June 2017

Leibniz Universität Hannover

-Institut für Theoretische Physik-

Bachelor’s Thesis

Bell Inequalities andGrothendieck’s Constant

Author:Sebastian Kinnewig(Sebastian.Kinnewig@gmail.com)

Supervisor:Dr. David Reeb

Advisor:Prof. Dr. Reinhard

Werner

Contents

1 Motivation 1

2 Theory 32.1 Correlation Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Classical Behaviour . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 Quantum Behaviour . . . . . . . . . . . . . . . . . . . . . . . 52.1.3 Alternative Description . . . . . . . . . . . . . . . . . . . . . . 6

2.2 The Bell Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 The CHSH Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 The Grothendieck Inequality . . . . . . . . . . . . . . . . . . . . . . . 142.5 Semidefinite Programming . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5.2 The Primal Problem . . . . . . . . . . . . . . . . . . . . . . . 182.5.3 The Dual Problem . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Computational Approach 213.1 Convex Hull Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Elimination of Duplications . . . . . . . . . . . . . . . . . . . . . . . 243.3 Testing the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.4 Computing the Upper Bound via SDP . . . . . . . . . . . . . . . . . 263.5 Computer Cluster LUIS . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Results 274.1 Previously Known Results . . . . . . . . . . . . . . . . . . . . . . . . 274.2 Our Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2.1 Case (m,n) = (4, 4) . . . . . . . . . . . . . . . . . . . . . . . . 294.2.2 Case (m,n) = (4, 5) . . . . . . . . . . . . . . . . . . . . . . . . 294.2.3 Case (m,n) = (4, 6) . . . . . . . . . . . . . . . . . . . . . . . . 304.2.4 Case (m,n) = (4, 7) . . . . . . . . . . . . . . . . . . . . . . . . 314.2.5 Case (m,n) = (5, 5) . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Résumé 36

Abstract

It was a landmark discovery by J. Bell that correlations obtainable from two quan-tum systems can be stronger than those from classical systems. This is famouslydemonstrated by the CHSH inequality, where each of two separated parties choosefrom m = n = 2 measurement settings and can achieve a maximal quantum valuelarger than its maximal classical value by a factor of

√2. We investigate the question

what the minimal number of measurement settings (m,n) is in order to achieve aquantum violation strictly larger than

√2, or equivalently, when the Grothendieck

constant K(m,n)G for matrix dimensions (m,n) exceeds

√2. To answer the question,

we first compute the tight Bell correlation inequalities for small (m,n) via large-scaleconvex hull computations, or extract it from known results in integer combinatorics.For each of the tight correlation inequalities we then compute the maximal quantumviolation by semidefinite programming. Beyond the values found in the literature, weobtain the new values K(4,4)

G = K(4,5)G = K

(4,6)G = K

(4,7)G = K

(5,5)G =

√2. Each found

tight correlation inequality has a maximal quantum violation smaller than 1.4 <√2,

except for the CHSH inequality.

CHAPTER 1

Motivation

Quantum mechanics and classical mechanics are in tension, as there are predictionsfrom quantum mechanics which appear to be ‘paradoxical’ from a classical pointof view. One difference between quantum mechanics and classical mechanics is thedescription of states for a certain system. Where a pure classical state completelydescribes a classical system such that if the state is known, all outcomes of measure-ments are predictable. A pure quantum state gives just a probability distributionfor the observables, i.e. measurements on the quantum system. A consequence ofthe quantum mechanical description of states is that group of particles can only bedescribed by one single quantum state and generally not by individual quantumstates. Those particles are called entangled. The possibility of entanglement is one ofthe greatest difference between classical and quantum mechanics. One consequenceof entanglement is, that we can obtain a stronger correlation between measurementoutcomes on quantum systems. A measurement can be seen as acting on a state,but in the case of an entangled quantum system it effects all entangled systemswhich happens instantaneously, even when the systems are arbitrary far apart fromeach other. But even so, this correlation cannot be used to transmit data withfaster-than-light speeds, because the correlation can only be observed when bothmeasurement results are known, and the transmission of the (classical) outcome ofthe measurements can only be done at maximum the speed of light.

This means that quantum mechanics violates the idea of local realism. Localmeans that two spacelike separated systems can not interact with each other. Inparticular if we act on the first system, the state of the second system remainsunchanged. Realism means that when all (possibly hidden) variables of a system areknown, the outcome of any measurement is predictable. That quantum mechanicscan violate local realism was already described in 1935 by a thought experiment fromAlbert Einstein, Boris Podolsky and Nathan Rosen [1] and is also known as the EPRParadox. Albert Einstein called this phenomenon ‘spooky action at a distance’ [2] ina letter to Max Born. In 1964 the Bell inequality [3] was formulated by John Bell,which draws a line between the achievable correlation between each the classicaland quantum physics. The Bell inequality only holds for local realistic (classical)models and can be violated by quantum mechanics. Therefore the Bell inequality

1. Motivationcan be used in so-called Bell experiments, which test if quantum systems in naturereally violate local realism. It is confirmed by various experiments that quantumsystems violate a Bell inequality. For example the experiment from Alain Aspect [4],which is seen as a confirmation of quantum mechanics. The most used Bell inequalityfor experiments is the CHSH Bell inequality which was named after John Clauser,Michael Horne, Abner Shimony, and Richard Holt [5].

The CHSH inequality is defined on a setting with two measurement devices ateach side. If we allow only classical states the upper bound in the correspondingCHSH inequality is always 1 but if we allow quantum states we obtain a maximalquantum violation of

√2, as shown by Boris Tsirelson [6]. By Tamas Vértesi [7] was

shown that with large numbers of measurement devices (m,n) = (14, 11) on eachside the quantum violation can exceed

√2. But it is still unknown, what the minimal

numbers (m,n) are, which breaks the√2 marker. Another rising question on this is

what other tight Bell inequalities exist for a given amount of measurement deviceson each side.

In the Second Chapter we take a closer look at correlation inequalities andintroduce mathematical descriptions for the set of local correlation matrices whichwe use to compute the tight Bell inequalities and the set of quantum correlationmatrices which we use to compute the maximal quantum violation. Furthermorewe introduce a mathematical definition of the Bell inequalities and we identify themaximal achievable quantum violation with the optimal Grothendieck constant [8].In order to compute the optimal Grothendieck constant we compute all tight Bellinequalities for given numbers of measurement devices (m,n) and compute from eachtight Bell inequality the maximal quantum violation by semidefinite programming.

In the Third Chapter the approach is introduced in detail along with the usedsoftware for the different steps.

In the Fourth Chapter the known results and the found results are discussed.

– 2 –

CHAPTER 2

Theory

2.1 Correlation Experiments

In this section it will be defined what a correlation experiment is, and we take a closerlook at the Bell inequality in respect to the correlation. Therefore we consider abipartite laboratory scenario. The first party (Alice) has measuring devices A(i) whichare enumerated by i = 1, 2, ...,m. The second party (Bob) has measuring devicesB(j) enumerated by j = 1, 2, ..., n. Each of them receives a (quantum or classical)particle and chooses one of their measurement devices to perform a measurement ontheir particle to obtain some classical outcome. We are interested in the correlationof these outcomes.

Now we want to compare two different scenarios, the classical and the quantumscenario. Therefore we consider the following behaviour of the outcomes:

P (a, b|A(i), B(j)) = ‘the probability to get the outcome (a, b) ∈MA ×MB with

the measurements A(i) on Alice side and B(j) on Bobs side.’

where MA is the set of possible outcomes of Alice and MB the one of Bob.When we are just interested in the correlation between the outcomes of Alice and

Bob, it suffices to look at the correlation coefficient, therefore we have to choose theoutcome set MA and MB as subset of the real numbers.

Ci,j :=∑a∈MA

∑b∈MB

a b P (a, b|A(i), B(j)) (2.1)

In this thesis we consider only two possible outcomes MA = MB = {±1}. SoCi,j = −1 stands for a perfect anti-correlation between the results of Alice and Bob,Ci,j = 0 stands for no correlation and Ci,j = 1 stands for a perfect correlation.

2.1.1 Classical Behaviour

For the classical model we want to preserve the concept of local realism.In this case the outcomes depends on some hidden variable λ. If the hidden variable

2. Theoryis known the outcome is predictable, which means that the classical behaviour isdeterministic. Therefore, the classical behaviour is more precisely called hiddendeterministic behaviour (HDB). The hidden variable λ can be interpreted as somekind of ‘state’, which the particles sent to Alice and Bob have in common andwhich is responsible for the correlation of the outcomes. If Alice or Bob perform ameasurement on their particle the state of the other particle remains unchanged.

source

A(i)

B(j)

a b

Alice Bob

λ

Figure 2.1: Alice and Bob each receive a (classical) particle with some inner ‘state’ λ outof a source and analyse their particle with some measuring device A(i) and B(j) to obtain aclassical outcome.

Therefore, we can describe the measurements by the quantities p(a|A(i), λ), theprobability of Alice to obtain a certain outcome ‘a’ by using the measurement deviceA(i), given the ‘hidden variable’ λ; and p(b|B(j), λ) analogously for Bob. A HDB isthen given by

PHDB(a, b|A(i), B(j)) =

∫dλp(λ)p(a|A(i), λ)p(b|B(j), λ) (2.2)

where p(λ) is the probability distribution for the hidden ‘state’ λ. Using this togetherwith equation (2.1) we can formulate the correlation coefficient for the hiddendeterministic behaviour

Ci,j =∑λ

p(λ)∑a∈MA

∑b∈MB

a b p(a|A(i), λ) p(b|B(j), λ)

=∑λ

p(λ) X(i)λ Y

(j)λ (2.3)

withX

(i)λ :=

∑a∈MA

a p(a|A(i), λ), Y(j)λ :=

∑b∈MB

b p(b|B(j), λ) ∈ [−1, 1]

X(i)λ and Y (j)

λ can be written in the form of some random variables X(i) whichtakes the value X(i)

λ by probability p(λ) and analogously for Y (i). So we can writeCi,j in the classical case as the expectation value of two real-valued random variableswhich only take values between 1 and −1:

Ci,j = E[X(i)Y (j)

]This leads us to the following main definition regarding the classical behaviour.

– 4 –

2. TheoryDefinition 2.1 (Local Correlation (LCm,n))

Let (Ci,j) ∈ Rm×n be a real matrix. (Ci,j) is called a local correlation matrix if andonly if there exist real-valued random variables

(X(i)

)mi=1

and(Y (j)

)nj=1

defined

on a common probability space which only take values between +1 and −1 suchthat

Ci,j = E[X(i)Y (j)

]∀ i ∈ {1, 2, . . . ,m}, j ∈ {1, 2, . . . , n}.

In the following we denote the set of all m×n local correlation matrices by LCm,n.(This definition is taken from [9, Definition 11.5])

2.1.2 Quantum Behaviour

In general the quantum case is described by the Quantum Behaviours, for which thehidden variable λ is replaced by some quantum state σ defined on the state spaceD(H), with H a Hilbert space. As shown later, this can result in a violation of localrealism. In the quantum case the measurement devices are mathematically realizedby observables.

p(z|Z) = Tr [σZz] = ‘probability to obtain "z" by using measuring device Z’,

where Zz is a Positive Operator Valued Measure (POVM).In the bipartite scenario the system is still described by one single state ρ ∈

D(HA⊗HB). When we perform measurements on each subsystem, the probability toobtain the outcome ‘a’ by using the measurement devices A(i) on the first subsystemand to obtain ‘b’ by using measurement devices B(j) on the second is given by

pQB(a, b|A(i), B(j)) = Tr[ρ(X(i)

a ⊗ Y(j)b

)], (2.4)

which is called the Quantum Behaviour. X(i) and Y (j) are sets of POVMs.

pair source

X(i)

Y(j)

a b

Alice Bob

ρ

Figure 2.2: Alice and Bob each receive a quantum particle in some common state ρ outof a pair source and analyse their part of the quantum system with some observables X(i),Y (j). Alice chooses from m measurement devices and Bob from n measurement devices.

The main difference between the classical case as introduced in Equation (2.2)and the quantum case as defined above is that in the classical case the outcome ofAlice depends only on her choice of the measurement device and the inner state λespecially Bobs choice of the measurement does not influence the outcome of Alice.When Alice or Bob do a measurement on their subsystem the inner state λ of thesubsystem from the other party remains unchanged. In the quantum case Alice

– 5 –

2. Theorymeasurement on her subsystem can effect the outcome of Bobs measurement andvice versa because the quantum state ρ, which the particles send to Alice and Bobhave in common, changes after a measurement was done.

If we chose as observable the expectation value operator

X(i) =∑a∈MA

a X(i)a , Y (j) =

∑b∈MB

b Y(j)b

for some outcome spaces MA and MB which are subsets of the real numbers, thenthe correlation coefficient is given by:

Ci,j =∑a∈MA

∑b∈MB

a b Tr[ρ(X(i)

a ⊗ Y(j)b

)]= Tr

[ρ(X(i) ⊗ Y (j)

)],

Now we summarise these considerations in the following definition

Definition 2.2 (Quantum Correlation (QCm,n))Consider a real matrix (Ci,j) ∈ Rm×n. (Ci,j) is called a quantum correlationmatrix if and only if there exists a state ρ ∈ D(Cd1 ⊗ Cd2) (for some d1 andd2) and self-adjoint operators

(x(i))mi=1

on Cd1 and(y(j))nj=1

on Cd2 satisfying‖x(i)‖∞, ‖y(j)‖∞ 6 1 such that

Ci,j = Tr[ρ(x(i) ⊗ y(j)

)]∀ i ∈ {1, 2, . . . ,m}, j ∈ {1, 2, . . . , n}

In the following we denote the set of all m× n quantum correlation matrices withQCm,n(This definition is taken from [9, Definition 11.6])

2.1.3 Alternative Description

The descriptions of the set of local correlation matrices (LCm,n) and the set of quantumcorrelation matrices (QCm,n) are intuitive, but we are interested in mathematicalsimpler descriptions of the sets LCm,n and QCm,n.

At first a simpler mathematical representation of the classical matrices which welater use to construct the set of tight Bell inequalities is given.

Proposition 2.1The set LCm,n can alternatively be described as

LCm,n = conv{(ξiηj)

m ni=1,j=1

∣∣∣ ξ ∈ {−1, 1}m, η ∈ {−1, 1}n}i.e. ξT = (ξ1, ξ2, . . . , ξm) and ηT = (η1, η2, . . . , ηn) are vectors with entries ±1.

Proof. From equation (2.3) follows that the set of matrices (Ci,j) forms a convex setbecause the probability distribution p(λ) is a partition of unity

(Ci,j) ∈ conv{(X(i)Y (j)

)m n

i=1,j=1

∣∣∣ ∣∣X(i) 6 1,∣∣Y (j)

∣∣ 6 1∣∣} . (2.5)

– 6 –

2. TheoryNow we use that any vector Z ∈ [−1, 1]d can be written as a convex combination ofthe vectors ξ ∈ {±1}d:

Z =∑

ξ∈{±1}dλξξ ∈ conv

{ξ|ξ ∈ {±1}d

}, λξ ∈ [0, 1] probability distribution. (2.6)

We use the short hand notationXT = (X(1), X(2), · · · , X(m)), Y T = (Y (1), Y (2), · · · , Y (n)).

⇒ LCm,n =conv{(X Y T

) ∣∣∣X ∈ [−1, 1]m, Y ∈ [−1, 1]n}

=

{∑k

λkXk YTk

∣∣∣∣Xk ∈ [−1, 1]m, Yk ∈ [−1, 1]n, λk partition of unity

}

With equation (2.6) we can write:

LCm,n =

∑k

λ(1)k

∑ξ∈{±1}m

λ(2)ξ ξ

∑η∈{±1}n

λ(3)η ηT

∣∣∣∣∣ λ(i) partition of unity,for i ∈ {1, 2, 3}

=

∑k

∑ξ∈{±1}m

∑η∈{±1}n

λ(1)k λ

(2)ξ λ(3)η ξ ηT

∣∣∣∣∣ λ(i) partition of unity,for i ∈ {1, 2, 3}

=conv

{(ξi ηj)

m ni=1,j=1

∣∣∣ξ ∈ {±1}m, η ∈ {±1}n}Remark 2.1 (Vertex Representation)The vertices of the set LCm,n are the m × n matrices given by ξT η for all vectorsξ ∈ {±1}m, η ∈ {±1}n. LCm,n has 2n+m−1 vertices because ξT η = ξT η only if

(ξ, η) =(ξ, η

)or (−ξ. − η) =

(ξ, η

)We call the set of m×n matrices ξ ηT ∀ξ ∈ {±1}m, η ∈ {±1}n vertex representationof the Bell polytope LCm,n.

Remark 2.2 (Cut Polytope)LCm,n is a convex polytope and is also called Bell polytope, it can be identified withthe cut polytope as described in the following.

Let G = (V,E) be an undirected Graph. The cut polytope CUTPG ⊆ R|E| of G isthe convex hull of all points δS ∈ R|E| with S ⊆ V with

δS(e) :=

{1, if exactly one vertex of e is in S0, if 0 or 2 vertices of e are in S

, e ∈ E.

– 7 –

2. TheoryIf Gm,n = (V,E) is the complete bipartite graph,with

V = {A1, A2, . . . , Am, B1, B2, . . . , Bn}

E ={(i, j)

∣∣i ∈ {A1, . . . , Am} , j ∈ {B1, . . . , Bn}}

then CUTPGm,n , or just CUTPm,n, can be identifiedwith the Bell polytope LCm,n by replacing δS(e)with (−2 · δS(e)) + 1.

A1 B1

A2 B2

A3 B3

Figure 2.3: A sketch of the com-plete bipartite graph G3,3.

The cut polytope is often considered in combinatorics. By the linear transfor-mation given above, the cut polytope can be translated into the Bell polytope,therefore it is equivalent to consider the Bell polytope or the cut polytope. Forfurther information on this topic see [10].

In the following we want to give a similar mathematical description for the setof quantum correlated matrices as for the local correlated matrices. The followingproposition is not trivial like Proposition 2.1. This notation for QCm,n was firstintroduced by B. Tsirelson in 1980 [6]. We will later use this proposition to com-pute the maximal quantum violation of a certain Bell inequality via semidefiniteprogramming.

Proposition 2.2The set QCm,n can alternatively be described as

QCm,n = conv{(〈xi, yj〉)m n

i=1,j=1

∣∣xi, yj ∈ Rd, d ∈ N, ‖xi‖ 6 1, ‖yj‖ 6 1}. (2.7)

x = (x1, x2, . . . , xm) and y = (y1, y2, . . . , ym) are sets of vectors out of a ddimensional (real) unit sphere with the scalar product 〈a, b〉 = aT b for vectorsa, b ∈ Rd.

Proof. First we show

QCm,n ⊆{(〈xi, yj〉)m n

i=1,j=1

∣∣∣xi, yj ∈ Bd(0, 1), d ∈ N}. (2.8)

Let (Ci,j) ∈ QCm,n be a quantum correlation matrix. Therefore there exists somestate ρ defined in the state space D(Cd1 ⊗ Cd2) and Xi ∈ Bsa(Cd1) with ‖Xi‖∞ 6 1and Yj ∈ Bsa(Cd2) with ‖Yj‖∞ 6 1 self adjoint operators, where Bsa denotes the setof self-adjoint operators, such that

Ci,j = Tr [ρ (Xi ⊗ Yj)]

holds, as Ci,j was defined in Definition 2.2. Therefore we define a bilinear β form onBsa(Cd1 ⊗ Cd2) by

β(S, T ) := Re (Tr [ρ S T ])) .

– 8 –

2. TheoryOne can show that β is positive semidefinite and symmetric. We denote the inducedform on the quotient space

V :=Bsa(H){

S ∈ Bsa(H)∣∣β(S, S) = 0

}by β as well. The vectors (Xi ⊗ 1) , (1⊗ Yj) out of the quotient space V satisfy

β (Xi ⊗ 1, 1⊗ Yj) = Re Tr [ρ (Xi ⊗ Yj)] = Ci,j

β (Xi ⊗ 1, Xi ⊗ 1) = Re Tr[ρ(X2i ⊗ 1

)]6 1 (2.9)

β (1⊗ Yj, 1⊗ Yj) = Re Tr[ρ(1⊗ Y 2

j

)]6 1.

Thus we set d := dim(V ), and identify V = Rd as a vector space with innerproduct 〈·, ·〉 = β(·, ·). Furthermore we define xi := Xi ⊗ 1 and yj := 1⊗ Yj, thenwith equations (2.9) follows that Ci,j = 〈xi, yj〉 and xi, yj ∈ Bd(0, 1). Therefore(Ci,j) ∈

{(〈xi, yj〉)m n

i=1,j=1

∣∣∣xi, yj ∈ Bd(0, 1), d ∈ N}

which proves equation (2.8).Now we show the reverse direction

QCm,n ⊇{(〈xi, yj〉)m n

i=1,j=1

∣∣∣xi, yj ∈ Bd(0, 1), d ∈ N}. (2.10)

The challenge is to implement xi, yj by quantum states. Be (xi)mi=1, (yj)nj=1 ∈ Bd

two sets of vectors. We define some Hermitian matrices that are multiples of unitarymatrices.

Ai =d∑

k=1

(xi)k Uk, Bj =d∑

k=1

(yj)k Uk

out of the d× d dimensional Hilbert space (for some d > 2d12de), with

Uk :=

{1⊗(k−1)2×2 ⊗ σx ⊗ σ⊗(k−1)y if k < d

2

1⊗(k−1)2×2 ⊗ σz ⊗ σ⊗(k−1)y else

σx, σy, σz the Pauli matrices. One can easy check that ‖Ai‖2 6√d , ‖Bj‖2 6

√d

and Tr [AiBj] = d 〈xi, yj〉 holds. Then for any maximal entangled state ρ = |ϕ〉 〈ϕ|holds

Tr[ρ(Ai ⊗BT

j

)]=

1

dTr [AiBj] = 〈xi, yj〉 .

One can easy check that ‖A‖∞ 6 1 and ‖B‖∞ 6 1, which means we have constructedself-adjoint operators Ai and Bj such that the equation (2.10) is satisfied.

Remark 2.3It always suffices to choose d = min {m,n} in Equation (2.7) to obtain the maximalquantum violation in Equation (2.14).

Consider W.L.O.G.m < n. It suffices to choose V ′ := span {X1 ⊗ 1, · · · , Xm ⊗ 1}such that the Equations (2.9) still holds.

– 9 –

2. TheoryThe value of β (Xi ⊗ 1, Xi ⊗ 1) remains unchanged therefore the inequal-

ity β (Xi ⊗ 1, Xi ⊗ 1) 6 1 still holds. If we consider the inner productβ (1⊗ Yj, 1⊗ Yj) on a smaller subspace V ′ it becomes smaller such thatβ (1⊗ Yj, 1⊗ Yj) 6 1 is still satisfied. The inner product β (Xi ⊗ 1, 1⊗ Yj)projects Y on X, such that n − m dimensions get projected onto the kernel soβ (Xi ⊗ 1, 1⊗ Yj) = Ci,j still holds.

Remark 2.4A direct consequence of this proof is that is suffices to choose the dimension of theHilbert space for the entangled quantum system as 2d

12de. As an example if Alice has

2 measurement devices and Bob 42 measurement devices. Alice and Bob can obtainthe maximal quantum violation by using just a qubit system. But 2d

12de is not the

optimal lower bound of the dimension for the quantum system.Even when Alice uses 3 measurement devices a qubit system would still suffices to

obtain the maximal quantum violation. Because the Pauli matrices σx, σy, σz spanthe space of observables of the 2 dimensional Hilbert space and the Pauli matriceshave span{σx, σy, σz} = 3.

2.2 The Bell Inequality

Bell inequalities, which are named after John Stewart Bell [3] describe the correlationof bipartite experiments as defined above. It only holds for local realistic models butit can be violated by quantum states. Originally Bell formulated the ‘EPR-Paradoxon’on a spin measurement of entangled electrons. He described the set for which thelocal realism holds via the Bell inequality.

We are interested in an mathematical definition such that we can compute thetight Bell inequalities

Definition 2.3 (Bell Inequalities)The set of Bell inequalities is the polar set LCom,n

LCom,n :=

{ϕ ∈M′

m,n

∣∣∣∣ supC∈LCm,n

ϕ(C) 6 1

},

withM′m,n the dual set from the m× n Matrices.

Remark 2.5Because the setM′

m,n is an isometric isomorphism toMm,n we can identify a Bellinequality as a m× n matrix M ∈Mm,n via the map

ϕ(C) := Tr[MTC

]=

m∑i=1

n∑j=1

Mi,jCi,j 6 1 ∀ C ∈ LCm,n.

– 10 –

2. TheoryRemark 2.6 (Halfspace Representation)Because LCm,n is a polytope with a finite number of vertices (see Remark 2.2)it can be completely described by a finite number of facets i.e tight inequalitiesext{LCom,n} the so-called halfspace representation or for short H-representation. TheH-representation gives us a list of the tight Bell inequalities, therefore we will computelater the H-representation with a convex hull algorithm.

Remark 2.7 (Orbit Representation)Another possible representation of the Bell polytope is the orbit representation. Whereeach facet i.e Bell inequality is represented by a list of vertices which vanishes on thatfacet. To obtain a Bell inequality from a list of vertices (which can be interpretedas a matrix) we just have to compute the nullspace from the corresponding matrix.M. Dutour Sikirić published a list of the orbit representations of the cut polytopes(defined in Remark 2.2) CUTP4,4, CUTP4,5, CUTP4,6, CUTP4,7 and CUTP5,5, whichwe use to compute the halfspace representation of the corresponding Bell polytopes.

The Bell inequality defined above is sometimes called correlation Bell inequalitybecause we only consider the correlation of measurement outcomes. For a fineranalysis of bipartite experiments one can make use of the individual probabilitiesP (a, b|A(i) B(j)) instead of the correlation Ci,j.

2.3 The CHSH Inequality

To get a better understanding for what a Bell inequality is, we take a closer lookat an example: The famous CHSH inequality. It is a proof that there are quantumcorrelations that can not be reproduced by a hidden variable model. Let be c ∈ LC2,2

then1

2(c1,1 + c2,2 + c2,1 − c2,1) 6 1

holds. We use for this the short notation described in Remark 2.5

Tr[MT c

]6 1 ∀c ∈ LC2,2 with M =

1

2

(1 −11 1

). (2.11)

The position of the ‘-1’ makes no difference as discussed in Section 3.2. One way tocheck that the CHSH inequality holds for the hidden deterministic behaviour is touse the mathematical description of LCm,n we have derivated in Proposition 2.1

LC2,2 = conv{(ξiηj)

m ni=1,j=1|ξ ∈ {±1}m, η ∈ {±1}n

}and test for all 22 · 22 = 16 possible choices of ξ1, ξ2, η1, η2 ∈ {±1} if the followinginequality is satisfied

2∑i=1

2∑j=1

Mi,jci,j =1

2(ξ1η1 + ξ2η2 + ξ2η1 − ξ1η2) 6 1, (ci,j) ∈ LC2,2

– 11 –

2. Theory(for a more general version see Section 3.3). But if we allow quantum correlationthere exists C ∈ QC2,2 such that

1

2(C1,1 + C2,2 + C2,1 − C1,2) > 1. (2.12)

In the following we will construct some maximal entangled quantum state ρ and someOperators X(1), X(2) for Alice and Y (1), Y (2) for Bob such that we obtain a violationof the CHSH inequality as in Inequality (2.12). Let us consider the following setupwhere Alice and Bob receives each a qubit which is a part of an entangled system(for example an electron, a photon or a cat) and do some measurements on them.Let the system be in the Bell state

ρ = |φ〉 〈φ|

with 〈φ| := 1√2(〈↑↓| − 〈↓↑|) = 1√

2((1 0)⊗ (0 1)− (0 1)⊗ (1 0))

⇒ ρ =1

2

0 0 0 00 1 −1 00 −1 1 00 0 0 0

Alice uses the expectation operator

X :=∑

a∈{±1}

a Xa =

(1 00 −1

), with X−1 :=

(1 00 0

), X1 :=

(0 00 1

)

To obtain X(1) and X(2) Alice applies some rotation α on her measurement device.So we obtain the following operator in dependence of the rotation angle.

X(α) =

(cosα sinαsinα − cosα

)·X ·

(cosα sinαsinα − cosα

)=

(cos 2 α sin 2 αsin 2 α − cos 2 α

)

Alice applies no rotation on her first measurement device α1 = 0 and on her secondmeasurement device she applies a rotation of α2 =

π4.

X(1) := X(0) =

(1 00 −1

), X(2) := X

(π4

)=

(0 −1−1 0

)analogously we obtain for Bob the operator in dependence of the angle β

Y (β) =

(cos 2 β sin 2 βsin 2 β − cos 2 β

)

– 12 –

2. TheoryBob applies to his first measurement device a rotation of β1 = 3

8π and to his second

measurement device he applies the rotation β2 = 38π + π

4.

Y (1) := Y

(3

8π

)=

1√2

(−1 −1−1 1

), Y (2) := Y

(5

8π

)=

1√2

(1 −1−1 −1

)Now we plug this into the CHSH inequality.∑

i,j

MijCi,j =∑i,j

Mij Tr[ρ(X(i) ⊗ Y (j)

)]=

1

2

(1√2+

1√2+

1√2−(− 1√

2

))=

4

2√2=√2

-1.5

-1

-0.5

0

0.5

1

1.5

0 π/8 π/4 3/8 π π/2 5/8 π 3/4 π 7/8 π π

γ

angle β

Figure 2.4: The quantum violation of the CHSH in respect to the angle β where Aliceuses the measurement devices X(1), X(2) as defined above and Bob uses the measurementdevices Y (1) = Y (β), Y (2) = Y (β + π

4 ).

The other tight Bell inequality beside the CHSH inequality in the case whereAlice and Bob each have two measurement devices is the trivial Bell inequality (seeRemark 3.2) which is of the form

Mtriv =

(±1 00 0

).

– 13 –

2. TheoryThe position and the sign of the 1 makes no difference (see Section 3.2), in the sensethat the maximal quantum violation is always kMtriv

= 1.

Remark 2.8This is not a proof that

√2 is the maximal achievable value. Via semidefinite

programming as described in Section 2.5 one can prove that√2 is actually the

maximal quantum violation of the CHSH inequality. But this example proves that ifwe allow Ci,j ∈ QC the Bell inequality can be violated.

Remark 2.9Most time in the literature the CHSH inequality is formulated as

c1,1 + c2,2 + c2,1 − c1,2 6 2, with (ci,j) ∈ LCm,n.

where both sides were multiplied by two such that the prefactor 12drops. Because

we will consider a lot of different Bell inequalities in this work we use the conventionthat a Bell inequality is always less or equal than one as we have it orignally definedin Definition 2.3.

2.4 The Grothendieck Inequality

In this section we will describe the relation between Grothendieck inequalities,Grothendieck constants and Bell inequalities along with their maximal quantumviolation. We will use this relation to phrase an optimization problem in respect tocompute the maximal quantum violation for a given number of measurement deviceson each side. In 1953 Alexander Grothendieck proved the following inequality.

Proposition 2.3 (The (Real) Grothendieck Inequality)There exists a k ∈ R such that for any m× n real matrix for any m, n ∈ N withthe property:

∑mi=1

∑nj=1Mi,jviwj 6 1 with |vi| 6 1 and |wj| 6 1 holds

m∑i=1

n∑j=1

Mi,j 〈Vi, Wj〉 6 k (2.13)

for all Vi, Wj ∈ Rd with ‖Vi‖ 6 1 and ‖Wj‖ 6 1 for any d ∈ N. This inequality iscalled Grothendieck Inequality, k is called the Grothendieck Constant.

Proof. see [8]

We are interested on the optimal k for fixed m, n. As described in the Motivationof this thesis it is a recent question what the minimal needed (m,n) is to obtain aquantum violation strictly larger than

√2. This question does not only arise in the

field of quantum information it is also an interesting question for analytics. K(m,n)G

is the most central value we want to compute in this thesis.

– 14 –

2. TheoryThe quantum violation can be identified with the real Grothendieck constant whichis a consequence of the Propositions 2.1 and 2.2.

m∑i=1

n∑j=1

Mi,jCi,j 6 k, (Mi,j) ∈ LCom,n, (Ci,j) ∈ QCm,n

Further we define the optimal Grothendieck constant

K(m,n)G := max

M∈LComax

C∈QCm,n

m∑i=1

n∑j=1

Mi,jCi,j. (2.14)

K(m,n)G ∈ R denotes the optimal Grothendieck constant for fixed matrix dimensions

(m,n).

It is useful, in order to compute the optimal Grothendieck constant, to define themaximal quantum violation of a certain Bell inequality

kM := maxC∈QC

m∑i=1

n∑j=1

Mi,jCi,j, (2.15)

where M ∈ LCom,n is a Bell inequality.

The maximum over LCom,n in Equation (2.15) can be replaced by the maximumover the set ext{LCom,n} because the maximum is assumed on the edge of LCom,n.The set ext{LCom,n} is the finite set of tight Bell inequalities. One can obtain theset of tight Bell inequalities for fixed matrix dimension (m,n) by computing thehalfspace representation of the Bell polytope (see Remark 2.6). In order to computethe optimal Grothendieck constant we rephrase the Grothendieck inequality as thefollowing optimization problem over the maximal quantum violation

K(m,n)G = max

M∈ext{LCom,n}

maxC∈QC

m∑i=1

n∑j=1

Mi,jCi,j = maxM∈ext{LCo

m,n}kM . (2.16)

First we have to compute all tight Bell inequalities. In the second step we computefrom each tight Bell inequality the maximal quantum violation, therefore we solve theoptimization problem formulated in Equation (2.15) via semidefinite programmingas described in the next section.

– 15 –

2. Theory

As discussed in Remark 2.2 the setLCm,n is a polytope with a finitenumber of facets. The facets canbe described by tight Bell inequal-ities M ∈ ext{LCom,n}. QCm,n isalso a convex set (but not nec-essary with a finite number ofvertices) and LCm,n is a subsetof QCm,n. This leads to anothergeometrical interpretation of theoptimal Grothendieck constant,namely the optimal Grothendieckconstant is the smallest real num-ber such that the argument

K(m,n)G LCm,n ⊇ QCm,n

holds.

LCm,n

QCm,n

tight Bell inequality

maximal quantum violation of a certainBell inequality

Figure 2.5: Geometrical interpretation ofthe Bell inequalities.

Of peculiar interest is the Grothendieck constant KG := limm→∞

limn→∞

K(m,n)G , for which

is known that

1.57079 ≈ π

26 KG 6

π

2 ln(1 +√2)≈ 1.78221.

The lower bound was proven by Grothendieck himself [8] and the upper bound wasproven by Krivine [11].

2.5 Semidefinite Programming

Constructing a proof that a given Bell inequality has a certain maximal quantumviolation can be rather complicated, an example is the proof for the maximal quantumviolation of the CHSH inequality K(2,2)

G =√2 done by B. Tsirelson [6]. But the

proof scheme used for the CHSH inequality cannot be easily applied on larger Bellinequalities. Another way to compute the maximal quantum violation of certainBell inequality is introduced in the work of S. Wehner [12], who used SemidefiniteProgramming (SDP) to compute the maximal quantum violation of a given Bellinequality. As the name says SDP is concerned with convex optimization problemsusing positive semidefinite matrices as variables. At next, the needed case to solve thisgiven problem is introduced (for further literature on this topic see [13]). AlthoughSDP is a numerical routine, we obtain a certified lower and upper bound for ourproblem. Another benefit of SDP is, that it can be solved in polynomial runtime.

– 16 –

2. Theory2.5.1 Preliminaries

Definition 2.4 (Gram Matrix)We call a matrix G ∈ Ms,s Gram matrix if and only if it can be written in theform:

G :=

〈z1, z1〉 · · · 〈z1, zs〉......

〈zs, z1〉 · · · 〈zs, zs〉

for some z ∈ Rd for some d ∈ N. Therefore G can be written as G = BTB forB ∈Md,s, where the i-th column of B is the vector zi:

B :=

| |z1 · · · zs| |

Proposition 2.4 (Cholesky Decomposition)

Let G ∈Ms,s be a symmetric matrix than holds

G > 0⇔ G = BTB, for some B ∈Md,s.

Proof. First we show that G > 0 ⇒ G = BTB, therefore we use the spectral

decomposition G =s∑r=1

λre(r) ·

(e(r))T where each λr is a real positive number.

⇒ Bi,j =s∑r=1

λr

(e(r) ·

(e(r))T)

ij=

s∑r=1

λre(r)i · e

(r)j

and define the vector zTi :=(√

λ1e(1)i

√λ2e

(2)i · · ·

√λne

(n)i

)⇒ 〈zi, zj〉 =

s∑r=1

√λre

(r)i ·

√λre

(r)j =

s∑r=1

λr

⟨e(r)i , e

(r)j

⟩= Gi,j

The other direction is trivial.

This shows that any positive semidefinite matrix can be written in the form of aGram matrix and conversely any Gram matrix is positive semidefinite. We usethis later to construct the SDP.

Remark 2.10A consequence of this proof is that is suffices to choose the dimension d 6 rank(G) 6 sfor the vectors zi in the proof above. We will see later that this implies that it sufficesto choose the vectors used in the Proposition 2.3 as Vi, Wj ∈ Rrank(G) to obtain themaximal quantum violation. The Gram matrix is not unique, hence it is possiblethat there exists a different Gram matrix with a smaller rank, therefore this alsogives just an upper bound for the minimal needed dimension of the correspondingvector space of the vectors Vi, Wj.

– 17 –

2. Theory2.5.2 The Primal Problem

The target is to rephrase the optimization problem from Equation (2.15) as anoptimization problem where the variable is a positive semidefinite matrix. For agiven Bell inequality M one can write:

maxVi,Wj∈Rd

∑mi=1

∑nj=1Mij 〈Vi,Wj〉 = p′

subject to ‖Vi‖ = 1 ∀i ∈ {1, 2, . . . ,m}, ‖Wj‖ = 1 ∀j ∈ {1, 2, . . . , n}

where p′ denotes the optimal solution of the primal problem. One could also claim‖Vi‖ , ‖Wj‖ 6 1 but it suffices to consider ‖Vi‖ , ‖Wj‖ = 1 because the maximum isassumed on the edge.

We denote the primal problem by

maxG>0, Gk,k=1

m∑i=1

n∑j=1

Mi,jGi+m,j = p′

in the following. Where G is the Gram Matrix from the vectorz = {V1, V2, · · · , Vm,W1,W2, · · · ,Wn} ∈ Rn+m:

G =

〈V1, V1〉 · · · 〈V1, Vm〉 〈V1,W1〉 · · · 〈V1,Wn〉...

......

...〈Vm, V1〉 · · · 〈Vm, Vm〉 〈Vm,W1〉 · · · 〈Vm,Wn〉〈W1, V1〉 · · · 〈W1, Vm〉 〈W1,W1〉 · · · 〈W1,Wn〉

......

......

〈Wn, V1〉 · · · 〈Wn, Vm〉 〈Wn,W1〉 · · · 〈Wn,Wn〉

This can be simplified.

maxG>0, Gk,k=1

m∑i=1

n∑j=1

Mi,jGi+m,j = p′

= maxG>0, Gk,k=1

m∑i=1

n∑j=1

(1

2Mi,jGi+m,j +

1

2MijGi,j+m

)

= maxG>0, Gk,k=1

1

2

m∑i=1

n∑j=1

(Mi,jGi+m,j +Mi,jGi,j+m)

= maxG>0, Gk,k=1

1

2Tr [GW ] , with W :=

(0m×m M

MT 0n×n

)

Now we have an easier formulation for the primal problem, which gives a lower boundfor the maximal quantum violation kM for a given Bell inequality M ∈ LCom,n.

– 18 –

2. Theory2.5.3 The Dual Problem

As next we derive the Dual Problem which gives an upper bound for the maximalquantum violation kM for a given Bell inequality M ∈ LCom,n.

Definition 2.5 (The Lagrange Dual Problem)The dual problem is given by:

minλk∈R

m+n∑k=1

λk = d′

subject to diag (λ) > W2

That the dual problem gives an upper bound to the primary problem is called weakduality which we want to prove in the following proposition.

Proposition 2.5 (Weak Duality)The duality gap between the upper and the lower bound is greater or equal tozero or just:

p′ 6 d′

Proof.

p′ = maxG>0, Gk,k=1

1

2Tr [GW ]

6 minA>0, λk∈R

maxG>0, Gk,k=1

(1

2Tr [GW ] + Tr [GA]−

n+m∑k=1

(Gk,k − 1)λk

)

= minA>0, λk∈R

maxG>0, Gk,k=1

(1

2Tr [GW ] + Tr [GA]− (Tr [G− 1]) diag (λ)

)= min

A>0, λk∈Rmax

G>0, Gk,k=1

(Tr

[GW

2

]+ Tr [GA]− Tr [G diag (λ)] + Tr [diag (λ)]

)6 min

A>0, λk∈R

(m+n∑k=1

λk

)+max

G>0Tr

[G

(W

2+ A− diag (λ)

)]

= minλk∈R

(m+n∑k=1

λk

)+

{0 ,if W

26 diag (λ)

∞ ,else

= minW26diag(λ), λk∈R

m+n∑k=1

λk

Proposition 2.6 (Strong Duality)When Slater’s conditions [13] hold for a SDP the duality gap between the primaland the dual problem is equal to zero or just

p′ = d′

– 19 –

2. TheoryProof. see [13, chapter 5.3.2]

Remark 2.11The dual problem gives an upper bond for the maximal quantum violation kMfor a given Bell inequality M ∈ LCom,n. Where λ = (λ1, λ2, . . . , λm+n) ∈ Rm+n isa (analytical) certificate for this upper bound because as we have proven aboveW26 diag (λ) holds which implies that for a given Bell inequality M ∈ LCom,n the

inequality kM − dλ 6 Tr [diag (λ)] holds where dλ is the to λ corresponding solutionof the dual problem.

Furthermore the strong duality holds for this SDP, which means that the dualitygap between the solution of the primal and the dual problem converges againstzero. The SDP will be solved by CVX a MATLAB package, which solves the SDPnumerical with a error of ε 6 10−7.

Since we have an easy formulation for the primal and the dual problem we cannow solve this SDP, as discussed in the next chapter.

– 20 –

CHAPTER 3

Computational Approach

To obtain the maximal quantum violation for fixed (m,n) i.e. the optimalGrothendieck constant, one has to solve the optimization problem formulated inSection 2.4

K(m,n)G = max

M∈ext{LCom,n}

maxC∈QCm,n

m∑i=1

n∑j=1

Mi,jCi,j

We separate this optimization problem for fixed (m,n) into three steps. The firststep is to compute all tight Bell inequalities M ∈ ext{LCom,n} this is possible becausethe number of tight Bell inequalities is finite as discussed in Remark 2.6. To obtainthe tight Bell inequalities we compute the dual from the Bell polytope. The onlyreasonable way to do this is by a convex hull algorithm. The second step is to computethe maximal quantum violation kM for all tight Bell inequalities M ∈ ext{LCom,n}which we computed in the first step. The maximal quantum violation of a given Bellinequality can be computed via SDP as described in Section 2.5. The last step is tofind the largest quantum violation from all tight Bell inequalities.

LC

The vertexrepresentation

of LC

mplrshalfspace

representationof LC

(Bell inequalities)removingdouble enterys

CVXK(m,n)

Gm,n

m,n

m,n

Figure 3.1: We begin with the construction of the vertex representation (see Remark 2.1)of the Bell polytope LCm,n, which will be translated into the halfspace representation of theBell polytope LCm,n by the convex hull algorithm mplrs, which gives us a list of all tightBell inequalities ext{LCom,n} (see Remark 2.6). From this list we remove the duplicationsup to symmetries (see Section 3.2) and compute from the remaining Bell inequalities themaximal quantum violation by formulating the optimization problem as a semidefiniteprogram which will be solved via the MATLAB package CVX.

3. Computational Approach

3.1 Convex Hull Algorithm

Computing the dual polytope is the analogous task to convert the vertex representa-tion (see Remark 2.1) into the halfspace representation which is the set of tight Bellinequalities as defined in Remark 2.6. There exist several algorithms to computethe dual polytope. The computation of the dual polytope is rather difficult, sincethe runtime of the computation of dual from the Bell polytope LCm,n seems to growexponentially in the dimension d = m n (see Figure 3.2). Therefore the first taskis to choose a suited algorithm for this computation. Some convex hull algorithmswhich also works in high dimension (dim > 10) are for example Qhull [14], mplrs[15], cdd [16] and Polyhedral [17].

The first naive try in Matlab to compute the dual of the Bell polytope LC4,4 tookabout 24 hours on a standard computer. For larger dimensional Bell polytopes theruntime is not reasonable. Matlab uses the Qhull algorithm (Short for quick hullalgorithm). In respect to this work, the second test was done by the c implementationof the Qhull algorithm with similar runtimes like the MATLAB implementation.The convex hull algorithm of cdd is suited for highly degenerative polytopes as theBell polytope. Another try was started with mplrs, which is implemented in c usingMPI so it works even on great cluster systems. Other benefits of mplrs are thatit works very stable even in high dimension (highest tested dimension in this workwas d = m n = 30) and it generates partial results as the computation proceeds.The computation of the dual from the Bell polytope LC4,4 took about 8 hours on astandard computer. Therefore mplrs was used for computations of dual polytopes inthis work.

Dutour Sikirić considerd the computaton of dual polytopes in [18] and alsopublished the convex hull algorithm polyhedral a package for GAP [19]. Polyhedraluses the symmetry of a given polytope to reduce the runtime. The runtime tocompute the dual Bell polytope LC4,4 is less than one minute on a standard computer.[18] which can be identfied by a linear map with the corresponding Bell polytope(see Remark 2.2).

– 22 –

3. Computational ApproachNow we take a look at the runtime of mplrs, therefore we plot the actual runtimeagainst the dimension d := m n. For the used hardware see Section 3.5.

10-1

100

101

102

103

104

105

106

8 10 12 14 16 18 20

runt

ime

t in

s

dimension d

Figure 3.2: The real runtime of the computation if the dual Bell polytopes LCm,n forthe cases (m,n) = (3, 3), (3, 4), (3, 5), (4, 4), (3, 6), (4, 5). Only the computations thatfinished were used for this figure.

Fit:

f(d) := 10(a·d+b), d ∈ Nwith:a = 0.545 ±0.037b = -5.001 ±0.572

The runtime of mplrs seems to be in O(2d) for the computation of the dual Bellpolytope LCm,n. This means increasing the dimension by one, the runtime increasesby the factor f(d)

f(d+1)≈ 3.55.

The runtime grows exponentially with the dimension from the Bell polytope,therefore it takes long to compute new tight Bell inequalities. For this work thelargest tested case was (m,n) = (6, 5) after 200 hours on 160 cores the only foundtight Bell inequalities from mplrs was the CHSH inequalities and no new tight Bellinequalities. Using the extrapolation function introduced earlier, the runtime for thecase (m,n) = (6, 7) would be higher than the age of the universe.

– 23 –

3. Computational Approach

3.2 Elimination of Duplications

For any Bell inequality exist some equivalent forms with the same maximal quantumviolation which we call duplications up to symmetries. This is recorded in thefollowing definition.

Definition 3.1For some given dimensions m and n let A, B ∈ LC o

m,n be two Bell inequalities.A and B are from the same equivalence class or for short A=B if A can betransferred into B by using line and/or column permutations as well as multiplyinglines and/or columns of A by ‘−1’.

Proposition 3.1Let be A, B ∈ LCom,n two Bell inequalities. If A=B

⇒ kA = kB,

where kA, kB are the to A and B corresponding maximal quantum violations asdefined in Equation 2.15

Proof. If we recall the definition of the Bell inequality we can interpret the permu-tation of columns as a renaming of Alice measurements devices and multiplying acolumn by ‘−1’ is just renaming the observables of the corresponding measurementdevice of Alice, analogous for Bob. Just renaming the measurement devices and theobservables does not change the maximal quantum violation.

Any equivalence class contains a lot of Bell inequalities but we are only interestedin one representative from each equivalence class. To get a those an algorithm thatremoves all duplications up to symmetries was written during the preparation forthis thesis.

Example 3.1For the CHSH inequality we obtain with the above given operations 8 different formswhich all have a maximal quantum violation of

√2 as mentioned in Remark 2.9 a

few examples are:

( yy−1 1

1 1

)→(1 −11 1

)∈ ext{LCo2,2}(−1 1

1 1

)| · (−1)

→(−1 1

−1 −1

)∈ ext{LCo2,2}

Remark 3.1It is possible to increase the dimension of a given Bell inequality by adding a columnand/or by adding a line of zeros at an arbitrary position. Further more if we adda line and/or a column of zeros on a tight Bell inequality we obtain a tight Bell

– 24 –

3. Computational Approachinequality for a higher dimension. For example if we consider the CHSH inequalitywhich is tight Bell inequality of the LCo2,2 polytope we obtain:

(−1 1 0

1 1 0

),

(−1 0 1

1 0 1

),

(0 −1 1

0 1 1

)∈ LCo2,3

−1 0 1

0 0 0

1 0 0

∈ LCo3,3

This can be applied analogous on all Bell inequalities to obtain a Bell inequality of ahigher dimension.

Remark 3.2 (Trivial Bell Inequality)The trivial case of the Grothendieck inequality is if we choose the matrix dimensionsm = n = 1. Then the Grothendieck inequality and the optimal Grothendieckconstant become

m V W 6 1, V,W ∈ R1 with ‖V ‖ , ‖W‖ 6 1⇒ K(1,1)G = 1

The trivial Bell inequality m = 1 is the only tight Bell inequality in the case(m,n) = (1, 1). By adding lines and/or columns of 0 as described in Remark 3.1 weobtain the (tight) trivial Bell inequalities for larger cases. As example for the Bellpolytope LC2,2 there are beside the 8 tight CHSH inequalities (see Example 3.1) 8trivial inequalities for example:(

−1 00 0

),

(0 10 0

), . . .

3.3 Testing the Results

To test if a computed Bell inequality M ∈ Rm,n is in the set LCm,n we considerξ ∈ {±1}m, η ∈ {±1}n

maxξi,ηj

m,n∑i,j=1

Mijξiηj = maxξi

m∑i=1

ξi

(maxηj

n∑j=1

Mijηj

)6 1, with ξi, ηj ∈ {−1, 1}

⇔Mij ∈ LCom,n

choose ξi := sgn

(maxηj

n∑j=1

Mijηj

)

⇒m=1∑i

∣∣∣∣∣maxηj

n∑j=1

Mijηj

∣∣∣∣∣ 6 1⇔Mij ∈ LCom,n (3.1)

As an example one can take a look at the CHSH inequality, only 22 = 4 possiblevectors ω ∈ {±1}2 exist, so it is easy to check that the inequality (3.1) holds for anychoice of ω

1

2|(ω1M1,1 + ω2M1,2 + ω1M2,1 + ω2M2,2)| =

1

2|(−ω1 + ω2 + ω1 + ω2)| 6 1.

– 25 –

3. Computational Approach

3.4 Computing the Upper Bound via SDP

Once the tight Bell inequalities are obtained, the maximal quantum violation from agiven Bell inequality using semidefinite programming as described in Section 2.5 canbe computed. Because the strong duality holds for this SDP it makes no differencewhether the primary or the dual problem is solved. There are several algorithm tosolve SDP’s. In the scope of the thesis CVX: Matlab Software for Disciplined ConvexProgramming, was used. Although SDP is a numerical routine, it gives a certificatefor the result. With the certificate the result can be analytically proven in aspectto a remaining error |ε| < 10−7. The runtime is polynomial in log

(1ε

)and in the

program description size.

3.5 Computer Cluster LUIS

The results presented here were carried out on the cluster system at the LeibnizUniversität Hannover, the Lower Saxony Ministry of Science and Culture (MWK)and the German Research Association (DFG). For this work the server Lena wasused, with 2x Intel Haswell Xeon E5-2630 v3 8-cores, 2.40 GHz, every node has20 MB cache, 16 cores and 64 GB memory.

– 26 –

CHAPTER 4

Results

4.1 Previously Known Results

For the trivial case (m,n) = (1, n) ∀n ∈ N, where m denotes the number of mea-surement devices on Alice side and n the number of measurement devices on Bobsside the optimal Grothendieck constant for the trivial case is K(1,n)

G = K(1,1)G = 1.

For higher dimensions the question after the optimal Grothendieck constant is moreinteresting. In the case (m,n) = (2, 2) there exist the two following tight Bellinequalities

The trivial Bell inequality with a maxi-mal quantum violation k = 1 (Remark3.2) (

1 00 0

)The CHSH Bell inequality with a max-imal quantum violation k =

√2 as

introduced in Section 2.3

1

2

(−1 11 1

)Where k is the shorthand notation of kM the maximal quantum violation ofa given Bell inequality M ∈ LCm,n as defined in Equation (2.15) and M is here theBell inequality alongside of k. The maximal quantum violation for CHSH inequalitywas already computed in 1980 by Tsirelson [6], so he proved K(2,2)

G =√2.

The only tight Bell inequalities which occur in the cases (m,n) = (2, n), (3, n)with n ∈ N are the trivial Bell inequality and the CHSH inequality as described abovefilled with zero lines and/or zero columns respective to (m,n) (as described in Remark3.1). Therefore the optimal Grothendieck constant in the cases (m,n) = (2, n), (3, n)with n ∈ N is the same as the Grothendieck constant for the case (m,n) = (2, 2), soK

(2,2)G = K

(2,n)G = K

(3,n)G =

√2, ∀n ∈ N holds. This was first mentioned in [20] but

not proven, a proof for this can be found in [10].Gisin computed two new Bell inequalities for the case (m,n) = (4, 4), but Gisin onlyconsidered Bell inequalities with coefficients less or equal than two. The complete listof tight Bell inequalities for the case (m,n) = (4, 4) was computed by Avis in 2006[10]. Therefore Avis computed the complete halfspace representation of the CUTP4,4

4. Resultspolytope with the convex hull algorithm cdd [16]. Avis proved that Gisin foundalready all tight inequalities for the case (m,n) = (4, 4). In the case (m,n) = (4, 4)two new Bell inequalities were also obtained during the work for this thesis. Themaximal quantum violation of the Bell inequalities in the case (m,n) = (4, 4) werecomputed analytically without SDP’s in the not jet published work by Ben Li [21].

Furthermore Avis published a partial result of the case (m,n) = (4, 5). Actu-ally Avis computed all tight Bell inequalities in the case (m,n) = (4, 5) as we willprove in Subsection 4.2.2. Dutour Sikirić computed with his package polyhedral theorbit representation of dual cut polytopes CUTP4,4, CUTP4,5, CUTP4,6, CUTP4,7

and CUTP5,5 [22]. Those can be easily transformed into the corresponding Bellpolytopes as discussed in Remark 2.7. The result published by Avis as well as theresults published by Dutour Sikirić agree with the results computed for this work,which suggest that all routines and algorithm used are working correct. The optimalGrothendieck constants K(m,n)

G for the cases (m,n) = (4, 5), (4, 6), (4, 7) and (5, 5)does not seem to have been computed earlier. The question what the smallestnumber of measurement devices (m,n) such that K(m,n)

G >√2 is not known. The

smallest known Bell inequality with a quantum violation larger than√2 is from

the case (m,n) = (14, 11) with a quantum violation of 1.419507 >√2 and was

shown in 2008 by Vértesi [7]. This improves the previous smallest known case with aquantum violation lager than

√2 with (m,n) = (20, 20) and a quantum violation of

10/7 ≈ 1.428571 >√2.

4.2 Our Results

For any tight Bell inequality we consider the following characteristic key quantities:

symbol meaningk maximal quantum violation

rrank of corresponding Gram matrixfrom the primal problem

onumber of vertices on which the Bellinequality is satisfied with equality

Figure 4.1: The characteristic key value quantities

As in the previous section we use the shorthand notation k for kM the maximalquantum violation for the Bell inequality alongside of k.

The rank r gives an upper bound for the minimal dimension of the vectorsVi, Wj ∈ Rrank(G) from Proposition 2.3 to obtain the maximal quantum violation (seeRemark 2.10). The rank was obtained by computing (numerically) the eigenvaluesof each Gram matrix and counting the non zero (> |10−5|) eigenvalues.

– 28 –

4. ResultsThe characteristic key quantity o is the number of vertices in which a certain

Bell inequality is satisfied with equality and represents how high the symmetry of acertain inequality is. The minimal needed number of vertices is d = m n to describea facet (i.e. inequality) of the Bell polytope.

4.2.1 Case (m,n) = (4, 4)

The both new inequalities are beyond the tight Bell inequalities from the case(m,n) = (2, 2):

1

10

2 −2 1 1−2 −1 2 11 2 1 21 1 2 −1

k = 1.3514r = 2o = 16

1

6

2 1 −1 01 −1 1 1−1 1 −1 10 1 1 0

k = 1.36083r = 4o = 24

Both new inequalities have an maximal quantum violation smaller then√2.

Therefore K(m,n)G =

√2 holds. The largest achievable quantum violation in the case

when Alice and Bob each use 4 measurement devices can be obtained by using theCHSH setting and ignore two measurement devices on each side. Furthermore it isremarkable that one of the new tight Bell inequalities above only has rank 2 and notrank 4.

4.2.2 Case (m,n) = (4, 5)

Computing the halfspace representation of the case (m,n) = (4, 5) is way morecomplicated. Where the computation of the halfspace representation from the case(m,n) = (4, 4) can be done in a reasonable runtime on a ‘standard’ PC (about 8to 16 hours using mplrs, depending on the used hardware), the case (m,n) = (4, 5)would take approximate 60 days on a ‘standard’ PC. Therefore the cluster systemfrom the Leibniz Universität Hannover was used. With 160 cores these computationcan be done in about 3 days. After removing the duplications up to symmetries thereare 4 new tight Bell inequalities beyond the already known tight Bell inequalitiesfrom the case (m,n) = (4, 4). Also here we can solve the optimization problem usingsemidefinite programming.

– 29 –

4. Results

1

8

−1 1 0 1 1−1 −1 2 1 −11 1 2 −1 1−1 1 0 −1 −1

k = 1.36931r = 4o = 32

1

15

−1 2 1 1 11 −1 1 1 −22 1 1 −1 10 2 −1 −1 −2

k = 1.354r = 3o = 24

1

6

1 0 0 0 11 0 1 1 −11 −1 0 −1 −11 1 −1 0 −1

k = 1.39443r = 4o = 40

1

10

1 2 1 2 0−1 −1 2 1 −11 1 2 −2 0−1 2 −1 −1 −1

k = 1.35485r = 3o = 26

The Grothendieck constant in the case (m,n) = (4, 5) is therefore K(4,5)G =

√2.

After we had found these 4 new tight Bell inequalities, we found that they agreewith the result from Avis [10] and also agree with the result from Dutour Sikirić [22]

4.2.3 Case (m,n) = (4, 6)

At this case only a partial result could be obtained using mplrs because the runtime isnot reasonable (the approximated runtime for the problem is about 200 days on 320cores). After removing the duplications up to symmetries the partial result containsthe following four new tight Bell inequalities beyond the tight Bell inequalities fromcase (m,n) = (4, 5):

1

10

−1 1 −1 0 −1 21 1 0 −2 −1 −1−1 −1 1 −1 −1 1−1 1 0 1 −1 −2

k = 1.36432r = 3o = 38

1

10

−1 1 0 1 −1 21 1 0 −2 −1 −1−1 −1 −1 −2 0 1−1 1 −1 1 0 −2

k = 1.35647r = 4o = 44

– 30 –

4. Results

1

10

−1 1 −1 1 −1 11 1 1 −1 −1 1−1 −1 −1 −2 −1 0−1 1 1 0 −1 −2

k = 1.29615r = 4o = 32

1

10

−1 2 −1 1 0 11 2 1 −2 0 0−1 −1 −1 −2 −1 0−1 1 1 1 −1 −1

k = 1.3711r = 3o = 40

That this is already the complete list of all new tight Bell inequalities for thecase (m,n) = (4, 6) which can be proven using the orbit representation from the dualCUTP4,6 Dutour Sikirić published on his website [22]. Therefore the Grothendieckconstant for this case is K(4,6)

G =√2 as well.

4.2.4 Case (m,n) = (4, 7)

For this case the runtime of mplrs would not be reasonable. These Bell inequalitieswere therefore computed from the orbit representation of the dual CUTP4,7 polytopepublished by Dutour Sikirić [22]. Beyond the tight Bell inequalities from the case(m,n) = (4, 6) 3 following new tight Bell inequalities were found:

1

10

0 −1 −1 1 −2 0 10 1 0 −1 −2 −1 −1−1 1 0 −1 −1 1 11 −1 1 −1 −1 0 1

k = 1.38219r = 3o = 64

1

10

0 −1 −2 1 −1 0 10 1 −1 −1 −1 −1 −1−1 1 0 −1 −1 1 11 −1 1 −1 −1 0 1

k = 1.36808r = 3o = 56

1

8

0 0 −1 1 −1 0 10 1 −1 −1 0 −1 0−1 0 0 −1 0 1 11 −1 0 −1 −1 0 0

k = 1.32288r = 4o = 56

The maximal quantum violation of all 3 new Bell tight inequalities is smallerthan

√2 which implies that the Grothendieck constant is K(4,7)

G =√2.

– 31 –

4. Results4.2.5 Case (m,n) = (5, 5)

At this case the number of inequalities dramatically increases. There are 1274 newtight Bell inequalities beyond the inequalities from the case (n,m) = (4, 5). Atthis amount of data, the runtime of mplrs is not reasonably any more. For thiswork a partial result for the case (m,n) = (5, 5) was computed using the clustersystem of the Leibniz Universität Hannover. The partial result already containsover 15000 tight Bell inequalities, after removing the duplications up to symmetriesabout 800 different tight Bell inequalities were found. The tight Bell inequality withthe largest quantum violation from this partial result, besides the CHSH inequality is:

1

8

0 1 1 −1 −11 1 1 0 10 0 −1 −1 0−2 1 0 1 01 1 −1 1 0

k = 1.38794r = 3o = 46

Dutour Sikirić also published the complete orbit representation of the dualpolytope from the CUTP5,5 on his website. As described in Remark 2.7 we cantransform the orbit representation into the halfspace representation of the CUTP5,5

and compute from this, as described in Remark 2.2, the halfspace representation ofthe Bell polytope LC5,5. Which gives us a complete list of the tight Bell inequalitiesin the case (m,n) = (5, 5).

The following inequality has the largest quantum violation besides the CHSHBell inequality

1

8

−1 1 1 0 −10 1 1 −1 1−1 0 −1 −1 1−1 1 −1 1 01 1 0 1 1

k = 1.39754r = 4o = 50

There exist only 3 Bell inequalities with rank r = 5, which are:

1

12

−1 1 −1 −1 −20 0 1 −1 00 0 −2 −2 4−1 −1 1 1 20 2 1 1 2

k = 1.32288r = 5o = 48

– 32 –

4. Results

1

8

−1 1 0 −1 −11 1 2 −1 11 1 −2 −1 1−1 0 0 0 10 1 0 1 0

k = 1.36931r = 5o = 64

1

12

−2 0 −1 −1 00 0 1 −1 02 2 −2 −2 2−1 −1 1 1 2−1 3 1 1 0

k = 1.32288r = 5o = 48

The Grothendieck constant in this case is therefore, attained by the CHSHinequality K(5,5)

G =√2. At next, a deeper look at the distribution of the maximal

quantum violation from each Bell inequality is taken.

0

10

20

30

40

50

60

70

1.2 1.25 1.3 1.35 1.4 √2 1.45

Num

ber

of B

ell I

nequ

alit

ies

Quantum Violation k

Figure 4.2: In the histogram the number of tight Bell inequalities from the case (m,n) =(5, 5) with a certain quantum violation is shown, the width of the bars is 0.002, the largestquantum violation of

√2 is obtained by the CHSH inequality.

It is remarkable that the second largest maximal quantum violation of a Bell

– 33 –

4. Resultsinequality is even smaller than 1.4. The average maximal quantum violation is1.3242.

There are just three tight Bell inequalitieswith rank 5 (see above). All other tightBell inequalities in the case (m,n) = (5, 5)have a smaller rank. And the matrix withthe largest quantum violation beside theCHSH inequality has only rank 4. Theviolation of the three Bell inequalities withrank 5 is near the average in the Figure4.2.

rank number of Bell inequalities5 34 293 8742 3751 1

Figure 4.3: The number of Bellinequalities with a certain rank.

0

20

40

60

80

100

120

140

160

180

200

0 12 24 36 48 60

Num

ber

of B

ell I

nequ

alit

ies

prefactor

Figure 4.4: The distribution of the prefactors of the tight Bell inequalities.

Beside the prefactor of the trivial Bell inequality, which is 1, the prefactors of allother tight Bell inequalities are even numbers. The largest occurring prefactor is 58.

– 34 –

4. Results

0

50

100

150

200

250

0 12 24 36 48 60

num

ber

of B

ell I

nequ

alit

ies

number of Bell inequalitys that satisfy with equality per vertex

Figure 4.5: This shows the distribution if the number of vertices that lie on each tightBell inequality i.e. ‘o’.

The minimal needed number of vertices to describe a Bell inequality via the orbitrepresentation (see Remark 2.7) in this case is given by d = n m, for this case is d= 25). The most Bell inequalities are described by a small number of facets. Thetrivial Bell inequality is satisfied with equality on 256 vertices, the CHSH inequalityis satisfied with equality on 80 vertices.

– 35 –

CHAPTER 5

Résumé

In the search for the answer what the largest achievable violation for correlationinequalities is, by using quantum states, we computed the halfspace representationi.e. the tight Bell inequalities of the Bell polytopes LC4,4, LC4,5, LC4,6, LC4,7 andLC5,5.

For the cases (m,n) = (4, 4), (4, 5) we computed all tight Bell inequalitiesand for the cases (m,n) = (4, 6), (5, 5) we computed a partial list of tight Bellinequalities via the convex hull algorithm mplrs with the cluster system from theLeibniz Universität Hannover and removed all duplications up to symmetries with anautographic algorithm. The main difficulty is that the runtime needed to computethe halfspace representation of a Bell polytope LCm,n seems to increase exponentiallywith the dimension d = m n. This makes it nearly impossible to compute thehalfspace representation in high dimensions.

For the cases (m,n) = (4, 6), (5, 5), (4, 7) we transformed the orbit representationof the CUTP4,6, CUTP4,7 and CUTP5,5 which Dutour Sikirić published on his website[22] to the halfspace representation of the Bell polytopes LC4,6, LC4,7 and LC5,5 whichgives the complete list of tight Bell inequalities for that cases. From those Bellinequalities the maximal quantum violation was computed along with the rank of thecorresponding Gram matrix and the number of vertices on which the Bell inequalityis satisfied with equality.

Our end result is that in all considered cases the maximal quantum violation is√2

only for the CHSH inequality, and smaller than 1.4 for all other tight Bell inequalities.So it was proven in the scope of the thesis, that the Grothendieck constants isK

(4,4)G = K

(4,5)G = K

(4,6)G = K

(4,7)G = K

(5,5)G =

√2, which was as far as we know not

known yet. On the minimal (m,n) such that K(m,n)G >

√2 is (m,n) = (11, 14) still

the smallest known case [7].Thus, further research on the smallest number of measurement devices needed on

each side to obtain a quantum violation larger√2 has to be done. This is also an

interesting question in the context of Grothendieck’s inequality. Further research onthe question if it is possible to use the high symmetry of Bell polytopes to reducethe computation time has to be done.

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