Optimization of statistical parameters of Eberhard inequality

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Master Thesis

Optimization of statistical parameters of Eberhard inequality

Author: Polina Titova

Supervisor: prof. Andrei Khrennikov Examiner: Dr. Astrid Hilbert Date: 2015-02-02

Course Code: 5MA11E Subject: mathematics

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Chapter 1 Introduction

One of the foundations of classical mechanics is the principle of local realism. Locality stands that any object can be aected only by its nearest local space and there is no action that can be transmitted faster than the speed of light. Realism refers to the philosophical position that the observed values are properties of objects, that is, they are objective and can be attributed to the physical systems before the measurement. These postulates are not only conrmed by classical physical experiments, intuitively they are plausible assumptions on the real world.

However, modern quantum mechanics is questioning the adequacy of the model of local realism for the description of the microcosm. According to the Heisenberg uncertainty principle there is a fundamental limit to the accuracy for the joint measurement of the position of a quantum particle and its momentum (as well as any other joint measurement of two observables, described by non-commuting operators). It means that for a quantum system, the values of position and momentum cannot be be assigned to it before the measurement. As a consequence of the Heisenberg principle, in quantum world there is no place for realism.

However, this position was criticised by Einstein, Podolsky and Rosen(EPR paradox)[8]. They showed using quantum mechanical formalism that it is possible to come to a contradiction with the above principle. They presented a hypothetical experiment such that one can accurately determine the position and momentum of a particle, taking measurements for particles in the entangled state, located far enough from each other. The discussion of this paradox raised the question of possible incompleteness of the quantum theory. Perhaps, quantum mechanics does not completely describe the state of the system, and there are still some unknown hidden parameters.

Another explanation of the paradox is the rejection of the locality principle.

Further evidence of contradiction between classical and quantum models of real world was given by Bell and other authors in the form of statistical inequalities which can be tested experimentally. At the very beginning the question about the coexistence of locality and realism with quantum theory seemed to be a philosophical discussion about the nature of reality, but with the aid of the Bell inequality it was formulated using mathematical formalism and then tested experimentally. Despite the fact that violations of Bell inequalities predicted by quantum mechanics were observed in most experiments [2], the process of verication is still not completed [18].

The main reason for performing further tests is that as any experiment, Bell's test cannot be performed with 100% eciency of detectors and without any noise in the detector response. An example of inequality that takes into account these types of errors is the Eberhard inequality derived in the article [7]. On its basis a number of experiments such as [19](see also [22, 6] on recent experimental studies on Bell test) were conducted, they also revealed the violation of the predictions of the classical theory.

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The main diculty in such experiments is that violations of inequality require a fairly high eciency of detectors, special initial state of a quantum system and installation angles of polarization beam splitters. These parameters should be such that the inequality is violated as much as possible. To achieve this result Eberhard used a simple brute-force optimization.

This work is devoted to mathematical modeling of the parameters of the Eberhard inequality using optimization techniques. One of the main purposes of this study is to consider the more general case when the detectors have dierent eciencies. In addition, in the article [7] the objective function has the meaning of a mathematical expectation. However, it is also useful to investigate the possible levels of variability of the results, expressed in terms of standard deviation. In this paper we consider the optimization of parameters for the Eberhard inequality using coecient of variation taking into account possible errors in the setup of angles during the experiment. We also model random uctuations of experimental settings, angles of polarization beam splitters. These uctuations also contribute to uncertainty in magnitude of the objective function. This problem of sensitivity of the degree of violation of the Eberhard inequality to the precision in the control of angles of polarization beam splitters is very important for experimenters.

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Chapter 2 Required elements of functional analysis

2.1 Hilbert spaces and linear operators on it

Here we present some basic denitions of functional analysis that can be used in quantum mechanics formalism.

Our presentation will be brief and not detailed, see, e.g. [27], for details.

Denition 2.1.1. For the complex vector space V an inner (scalar) product h·, ·i is a C-valued map on V ×V → C that satises the following conditions:

1. Conjugate symmetry - hx, yi = hy, xi,

2. It is positively dened - hx, yi ≥ 0 and hx, yi = 0 ⇔ x = y,

3. Linearity - hx, ayi = ahx, yi, where a ∈ C, and hx, y¹+ y₂i = hx, y¹i + hx, y²i.

Denition 2.1.2. A Hilbert space H is a complex vector space with an inner product such that the space H is complete with respect to the norm kxk = phx, xi

The space H = Cⁿ with the inner product dened by hψ, ϕi = Pⁿi=1ψ_iϕ_i is Hilbert space. A well-known example of a Hilbert space H = L2(Rⁿ, dx)of square integrable function with respect to the Lebegue measure. In this space the inner product is dened by the equality hψ, ϕi = R_Rnψ(x)ϕ(x)dx.

Another important notion which is widely used in quantum mechanics is the notion of a linear operator.

Denition 2.1.3. Linear operator A on a Hilbert space H is a map H → H that has the following properties:

1. ∀ψ¹, ψ2∈ H : A(ψ¹+ ψ2) = Aψ1+ Aψ2

2. ∀λ ∈ C, ψ ∈ H : A(λψ) = λAψ

To simplify further considerations let us assume that the Hilbert H space has nite dimension.

Denition 2.1.4. For a linear operator A, its adjoint operator A^∗is dened with the aid of the equality hAψ, ϕi = hψ, A^∗ϕi, ψ, ϕ ∈ H. Operator A is called self-adjoint if A^∗= A.

Denition 2.1.5. Let V be a vector space over C and let A be an operator on V .

1. A scalar λ ∈ C is an eigenvalue of A if there exists a nonzero vector ψ ∈ V such that

Aψ = λψ.

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2. Such a vector ψ is called an eigenvector of A.

3. In some cases there are several dierent eigenvectors associated with the same eigenvalue λ. The eigenvectors to a given eigenvalue λ together with the zero vector form a subspace of V called the eigenspace of λ.

4. The set of all eigenvalues of an operator is called the spectrum.

For self-adjoint operators in a Hilbert space the following theorem holds.

Theorem 2.1.1. For every self-adjoint operator A in a complex Hilbert space H:

1. All its eigenvalues are real,

2. In H there is an orthonormal basis consisting of its eigenvectors.

Denition 2.1.6. A self-adjoint operator is called positive if it has only non-negative eigenvalues in its spectrum.

Another useful operator on Hilbert space is a projector operator.

Denition 2.1.7. Let H0 be a linear subspace of H. The operator π : H → H0 is called a projector if π^∗= π and π²= π. It projects H orthogonally onto H0= π(H).

Let {e¹, . . . , en} be a basis in H and {e¹, . . . em} a basis in H⁰. If ej ∈ H⁰ then πej = ej and if ej ∈ H0^⊥ then πej = 0. For an arbitrary element ψ ∈ H, the projection operator acts as:

πψ =

m

X

i=1

z_ie_i, where zi=heⁱ, ψi.

2.2 Tensor product

Let H1 be a vector space with an orthogonal basis system {e¹, . . . en} and let H² be another vector space with an orthogonal basis {f¹, . . . fm}. We will use the formal symbol ⊗ to construct a new orthogonal basis from the set of ordered pairs (ei, fj)in a new vector space:

{eⁱ⊗ f^j|eⁱ∈ H¹, f_j∈ H²}. (2.2.1)

Denition 2.2.1. The set of formal sums of the form

ψ =X

i,j

zi,jei⊗ f^j, zi,j∈ C (2.2.2)

with the naturally dened operation of addition and multiplication by scalars is called the tensor product of H1 and H2 and denoted by the symbol H = H1⊗ H².

If H1 and H2 are Hilbert spaces then H = H1⊗ H²is also a Hilbert space with the inner product dened by

hψ, ϕi = hX

i,j

zi,jei⊗ f^j,X

k,l

vk,lek⊗ f^li =X

i,j

X

k,l

zi,jvk,lheⁱ⊗ f^j, ek⊗ f^li,

where a scalar product of basis vectors is dened by

heⁱ⊗ f^j, ek⊗ f^lidef= heⁱ, eki · hf^j, fli = δ^i,kδj,l.

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For the elements ψ1, ψ2 from the spaces H1 and H2, correspondingly, the tensor product is given by

ψ1⊗ ψ²= X

i

xiei

!

⊗



 X

j

yjfj



def= X

i,j

xiyj· eⁱ⊗ f^j ∈ H¹⊗ H². (2.2.3)

Elements of H = H1⊗ H²that can be written in the form (2.2.3) are called factorizable elements. But space the H does not consist only of factorizable elements. Here we construct an example of a vector that cannot be written in the form (2.2.3).

Consider Hilbert space H1⊗ H², where Hi are Hilbert spaces with the basis {e¹, e2}. Let us show that the following quantum state ψ is not factorizable

ψ = e1⊗ e²+ e2⊗ e¹

2 6= ψ¹⊗ ψ² for any pair ψ1, ψ2.

Proof. Let us assume that ψ can be represented by ψ = ψ1⊗ ψ², where ψ1and ψ2can be written as weighted sums of basis vectors:

ψ₁= a₁e₁+ a₂e₂, a_i ∈ C, |a¹|²+|a²|²= 1 ψ2= b1e1+ b2e2, bi∈ C, |b¹|²+|b²|²= 1.

Such a ψ can be represented by the following equations:

ψ₁⊗ ψ² = (a₁e₁+ a₂e₂)⊗ (b¹e₁+ b₂e₂) =

= a1b1· e¹⊗ e¹+ a1b2· e¹⊗ e²+ a2b1· e²⊗ e¹+ a2b2· e²⊗ e². One should solve the following system of equations to nd coecients a1, a2, b1, b2 satisfying:











a1b2=¹₂ a2b1=¹₂ a1b1= 0 a₂b₂= 0.

The system of equations is inconsistent, therefore ψ is not factorizable.

Denition 2.2.2. Consider two linear operators A1: H1 → H¹ and A2: H2 → H². The tensor product of these operators is dened by the equality

(A₁⊗ A²)(ψ₁⊗ ψ²) = A₁ψ₁⊗ A²ψ₂.

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Chapter 3 The mathematical formalism of quantum mechanics

3.1 Postulates of quantum mechanics

The mathematical formalism of quantum mechanics can be formulated as a list of postulates [17] (see also [16]) based on the theory of self-adjoint operators on a complex Hilbert space H. We will use the term quantum system to describe a part of physical universe which is selected for the analysis using the formalism of quantum mechanics.

The following list of postulates and principles is called Copenhagen interpretation of quantum mechanics.

Postulate 1: A quantum state ψ is a vector of the complex Hilbert space such that hψ, ψi = 1. This vector completely describes the state of the quantum system. Two vectors which dier by a factor of γ = e^iϕ describe the same quantum state.

Postulate 2: A physical observable a is represented with the aid of a self-adjoint operator A in H. Dierent observables are represented by dierent operators.

Postulate 3: If an observable is represented by the operator A then results of observations are given by the spectrum of A.

Any self-adjoint operator can be written in the form A =X

m

a_mπ_m^A,

where πm^A is an orthogonal projector onto an eigenspace corresponding to the eigenvalue am.

Postulate 4: Born's rule - If an observable is represented by the operator A then the probability to obtain an eigenvalue amafter measurement can be calculated by using the formula

P (A = am) =kπ^Amψk².

The eigenvectors of A for a particular eigenvalue am form a basis of a subspace Hm of the Hilbert space H:

H_m={ϕ : Aϕ = a^mϕ}. Then the space H can be represented as H = H¹⊕ H²⊕ . . . H^k. Projector π^Am: H → H^m equals

π^A_m=

nm

X

l=1

hψ, ϕ^mliϕ^ml,

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where ϕml - l-th eigenvector from Hm.

Postulate 5: Consider a quantum state ψ and a self-adjoint operator A. Then after a measurement with a result A→ a^mthe quantum state ψ collapses to state ψm:

ψ_m= π^A_mψ kπ^Amψk.

Postulate 6: The time evolution of the quantum state ψ is described by the Schrodinger equation

i~d

dtψ(t) =Hψ(t)

with the initial condition ψ(0) = ψ0, where H is a self-adjoint positive operator representing the energy of the system(Hamiltonian).

Postulate 7: If there are two quantum systems with Hilbert state spaces H1 and H2, then the state space of the compound system is given by H1⊗ H².

Denition 3.1.1. If a state ψ ∈ H1⊗ H² is not factorizable in the form ψ = ψ1⊗ ψ², where ψ1∈ H¹, ψ₂∈ H² then it is called entangled.

Consider 2-dimensional Hilbert spaces H1= H2 the basis {e¹, e2}. One of the examples of entanglement is

ψ = e1⊗ e²+ e2⊗ e¹

2 .

It was shown before that in that state ψ cannot be factorizable as ψ = ψ1⊗ ψ². Another example is ψ = e1⊗ e²+ e1⊗ e¹+ 2e2⊗ e¹.

Proof. Let us assume that ψ is represented by ψ = ψ1⊗ ψ², where ψ1and ψ2can be written as a weighted sum of basis vectors:

ψ₁= a₁e₁+ a₂e₂, a_i ∈ C, |a¹|²+|a²|²= 1 ψ2= b1e1+ b2e2, bi∈ C, |b¹|²+|b²|²= 1.

By using the rules of tensor multiplication of vectors we have:

ψ₁⊗ ψ² = (a₁e₁+ a₂e₂)⊗ (b¹e₁+ b₂e₂) =

= a1b1· e¹⊗ e¹+ a1b2· e¹⊗ e²+ a2b1· e²⊗ e¹+ a2b2· e²⊗ e². To nd coecients a1, a2, b1, b2 one should solve the following system of equations:











a₁b₂=^√¹

6

a2b1=^√²

6

a1b1=^√¹

6

a₂b₂= 0 ,

The system is inconsistent. Therefore the assumption is false and ψ is not factorizable.

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3.2 Density operator

To describe the behaviour of an ensemble of quantum states, where each state is obtained with some probability, we will use the notion of a density operator, see, e.g., [17,16].

For a pure state ψ we can dene an orthogonal projection operator: Pψ onte the state ψ: Pψϕ =hψ, ϕiψ. It has following properties:

1. Pψ is hermitian.

Proof. hP^ψϕ, vi = hhψ, ϕiψ, vi = hψ, ϕihψ, vi = hψ, vihϕ, ψi = hϕ, hψ, viψi = hϕ, P^ψvi

2. Pψ≥ 0.

Proof. hPψϕ, ϕi = hψ, ϕihψ, ϕi = |hψ, ϕi|²≥ 0.

3. Tr Pψ= 1, where Tr A = P_khAe^k, eki and {e^k} is an orthogonal basis.

Proof. Tr Pψ=Pn

k=1hhψ, e^kiψ, e^ki =P

khψ, e^kihψ, e^ki =P

kψ_k²= 1. 4. Pψ²= Pψ.

Proof. Pψ²ϕ =hψ, hψ, ϕiψiψ = hψ, ϕihψ, ψiψ = hψ, ϕiψ = P^ψϕ.

For an ensemble of states the density operator is dened by

ρ =X

i

p_iP_ψ_i,

where pi> 0, P

ipi= 1, represents the probability to obtain ψi after a measurement.

One can easily show that the operator ρ satises properties 1-3 (using corresponding properties 1-3 for each operator Pψ_i). In the general case the property 4 of a projection operator is violated. For example, let us take the density operator ρ = 1

2P_ψ₁+ 1

2P_ψ₂, where ψ1= (1, 0)^T and ψ2= (0, 1)^T. Let us nd the square of this operator:

ρ²ϕ =1

2hψ¹, ρϕiψ¹+1

2hψ², ρϕiψ²=

= 1 2hψ¹,1

2hψ¹, ϕiψ¹+1

2hψ², ϕiψ²iψ¹+1 2hψ²,1

2hψ¹, ϕiψ¹+1

2hψ², ϕiψ²iψ²=

=1

4hψ¹,hψ¹, ϕiψ¹iψ¹+1

4hψ¹,hψ², ϕiψ²iψ¹+1

4hψ²,hψ¹, ϕiψ¹iψ²+1

4hψ²,hψ², ϕiψ²iψ². The nal result is:

ρ²ϕ = 1

4hψ¹, ϕiψ¹+1

4hψ², ϕiψ²= 1 2ρ6= ρ.

Density operator ρ can also be written in the form ρψ = P_ip_iheⁱ, ψieⁱ.

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3.3 Elements of quantum probability theory

In quantum mechanics the result of a measurement depends on the state in which the system occupied before it collapsed to some pure state. It means that values of quantum probabilities, expectations, standard deviations etc.

depend not only on an observable, but also on the initial state of a system described by a density operator.

Let A be an observable with eigenvalues {a¹, . . . an} and eigenvectors {f¹, . . . fn}, where ψ describes the state of the system. Then the probability to obtain ai after a measurement is given by

P (A = a_i) =|hψ, fⁱi|²= Tr ρ_ψA_i, where Ai=|fⁱihfⁱ|.

Proof. Let us start on the right hand side of the chain of equalities. By denition we have:

Tr ρψAi=X

j

hρ^ψAifj, fji.

Then, using the denition of A:

X

j

hρ^ψA_if_j, f_ji = hhρ^ψ, f_ii, fⁱi = hhψ, fⁱiψ, fⁱi =

=hψ, fⁱihψ, fⁱi = |hψ, fⁱi|².

For the mixed state, by denition the probability is given by the same formula

P (A = ai) = Tr ρ|fⁱihfⁱ|. (3.3.1)

Quantum expectation value of A onto the density operator ρ is dened by the following formula

Aρ≡ hAρ, ρi = Tr ρA. (3.3.2)

Sometimes the quantum expectation operator of A is written as hAi.

The quantum dispersion is dened in the same way it is dened in the classical probability theory:

σ²_A_ρ = Aρ− A^ρ²

= A²_ρ− A^ρ². (3.3.3)

3.4 Heisenberg's uncertainty principle

The uncertainty principle of quantum mechanics states that there is a limit to the precision with which some of physical parameters of one system can be known simultaneously. This limit does not depend on the precision of used measurement devices or on the level of technology, it is fundamental.

For the rst time the uncertainty principle was formulated by Werner Heisenberg in 1927. He discovered that the more precisely the position of a particle can be measured, the less precisely its momentum can be determined.

In 1928 it was formulated as an inequality:

σ_xσ_p≥ ~ 2,

where σx is the standard deviation of position, σp is the standard deviation of momentum and ~ is Planck's constant.

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The most general form of the uncertainty principle is given by the Schrodinger inequality [29]:

σ_A²σ_B² ≥ 1

2h{A, B}i − hAihBi

2

+ 1

2ih[A, B]i

2

, (3.4.1)

where A and B are self adjoint operators, [A, B] = AB − BA - commutator operator and {A, B} = AB + BA - anti-commutator operator.

Proof. For the derivation of the inequality we will use Cauchy-Schwartz inequality for scalar products:

|hf, gi|²≤ hf, fihg, gi. (3.4.2)

The quantum dispersion σA², that depends on a state ψ, of a self-adjoint operator A can be found using the following formula:

σ²_A=h(A − hAi)²ψ, ψi = h(A − hAi)ψ, (A − hAi)ψi.

We denote f = (A − hAi)ψ and g = (B − hBi)ψ, then for the left-hand side of (3.4.2) we obtain:

|hf, gi|²=|h(A − hAi)ψ, (B − hBi)ψi|²=|h(B − hBi)(A − hAi)ψ, ψ|²=

=|hBAψ, ψi − hBihAi − hAihBi + hAihBi|²=|hBAi − hAihBi|². As any other complex number the left-hand side of the inequality can be written as

|hf, gi|²= 1

2(hf, gi + hf, gi)

2

+ 1

2i(hf, gi − hf, gi)

2

, where hf, gi is given by

hf, gi = hBAψ, ψi − hAihBi = hψ, BAψi − hAihBi = hABψ, ψi − hAihBi.

For the right-hand side, recalling the denition of the quantum dispersion, we obtain:

hf, fihg, gi = σ²Aσ²_B.

Putting it all together to the inequality (3.4.2) we obtain the Schrodinger inequality (3.4.1):

1

2(hBAi − hAihBi + hABi − hAihBi)

2

+

1

2i(hBAi − hAihBi − hABi + hAihBi)

2

≤ σ²Aσ_B²

⇔ 1

2h{A, B}i − hAihBi

2

+ 1

2ih[A, B]i

2

≤ σ²Aσ_B².

In quantum mechanics the position operator is dened by the formula:

(ˆxf ) = x· f(x) and momentum operator is dened by

(ˆpf ) = ~ i

df dx.

It the general case the Robertson inequality [26] can be derived from (3.4.1):

σ_A²σ²_B≥ 1

4|h[A, B]i|².

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The commutator operator of position and momentum is

[ˆx, ˆp]f (x) = (ˆxˆp− ˆpˆx)f(x) = x~ i

df (x) dx −~

i dxf (x)

dx = i~If (x),

where I is the identity operator. Then in this case the Robertson inequality becomes the Heisenberg inequality:

σ_A²σ²_B≥ 1 4~².

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Chapter 4 Einstein-Podolsky-Rosen paradox

The Einstein-Podolsky-Rosen paradox(EPR-paradox, [8]) was published in 1935 as the criticism of some statements of the Copenhagen interpretation of quantum mechanics. The main principle of this interpretation states that for any quantum system the wave function presents the nest possible description of its state for any given time and after a measurement it collapses to one of its eigenstates with some probability. Moreover, Heisenberg's uncertainty principle means that the position of a particle and its momentum cannot be measured simultaneously with arbitrarily good precision. Einstein and his colleagues [8] did not agree with probabilistic measurement outcomes.

The article [8] is based on a thought experiment which can be presented as follows. Consider two systems S1, S₂, both with the state space L2([0, 1])and a source which produces pairs for these systems. The state of both systems S = (S1, S₂)can be represented by a quantum state from the tensor product of its component Hilbert spaces. Take a self-adjoint operator A with eigenvalues {ak} and corresponding eigenvectors {ϕ^k(x₁)} which are functions of one variable x1. It denes an observable on S1. Consider also a self-adjoint operator B with eigenvalues {b^k} and eigenvectors {ψ^k(x1)}. It denes another observable on the same system.

In the general case of two Hilbert spaces H1 and H2 and with orthonormal bases {e^k} and {f^k}, respectively, an arbitrary state ϕ ∈ H¹⊗ H² can be represented as:

ϕ =X

k,m

c_k,me_k⊗ f^m.

If H1= H₂= L₂([0, 1]), then their tensor product is L2([0, 1])⊗ L²([0, 1]) = L([0, 1]²). For this space any state ϕ can be represented by the sum:

ϕ(x1, x2) =X

k,m

ck,mek(x1)fm(x2), (4.0.1)

or, for the continuous case, it has the form:

ϕ(x₁, x₂) = Z

u(x, x₁)v(x, x₂)dx. (4.0.2)

Suppose we want to execute a measurement of A. Before this the particle is assumed to have the state ψ(x1, x2) =P

kvk(x2)ϕk(x1)where vk(x2) =P

mck,mfm(x2)from (4.0.1) . According to the projection postulate after the measurement the state of the system will collapse to a pure state ψ = ϕm(x1)vm(x2). This implies that at the same time the second system will also have the determinate state vm(x2). But if two detectors are situated far enough from each other(so they cannot have an impact on each other according to the locality principle), then the second system has to have such a state vm(x₂)even before measurement.

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Suppose after all that we have changed our decision and we want to measure B on S1, instead of A. By analogy with the previous consideration, after the measurement the state of the system will collapse to the state ψ = ζ_n(x₁)u_n(x₂) for some n ∈ N and in this case the second system will have the state uⁿ(x₂). It has to have such state independently of the measurement on the rst system i.e. it stands in this state even before any measurements.

By taking into account this thought experiment we can conclude that the second system has 2 wave-functions vm(x2) and un(x2). The paradox is that we can construct such states that such an assignment is impossibel according to the Heisenberg's principle. Here we present the example from the original article [8].

Consider the state ψ(x1, x2) =R e^ip^~^(x¹^+x²^−x⁰⁾dp, where x0is a constant. On the one hand, it can be represented by the following formula:

ψ(x1, x2) = Z

ϕ(p, x1)v(p, x2)dp, where

ϕ(p, x₁) = e^ip^~^x¹ and

v(p, x2) = e^ip^~^(x²^−x⁰⁾. The momentum operator is dened by ˆp = ~

i d

dx. Its eigenfunction is ψ = eîλ^~^x for eigenvalue λ ∈ R. After the measurement of A on the rst system its state will collapse to a particular eigenfunction ϕ(p, x1) = eîp^~^x¹. The state of the second system v(p, x2) = eîp^~^(x²^−x⁰⁾ is an eigenfunction of the momentum operator, corresponding to the eigenvalue −p.

On the other hand, the state under consideration can be represented as ψ(x1, x2) = δ(x1+ x2− x⁰) = ~R δ(x − x₁)δ(x− x²+ x₀)dx =R ζ(x, x1)u(x, x₂)dx.

The position operator for the second system is dened by ˆx₂f = x₂f. Its eigenfunction is δ(x − λ) for the eigenvalue λ. After the measurement of B on the rst system its state will collapse to the eigenfunction ζ(x, x1) = δ(x−x¹). The state of the second system u(x, x2) = δ(x−x²+ x₀)is also an eigenfunction of the position operator, corresponding to the eigenvalue x + x0.

At this point the authors conclude that both position and momentum of the second system are elements of reality, since they couldn't be aected by the measurements on the rst system. But Heisenberg's principle demands that they cannot be known both i.e. they cannot both be elements of reality simultaneously.

The following question arose as a consequence of this paradox. Does quantum mechanics provide a complete description of the physical reality? Another possible explanation of this paradox can be found by rejecting locality.

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Chapter 5 Bell inequalities

One of the possible solutions of the EPR-paradox is that the quantum mechanics theory is not complete, the actual state of a system is described not only by its quantum state ψ but also by some hidden, i.e. yet unknown, variables.

In this case all probabilistic predictions of quantum mechanics can be explained by the existence of some unknown degrees of freedom. Using that assumption one can conclude that reality can still have a deterministic nature as well as a probabilistic one.

John Bell in his article [4] assumed that there are hidden parameters ω and all measurement results are random variables depending on them. He formulated a statistical inequality that contradicts to the quantum mechanical predictions.

The Bell inequality is formulated in the following theorem. For two classical random variables ξ and η we set

hξ, ηi = Z

Ω

ξ(ω)η(ω)dP (ω).

Theorem 5.0.1 (Bell inequality). Let ξa(ω), ξb(ω), ξc(ω)be discrete random variables taking the values ±1. Then the following inequality holds:

|hξ^a, ξbi − hξ^c, ξbi| ≤ 1 − hξ^a, ξci.

Proof. We have:

|hξ^a, ξ_bi − hξ^c, ξ_bi| = Z

Ω

ξ_aξ_bdP − Z

Ω

ξ_cξ_bdP

= Z

Ω

(ξ_a− ξ^c)ξ_bdP . After multiplying the last formula by ξa²= 1we have:

Z

Ω

(1− ξ^aξ_c)ξ_aξ_bdP . And nally, using that |ξⁱ| = 1 we get the desired result:

Z

Ω

(1− ξ^aξc)ξaξbdP ≤

Z

Ω

(1− ξ^aξc)dP

= 1− hξ^aξci.

After Bell's article some other inequalities were derived. For example, the Wigner inequality is more suitable for testing, because it contains probabilities instead of covariations.

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Theorem 5.0.2 (Wigner inequality). Let the random variables satisfy the conditions of Theorem 5.0.1. Then the follwoing inequality holds:

P (ξ_a = +1, ξ_b= +1) + P (ξ_b=−1, ξ^c= +1)≥ P (ξ^a=−1, ξ^c = +1).

Proof. The rst probability can be written as

P (ξa = +1, ξb= +1) = P (ξa= +1, ξb= +1, ξc = +1) + P (ξa= +1, ξb= +1, ξc=−1), analogously, the second one can be written as

P (ξb=−1, ξ^c= +1) = P (ξa = +1, ξb=−1, ξ^c= +1) + P (ξa=−1, ξ^b =−1, ξ^c= +1).

Then

P (ξ_a = +1, ξ_b= +1) + P (ξ_b=−1, ξ^c= +1) = P (ξ_a= +1, ξ_b= +1, ξ_c= +1)+

+P (ξ_a = +1, ξ_b= +1, ξ_c=−1) + P (ξ^a= +1, ξ_b =−1, ξ^c= +1) + P (ξ_a=−1, ξ^b=−1, ξ^c = +1) =

= P (ξ_a = +1, ξ_c= +1) + P (ξ_a= +1, ξ_b = +1, ξ_c=−1) + P (ξ^a=−1, ξ^b=−1, ξ^c = +1)≥

≥ P (ξ^a= +1, ξc = +1).

Another example of an inequality of the Bell type is the Clauser-Horne-Shimony-Holt inequality.

Theorem 5.0.3 (Clauser-Horne-Shimony-Holt inequality). For all random variables ξj(ω) and ξj⁰(ω) such as

|ξ^j(ω)| ≤ 1 and |ξj⁰(ω)| ≤ 1 the following inequality holds:

hξ¹, ξ₁⁰i + hξ¹, ξ₂⁰i + hξ², ξ₁⁰i − hξ², ξ₂⁰i ≤ 2 Proof. For real numbers bounded by 1 the following inequality holds:

ξ1ξ₁⁰ + ξ1ξ⁰₂+ ξ2ξ₁⁰ − ξ²ξ₂⁰ ≤ 2.

After integrating it we obtain the Clauser-Horne-Shimony-Holt inequality.

One of the main points why the Bell inequality is so interesting is that such statistical inequalities can be tested in experiments. Moreover, if classical and quantum predictions are incompatible one can check whether experimental results match with one of the predictions. But if one wants to compare results of two theories one should present a mechanism of mapping between the two models. To connect quantum mechanical predictions and classical probability theory, Bell made some assumptions. Under these assumption predictions on quantum mechanics do not match with the Bell type inequalities. Some experiments were performed and their results also violate the Bell type inequalities [2].

Let us construct a theoretical example of such violation from [17]. Consider a two-particle system in the state ψ = ^√¹

2(| + −i − | − +i) and the spin operator which measures the spin of one particle σ(θ) = cos θσz+ sin θσx,

Optimization of statistical parameters of Eberhard inequality

Master Thesis