Ordering of Entangled States with Different Entanglement Measures

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Ordering of Entangled States

with Different Entanglement Measures

Ordning av Sammanflätningsgrad hos Kvantmekaniska Tillstånd för Olika Mätmodeller

Jennie Sköld

Faculty of Science and Technology Subject: Bachelor Thesis in Physics Credits: 15 ECTS

Supervisor: Jürgen Fuchs Examiner: Claes Uggla

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Abstract

Quantum entanglement is a phenomenon which has shown great potential use in modern technical implementations, but there is still much development needed in the field. One major problem is how to measure the amount of entanglement present in a given entangled state. There are numerous different entanglement measures suggested, all satisfying some conditions being of either operational, or more abstract, mathematical nature. However, in contradiction to what one might expect, the measures show discrepancies in the ordering of entangled states. Concretely this means that with respect to one measure, a state can be more entangled than another state, but the ordering may be opposite for the same states using another measure. In this thesis we take a closer look at some of the most commonly occurring entanglement measures, and find examples of states showing inequivalent entanglement ordering for the different measures.

Sammanfattning

Kvantmekanisk sammanflätning är ett fenomen som visat stor potential för framtida tekniska tillämpningar, men för att kunna använda oss av detta krävs att vi hittar lämpliga modeller att mäta omfattningen av sammanflätningen hos ett givet tillstånd. Detta har visat sig vara en svår uppgift, då de modeller som finns idag är otillräckliga när det gäller att konsekvent avgöra till vilken grad olika tillstånd är sammanflätade. Exempelvis kan en modell visa att ett tillstånd är mer sammanflätat än ett annat, medan en annan modell kan visa på motsatsen - att det första tillståndet är mindre sammanflätat än det andra. En möljig orsak kan ligga i de olika modellernas deifnition, då vissa utgår från operativa definitioner, medan andra grundas på matematiska, abstrakta villkor. I denna uppsats tittar vi lite närmre på några av de mätmodeller som finns, och hittar exempel på tillstånd som uppvisar olika ordning av sammanflätningsgrad beroende på vilken modell som används.

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Acknowledgmenets

I would like to thank all my family and friends for believing in me, your encouragement and support has been invaluable and kept me going even if the road has been rough at times. A special thanks to Jürgen Fuchs, my supervisor, for your commitment. It is hard to find such a dedicated person and it has been a privilege to work with you. Thank you!

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1 Introduction

The counterintuitive world of quantum mechanics is fascinating in many ways. One of its more striking features is the phenomenon called entanglement which is far from completely understood [13]. Here, separated quantum entangled particles seems to interact in a way which cannot be explained by classical physics; by conducting measurements on one of the particles, we directly know the results for measuring the other particle [5].

Quantum entanglement has shown to be an invaluable resource in modern technical applications such as quantum information theory and teleportation [19, 23], and for this reason, the edge of research is now making efforts trying to understand more about its peculiar nature. In order to fully understand and develop theories around such applications, we need to face the challenge of finding efficient methods of quantifying, or measure, the amount of entanglement in a given state [13].

There are a number of proposed entanglement measures. Some of them are defined in terms of state conversions, in an operational sense, such as concentrating a number of partially entangled input states into a smaller number of maximally entangled states using local operations. Other measures take a more abstract form. These measures are required to be based on natural properties, such as the inability to create entanglement without the particles interacting directly [13]. Quantifying entanglement is a challenge, and as will be illustrated in this thesis, there is no unique measure for entanglement for systems in mixed states. There are also contradictions in the ordering of the amount of entanglement of certain states for different measures [27, 19].

The von Neumann entropy is the widely accepted, unique measure for a bipartite system in a pure state [19]. However, there are numerous other ways to measure entanglement if the two systems are in a mixed state. In this paper, the focus will be on entanglement of formation, entanglement distillation, entanglement cost and relative entropy of entanglement, along with the von Neumann entropy. There is also a quantity named the modified Horodecki measure that is presented, which is a measure of the degree of violation of Bell’s inequality, and hence the degree of entanglement [19].

Fascinating as it might seem, far from everyone believed in the peculiar concept of entanglement.

Among the doubters were Albert Einstein, Boris Podolsky and Nathan Rosen, who were sceptical to nonlocality and the inability to make deterministic predictions in quantum mechanics. Together they published an article in 1935, where they questioned the physical reality of quantum mechanics, and suggested there must exist hidden variables, which when found would make the theory complete and hence deterministic [9].

The concept can be compared with statistical mechanics. In order to describe e.g. a gas, a statistical approach is used where probability distributions predict a likely result of the behaviour of the gas. It is however in principle possible to consider each particle within the gas, to measure their individual velocities and collisions, and by simple mechanics predict exactly where they will be going and how the gas will behave as a whole. Although, due to huge numbers of particles, this is impossible in practice. The inability of determining the individual parameters of the atoms in the gas, is referred to as classical indeterminism. This might have been what motivated Einstein, Podolsky and Rosen to suspect that the same mechanisms, or hidden variables, could explain the indeterminism related to quantum systems [6].

There were many who tried to prove E, P and R to be wrong in their assumptions regarding the hidden variables. The theory that later came to be the proof that a hidden variable theory was incorrect, and that quantum mechanics is indeed non-deterministic, was laid down by J.S. Bell in 1964 in his famous paper "On the Einstein Podolsky Rosen paradox" [3]. He mathematically showed

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that there are conditions that must be satisfied by all local, deterministic theories [6], and these conditions are not fulfilled for all quantum mechanical phenomena. By considering various systems, among them two spin-¹₂-particles in a singlet state, he showed that if the two quantum systems are placed far from each other, they still cannot be considered as two independent systems. If individual measurements are performed on each of the systems, the results will be depending on each other, and the systems are said to be correlated. Such correlations cannot be explained by any local model [23].

Since the technology to perform the necessary experiments to test his calculations were not available at the time, it took some time until the theories were experimentally confirmed. One of the first experiments was performed by Freedman and Clauser in 1972 [14], and the first convincing test was carried out by Aspect, Grangier and Roger in 1981 [1].

The main purpose of this thesis is to try to walk through some of the most common entanglement measures by describing their definitions, and in a graspable way describe what they do and how they are related. Then, by using the most commonly occurring measures in different articles, in which entanglement ordering between different measures are compared [18, 19], calculations for some chosen states will be carried out in order to check whether the results agree with what is stated in the articles.

Section 7 contains calculations for some arbitrarily chosen states, and for some states fulfilling certain conditions. The purpose is to check the statement in [19], that different entanglement measures can result in different ordering of the entangled states. In other words, we investigate whether, for two different measures E⁰ and E⁰⁰, it can happen that the equivalence

E⁰(ρ₁) < E⁰(ρ₂) ⇔ E⁰⁰(ρ₁) < E⁰⁰(ρ₂)

is violated for some states ρ₁ and ρ₂. Throughout, we limit ourselves to calculations concerning the two-qubit case, since a general theory involving multipartite states is still far from complete [22].

We start out with some key concepts for the subject of entanglement and entanglement measures in section 2. Then we move on to the particularly important concept of entropy in section 3. Entropy is essential in defining the unique measure for entanglement for pure states; the von Neumann entropy which will be discussed in section 6. In the same section, definitions and discussions of all entanglement measures relevant in this thesis are found. Bell’s ideas are presented briefly along with his famous inequality in section 4. The definitions of entanglement for pure and mixed states, together with some criteria for identifying separability, are found in section 5. In section 7, the different conditions for states showing various ordering of entanglement will be presented, together with results from calculating the same from suitable states.

Finally, in the Appendix there are some key calculations presented. Far from all calculations performed in the making of this thesis are included. The reason for this is that many calculations are very similar, and including all of them would both be time consuming and unnecessary, since they would not contribute significantly to the reader’s understanding of the subject.

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2 Key concepts and definitions

2.1 Hilbert space

A Hilbert space H is a frequently occurring notion in physics and mathematics, and is the generalization of the Euclidian space into an abstract vector space of infinite dimension. Hilbert spaces are used when describing the different spaces to which the quantum mechanical systems in question belong.

It is defined as a real or complex vector space with an inner product, hx, yi, where x and y are elements of H. The inner product satisfies the following properties:

• The inner product of x and y, is equal to the complex conjugate of the inner product of y and x

hx, yi = hy, xi. (2.1)

• For complex numbers a and b, the inner product is linear in its first argument, and antilinear in its second argument

hax1+ bx2, yi = ahx1, yi + bhx2, yi, hx, ay1+ by₂i = ahx, y1i + bhx, y2i.

(2.2)

• An element’s inner product with itself is positive definite

hx, xi ≥ 0. (2.3)

In the context of quantum mechanics, one often deals with tensor products of Hilbert spaces, where by combining two Hilbert spaces, one obtains another Hilbert space [30].

2.2 Tensor product

When describing the state spaces of composite systems used in the context of quantum entanglement, the proper mathematical concept used is called tensor product, or direct product of vector spaces. It is denoted by ⊗, and defined as an operation that combines two vector spaces, S of dimension n and T of dimension m, into a new vector space, S ⊗ T of dimension n × m [31].

Concerning entanglement, the tensor product is indispensable when talking about different quantum mechanical systems described by Hilbert spaces. If there are two Hilbert spaces, denoted H_A and HB, and the vectors |ψAi and |ψBi belong to HA and HB respectively, then the vectors |Ψi of the composite system H = HA⊗ HB are represented by linear combinations of vectors of the special form |ψAi ⊗ |ψBi [17].

For example, a system consisting of the spin up- and down vector basis, denoted | ↑i, | ↓i has the composite system representation | ↑i ⊗ | ↓i. For simplicity, a widely accepted simplification of this notation is adopted, namely

| ↑i ⊗ | ↓i = | ↑↓i, (2.4)

which will also be used for calculations in this thesis [11].

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2.3 Qubits

A qubit is the fundamental unit of quantum information transmission. It is made up by a two-state quantum system, such as a spin ¹₂-particle. The qubit differs from the standard bit in information theory and computing, in the sense that instead of just representing either 1 or 0, it can also exist as a superposition of 1 and 0 (figure 1). Furthermore, it can also be entangled with other qubits [4].

Mathematically, a qubit is described as a 2-dimensional complex vector space.

When talking about entanglement concerning a system of two qubits, it is often referred to as a bipartite system, or bipartite state. This is the simplest non-trivial case of entanglement for which there are well defined measures [19]. Concerning entanglement measures of multipartite states, there is still a lot of development needed in order to find effective methods.

Figure 1: A classical bit compared with a qubit. The graphical representation of a qubit is known as a Bloch sphere. Image from: http://qoqms.phys.strath.ac.uk/research_qc.html

2.4 Bell states

There is a set of maximally entangled states of two qubits, which often occur in the context of entanglement, called the Bell states or EPR states. One of the Bell states is the singlet Bell state, or singlet state

|Ψ⁻i = 1

√2(| ↑↓i − | ↓↑i), (2.5)

where | ↑i and | ↓i represents the spin up and spin down vectors respectively, and | ↑↓i denotes the tensor product between spin up and down, and vice versa. Another commonly used notation is |0i for spin up, and |1i for spin down. When talking about multiple singlet states in the context of LOCC-operations (see below), they are sometimes referred to as singlets.

The Bell states can be described as a basis spanning a combined, 4-dimensional Hilbert space H = H1⊗ H2, and since H is 4-dimensional, there is a total of four orthogonal Bell states. Along

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with (2.5), the states are [14]:

|Ψ⁺i = 1

√

2(| ↑↓i + | ↓↑i),

|Φ⁻i = 1

√2(| ↑↑i − | ↓↓i),

|Φ⁺i = 1

√2(| ↑↑i + | ↓↓i).

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2.5 Convex roof construction

There are many strategies to define entanglement measures, one of them being the convex roof construction where the entanglement of formation in section 6.4 is a good example of the concept. It is defined as the convex roof of the von Neumann entropy discussed in section 6.2.

If a function f constitutes, for all arguments, the upper bound for a function ˆf , and ˆf is the largest convex function bounded by f , then ˆf is defined as the convex roof of the function f [22]. The convex roof of a function f defined on a convex subset of Rⁿ, can in general be constructed via

f (x) =ˆ inf

x=P

ip_ix_i

X

i

pif (xi), (2.7)

which is a variational problem that takes the infimum over all possible probability distributions p_i, and over all xi such that x =P

ipixi.

A simple example of the convex roof of a single variable function f (x) = x⁴− 2α²x², is presented in [22], namely

f (x) =ˆ

( x⁴− 2α²x² for |x| ≥ α

−α⁴ for |x| ≤ α . (2.8)

This method can be applied to construct entanglement measures from concave, unitarily invariant functions of density matrices.

2.6 LOCC-operations

A frequently occurring concept in the context of quantum entanglement and information theory is the notion of LOCC-operations, which stands for "Local Operations and Classical Communication". The underlying idea is that quantum systems in entangled states, such as qubits, can be shared between two different locations. On each location, we place an observer. Let’s call them Alice and Bob, and denote the locations A and B, respectively. If we then say that Alice owns one part of the qubit in her lab at A, then Bob analogously holds the other part in his lab at B. Alice and Bob can use classical communication (CC) to coordinate measurements and local operations (LO) performed in each of their laboratories. On the other hand, they are not allowed to send entangled particles between them, meaning that between the two systems, entanglement cannot be created [24].

An example of an LOCC operation is how to concentrate a number, say n pairs, of partially entangled states, into another number, say m pairs, of maximally entangled states. Alice and Bob each hold a copy of each pair in their labs, and by local operations on their respective particles, they can

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transform their pairs into maximally entangled states such as the so called Bell singlet state. In these operations, they can never create a larger number of maximally entangled states than the number of initial states, since entanglement would then have to be created by local operations, and this is impossible. Hence, n > m. This kind of process is useful when transmitting quantum information over so called noisy quantum channels, where entanglement is lost during the transmission process.

By communicating classically, Alice and Bob can coordinate their measurements and operations in order to "re-create" maximally entangled states [4].

3 Entropy

When searching through literature dealing with quantum entanglement and entanglement measures, the notion of entropy is commonly occurring in the context. Since it seems to be a rather important linkage between classical statistics and quantum statistics, a brief explanation on what entropy actually is, is in order.

3.1 What is entropy?

Entropy in statistical physics is a measure for the number of specific states in which a thermodynamical system may be arranged. The word entropy, referring to the greek words for "energy" and

"tropos" (the latter meaning "turning point"), was created by Rudolph Clausius in 1864, when he published his work "Abhandlungen über die mechanische Wärmeteorie". His thoughts laid the foun- dation of classical thermodynamics. He also postulated that the entropy of a closed system could never decrease. This statement is generally known as the second law of thermodynamics [21].

Ludwig Bolzmann was the one to really clarify the concept, when he in the 1870’s related the macroscopic concept of entropy to the molecular properties of a system [15]. Assume a large number of microscopic systems constituting a bigger, macroscopic system, like the particles in a gas. Should the individual energies of the particles be assumed to be quantized (with respect to our quantum mechanical point of view), they would each have one of the energies E1< E2< ... < Em. There are Ni particles in energy level Ei, andX

iNi = N is the total number of particles in the system.

The number of particles N_i at a certain energy level E_i is called the occupation number. Together, they make up a macrostate of the system, for which the total energy E =P

iN_iE_i. The N =P

iN_i particles can produce a large number of configurations which constitute a macrostate, in which each microstate possesses energy Ei [10].

There are many ways in which the microstates can arrange themselves in order to form the same macrostate, more exactly

N N₁, N₂, ..., N_m

!

:= N !

N1!N2! · · · Nm! (3.1)

ways [21]. Assuming that all configurations of microstates that make up a macrostate are equally likely, this multinomial coefficient is proportional to the probability of that specific macrostate. This was referred to by Bolzmann as the thermodynamical probability of the macrostate.

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For N → ∞, also known as the thermodynamic limit, it is more convenient to use relative numbers p_i := ^N_Nⁱ to label the macrostate, and ^E_N =P

ip_iE_i for the average energy per particle. Here, we see that P

ipi = _N¹ P

iNi= ^N_N = 1. In order to find the most probable macrostate, (3.1) needs to be maximized. Using Stirling’s approximation of the factorials, we find that

1

N log N

N1, N2, ..., Nm

!

= −X

i

pilog pi+ O(1

N log N ), (3.2)

where we set

−X

i

pilog pi =: H(p1, p2, ..., pm). (3.3)

For large N , it is sufficient to maximize (3.3), which is commonly known as the Shannon entropy, or the Shannon mixing entropy. This quantity is used to describe the degree of randomness (or disorder) of a given probability distribution [16].

3.2 Entropy and quantum mechanics

In quantum mechanics, we come across a peculiar phenomenon called entanglement, in which particles are correlated. In other words, if an individual measurement is performed on one of the particles, the outcome when measuring the other particle is already determined. This has turned out to be of great potential use in many technical implementations, such as quantum engineering, quantum communication, quantum computation and quantum information [19].

It is by the laws of nature not possible to create entanglement by local operations (LO) at a distance, but it can be preserved or destroyed. This is the quantum analogue to the second law of thermodynamics [24], and this is what makes entropy so useful when dealing with different measures of entanglement.

The entanglement of a system can be expressed in several ways, one of them being the von Neu- mann entropy which will be discussed in section 6.2. It can be viewed as the quantum counterpart of the Shannon entropy (3.3) [16], and for bipartite systems in a pure state, the von Neumann entropy of one of the systems is the unique measure of the entanglement of the two systems.

4 Bell’s inequality and the CHSH-inequality

The Bell inequalities are a set of inequalities that were introduced to prove that the local hidden variable (LHV) model, suggested by Einstein, Podolsky and Rosen, is wrong. The first of these inequalities was published by Bell in his work in 1964 [3], where he mathematically formulated the ideas of a hidden variable theory, and derived an inequality that cannot be violated if such a theory would be correct.

Bell based his derivation on E, P and R’s argument, which can be stated as follows. Assume there is a bipartite system, for example a pair of spin one-half particles in the singlet state moving in opposite directions. To each part measurements are performed separately, one option being to measure the components of the spins on each of them using Stern-Gerlach magnets. If one of the magnets measures the value +1, then the other one will, according to quantum mechanics, yield the value -1. Then, it

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is assumed that the results were already determined locally in both parts before the measurements were carried out. From these assumptions, he derived the following inequality [3]:

1 + P (~b, ~c) ≥ |P (~a,~b) − P (~a, ~c)|, (4.1) in which ~a,~b and ~c are unit vectors of the directions in which spin components are measured, and P (~a,~b), P (~b, ~c), P (~a, ~c) are the quantum mechanical expectation values of the spin components. These are found to be [3, 6]

P (~a,~b) = −~a · ~b = −cos φ, (4.2)

where φ is the angle between ~a and ~b. The formula (4.2) analogously holds for vectors ~b and ~c, and ~c and ~a. The inequality (4.1) is satisfied for any local theory, and should also be satisfied if quantum mechanics were govered by some hidden variables. This was shown in [3] not to be the case.

Experimental verification of (4.1) is difficult, since Bell assumed perfect correlations which is prac- tically impossible to obtain. There are however numerous other inequalities inspired by Bell’s work.

For example, consider two separated subsystems, the first in which quantities (A₁, A₂) are measured, and the second in which (B1, B2) are measured. The results for measuring A1, A2, B1 and B2 are denoted a1, a2, b1and b2, and assumed to take numerical vales ±1. The constraint for this situation obtained by Clauser, Horne, Shimony and Holt, known as the CHSH-inequality, states that [11]

hA1B1i + hA2B1i + hA1B2i − hA2B2i ≤ 2, (4.3) where hAiBji represents the average measurement results. This inequality is applicable to real experiments and a generalization of Bell’s theorem [7], and can also be written in terms of the measurement results:

a₁b₁+ a₂b₁+ a₁b₂− a2b₂≤ 2. (4.4)

5 Entanglement

5.1 Entanglement for pure states

Consider a combined Hilbert space H, described by the tensor product of two systems H_A and H_B. Vectors in H_A⊗ H_Bthat can be written on the form

|Ψi = |ψ^Ai ⊗ |ψ^Bi, (5.1)

where |ψ^Ai ∈ HA and |ψ^Bi ∈ HB, are called decomposable tensors [12], product states or separable [11]. These can be viewed as the algebraic analogue of separable solutions to a differential equation.

Most solutions to differential equations are non-separable, and this is also the case for most tensors.

However, general tensors can be written as linear combinations of decomposable tensors, and this is what leads to the phenomenon called entanglement.

Consider a quantum system that is obtained by combining two systems, A and B, where A stands for "Alice" and B stands for "Bob". If the states of A form a Hilbert space H_A, of dimension d_A, and analogously, HB of dimension dB for B, then the composite system is described by the tensor product H = H_A⊗ H_B. Any vector |Φi in H can thus be expressed as [11]

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|Φi =

d_A,d_B

X

i,j=1

cij|aii ⊗ |bji ∈ H_A⊗ H_B, (5.2)

where |aii and |bji are basis vectors of H_A and H_B, respectively, and the coefficients cij form a complex dA× dB matrix.

If we can find states |φ^Ai ∈ HA and |φ^Bi ∈ HB, such that |Φi = |φ^Ai ⊗ |φ^Bi, where |Φi is a pure state, we end up in the situation of (5.1), namely separable, uncorrelated states. The definition of pure product states means that since the systems are uncorrelated, i.e. independent of one another, the state can be produced locally. This means that Alice and Bob can create "their" own states |ψ_Ai, and |ψ_Bi, respectively, in their own separate labs.

5.2 Entanglement for mixed states

When the situation is such that there is not a single pure state, but a mixture, or a statistical ensemble of pure states, it is called a mixed state, which is a more general situation than pure states. Mixed states cannot be written as linear combinations of pure states as in (5.2); they need to be described by a density matrix; ρ [29].

The underlying idea of the density matrix is that the exact state of the quantum system is un- known, and therefore the probability pifor finding the system in one of some states |Φii ∈ H is used, and we get

ρ =X

i

pi|ΦiihΦi|, (5.3)

with

X

i

p_i= 1, p_i≥ 0. (5.4)

The density matrix representation for pure states has only one term in the sum (5.4), and is then expressed as ρ = |ΨihΨ|, since there is only one state and the probability for that state is p1= 1.

When defining separability and entanglement for mixed states, the idea is the same as for pure states, namely that if the states can be produced locally, in separate labs, they are separable. If not, they are entangled. In other words, if there exist states ρ^A for Alice, and ρ^B for Bob, and if

ρ = ρ^A⊗ ρ^B, (5.5)

then ρ is a product state. Furthermore, if there are convex weights ωi, and product states ρ^A_i and ρ^B_i such that

ρ =X

i

ω_iρ^A_i ⊗ ρ^B_i , (5.6)

then the state is separable [11].

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As an illustration, the density matrix for a mixed state consisting of the pure states

|Ψ1i = 1 2| ↑↓i −

√ 3 2 | ↓↑i,

|Ψ2i = 1

√

2| ↑↓i − 1

√ 2| ↓↑i,

(5.7)

is given by

ρ =X

i

p_i|Ψ_iihΨ_i| = p₁|Ψ₁ihΨ₁| + p₂|Ψ₂ihΨ₂|. (5.8)

Writing out (5.8) in terms of the probabilities p1 and p2, where p1+ p2= 1, one obtains

ρ =







0 0 0 0

0 (^p₄¹+^p₂²) −(

√3p1

4 +^p₂²) 0 0 −(

√3p₁

4 +^p₂²) (^3p₄¹ +^p₂²) 0

0 0 0 0







. (5.9)

It is clear that the entries in the density matrix depend on the probabilities for the state to be found in either of the pure states |Ψ1i or |Ψ2i. Details on how ρ is found are described in the first half of Appendix A.3.

5.3 Identifying separable states

To find out whether a given state is separable or not, is not always an easy task. The criteria for separability is only that there exists some decomposition for the certain state into product states for pure states. For mixed states, a decomposition into a convex sum of tensor products is required.

It is not always clear by inspection whether a state is decomposable or not, since for example

|Ψi =

√3| ↑i + | ↓i

2 ⊗

√2| ↑i +√ 2| ↓i

2 (5.10)

also can be written on the form

|Ψi =

√6| ↑↑i +√

6| ↑↓i +√

2| ↓↑i +√ 2| ↓↓i

4 . (5.11)

In (5.11) it is much less obvious that |Ψi is separable. As is seen for this case, it is easier to determine whether the state is separable if it is expressed in the bases n

|ψ⁺₁i =

√3|↑i+|↓i

2 , |ψ⁻₁i =

√3|↑i−|↓i 2

o of H₁, and n

|ψ₂⁺i =

√2|↑i+√ 2|↓i

2 , |ψ₂⁻i =

√2|↑i−√ 2|↓i 2

o of H₂, than if it is expressed in the basis {| ↑i, | ↓i}.

Any state can be presented in terms of some base vectors which directly shows whether the state is entangled or not, and this representation is called the Schmidt decomposition of the state [17].

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Schmidt decomposition

In the combined Hilbert space H = H1⊗ H2, the arbitrary bases for each subspace can be denoted

ψ¹_i and ψ²_j respectively, and any pure state in H can thus be expressed as

|Ψi =X

ij

dij|ψ_i¹i ⊗ |ψ_j²i, (5.12)

where the coefficients d_ij, also known as the expansion coefficients, are given by d_ij= hψ¹_i| ⊗ hψ²_j|Ψi, or the overlap of the state |Ψi with the basis vectors.

If there are arbitrary, local unitary transformations on H1 and H2 denoted U and V respectively, and a change of basis is defined as

| eψ¹_ii = U |ψ¹_ii,

| eψ²_ji = V |ψ_j²i, (5.13)

then the coefficients dij change according to

=X

p,q

hψ¹_i| U^†|ψ_p¹i hψ_p¹| ⊗ hψ²_j| V^†|ψ_q²i hψ_q²|Ψi =

=X

p,q

hψ¹_i| U^†|ψ_p¹i hψ_j²| V^†|ψ²_qi hψ¹_p| ⊗ hψ_q²|Ψi =

=X

p,q

uipvjqdpq.

(5.14)

Here, u = (uip) and v = (vjq) are unitary matrices defined as

u_ip= hψ¹_i| U^†|ψ_p¹i, v_jq = hψ²_j| V^†|ψ_q²i. (5.15) We have by this procedure obtained the singular value decomposition of the matrix dij. The diagonal entries, or eigenvalues, of d are called the singular values of d. The singular values are denoted S_i and are real and non-negative.

This gives the new expression for |Ψi:

|Ψi =X

ij

deij| eψ¹_ii ⊗ | eψ²_ji =X

i

X

j

X

pq

uipdpqvqj| eψ¹_ii ⊗ | eψ²_ji. (5.16)

We can define new bases

|αpi =X

i

uip| eψ¹_ii, and |βqi =X

j

vqj| eψ_j²i. (5.17)

With these new bases, we finally arrive at the state

|Ψi =X

p,q

d_pq|α_pi ⊗ |β_qi. (5.18)

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Since dpq is a diagonal matrix, (5.18) can be rewritten as

|Ψi =X

i

dii|αii ⊗ |βii. (5.19)

Thus we have learned that, if |Ψi is any vector in H, then there exist local orthonormal bases |α_ii and |βii in H1and H2 respectively, such that

|Ψi =X

`

pλ_`| α_`i ⊗ | β_`i, (5.20)

where the λ`= S_`² are known as the Schmidt coefficients. The set of separable states {|α`i ⊗ |β`i} is known as the Schmidt basis. If there is only one eigenvalue, λ₁= 1, the state is separable.

The Schmidt decomposition will be important when talking about the definition of the von Neu- mann entropy in section (6.2). However, this discussion only holds for pure states. For mixed states, the Schmidt decomposition does not provide a sufficient criterion to define separability. In order to evaluate mixed states, one needs to move on to other tools. There are many such tools; two of the more prominent ones are entanglement witnesses and positive maps. Here, we will take a closer look at one of them, namely the positive map known as the PPT criterion [17].

The PPT criterion

The PPT criterion says that any separable state stays positive under the transposition of one of its subsystems, while entangled states doesn’t. In other words; when partially transposing an entangled state, the eigenvalues will remain positive if the state is pure, whereas if the state is entangled, at least one of its eigenvalues will be negative [17]. The PPT criterion is important when discussing the negativity of a state, which will be considered in section 6.3.

Let us consider a separable density matrix ρ. As seen in section 5.2, ρ can be written as ρ =X

i

ωiρ^A_i ⊗ ρ^B_i , (5.21)

where P

iω_i= 1, if it fulfils the separability criterion. This is defined as when the matrix obtained under partial transposition consists of positive eigenvalues.

The partial transposition is defined as the transposition with respect to one of the subsystems in (5.21). Writing out ρ in terms of matrix elements, one obtains

ρmµ,nν =X

i

ωi(ρ^A_i )mn(ρ^B_i )µν. (5.22)

The partial transpose with respect to subsystem A then reads ρ^T_mµ,nν^A = ρ_nµ,mν =X

i

ω_i(ρ^A_i)_nm(ρ^B_i )_µν=X

i

ω_i(ρ^A_i)^T_mn(ρ^B_i )_µν, (5.23)

where the indices m and n are interchanged [20].

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6 Entanglement measures

An entanglement measure is defined as a function that, for any given state, quantifies the amount of entanglement present [11]. It is commonly denoted by E(ρ), where ρ is the density matrix for the system on which the measurements are performed. The unit used for measuring entanglement is defined by the amount of entanglement for a maximally entangled state, such as the singlet state, and is called an ebit [4]. Since entanglement is a very complex property, there is no unique measure for arbitrary states [19]. In this text, the five most prominent entanglement measures will be discussed, where the first one is:

• The von Neumann entropy (SvN)

It is particularly important since it is the unique measure for pure states, meaning that all other measures will coincide with the von Neumann entropy when evaluated for pure states.

The remaining four measures are:

• Entanglement cost (E_C), and particularly the related PPT entanglement cost (E_{P P T}).

• Entanglement of distillation (ED)

• Entanglement of formation (EF)

• Relative entropy of entanglement (ER)

The first two will be considered together in section 6.3, since they are closely related in a way that will become clear later.

In addition to these five measures, there is another quantity which will be presented, namely the Modified Horodecki measure (B), or the nonlocality measure. It is used to describe the degree of violation of the CHSH-inequality (4.3) [19], and shows some interesting results which will be presented at the end of section 7.

There are several properties that entanglement measures should fulfil [11]. Interestingly, these vary depending on the situation and the measure itself. Some measures only fulfil a few of the requirements, but there are two that need to be fulfilled by all of them. The reason for this seemingly contradictive statement is likely the fact that there is not yet a fully developed theory concerning entanglement. Some of the measures are defined in terms of state conversions, in a more operational sense for performing tasks such as teleportation, whereas others are defined in a more abstract, mathematical sense based on some natural properties [13]. There is still no measure valid for all entangled states, and the most suitable for one certain situation might not be ideal in another. Unless of course we are in the situation of pure states, where the von Neumann entropy is the known measure with which all other measures coincide.

6.1 Conditions for entanglement measures

The first two requirements, which all measurements are expected to fulfil are:

1. The entanglement measure E(ρ) should vanish when ρ is separable, i.e.

E(ρ_SEP) = 0. (6.1)

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2. E(ρ) cannot increase under LOCC-transformations, since entanglement cannot be created under such operations. Hence, for any LOCC operation Λ,

E[Λ(ρ)] ≤ E(ρ). (6.2)

In words, if a quantum state ρ can, by LOCC operations, be transformed into another state Λ(ρ), then one ρ must be at least as entangled as Λ(ρ) [22]. This condition is also referred to as the monotonicity condition, and is very important. Some authors even propose that this is the only necessary condition, and that others would follow from this, or should be considered optional [14].

A similar condition that is sometimes used, is that E(ρ) should not increase on average under LOCC.

As mentioned, conditions (1) and (2) are considered to be the most important but there are however a number of other conditions proposed that can be useful depending on the situation. Some of them are:

3. Convexity, meaning that E cannot increase under mixing of two or more states:

E(X

k

p_kρ_k) ≤X

k

p_kE(ρ_k). (6.3)

Expanding (6.3) for bipartite states gives

E(p1ρ1+ (1 − p1)ρ2) ≤ p1E(ρ1) + (1 − p1)E(ρ2). (6.4)

In (6.3), pk≤ 1 is the rate at which the selection of states ρkappears, and going from a selection of such states to a mixture ρ =P pkρk will lead to a loss of information. This property is not fulfilled by all measures, but it is very common, as all measurements obtained by the convex roof construction are convex.

4. Invariance under local unitary transformations, i.e. local unitary operations only causes a change of basis, the quantum correlations remain unchanged [26].

E(ρ) = Eh

(UA⊗ UB) ρ (U_A^† ⊗ U_B^†)i

. (6.5)

5. Should multiple, say n, copies of a state exist, it would be reasonable to require additivity, i.e.

E(ρ^⊗n) = nE(ρ). (6.6)

An even stronger requirement is full additivity, that is

E(ρ1⊗ ρ2) = E(ρ1) + E(ρ2), (6.7) for any pair ρ₁, ρ₂. Both additivity requirements are not always fulfilled, or difficult to prove for a given measure.

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6.2 Von Neumann entropy

As seen in section 5.3, any pure state |Ψi of a general bipartite system consisting of two subsystems A and B can be expressed as

|Ψi =

n

X

i=1

c_i|ψ_i^Ai ⊗ |ψ^B_i i, (6.8)

where the density operator for the state is described by ρ = |ΨihΨ|, and the sets of states {|ψ₁^Ai, . . . , |ψ^A_ni}, {|ψ₁^Bi, . . . , |ψ_n^Bi} for subsystems A and B, respectively, are orthonormal [28]. Should there be just one term in the sum, it would be a pure product state, should there be at least two, the state would be entangled.

The coefficients ci in (6.8) are the square roots of the Schmidt coefficients that we defined in section 5.3, and they do not change under unitary transformations. It is therefore reasonable to define the entanglement of |Ψi as depending only on these quantities, namely the widely accepted and unique measure of entanglement for pure states:

SvN(ρ) = −

n

X

i=1

c²_ilog₂c²_i, (6.9)

where c²_i are the non-zero eigenvalues for the density matrix of either A or B. This is known as the von Neumann entropy SvN of the density matrix associated with either of the two subsystems A and B. This means that in order to obtain the von Neumann entropy for a certain pure state, one needs to trace out one of its subsystems and thereafter perform the necessary calculations.

Another common way of expressing the von Neumann entropy is

S_vN(ρ) = −tr (ρ log ρ), (6.10)

but using this relation for computations is rather cumbersome since it involves the logarithm of a matrix (ρ).

The von Neumann entropy is the unique entanglement measure for pure states [28], with which all other entanglement measures coincide. But it is not as useful when evaluating entanglement for mixed states. The reason for this is that tracing out one of the subsystems (A), may not lead to the same value of entanglement as tracing out the other subsystem (B) would [8].

The von Neumann entropy for the singlet Bell state

To illustrate how the von Neumann entropy is calculated for a simple system, consider the maximally entangled singlet state |Ψ⁻i in (2.5), with density matrix:

ρ(|Ψ⁻i) = 1 2

"

1 0 0 0

!

⊗ 0 0

0 1

!

− 0 1

0 0

!

⊗ 0 0

1 0

!

− 0 0

1 0

!

⊗ 0 1

0 0

!

+ 0 0

0 1

!

⊗ 1 0

0 0

!#

.

(6.11)

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Tracing out the subsystem to the right, let’s denote it B, we obtain the reduced density matrix:

Tr_B(ρ) = 1 2

"

1 0 0 0

!

+ 0 0

0 1

!#

=

1 2 0 0 ¹₂

!

. (6.12)

Using (6.9) we obtain

SvN(ρ) = −1 2

log₂(1

2) + log₂(1 2)

= −log₂(1

2) = log₂(2) = 1 ebit, (6.13) which shows that the singlet state is maximally entangled.

6.3 Entanglement distillation and Entanglement cost

Entanglement distillation and entanglement cost are operationally defined extreme measures, provid- ing upper, and lower bounds respectively for other entanglement measures¹. Provided these measures coincide for pure states (the von Neumann entropy), all other measures will do the same [8].

Entanglement distillation

Let Γ be an LOCC trace-preserving operation, and ∆(K) be the density operator for the maximally entangled state vector, where K is the dimension, such that ∆(K) = |Ψ⁻_KihΨ⁻_K|. The entanglement of distillation (ED) can then be formally defined as:

ED(ρ) = sup{r : lim

n→∞

h infΓ tr

Γ(ρ^⊗n) − ∆(2^rn) i

= 0}, (6.14)

and is described as the rate at which noisy mixed states (ρ) can be converted into maximally entangled singlet states via LOCC operations. Another, more compact way of expressing (6.14) is given by [11]:

ED(ρ) = sup

LOCC

n_outlim→∞

nout

n_in , (6.15)

where n_ininput copies of ρ are mapped onto n_out copies of a maximally entangled state, such as the singlet state, in the asymptotic regime where nout→ ∞.

The entanglement distillation is additive, but it is not yet known whether it fulfils the convexity condition or not [8].

Entanglement cost

Conversely, the entanglement cost (E_C) can be defined as the maximal rate at which maximally entangled singlet states can be converted into some desired output state (ρ). The entanglement cost fulfils both the additivity and the convexity conditions, and is formally defined as [22]:

EC(ρ) = inf{r : lim

n→∞

h

infΓ tr| ρ^⊗n− Γ(∆(2^rn) |i

= 0}. (6.16)

or

EC(ρ) = inf

LOCC lim

nout→∞

n_in nout

. (6.17)

1These measures both have very complex, variational definitions and are difficult to evaluate. For this reason, we present them here briefly without analysing them any further.

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Here, nininput singlets are mapped onto nout copies of the output state ρ in the asymptotic regime.

The PPT entanglement cost

The characterization of LOCC-operations is far from well-understood, and sometimes it is easier to adopt a set of operations that are closely related, although much easier to characterize. These are called PPT operations, and are defined as operations preserving the positivity of the partial transpose, i.e. any state with positive partial transpose is mapped onto another state with positive partial transpose [2].

The PPT entanglement cost can be defined as [19]:

E_{P P T}(ρ) = lg[N (ρ) + 1], (6.18)

where N denotes the negativity

N (ρ) = 2X

j

max(0, −µ_j), (6.19)

which, as opposed to the entanglement cost, is easy to compute. The negativity can be viewed as the violation of the PPT criterion [11] discussed in section (5.3). In (6.19), µj are the eigenvalues of the partial transpose of ρ. Since the negativity is significantly easier to calculate for mixed states, it is usually the measure used in literature focusing on comparing different measures, for example [18]

and [19].

The negativity for the singlet Bell state

If, instead of tracing out subsystem B in (6.11) for |Ψ⁻i as we did when looking at the von Neumann entropy, we choose to transpose the same subsystem, we obtain:

ρ^T^B =1 2







0 0 0 1

0 1 0 0

0 0 1 0

1 0 0 0







. (6.20)

The resulting eigenvalues for ρ^T^B are

µ_1,2,3=1

2, and µ₄= −1

2. (6.21)

As we see, µ₄ is negative, which gives the negativity

N (|Ψ⁻i) = 1 (6.22)

according to (6.19). Inserting into (6.18), we obtain entanglement EP P T(ρ) = 1 ebit. Hence, the singlet state is maximally entangled with respect to the negativity as well as for the von Neumann entropy.

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6.4 Entanglement of formation

The entanglement of formation is a mathematically defined measure [8], that can be physically in- terpreted as the minimum number of singlets needed in order to create a single copy of the state ρ [28, 11]. It is also defined as the convex roof of the von Neumann entropy (6.9), and therefore fulfils the convexity condition. Furthermore, it is invariant under local unitary transformations [8], and is expressed as:

E_F(ρ) = infX

i

p_iS_vN(ψ_i). (6.23)

In (6.23), ρ =P

ip_i|ψ_iihψ_i|, p_i≤ 0 andP

ip_i = 1 [14], meaning the largest convex function smaller than S_vN is found among all pure-state decompositions of ρ [28].

For two qubits, the entanglement of formation can be expressed in terms of a measure called concurrence. In fact, the bipartite qubit-situation is the only situation where there is a general formula for the concurrence. There are ideas of how to extend the formula for larger systems, but this is beyond the scope of this thesis and will not be considered since the focus here will be on the two-qubit case.

In order to describe the entanglement of formation of a two-qubit state ρ, we use the concurrence C as defined by Wootters [19, 28]:

C(ρ) = max ( 0 , 2 max

i Ωi−X

i

Ωi) = max ( 0 , Ω1− Ω2− Ω3− Ω4). (6.24)

The Ωi’s in (6.24) are the square roots of the eigenvalues of ρ ˜ρ, where ˜ρ is equal to the spin-flip operator applied to ρ:

ρ ˜ρ = ρ(σ_y⊗ σ_y)ρ^∗(σ_y⊗ σ_y), (6.25) where

σ_y = 0 −i i 0

!

(6.26) is the second Pauli spin matrix.

The concurrence in this case is a simple measure of entanglement, and is connected with E_F by the following formula

EF(ρ) = h(ξ) = h(1 2[1 +p

1 − C(ρ)²]), (6.27)

where

h(ξ) = −ξlog₂ξ − (1 − ξ)log₂(1 − ξ) (6.28) is the binary entropy function.

The concurrence for the singlet Bell state

Again considering the singlet Bell state |Ψ⁻i, and using (6.25), we get that:

ρ (σy⊗ σy)ρ^∗(σy⊗ σy) =







0 0 0 0

0 1/2 −1/2 0

0 −1/2 1/2 0

0 0 0 0







, (6.29)

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with the resulting eigenvalues

λ₁= 1, λ_2,3,4= 0, (6.30)

which, according to (6.24), leads to the concurrence

C(|Ψ⁻i) = 1. (6.31)

Again, |Ψ⁻i is shown to be completely entangled. For detailed calculations, see Appendix A.2.

6.5 Relative entropy of entanglement

The relative entropy of entanglement is a distance measure, i.e. it uses the distance of entangled states to separable states as a measure for entanglement (illustrated in figure 2) [11]. Just as E_F, E_REEfulfils the convexity condition, as well as being invariant under local unitary transformations [8].

If ρ represents the entangled state lying in the set of all states, and σ denotes the separable states lying in the subset X defined as the set of all separable states, then the relative entropy of entanglement with respect to X is defined as [22]

E_R^X(ρ) := inf

σ∈XS(ρ||σ), (6.32)

where

S(ρ||σ) := tr{ρ log ρ − ρ log σ}. (6.33)

(6.33) is known as the quantum relative entropy, and is a measure of distinguishability between quantum states. The relative entropy of entanglement is in general not additive for bipartite states.

E_REE is not easy to compute for arbitrary mixed states. In fact, there are only a small number of specific, high-symmetry sets of states for which there are known analytical formulas [19]. For this reason, numerical methods are needed for computations, and will not be considered in this thesis.

Figure 2: The relative entropy of entanglement uses the distance between the entangled state ρ and the closest separable state σ as a measure for entanglement.

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6.6 The modified Horodecki measure

The degree of violation of the Bell inequality (4.3) for a state σ is described by the modified Horodecki measure B, also known as the nonlocality measure [19]. This is defined as

B(σ) =p

max [ 0, M (σ) − 1]. (6.34)

M (σ) is the Bell inequality violation parameter, given by the sum of the two largest eigenvalues (Ωi) of TpT_p^†, where Tp is the 3 × 3 matrix:

Tp=







Tr(σσ_x⊗ σx) Tr(σσ_x⊗ σy) Tr(σσ_x⊗ σz) Tr(σσy⊗ σx) Tr(σσy⊗ σy) Tr(σσy⊗ σz) Tr(σσz⊗ σx) Tr(σσz⊗ σy) Tr(σσz⊗ σz)





, (6.35)

where σx,y,z are the Pauli spin matrices [18].

The maximum possible violation of the CHSH inequality (4.3) for a two-qubit state ρ is denoted hBiρ, and related to M (ρ) by

maxB hBiρ= 2p

M (ρ), (6.36)

where B is the Bell operator.

The nonlocality measure B(σ) is equal to the concurrence and negativity for pure states. Here, we check if this is true for the singlet state.

The nonlocality measure for the singlet state Using (6.35) on the density matrix ρ for |Ψ⁻i, we obtain

Tp=







−1 0 0

0 −1 0

0 0 −1





, (6.37)

resulting in

TpT_p^†=







1 0 0 0 1 0 0 0 1





, (6.38)

with the sum of the two largest eigenvalues M (ρ) = 2. Inserted into (6.34), we obtain

B(ρ) = 1, (6.39)

which shows maximum entanglement. It also agrees with the statement that the nonlocality measure coincides with the concurrence and negativity for pure states.

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7 Comparing the concurrence, negativity and violation of the Bell inequality for specific states

In [19], measurements of the entanglement of a two-qubit system were performed, using three different methods, namely the negativity, concurrence and the relative entropy of entanglement. They were then compared in order to show that different measurements show different ordering of the entangled states. They also found that two Bell diagonal states with the same entanglement according to the three measures, showed different degrees of violation of Bell’s inequality, or nonlocality.

In the literature ([18, 19]), the states analysed where generated by so called Monte Carlo simula- tion, which is a numerical tool that can be used to study many-particle quantum systems. The method in general is used to simulate many-particle systems based on random numbers [25].

Two different entanglement measures, say E⁰ and E⁰⁰imply the same ordering of states if and only if E⁰(ρ1) < E⁰(ρ2) ⇔ E⁰⁰(ρ1) < E⁰⁰(ρ2) (7.1) is valid for arbitrary states ρ₁and ρ₂. These inequalities are always fulfilled for qubits in pure states (since all good measurements are equivalent here), but not for all entanglement measures.

The property of different entanglement measures implying different ordering of states might seem counterintuitive. This is mentioned in many texts [22, 27, 14], and one reason for this might be the different ways different measures are defined.

In this thesis, the main focus is put on the concurrence and the negativity, since the relative entropy of entanglement requires complicated numerical methods for calculation.

7.1 Comparison of different measures for pure states

All entanglement measures are required to coincide with the von Neumann entropy for pure states.

We have already seen that this is true for the Bell singlet state, but it also includes states who are not maximally entangled. To check if this is indeed the case, we used that partly-entangled, normalized states can be obtained from the relation [4]

|Ψi = cos (θ)| ↑↓i − sin(θ) | ↓↑i, (7.2)

to construct the state

|Ψi = 1

2| ↑i ⊗ | ↓i −

√3

2 | ↓i ⊗ | ↑i, (7.3)

for which EvN, EF and EP P T were calculated. The concurrence and negativity for |Ψi were found to be

C(ρ) = N (ρ) =

√3

2 . (7.4)

Computing EvN and EF, we obtain

E_vN = E_F = −3

4log₂(3) + 2 ≈ 0.811278 ebits, (7.5)

Ordering of Entangled States with Different Entanglement Measures