 2
c−2q
.
C Proof of Theorem 6
Proof. Let us first set B
z,x,y
= B
a,b,x,y
for all a ⊕ b = z. Let ` ∈ L
⊥
det
. Then, we have:
B(`) =
X
(x,y)∈R
B
z,x,y
+
X
(x,y)∈S
B
z,x,y
where R and S are the two rectangles where ` outputs z. Let us take a rectangle R. Then
:
X
(x,y)∈R
B
z,x,y
=
1
2 · g(n)
X
j
v
j
µ(V
j
∩ R) −
X
i
u
i
µ(U
i
∩ R)
≤ 1/2
with the inequality following from (1). This proves (3).
Let us now compute B(p
f
). By linearity of B and the definition of its coefficients, we
have:
B(p
f
) =
X
a,b,x,y
B
a,b,x,y
p
f
(a, b|x, y)
=
1
2
X
(x,y)∈f
−1
(z),a,b
B
a,b,x,y
χ
{z}
(a ⊕ b) +
1
2
X
(x,y)∈f
−1
(¯z),a,b
B
a,b,x,y
χ
{¯z}
(a ⊕ b)
= 1/2
X
j
X
(x,y)∈V
j
v
j
g(n)
−1
µ(x, y)
=
1
2 · g(n)
X
j
v
j
µ(V
j
)
(for the third equality we used the fact that B
a,b,x,y
= 0 when a ⊕b = ¯z). This proves (4).
Moreover, for any family of additive error terms ∆(a, b|x, y) ∈ [−1, 1] such that
X
a,b
|∆(a, b|x, y)| ≤ ∀x, y ∈ X × Y,
Accepted in Quantum 2018-05-12, click title to verify 21
denoted collectively as ∆, we have
|B(∆)| =
X
a,b,x,y
B
a,b,x,y
∆(a, b|x, y)
=
1
2 · g(n)
X
a,b : a⊕b=z
X
i
X
(x,y)∈U
i
(−u
i
)µ(x, y)∆(a, b|x, y) +
X
j
X
(x,y)∈V
j
v
j
µ(x, y)∆(a, b|x, y)
≤
1
2 · g(n)
X
i
X
(x,y)∈U
i
|u
i
|µ(x, y)
X
a,b
|∆(a, b|x, y)|
+
X
j
X
(x,y)∈V
j
|v
j
|µ(x, y)
X
a,b
|∆(a, b|x, y)|
≤
2 · g(n)
X
i
|u
i
|µ(U
i
) +
X
j
|v
j
|µ(V
j
)
From this calculation and (4), we obtain, for p
0
= p
f
+ ∆ :
B(p
0
) = B(p
f
) + B(∆) ≥
1
2 · g(n)
X
j
v
j
µ(V
j
) −
X
j
|v
j
|µ(V
j
) +
X
i
|u
i
|µ(U
i
)
,
which proves (5).
D Explicit examples
Let us formulate a special case of Theorem 6 that will be useful in the examples. Here
there is just one subset in f
−1
(0) and one in f
−1
(1).
Corollary 2. Let f be a (possibly partial) Boolean function on X × Y, where X, Y ⊆
{0, 1}
n
. Given γ ∈ (0, 1) and g : → (0, 1), suppose that there is a distribution µ on
X × Y such that: for any rectangle R ⊆ X × Y,
µ(R ∩ f
−1
(1)) > γµ(R ∩ f
−1
(0)) − g(n). (10)
Then µ satisfies (1) with z = 0, i = j = 1, U
1
= f
−1
(1), V
1
= f
−1
(0), u
1
= 1, v
1
= γ. Let
B be defined by (2), that is: for all a, b, x, y ∈ {0, 1} × {0, 1} × X ×Y,
B
a,b,x,y
=
−
1
2·g(n)
µ(x, y) if f(x, y) = 1 and a ⊕ b = 0
γ
2·g(n)
µ(x, y) if f(x, y) = 0 and a ⊕ b = 0
0 otherwise.
Then, B satisfies
B(`) ≤ 1, ∀` ∈ L
⊥
det
,
B(p
f
) =
γ
2 · g(n)
µ(f
−1
(0))
and for any p
0
∈ P such that |p
0
− p
f
|
1
≤ :
B(p
0
) ≥
1
2 · g(n)
h
γµ(f
−1
(0)) −
γµ(f
−1
(0)) + µ(f
−1
(1))
i
.
Accepted in Quantum 2018-05-12, click title to verify 22
D.1 Disjointness
In [46], Razborov proved the following.
Lemma 2 ([46]). There exist two distributions µ
0
and µ
1
with supp(µ
0
) ⊆ DISJ
−1
n
(1) and
supp(µ
1
) ⊆ DISJ
−1
n
(0), such that: for any rectangle R in the input space,
µ
1
(R) ≥ Ω(µ
0
(R)) − 2
Ω(n)
.
Following his proof, one can check that we actually have:
µ
1
(R) ≥
1
45
µ
0
(R) − 2
−n+log
2
(2/9)
.
So, letting µ := (µ
0
+ µ
1
)/2,
µ(R ∩ f
−1
(0)) ≥
1
45
µ(R ∩ f
−1
(1)) − 2
−n+log
2
(4/9)
. (11)
Remark 1. Actually, supp(µ
1
) = A
1
:= {(x, y) : |x| = |y| = m, |x∩y| = 1} ⊆ DISJ
−1
n
(0).
Note that by this construction, µ(f
−1
(0)) = µ(f
−1
(1)) = 1/2. Combining (11) with
Corollary 2 (with g(n) = 2
−n+log
2
(4/9)
), we obtain:
Corollary 3. There exists a Bell inequality B satisfying: ∀` ∈ L
⊥
det
, B(`) ≤ 1,
B(p
DISJ
n
) =
1
90
2
n−log
2
(4/9)
,
and for any distribution p
0
∈ P such that |p
0
− p
DISJ
n
|
1
≤ ε,
B(p
0
) ≥ 2
n−log
2
(4/9)
1 − 46
90
.
More precisely, Theorem 6 gives an explicit construction of such a Bell inequality: we
can define B as:
B
a,b,x,y
=
−2
n−log
2
(4/9)
µ(x, y) if DISJ
n
(x, y) = 0 and a ⊕ b = 1
1
45
2
n−log
2
(4/9)
µ(x, y) if DISJ
n
(x, y) = 1 and a ⊕ b = 1
0 otherwise.
To obtain Proposition 5, we use Corollary 1 together with the fact that Q
ε
0
(DISJ
n
) =
O(
√
n).
D.2 Tribes
Let n = (2r + 1)
2
with r ≥ 2 and let TRIBES
n
: {0, 1}
n
× {0, 1}
n
→ {0, 1} be defined as:
TRIBES
n
(x, y) :=
√
n
^
i=1
√
n
_
j=1
(x
(i−1)
√
n+j
and y
(i−1)
√
n+j
)
.
In [19][Sec. 3] the following is proven:
Accepted in Quantum 2018-05-12, click title to verify 23
Lemma 3. There exists a probability distribution µ on {0, 1}
n
× {0, 1}
n
for which there
exist numbers α, λ, γ, δ > 0 such that for sufficiently large n and for any rectangle R in
the input space:
γµ(U
1
∩ R) ≥ αµ(V
1
∩ R) − λµ(V
2
∩ R) − 2
−δn/2+1
where U
1
= TRIBES
−1
n
(0), {V
1
, V
2
} forms a partition of TRIBES
−1
n
(1) and µ(U
1
) =
1 − 7β
2
/16, µ(V
1
) = 6β
2
/16, µ(V
2
) = β
2
/16 with β =
r+2
r+1
.
In [19], the coefficients are α = 0.99, λ =
16
3(0.99)
2
and γ =
16
(0.99)
2
(the authors say these
values have not been optimized).
Combining this result with our Theorem 6 (taking z = 1, i = 1, j = 2, U
1
, V
1
, V
2
as in
Lemma 3, u
1
= γ, v
1
= α, v
2
= −λ, and g(n) = 2
−δn/2+1
), we obtain:
Corollary 4. There exists a Bell inequality satisfying: ∀` ∈ L
⊥
det
, B(`) ≤ 1,
B(p
TRIBES
n
) = 2
δn/2−1
β
2
16
(6α − λ),
and for any distribution p
0
∈ P such that |p
0
− p
TRIBES
n
|
1
≤ ε,
B(p
0
) ≥ 2
δn/2−1
"
β
2
16
(6α − λ) − (γ(1 − 7β
2
/16) + λβ
2
/16 + α6β
2
/16)
#
.
More precisely, Theorem 6 provides a Bell inequality B yielding this bound, defined
as:
B
a,b,x,y
=
−γ2
δn/2−1
µ(x, y) if (x, y) ∈ U
1
and a ⊕ b = 1
α2
δn/2−1
µ(x, y) if (x, y) ∈ V
1
and a ⊕ b = 1
−λ2
δn/2−1
µ(x, y) if (x, y) ∈ V
2
and a ⊕ b = 1
0 otherwise.
To obtain Proposition 6, we use Corollary 1 together with the fact that Q
ε
0
(TRIBES
n
) =
O(
√
n(log n)
2
).
D.3 Gap Orthogonality
Let f
n
be the partial functions over {−1, +1}
n
×{−1, +1}
n
by f
n
(x, y) = ORT
64n
(x
64
, y
64
),
that is:
f
n
(x, y) =
(
−1 if |hx, yi| ≤
√
n/8
+1 if |hx, yi| ≥
√
n/4.
In [48], Sherstov proves the following result.
Lemma 4 ([48]). Let δ > 0 be a sufficiently small constant and µ the uniform measure over
{0, 1}
n
× {0, 1}
n
. Then, µ(f
−1
n
(+1)) = Θ(1) and for all rectangle R in {0, 1}
n
× {0, 1}
n
such that µ(R) > 2
−δn
,
µ(R ∩ f
−1
n
(+1)) ≥ δµ(R ∩f
−1
n
(−1)).
This implies that if we put uniform weight on inputs of ORT
64n
of the form (x
64
, y
64
)
and put 0 weight on the others, we get a distribution µ
0
satisfying the constraints of
Corollary 2 for ORT
64n
together with γ = δ from Lemma 4 and g(64n) = 2
δn
.
Accepted in Quantum 2018-05-12, click title to verify 24
To get a distribution satisfying the constraints of Corollary 2 on inputs of ORT
64n+l
for all 0 ≤ l ≤ 63 we extend µ
0
as follows:
˜µ(xu, yv) =
µ
0
(x, y) if u = +1
l
, v = −1
l
and
hx, yi < −
√
64n or 0 ≤ hx, yi ≤
√
64n
µ
0
(x, y) if u = +1
l
, v = +1
l
and
−
√
64n ≤ hx, yi < 0 or hx, yi >
√
64n
0 otherwise
Using this distribution ˜µ together with γ = δ from Lemma 4 and with g(n) = 2
−δn
we
obtain, from Corollary 2, a Bell inequality violation for ORT
64n+l
for all 0 ≤ l ≤ 63:
Corollary 5. There exists a Bell inequality B satisfying: ∀` ∈ L
⊥
det
, B(`) ≤ 1,
B(p
ORT
64n+l
) = 2
δn
δ˜µ(ORT
−1
64n+l
(−1)),
and for any distribution p
0
∈ P such that |p
0
− p
ORT
64n+l
|
1
≤ ε,
B(p
0
) ≥ 2
δn
δ˜µ(ORT
−1
64n+l
(−1)) −
δ˜µ(ORT
−1
64n+l
(−1)) + ˜µ(ORT
−1
64n+l
(+1))
.
More precisely, Theorem 6 gives an explicit construction of such a Bell inequality: we
can define B as:
B
a,b,x,y
=
−2
δn
˜µ(x, y) if (x, y) ∈ ORT
−1
64n+l
(+1) and a ⊕ b = −1
δ2
δn
˜µ(x, y) if (x, y) ∈ ORT
−1
64n+l
(−1) and a ⊕ b = −1
0 otherwise.
To obtain Proposition 7, we use Corollary 1 together with the fact that Q
ε
0
(ORT
n
) =
O(
√
n log n).
E Equivalent formulations of the efficiency bounds
In [31], the zero-error efficiency bound was defined in its primal and dual forms as follows
Definition 4 ([31]). The efficiency bound of a distribution p ∈ P is given by
eff(p) = min
ζ,µ
`
≥0
1
ζ
subject to
X
`∈L
⊥
det
µ
`
`(a, b|x, y) = ζp(a, b|x, y) ∀(a, b, x, y) ∈ A×B×X×Y
X
`∈L
⊥
det
µ
`
= 1
= max
B
B(p)
subject to B(`) ≤ 1 ∀` ∈ L
⊥
det
The -error efficiency bound was in turn defined as min
p
0
∈P|p
0
−p|
1
≤
eff(p
0
). In this
appendix, we show that this is equivalent to the definition used in the present article
(Definition 2). In the original definition, the Bell functional could depend on the particular
p
0
. We show that it is always possible to satisfy the constraint with the same Bell functional
for all p
0
close to p.
In order to prove this, we will need the following notions.
Accepted in Quantum 2018-05-12, click title to verify 25
Definition 5. A distribution error ∆ is a family of additive error terms ∆(a, b|x, y) ∈
[−1, 1] for all (a, b, x, y) ∈ A×B×X×Y such that
X
a,b
∆(a, b|x, y) = 0 ∀(x, y) ∈ X × Y.
For any 0 ≤ ≤ 1, the set ∆
is the set of distribution errors ∆ such that
X
a,b
|∆(a, b|x, y)| ≤ ∀(x, y) ∈ X × Y.
This set is a polytope, so it admits a finite set of extremal points. We denote this set by
∆
ext
.
We will use the following properties of ∆
ε
.
Fact 1. For any distribution p ∈ P, we have
{p
0
∈ P| |p
0
− p|
1
≤ } ⊆ {p + ∆| ∆ ∈ ∆
ε
}
The reason why the set on the right-hand side might be larger is that p + ∆ might not
be a valid distribution. In order to ensure that this is the case, it is sufficient to impose
that all obtained purposed probabilities are nonnegative, leading to the following property.
Fact 2. For any distribution p ∈ P, we have
{p
0
∈ P| |p
0
− p|
1
≤ } = {p + ∆| ∆ ∈ ∆
ε
& p(a, b|x, y) + ∆(a, b|x, y) ≥ 0 ∀a, b, x, y}
We are now ready to prove the following theorem.
Theorem 7. Let p ∈ P be a distribution, eff
ε
(p) be defined as in Definition 2 and eff(p)
be defined as in Definition 4. Then, we have
eff
ε
(p) = min
p
0
∈P:|p
0
−p|
1
≤
eff(p
0
).
Proof. Let eff
ε
(p) = min
p
0
∈P:|p
0
−p|
1
≤
eff(p
0
). We first show that eff
ε
(p) ≤ eff
ε
(p). Let
(B, β) be an optimal feasible point for eff
ε
(p), so that
eff
ε
(p) = β,
B(p
0
) ≥ β ∀p
0
s.t. |p
0
− p|
1
≤ ,
B(`) ≤ 1 ∀` ∈ L
⊥
det
.
Therefore (B, β) is also a feasible point for eff (p
0
) for all p
0
∈ P such that |p
0
− p|
1
≤ ,
so that eff(p
0
) ≥ β for all such p
0
, and
eff
ε
(p) ≥ β = eff
ε
(p).
It remains to show that eff
ε
(p) ≥ eff
ε
(p). In order to do so, we first use the primal
Accepted in Quantum 2018-05-12, click title to verify 26
form of eff(p
0
) in Definition 4 to express eff
ε
(p) as follows
eff
ε
(p) = min
p
0
∈P
s.t |p
0
−p|
1
≤
eff(p
0
)
= min
ζ,µ
`
≥0,p
0
∈P
1
ζ
subject to
X
`∈L
⊥
det
µ
`
`(a, b|x, y) = ζp
0
(a, b|x, y) ∀(a, b, x, y) ∈ A×B×X×Y
X
`∈L
⊥
det
µ
`
= 1, |p
0
− p|
1
≤
= min
ζ,µ
`
≥0,∆∈∆
ε
1
ζ
subject to
X
`∈L
⊥
det
µ
`
`(a, b|x, y) =
ζ[p(a, b|x, y) + ∆(a, b|x, y)] ∀(a, b, x, y) ∈ A×B×X×Y
X
`∈L
⊥
det
µ
`
= 1,
where the last equality follows from Fact 2 and the fact that the first condition of the
program imposes that p(a, b|x, y) + ∆(a, b|x, y) is nonnegative (since
P
`
µ
`
`(a, b|x, y) is
nonnegative). Since ∆
ε
is a polytope, eff
ε
(p) can be expressed as the following linear
program
eff
ε
(p) = min
ζ,µ
`
≥0,ν
∆
≥0
1
ζ
subject to
X
`∈L
⊥
det
µ
`
`(a, b|x, y) = ζ[p(a, b|x, y)+
X
∆∈∆
ext
ε
ν
∆
∆(a, b|x, y)] ∀(a, b, x, y) ∈ A×B×X×Y
X
`∈L
⊥
det
µ
`
= 1,
X
∆∈∆
ext
ε
ν
∆
= 1.
Note that this can be written in standard LP form via the change of variables ζ
0
=
1/ζ, µ
`
= ζw
`
. By LP duality, we then obtain
eff
ε
(p) = max
B,β
β
subject to B(p + ∆) ≥ β ∀∆ ∈ ∆
ε
,
B(`) ≤ 1 ∀` ∈ L
⊥
det
.
Comparing this to the definition of eff
ε
(p) (Definition 2) and together with Fact 1, we
therefore have eff
ε
(p) ≤ eff
ε
(p).
Accepted in Quantum 2018-05-12, click title to verify 27