Holographic vs. non-holographic inequalities.

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Report Bachelor Project Physics and Astronomy (15EC) conducted between 01-04-2018 and 01-07-2018

Holographic vs. non-holographic inequalities.

Author: Linda M. van Manen, 11153989

Supervisor: dhr. dr. B.W. Freivogel

Examiner: dhr. dr. J.P. van der Schaar

Institute for Theoretical Physics Amsterdam

Faculty of science, University of Amsterdam

Amsterdam, the Netherlands

Faculty of science, VU University

Amsterdam, the Netherlands

A thesis submitted in fulfillment of the requirements for the degree of Bachelor of

Science.

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1

Samenvatting

In Einsteins algemene relativiteitstheorie wordt zwaartekracht beschreven als kromming van ruimte-tijd door massa en energie. De ruimteruimte-tijd kan zelf ook gekromd zijn. Ook als het geen massa bevat, dus wanneer het een vacuüm is. Uit Einsteins vergelijkingen blijkt dat een vacuüm vlak, positief-, of negatief gekromd kan zijn. De negatieve gekromde ruimte wordt Anti-deSitter ruimtetijd (AdS) genoemd. Anti-deSiter ruimtetijd wordt gebruikt bij zwaartekracht theorieën. Er bestaat een du-aliteit tussen deze Anti-deSitter ruimtetijd en kwantumvelden theorie die deeltjes beschrijft. Een dualiteit houdt in dat een grootheid beschreven in een ruimte, ook beschreven kan worden in een andere ruimte. Dit maakt het mogelijk om grootheden die lastig zijn te berekenen in een ruimte, makkelijker te berekenen zijn in een andere ruimte. Een mogelijk bekende dualiteit is de Heisen-berg onzekerheids relatie. Deze relatie stelt dat de impuls niet exact in positie ruimte kan worden beschreven, maar wel exact in impuls ruimte kan worden beschreven. Eveneens kunnen kwan-tum toestanden die lastig in kwankwan-tumvelden theorie te beschrijven zijn, makkelijk in Anti-deSitter ruimtetijd beschreven worden. Deze dualiteit is bekend als de “Anti-deSitter spacetime/conformal field theory conjecture”, oftewel de AdS/CFT conjecture en betreft een sterke-zwakke koppeling dualiteit. Een systeem met een sterke koppelingskonstante is beschouwd als een systeem met een hoog kwantum karakter, en een systeem met een zwakke koppelingskonstante is een klassiek systeem. De dualiteit is dus een kwantum/klassieke dualiteit, die de klassieke geometrie van Anti-deSitter ruimtetijd relateert aan kwantum systemen in kwantumvelden theorie. De AdS/CFT con-jecture is het eerste voorbeeld van het holografische principe. Het holografische principe verteld dat alle informatie in een ruimte gecodeerd is op de rand van die ruimte. In dit geval ligt alle informatie over zwaartekracht in Anti-deSitter ruimtetijd gecodeerd in kwantumveldentheorie op de rand van de Anti-deSitter ruimtetijd. In context van de AdS/CFT conjecture is er ook een dualiteit bedacht tussen verstrengeling entropie, een methode om verstrengeling te kwantificeren, en de minimale oppervlakte in Anti-deSitter ruimtetijd. In een twee dimensionale AdS kan de minimale oppervlakte gezien worden als de kortste lijn door de AdS, waar de uiteinden eindigen op de rand van een sub gebied in het kwantumveld.

Als entropie in kwantumvelden theorie een dualiteit heeft in Anti-deSitter ruimtetijd, dan moeten de eigenschappen van entropie ook terug te vinden zijn in de geometrie van Anti-deSitter ruimtetijd. In dit onderzoek hebben we laten zien dat dit inderdaad het geval is. Verder laten we zien dat de entropie in termen van de minimale oppervlakte in AdS, oftewel de holografische entropie, ook voldoet aan een eigenschap bekend als monogaam gedeelde informatie tussen drie sub gebieden in een kwantumveld. Deze eigenschap van monogaam gedeelde informatie geldt niet altijd. Zo zijn kwantum correlaties wel monogaam, maar klassieke correlaties niet. Dit past weer bij het idee dat de dualiteit een kwantum/klassieke dualiteit is. De klassieke holografische entropie is hier gerelateerd aan kwantum correlaties, en niet aan klassieke correlaties. De kwantum toestanden met klassieke correlaties zijn hoogstwaarschijnlijk gerelateerd aan onderliggende kwantum effecten in AdS.

Entropie is een methode om verstrengeling te kwantifiseren voor zogeheten pure toestanden. Een kwantum toestand is puur als alle information over de toestand tot beschikking is. Als dit niet het geval is dan is de toestand een gemixte toestand. Verstrengeling tussen gemixte toestanden kunnen worden gekwantifiseerd door verstrengeling negativiteit. Er is voorgesteld dat deze negativiteit in Anti-deSitter ruimtetijd gelijk is aan driekwart de gedeelde informatie tussen twee sub gebieden van een kwantumveld. Uit ons onderzoek is gebleken dat driekwart van de gedeelde informatie, de maximale waarde is die negativiteit kan hebben voor een systeem met twee qubits, mits het systeem puur is bij een temperatuur van ongeveer nul graden Kelvin. De negativiteit voor sytemen met (veel) meer dan twee qubits lijkt een maximale waarde van een halve gedeelde informatie te hebben. De dualiteit relateert AdS aan een kwantumveld en een veld is niet vergelijkbaar met twee qubits, maar eerder aan een hele hoop qubits. Het blijft dus een vraag waarom de holografische negativiteit daadwerkelijk gelijk is aan driekwart van de gedeelde informatie.

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2

Abstract

The duality between quantum field theories and theories of gravity in negatively curved space, known as Anti-deSitter space is a first example of the holographic principle. In context of this duality Ryu and Takayanagi proposed a duality between entanglement entropy and the minimal surface in Anti-deSitter space, or a holographic entanglement entropy. Entanglement entropy obeys certain inequalities such as Araki-Lieb and (strong) subadditivity and these inequalities are likewise obeyed by holographic entanglement entropy. Besides the general inequalities of entropy, the mutual information is monogamous in the holographic dual, whereas this is not true in general. For instance, classical correlated quantum states, such as a Markov chain or the GHZ state with qubits traced out, do not obey the monogamy inequality. The duality is a “classical/quantum” duality, hence there is no reason why the classical holographic mutual information should impose monogamy on classical correlated states. These classical correlated states are expected to be dual to the corrections terms on holographic entropy, which arise from underlying quantum effects inside anti-deSitter space.

Entanglement entropy is a measure for pure states. For mixed states, an effective measure is the entanglement negativity. It has been proposed that the holographic negativity is equal to three-quarters of the mutual information. In this thesis an attempt has been made to find a pattern in quantum states for which the relation holds, by numerically computing the ratio negativity/mutual information with a program written in Mathematica. In general, this relation is only found as upper bound for two qubit system, which are maximally entangled in the groundstate. The negativity for multipartite systems are seemingly bounded by one-half of the mutual information.

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CONTENTS 3

1 Introduction

1.1 Entanglement

In the early days, the concept of entanglement was mainly perceived as a qualitative aspect of quantum theories that distinguishes it from our classical intuition. This distinction became quantitative with Bell’s inequality, which was established after the EPR paradox. Bell’s inequality can be seen as an early attempt to quantify quantum correlations and gave means to experimentally verify the aspects of quantum theory [18][13].

Nowadays, we are able to coherently prepare, manipulate, and measure individual quantum systems, as well as create controllable quantum correlations. With all these developments, quan-tum correlations became a physical resource that may be used to perform tasks that are either impossible or considerably inefficient with classical resources, such as computational, and crypto-graphic tasks [18][13]. But, what is a quantum correlation?

In the context of quantum information, classical correlations are defined as correlations that can be generated by local operations and classical communications (LOCC) [18]. Correlations in a quantum state, that cannot be simulated classically, are quantum correlations. A type of quantum correlation between particles is entanglement of states. A practical application of en-tanglement is quantum communication, which is established by correlations between separated quantum systems, or entangled systems. When Alice holds one end of the entangled system and Bob the other, the system can be used as a quantum information channel. Through this channel they may exchange quantum states. If the systems of Alice and Bob are not maximally entangled, their systems may be entangled with the environment. This result in information being leaked, which impairs our ability to send quantum states over long distances [18]. To prevent this in-formation leak, one can perform entanglement distillation, with LOCC. Entanglement distillation transforms N copies of an arbitrary entangled state to M < N copies of maximally entangled states, or so-called Bell pairs. The rate at which this transformation occurs, is measured by dis-tillable entanglement [18]. It is, therefore of interest to study an entanglement measure, known as the logarithmic negativity. The logarithmic negativity provides an upper bound to distillable entanglement [3], and so, a measure of which states are most suitable for quantum communication. In 2001, a duality was proposed between negatively curved spacetime, or “bulk”, and quantum field theory. This conjecture is known as the Anti-deSitter spacetime/conformal field theory con-jecture, and it allow us to study quantum correlations which are otherwise impossible or extremely challenging to study. In context of this conjecture we are able to study entanglement measures, such as negativity, in classical theories of gravity [22]. An entanglement measure, suitable for pure states, called entanglement entropy is already widely studied in context of this duality [21]. The significant number of studies have provided support for the duality between entropy and space-time. In the beginning of this year, Chaturvedi et al. proposed a relation between entanglement negativity and mutual information in context of the duality [3]. Later, Jain et al. computed this relation for two adjacent subregions [12]. Besides entropy it is of interest to study negativity in context of this duality too, since negativity is an effective measure for mixed states.

In this thesis, a short introduction to the Anti-deSitter spacetime/conformal field theory con-jecture is given, before the entanglement entropy in context of this duality is evaluated. We find that inequalities as Araki-Lieb, and (strong) subadditivity, obeyed by entropy in non-holographic theory are likewise obeyed by the area law in holographic theory[8]. Secondly, we find that the area law imposes monogamy on mutual information between quantum subsystems A and B [8]. This monogamous mutual information is not a general inequality obeyed in quantum theory. For instance, the Markov chains and GHZ state with qubits traced out, have classical correlations, but no entanglement. Classical correlations are not monogamous. This becomes apparent, when considering Alice, Bob, Carla, and Diego, reading the same newspaper. Afterwards they are able to all contain all information from the newspaper.

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2. ANTI-DESITTER SPACE/CONFORMAL FIELD THEORY (ADS/CFT) CONJECTURE 5

Since the area law is classical and correlated to highly quantum states [11], there is no reason to assume that mutual information is monogamous for classical correlated quantum states. The natural question that follows is if the quantum corrections made to the area law by Faulker et al. [5], imposes any monogamy inequality on classical correlated states. It turned out that answering this question is a challenging task, since computing mutual information in quantum field is not trivial to perform.

After evaluating entanglement entropy, the proposed relation between entanglement negativity and mutual information in holographic theory, is numerically computed for the Werner state and random thermal states. We have discovered that the relation does not hold in general but is seemingly an upper bound for states, with entanglement between an equal number of qubits, and are (nearly) maximally entangled in the groundstate.

2 Anti-deSitter space/conformal field theory (AdS/CFT)

conjecture

In his world-famous relativity theory, Albert Einstein describes gravity as a local curvature of spacetime, due to mass and energy. This curvature of spacetime is described by the stress tensor Tµν in Einstein’s field equation’s:

Gµν+ Λgµν =

8πG c4 Tµν.

The geometry of spacetime is described by the vacuum solution of Einstein’s field equations, ob-tained by setting Tµν to zero. The spacetime can be a flat-, positive curved-, or negative curved

manifold, and depends on the cosmological constant Λ being zero, positive or negative. Here, flat space is the familiar Minkowski spacetime. The positive and negative curved spaces are referred to as deSitter and Anti-deSitter spacetime, respectively

In 2001, a duality between Anti-deSitter (AdS), in which quantum gravity can be described and a (non-gravitational) quantum field on the boundary of AdS was proposed [11]. The field is a conformal field, meaning it is invariant under conformal transformations1. The Anti-deSitter space/conformal field theory (AdS/CFT) conjecture is an example of the Holographic principle, proposed by ’t Hooft [22]. ’t Hooft suggested that quantum gravity can be described as a topolog-ical quantum field theory, where all degree of freedom can be projected onto the boundary. The result is that all information about a higher dimensional space is encoded on a lower-dimensional boundary of that space [22][11].

In textbooks, the AdS/CFT conjecture, is usually portrayed as a cylinder, where time runs along the length of the cylinder, and a slice perpendicular to the time axes, a time-slice, is the hyperbolic space on a moment in time (See figure 1). The angels and demons on a time-slice, are continuously halved, when getting closer to the boundary and such never reach the boundary. This plane can be seen with equally sized angels and demons and the boundary is infinitely far away.

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3. ENTANGLEMENT ENTROPY 6

Figure 1: Visualization of the AdS/CFT conjecture. Inside the cylinder is the anti-deSitter space. All information in AdS is encoded in the CFT at boundary infinitely far away. This is portrait as angels and demons that are continuously halved and such never reach the boundary. The plane with angels and demons can be seen as a hyperbolic plane, where the angels and demons are all the same size.

Furthermore, AdS/CFT is a strong-weak coupling duality [11]. A system with a high coupling constant is considered highly quantum, while systems with weak coupling constants are classical systems. AdS/CFT is then a “classical/quantum” duality, where for instance, classical geometries inside the bulk are dual to quantum effects, like entanglement, on the boundary.

3 Entanglement entropy

A quantum system split up into two subsystems is called a bipartite system and described by the Hilbert space H = HA⊗ HB. At zero temperature, the full system will be in the groundstate. If

we assume that the groundstate is not degenerate, the full system is considered a pure state. A pure state is described by the state vector:

|Ψi =X

i,j

cij|ψiiA⊗ |φji_B,

where |ψiiA, and |φji_B are the basis vectors of HA, and HB, respectively. If the state can be

written as a product state:

|Ψi =X i ci|ψiiA⊗ X j cj|φji_B,

then the state is considered a separable state. In other words, a state is entangled if the state cannot be written as a product state.

Occasionally, not all information about a state is accessible for an observer. Consider a ther-mal gas, where the state of every single particle is unknown. The best one can do is describe the behaviour of many particle, by a probability distribution of the states that the particles can adopt. Equally, the state of a system can be described as a canonical ensemble of pure states. These type of states are called mixed states, and are represented by a density matrix: ρ =P

ipk|Ψki hΨk|.

Note, that a pure state can also we written as a density matrix. A density matrix of a pure state has one probability of 1, and the rest zero’s.

Mixed states arise from bipartite, entangled states, where the observer can only observe and perform operations on a subsystem, let’s say A. The subsystem is then described by the reduced density matrix, which is obtained by tracing out HB: ρA ≡ TrBρ. Likewise, an observer in HB,

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3. ENTANGLEMENT ENTROPY 7

will only be able to access ρB, obtained by taking the trace with respect to A.

The observer in A is able to quantify the entanglement between the subsystems A and B, by performing an entanglement measure. The entanglement entropy is one such measure and is defined as the von Neumann entropy:

S(ρA) = −TrAρAlog ρA (1)

The entropy has certain properties, for example, if the full system is in a pure state, then the entropy SAB equals zero, and the subsystems SA and SB are equal to each other.

3.1 Properties of entanglement entropy

In this paragraph, some inequalities, obeyed by entanglement entropy are discussed and proven. The first inequality is known as the subadditivy inequality:

S(AB) ≤ S(A) + S(B), (2)

which hold if and only if the systems A and B are uncorrelated, that is if ρAB = ρA⊗ ρB.

Here, we’ve used the notation S(A) ≡ S(ρA), S(B) ≡ S(ρB), and S(AB) ≡ S(ρAB).

Subadditivity can be proven with an application of Klein’s inequality [15]. The Klein’s inequality states:2

Tr [f (ρ) − f (σ) − (ρ − σ)f0(ρ)] ≥ 0

Where ρ ≡ ρABand σ ≡ ρA⊗ρBare two positive definite, hermitian matrices, and f : (0, ∞) →

R is a differentiable convex function.3

Theorem 1. Suppose S(AB) is the von Neumann entropy for a state ρAB, where ρABis a positive

definite, hermitian density matrix and ρAB = ρA⊗ ρB, then

S(AB) ≤ S(A) + S(B)

Proof. Consider the Klein’s inequality with f (A) = A log A, A ∈ {ρ, σ}: Tr [ρ log ρ − σ log σ − (ρ − σ)(log σ + 1)] ≥ 0

Tr [ρ log ρ − ρ log σ − ρ + σ] ≥ 0

Tr [ρ log ρ] + Tr [σ] ≥ Tr [ρ log σ] + Tr [ρ] Tr [ρ log ρ] ≥ Tr [ρ log σ]

Given that ρ = σ. then:

S(AB) = −Tr [ρ log ρ] ≤ −Tr [ρ log σ]

≤ −Tr [ρAB(log ρA+ log ρB)]

≤ −Tr [ρAlog ρA] − Tr[ρBlog ρB]

≤ S(A) + S(B)

2_{For a proof of Klein’s inequality see [2]}

3_{A function is called convex when a line segment between two random points on the graph of the function are}

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4. ENTANGLEMENT ENTROPY IN QUANTUM FIELD THEORY 8

The second inequality is the triangle inequality, known as the Araki-Lieb inequality:

S(AB) ≥ |S(A) − S(B)| (3)

Proof. (Araki-Lieb inequality)

Consider two system A and B, and let C be the complement. Hence C purifies the system AB. With the subadditivity inequality we have:

S(C) + S(A) ≥ S(AC) (4)

S(ABC) is a pure state, hence the subsystems are equal: S(AC) = S(B) S(C) = S(AB)

By plugging these into (4) and rearraging the inequality, we find: S(AB) ≥ |S(A) − S(B)|

So far, we’ve partitioned the systems into two subsystems, but the system may also be parti-tioned into three subsystems A, B and C. This system is denoted as a tripartite system. For a tripartite system, the entropy obeys the strong subadditivity inequality:

S(ABC) + S(B) ≤ S(AB) + S(BC)

The strong subadditivity can also be expressed in another form when A and B are two depen-dent, overlapping subsystem:

S(A ∪ B) + S(A ∩ B) ≤ S(A) + S(B) (5)

Throughout this thesis, we shall use expression (5).

The proof for strong subadditivity is based on Lieb’s theorem, and is a surprisingly complex proof. Due to the complexity, it shall be disregarded here. For further information see Nielsen and Chuang [15].

4 Entanglement entropy in quantum field theory

In classical mechanics, particles are described as a zero-dimensional point in space. The dou-ble slit experiment has shown, however, that this concept is inaccurate. Particles have wave-like characteristics and for an accurate description, one needs to consider particles as excitations in quantum fields [14]. For instance, the first quantum field theory is quantum electrodynamics, which described the interactions between electron- and electromagnetic fields. Here, an electron is an excitation in an electron field and photons are excitations in electromagnetic fields [14].

To visualize a field φ(~x, t) describing a physical system, we can visualize degrees of freedom (dof) on each point in space [14]. In a discreet system, this would be a lattice with a finite Hilbert space Hα on each site α [20]. Taking the continuum limit, by letting the lattice spacing go to

zero, we find a field described by an infinite set of dof, with an infinite energy [14]. This infinity is referred to as UV divergence and can be regulated by placing an UV cut-off [20]. Regulating the field, with an UV cut-off is placing a finite distance between each point in the field, quantizing

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4. ENTANGLEMENT ENTROPY IN QUANTUM FIELD THEORY 9

the field [20].

In quantum field theory (QFT) we are not interested in the entropy between two points in the field. If one would try to trace out every point in a field, except for two, the system will decohere extensively, and any entanglement that was present would get destroyed. Instead we factorize the field into two subregions A and its complement Ac, which decomposes the lattice Hilbert space into: H = HA⊗ HAc [20]. We may now construct the reduced density matrix ρ_A by tracing out

HAc and compute the entropy as was given in (1) [20].

Let us consider the field in the groundstate, i.e. the field is in a pure state. It follows that the entropy of two subregions A and Ac _{are equal to each other. Note that the only space shared by}

A and Ac _{is the boundary of A, denoted as the entangling surface. Hence the entropy is defined}

locally at the boundary and can be approximated by the number of links that are cut by the entangling surface. When taking the continuum limit, the entangling surface no longer clearly cuts through links between points inside A and point outside A [20]. Deciding which point is in A and which is in its complement has become ambiguous, and is solved by placing a UV cut-off [20]. Hence, we consider a lattice in the groundstate with lattice spacing and the lattice is divided into two regions A, and its complement Ac. (See figure 2a). The number of dof on every lattice site is given by the central charge c.

A

∂A

(a) Lattice with lattice spacing and number of degree of freedom c on ev-ery lattice site. The lattice is divided in a subregion A and its complement Ac. The system is considered in its groundstate, hence we can approxi-mate the entropy by counting corre-lations between point in A and Ac_at

the entangling surface ∂A.

`

A

∂A

(b) Quantum fluctuations connect points inside A with point outside A at different length scales `. By taking these fluctuations in consideration, a better approximation is found for the entropy.

The entropy is now given by the number of links between points inside A and outside A, that are cut by the entangling surface [20]. This is given by the length of the entangling surface divided by the lattice spacing. The entropy is then approximately: S(a) ∝ Length(∂A)

A better approximation is gained by considering quantum fluctuations on the entangling sur-face, connecting points inside A with points outside A at different length scales ` (See figure 2b). These quantum fluctuations all contribute a term Length (∂A)_` to the entropy. The length of the fluctuations extends to the correlation length ξ and its minimal length is cut off at . The entropy is then found by integrating over all length scales:

SA= c

Z ξ

d`Length/ Area/ V olume(∂A) `d−1

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5. HOLOGRAPHIC ENTANGLEMENT ENTROPY 10 t ρ θ A B γA

Figure 3: AdS3, with CFT2 at the boundary. The CFT is split up into two subregions A and B.

The minimal surface γA is the geodesic through AdS which ends on the boundary of A.

In CFT2 the entropy is [20]: SA= c 3log ξ

and is dual to a geometry in AdS3, as we will present in the next paragraphs.

5 Holographic entanglement entropy

In the seventies, a relation was discovered for what is now called the Bekenstein-Hawking entropy, or black hole entropy. It relates the thermal entropy to the surface area of the black hole. Analo-gous to the Bekenstein-Hawking entropy, Ryu and Takayanagi proposed an area law in context of AdS/CFT. Their proposition stated that the entanglement entropy in CFTd+1 is related to the

minimal area surface (γA) in AdSd+2 [21]:

SA=

Area (γA)

4Gd+2n

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where Gd+2n is the Newton constant, and γAis the d-dimensional minimal surface in AdSd+2.

In two spatial dimensions the minimal surface then becomes the shortest line through AdS3, which

begins and ends at the boundary of the subregion A in CFT (See figure 3).

5.1 The minimal surface in AdS

3

The entropy of CFT2 that was given in 4, can now also be derived by computing the geodesic in

AdS3 at a constant time. This can be done in the global co¨ordinates (t, ρ, θ) (See figure 3), and

in Cartesian coördinats, by considering an AdS as the upper half of a hyperbolic plane [21]. The geometry of this plane is the Poincaré half-plane model and is described by the Poincaré metric:

ds2=R

2

z2(dx

2_{+ dz}2_),

where R is the ADS radius. A path through this plane is given by the action integral: s = Z ds = Z _Rdz z p x02_{+ 1 ≡} Z L(z, x(z), x0(z))dz (7)

with L(z, x(z), x0(z)), the Lagrangian. The minimal length is found by minimizing the path, with the Euler-Lagrange equation

d dxL(z, x(z), x 0_{(z)) =} d dz d dx0L(z, x(z), x 0_(z))

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5. HOLOGRAPHIC ENTANGLEMENT ENTROPY 11 z x A CFT AdS

(a) Geodesic through AdS3, which

ends on the boundary of subregion A in CFT2 on the x axes. The

boundary of AdS is infinite far away from every point in AdS, hence the geodesic is infinite. z x A CFT AdS `/2

(b) Geodesic through AdS3, with an

UV cutoff . With the cutoff, we obtain a finite sized subregion A in CFT2 and finite geodesic.

Figure 4: Representation of the Poincar´e half-plane model. The upper half plane (z ≥ 0) is a hyperbolic plane, representing AdS3. The CFT2at the boundary of AdS3is placed at the x axes.

The solution, x2_{+ z}2_{= r}2_{, is a semi-circle with z ≥ 0.}

The geodesic is parameterized by:

x = r cos(θ) z = r sin(θ). By plugging these in (7) we find the length of the geodesic:

γ = R Z π 0 dθptan(θ) 2_{+ 1} tan(θ) = R Z π 0 dθ cosec(θ) (8)

The integral is divergent, which is expected, since the boundary of ADS3is an infinite distance

away from every point in the bulk. This divergent is the infrared divergent and is dual to the UV divergent in QFT, and likewise, the integral can be regulated by restricting the size of AdS with an infrared cut-off [21].

With the infrared cut-off we obtain a finite sized region A with length `, such that x ∈ −`

2 , ` 2

and a geodesic running from {R, θ} to {R, π − θ}, where θ is small and thus can be approximated as: θ = 2_` (See figure 4b).

By plugging these integral boundaries in (8), the geodesic can be obtained in terms of the UV cut-off and length ` of A:

γ = R Z π−2_` 2 ` dθ cosec(θ) = 2R log cot` ≈ 2R log` (9)

The holographic entropy as was given in (6) has now become: S(A) = R 2Gn log` = c 3log ` , with c = 3R 2Gn .

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5. HOLOGRAPHIC ENTANGLEMENT ENTROPY 12

5.2 Properties of holographic entanglement entropy

If states in CFT have a dual in classical theory of gravity, then the inequalities, discussed in section 3.1 must also hold in said theories of gravity [8]. In this section we will evaluate the inequalities obeyed by entanglement entropy and show they are also obeyed by the area law.

The first one is a general property of entropy, which is trivial, but added for consistency. (i) S ≥ 0

Proof. This is trivial. An area is always positive. (ii) Subadditivity: S(A) + S(B) ≥ S(A ∪ B)

Proof. Let Ab and Bb be two regions on a time-slice of a static AdS, and A and B be two

subsets on the boundary of the AdS. The regions Ab and Bb are adjacent to A and B,

respectively, and the boundary of the regions, ∂Aband ∂Bbare minimal surfaces, which end

on the boundary of A and B, respectively. Furthermore, let γA∪B be the minimal surface,

ending at the boundary of A ∪ B. (see figure 5).

Notice that ∂(Ab∪ Bb) ends on ∂(A ∪ B), and by definition of the minimal surface γA∪B ≤

any other path between ∂(A ∪ B).

Secondly, ∂(Ab∪ Bb) = γA+ γb− ∂(Ab∩ Bb), with 0 ≤ ∂(Ab∩ Bb) ≤ M in{γA, γB}. Thus

γA+ γb− ∂(Ab∩ Bb) ≥ γ(A∪B), and then also:

γA+ γB≥ γA∪B

(iii) Araki-Lieb: |S(A) − S(B)| ≤ S(A ∪ B)

Proof. Note that with subadditivity, the Araki-Lieb inequality can be rewritten as: |S(A) − S(B)| ≤ S(A) + S(B)

With S ≥ 0, this is always true.

(iv) Strong subadditivity: S(A) + S(B) ≥ S(A ∪ B) + S(A ∩ B)

Proof. Consider the same setup as is shown in figure 5. From the proof for subadditivity, we have ∂(Ab∪ Bb) = γA+ γb− ∂(Ab∩ Bb), and let γA∩B be the minimal surface, ending

on ∂(A ∩ B). Notice that ∂(Ab∩ Bb) and γA∩B, both end on ∂(A ∩ B). By definition of the

minimal surface, we have ∂(Ab∩ Bb) ≥ γA∩B. Since γA+ γb− ∂(Ab∩ Bb) ≥ γ(A∪B) is true

then also:

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6. MUTUAL INFORMATION 13 Ab Bb ∂(Ab∪ Bb) γA γB γA∪B A B

Figure 5: Two subsystems A and B on the boundary of a time slice of AdS3. Inside the bulk, the

minimal surfaces γAand γBenclose the regions Aband Bb, respectively. Any path from one end of

∂(A∪B) to the other end is larger or equal then the minimal surface γA∪B, thus ∂(Ab∪Bb) ≥ γA∪B

In [8], Headrick et al. has proven that all known general properties of entanglement entropy are obeyed by the area law. Besides the general properties of entanglement entropy, the area law also obeys an inequality, known as monogamy of mutual information [8]. For a CFT to have a dual in classical theories of gravity, the CFT is acquired to obey this monogamy inequality. This is however, not always true, as will become clear in the next section.

6 Mutual information

For two disjoint regions A and B, we can create a linear formation of the entropy, such that it cancels the UV divergences [6]. The linear formation:

I(A : B) := S(A) + S(B) − S(AB) (10)

is then a finite quantity known as the mutual information. It measures both classical and quantum correlations between A and B [6]. A measure of extensively of mutual information (MI) is the tripartite information4 [6].

I3(A : B : C) :=S(A) + S(B) + S(C) − S(AB) − S(BC) − S(AC) + S(ABC)

=I(A : B) + I(A : C) − I(A : BC)

The holographic mutual information is always extensive, i.e. I3 ≤ 0 [6]. This is because

holographic mutual information quantifies quantum correlations [5] and a fundamental property of entanglement is monogamy e.g. if qubit A and B are maximally entangled, then A or B cannot be entangled with qubit C. The result is monogamy of mutual information. In non-holographic theories, however, the mutual information I(A : B) quantifies both classical correlations and quantum entanglement [6]. Classical correlations are not monogamous hence the monogamy of mutual information (MMI) is a property of holographic theory, which does not hold in general [6].

The proof of monogamous mutual information in AdS3/CFT2is visualized in figure (6).

Proof. (Monogamous mutual information)

Let γA, γB and γC be three geodesics in AdS3, anchored to the boundary of subsets A, B

and C in CF T2, respectively (See figure (6)). Furthermore, let `A, `B and `C be the lengths of

4_{In classical information theory, the tripartite information is also called the I-measure or the interaction}

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6. MUTUAL INFORMATION 14

the subsets A, B and C, and `AB, `BC and `AC be the lengths of the subsets AB, BC and AC,

respectively.

Then À ≤ ÀB, `C ≤ `BC and `B+ ÀBC = 2 `B+ ÀC ≤ ÀC. Since γ = 2R log`, we can

rewrite this to:

γA≤ γAB γC≤ γBC γB+ γABC≤ γAC and such: γA+ γB+ γC+ γABC ≤ γAB+ γBC+ γAC A B C A B C

S(AB) + S(AC) + S(BC) S(A) + S(B) + S(C) + S(ABC)

Figure 6: A tripartite system in CFT2, with geodesics anchored to the boundaries of the subregions

A, B and C, extending into AdS3. From the figure its clear that S(AC) = S(B) + S(ABC),

S(AB) > S(A) and S(BC) > S(C), thus S(AB) + S(AC) + S(BC) ≥ S(A) + S(B) + S(C) + S(ABC).

As mentioned before, the monogamy inequality does not hold for general quantum systems. Two often mentioned examples that do not obey MMI are non-trivial Markov states and GHZ state, with 4 or more qubits. In the next section, we will discuss these two states.

6.1 Markov states and GHZ states

6.1.1 Markov chains

Classical Markov chains: A Markov chain is a stochastic process, with so called Markov properties. The stochastic process has Markov properties when the status of a system at a point in time is sufficient to determine the conditional probability distribution of the status of the system at a next time step. In other words, three jointly distributed random variables form a classical Markov chain in order X − Y − Z, if X is the probability distribution of Y , when X is known, and Y is the probability distribution of Z, after Y has been determined. One of the characteristics of a Markov chain is that the conditional mutual information vanishes:

X − Y − Z is a Markov chain ↔ I(X : Y |Z) = 0 with I(X : Y Z) = H(XY ) + H(Y Z) − H(XY Z) − H(Y ) and H(X) = −P

x∈XP (x) log p(x) is

the Shannon entropy.

Quantum Markov chains: For a quantum mechanical Markov chain, the variables are re-placed by density matrices. A tripartite state ρABC on HA⊗ HB⊗ HCis a Markov chain in order

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6. MUTUAL INFORMATION 15

A − B − C if the correlation between A and C are mediated through B [6]. In this case, there exist a positive, trace-preserving map E : B → B ⊗ C, such that [6]

ρABC= (IA⊗ E)ρAB

This map shows that C can be constructed by operations solely on B. All Markov states in a finite dimension, acquire the form [7]:

ρABC=

M

i

piρAbL

i ⊗ ρbRiC (11)

where HB has been decomposed into HB=LiHbL

i ⊗ HbRi and with states ρAbLi on HA⊗ HbLi

and ρ_bR

iC on HbRi ⊗ HC.

It has been proven by Petz [17] that just like the classical case, the quantum condition infor-mation vanishes for a quantum Markov state:

I(A : BC) = S(AB) + S(BC) − S(ABC) − S(B) = 0

By rewriting the monogamy inequality as I(A : C|B) ≥ I(A : C), it becomes clear that quan-tum Markov chain can only obey MMI if I(A : C) = 0 [6].

Expression (11) shows there is no entanglement between A and C, for all Markov chains. This means quantum Markov chains must have classical correlations between A and C, otherwise I(A : C) = 0 and Markov chains would not violate the monogamy inequality. When considering a quantum walk, an example of a Markov chain, it is clear that the system at t = 0 holds informa-tion about the locainforma-tion of the system at t = 2, and thus there is classical correlainforma-tions between the system at t = 0 and the system at t = 2. The quantum walk is therefore an example of a Markov chain that does not obey the monogamy inequality.

6.1.2 GHZ state

The Greenberger–Horne–Zeilinger (GHZ) state is non-biseparable state of at least 3 qubits, written as [4]: |GHZi = |0i ⊗M + |1i⊗M √ 2 , M ≥ 3

The GHZ state is known to be maximally entangled [4]. Once one or more qubits are traced out the entanglement is destroyed and only classical correlations are left [4]. It follows that once a trace is taken over a n-qubit GHZ state with n ≥ 4, only classical correlations between 3 or more qubits are left. Since classical correlation are not monogamous, this violates the monogamy inequality. The violation can easily be shown, by calculating the tripartite information:

Consider a 4-qubit GHZ state. When tracing out a qubit, we find: ρABC =

1

2[(|000i h000|) + (|111i h111|)]

with eigenvalues 1₂(2x) and 0 (6x). This density matrix is invariant under partial transpose, hence the eigenvalues of ρTn

ABC with n ∈ {A, B, C} remain the same. One can check that for this state,

the sum of the eigenvalues are the same for all ρmwith m ∈ {A, B, C, AB, BC, AC, ABC}. Finally,

with eq (1) we find S(m) = log 2 and the tripartite information becomes: I3= log 2 ≥ 0

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7. ENTANGLEMENT NEGATIVITY 16

6.2 Monogamy inequality for corrections to holographic entropy

Both the Markov state and the GHZ state as described above, violate the monogamy inequality due to classical correlations. The area law is a classical geometry and since AdS/CFT is a strong-weak coupling duality, related to quantum correlations in CFT. It makes sense that the area law does that impose monogamy of mutual information on classical correlated states.

Besides the classical physics in AdS, there are quantum effects inside AdS [5], dual to classical quantities in CFT. These quantum effects give correction terms to the area law.

6.2.1 Corrections to holographic entanglement entropy

The quantum effects inside AdS, influence the overall entanglement entropy between a region A and its complement [5]. These quantum effects inside AdS give correction terms of order G0

N to

the area law [5]. The area law can be seen as the leading term of the GN expansion [5]. The full

equation for holographic entanglement entropy then becomes [5]:

S(A) = Scl(A) + Sqm(A) + O(GN) (12)

where Scl(A) is the area law (6), and Sqm is the entropy from quantum effects inside the bulk:

Sqm= Sbulk+

δA 4GN

+ h∆SW aldi + Scounterterms (13)

The first term is the entanglement entropy between regions Ab and Acb which are separated

by the minimal surface [5]. When calculating the bulk entropy, the bulk can be considered as an effective field theory, in which Sbulk can be computed as normally is done in any QFT [5]. The

second term in (13) is for the shift in the classical background, due to the quantum corrections, causing a change in the area [5]. The term h∆SW aldi, is the quantum expectation value of the

Wald-like entropy and Scounterterms is the counter terms needed for finite computations [5].

A natural question that follows is if these quantum effects inside AdS, impose any monogamy inequality for classical correlated states. To answer this question, we want to compute the mutual information solely from bulk quantum entanglement, which is done by considering a disjoint system of regions A, B and C, where A, B and C are far apart from each other. This, because the minimal surfaces γA+γBequals γAB, when A and B are far apart, and thus the mutual information between

A and B is zero, when they represent spacial regions. It follows that the mutual information and the tripartite information for the area law is zero. Furthermore, the last three terms in (13) also cancel for a disjoint region [5]. On the other hand, the tripartite information is not zero for the quantum entanglement in the bulk [5]. Thus, the bulk entanglement is the only part that contributes to the tripartite information, hence:

I3(A : B : C) = I3bulk(Ab : Bb: Cb)

Computing the mutual information for a QFT appeared an overall challenging task, and shall therefore be omitted here.

7 Entanglement negativity

Entanglement negativity is one of the entanglement measures proficient in quantifying multipartite mixed states. What distinguishes negativity from the other suggested measures for mixed states, is that negativity is efficiently computable [23]. Negativity is monotone under LOCC operations, convex but not additive. It provides an upper bound to teleportation capacity, but has further no interpretation [23]. It is defined as:

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7. ENTANGLEMENT NEGATIVITY 17

N (ρ) ≡ ||ρ

TA_||

1− 1

2 (14)

Negativity is based on the trace norm of the partial transpose of ρ. For a bipartite system shared by Alice and Bob, the partial transpose ρTA _{is defined as [16]:}

hiAjB| ρTA|kAlBi ≡ hkAjB| ρ |iAlBi

whereas the trace norm of a hermitian operator O is defined as ||O||1= Tr|O| ≡ Tr

√

O†_{O. The}

normalized partial transpose satisfies TrρTA_{= 1, hence [23]:}

TrρTA₌ X λi>0 λi+ X λi<0 λi= 1 → X λi>0 |λi| = X λi>0 λi= 1 − X λi<0 λi (15)

By writing out the trace norm of the partial transpose of ρ and using eq (15), we obtain: ||ρTA_|| 1= X λi>0 |λi| + X λi<0 |λi| = 1 + 2 X λi<0 |λi| ≡ 1 + 2N (ρ) .

Negativity essentially quantifies the Positive Partial Transpose (PPT) criterion5 for separa-bility, by measuring the degree in which in the eigenvalues of ρTA _{are negative [23]. The PPT}

criterion is sufficient in determining separability if dim(HA⊗ HB) ≤ 6 [10]. In higher

dimen-sions, there may exist entangled states with positive partial transposed [10]. These states are bound entangled, and cannot be distilled for quantum communication purposes [19]. For quantum communication, we desire free entangled states, which have negative partial transposed [19]. One may notice that negativity vanishes for separable- and bound entangled states, since ||ρTA_||

1= 1.

Hence, negativity identifies free entangled states.

7.1 Logarithmic negativity

A second quantity, which will be focused on in holographic theory, is the logarithmic entanglement, defined as [23]:

E(ρ) = log ||ρTA_||

1 (16)

The logarithmic negativity is monotone under deterministic LOCC and is additive [23]. It provides an upper bound on distillable states [23]. It does have some drawbacks. For instance, it is not convex and it does not reduce to the entropy entanglement for pure states [23]. The logarithmic negativity is usually referred to as the ”negativity”, and we shall do likewise in the rest of this thesis.

The negativity in CFT is derived with the replica technique. With this technique, the negativity for two adjacent systems becomes [1]:

E = c 4ln

`1`2

`1+ `2

+ cnst,

where `1 and `2 are the lengths of subregions 1 and 2, respectively.

5_{The PPT criterion by Peres provides a necessity for separable states. Peres states that if the partial transpose}

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8. HOLOGRAPHIC ENTANGLEMENT NEGATIVITY IN ADS3/CFT2 18

8 Holographic entanglement negativity in AdS

3

/CFT

2

A holographic prescriptions for the negativity has been proposed by Chaturvedi et al. [3], for a bipartite system in AdS3/CFT2:

E = lim

B→Ac

3 16G(3)_N

(2LA+ LB1+ LB2− LA∪B1− LA∪B2)

where L is the minimal surface in AdS3, and the Brown-Henneaux formula c = _2G3R(3) is used. The

CFT is divided into a subregions A, B1, and B2, where B1 and B2 extend to infinity. With the

area law (eq (6)), the equation reduces to: E = lim B→Ac 3 4(2SA+ SB1+ SB2− SA∪B1− SA∪B2) (17) = lim B→Ac 3 4(I(A : B1) + I(A : B2)) (18)

Later, a holographic description for a adjacent system was proposed by Jain et al. [12]:

E = 3

16G(3)_N

LA1+ LA2− LA1∪A2 =

3

4I(A1: A2)

This relation between negativity and mutual information is a holographic relation, which does not hold for all states in general. In this thesis an attempt has been made to find a pattern in quantum states for which the relation holds, by numerically computing the ratio negativity/mutual information (E_I) with a program written in Mathematica (See Appendix).

First, notice that the negativity is an entanglement measure, designed for mixed states. Sec-ondly, states with more (quantum) correlations have per definition higher mutual information and one might expect negativity also to be higher for state with more quantum correlation. An inter-esting question is if states with highest correlations also provide a bound on the ratio. Intuitively, one might also expect robust states to maintain higher negativity at finite temperatures, thus for mixed states. Hence robust entangled states might be of interest. Due to these two statements, we decided to first evaluated the ratio E_I with Werner states, and continued with general, random states.

8.1 Numerical results

8.1.1 The Werner state

The Werner state belongs to the family of maximally entangled mixed states [9], and gives a useful spectrum from maximally mixed, separable states to maximally entangled pure states. The Werner state for two qubits is given by:

ρw= p |Ψ−i hΨ−| +

1 − p

4 I4, (19)

where |Ψ−i =_√1

2(|10i − |01i) is a Bell pair. The type of Bell pair can be chosen freely, since all

Bell pairs give equal results for the ratio E_I. The Werner state reduces to the Bell pair for p = 1, is separable for p ≤ 1₃, and maximally mixed (identity matrix) for p = 0.

The results for the two qubit Werner state are displayed in figure (7). Remarkably we find the two qubit Werner state to be maximal at p = 0.59 with a ratio of E_I = 0.755.

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The Werner state can be generalized to multipartite systems: ρw= p |Ψi hΨ| +

1 − p d Id,

where |Ψi is a multipartite system, and d = 2q_{, where q is the number of qubits. Here, the GHZ}

state or the W state6 _{is used for |Ψi. The three and four qubit states are pure for p = 1, and}

separable for p = 1₅, and p = 1₉, respectively.

The Werner state for three, four and six qubits are plotted in figure (7). For the four qubit Werner state, there has been made a distinction between entanglement between qubit a & bcd, and ab & cd. By evaluating the graphs, one can see that the ratio for entanglement between qubit a and the rest proceed the 3₄. The ratio for both the GHZ- and the W state with entanglement between ab & cd, is equal for every p, and have seemingly an upper bound at 3₄.

(a) Entanglement between equal amount of

qubits (b) Entanglement between qubit a and the rest

Figure 7: The graphs of the ratio negativity/mutual information versus different values of p in the Werner state. The terms GHZ6, GHZ4, W4, GHZ3, and W3 in the legends denote the Werner state with the GHZ-, or W state, with 3, 4 or 6 qubits. The” Bell pair” is the 2 qubit Werner state with the Bell pair, as was given in (19). The a|bcd, ab|cd, indicates entanglement between a & bcd, and ab & cd, respectively. Noteworthy are the states with entanglement between equal amount of qubits, due to the resulting E_I ≈3

4, as seemingly upper bound for these states.

8.1.2 Random thermal states

Based on the result of the Werner states, the focus remained on entanglement between equal amounts of qubits, but was extended to general random states, at finite temperatures, and with more qubits. This was realized by constructing random Hamiltonians. Here, the Hamiltonians are symmetric matrices with random real entries r, where −1 < r < 1. From the random Hamiltonian a thermal density matrix was constructed:

ρ =X

i

ekB tEi _|ψ_i_{i hψ}_i_{| ,}

where kB = 1, and Ei, and |ψii are the eigenvalue, and eigenstate of the Hamiltonian respectively.

First, the two qubit states, were constructed with temperature ₁₀1 ≤ t ≤ 1. The mutual information and negativity have an upper limit of 2 log 2 ≈ 1.4 and log 2 ≈ 0.7, respectively. These upper limits are reached when the state is a maximal entangled pure state. This gives

6_{The W state is the second non-bi-separable state, besides the GHZ state. The W state for n qubits is given}

by: |W i = _√1

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a ratio of E_I = 1₂ for maximally entangled pure states. When correlations between the qubits decreases, the mutual information and negativity decreases, such that the ratio increases (See figure 8). The highest ratio found for pure states was E_I ≈ 100, whereas it dropped instantly to

E

I ≈ 3, for mixed states. After evaluation of the plots in figure (8), it became clear that these

high ratios occur from states with virtually nil correlations, and are therefore of no interest for the purpose of quantum channels.

Figure 8: The ratio negativity/mutual information E_I for two qubit states versus the highest eigenvalue the density matrix at temperatures 1/10 ≤ t ≤ 1. Figure (d) contains all states where the mutual information I ≥ 0.2 and negativity E ≥ 0.1 at t = 1/10, whereas the states in figure (a) have I ≥ 1.35 and E ≥ 0.65 at t = 1/10. The mutual information and negativity have an upper limit of 2 log 2 ≈ 1.4 and log 2 ≈ 0.7, respectively. These upper limits are reached when the states are maximally entangled. Hence the states in (a) are (nearly) maximal entangled states at t = 1/10. Pure states that are maximally entangled in figure (a) are seemingly bounded by 3₄.

For this reason, every state with a negativity and mutual information lesser or equal then 0.1, has been ignored in the result for two qubits, given in figure(9). Interestingly, the plots in figure(9) show immediately an “arc” appearing at higher temperatures, with an extremum at E_I ≈ 3

4. From

figure (8), it also becomes clear that the states in the arc are states which are high to maximal correlated at a temperature t ≤ ₁₀1. The arc coincides with the results from the Werner state.

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Figure 9: The ratio negativity/mutual information (E_I) for two qubit systems, with density matrix ρ =P

ie

Ei

t |ψ_ii hψ_i|, at different temperatures t. The ratio is plotted versus the highest eigenvalue

of the density matrix. A pure state has a highest eigenvalue of 1, whereas a maximally mixed, separable state has a highest eigenvalue of 0.25. At low temperatures, the ratio reached high values for pure states, shadowing the result for mixed states. The states with a high ratio, have low mutual information and negativity and are such of no interest for quantum communication. Hence, states with a mutual information and negativity below 0.1 have been ignored. As the temperature increases, an ”arc” occurs, with an extremum at E_I ≈ 3

4. The arc coincides with the

result from the Werner state (figure 7).

When considering multipartite systems with four, six or higher number of qubits, the ratio for pure states are seemingly more concentrated on E_I ≈ 1

2. Furthermore, the higher amount of

qubits, the more concentrated on 1₂ it gets. This because two qubits are more likely to have lesser correlation, then higher amount of qubits. At higher temperature, the ratio remains approximately the same, until it has reached the point where the state is maximally mixed and all entanglement gets destroyed. Here, the ratio drops to zero. It is expected that for high number of qubits, the ratio for pure states at low temperature, will always be 1₂, which will also be the upper bound on

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the ratio. Hence, E_I =3₄ is only found as upper bound for two qubit states.

Figure 10: The ratio of entanglement negativity and mutual information for random four qubit system, with entanglement between two and two qubits. Pure states are more concentrated at one half. When the temperature increases, the ratio remains approximately the same, until it has reached the point where the state is maximally mixed and all entanglement gets destroyed.

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Figure 11: The ratio of entanglement negativity and mutual information for random six qubit system, with entanglement between three and three qubits. Pure states are more concentrated at one-half as is for states with lesser qubits. When the temperature increases, the ratio remains the same until the state is maximally mixed. Here, all entanglement gets destroyed and the ratio drops to zero.

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9. CONCLUSION AND DISCUSSION 24

9 Conclusion and discussion

We have given a short introduction to a conjecture about a duality between geometry of space-time and conformal field theories. The conjecture might help us in formulating the emergence of spacetime and allows us to examine strong-coupling regimes in QFT, which are otherwise not possible or extremely challenging. For a CFT to have a dual in classical theories of gravity, the CFT is acquired to obey the same inequalities imposed by the theory of gravity.

The area law by Ryu and Takayanagi, is a specific duality in context of AdS/CFT and we have seen that the minimal surface in AdS obeys a number of inequalities such as (strong) subadditivity and monogamous mutual information. However, mutual information is not monogamous for clas-sical correlated states, and is presumed the result of the duality stating that clasclas-sical geometry of AdS is encoded in highly quantum states at the boundary. Thus, there is no duality between classical gravity and classical correlated quantum states. The classical geometry of spacetime is, however, influenced by underlying quantum effects, such as quantum entanglement between re-gions inside the bulk. In context of the strong-weak coupling duality, these quantum effects are presumably dual to classical correlated states. At this moment it remains a future study subject, if bulk entanglement imposes any monogamy- or other inequalities on classical correlated states.

Entanglement entropy quantifies pure states. To learn about mixed states, it is interesting to study an effective entanglement measure for mixed states, such as entanglement negativity. A duality between AdS and entanglement negativity might also provide usefulness in practical ap-plications as quantum communication and teleportation. This is due to the logarithmic negativity being an upper bound on distillable entanglement.

The holographic negativity is equal to three-quarters of the mutual information of two adjacent subregions in a CFT. In general, non-holographic theory, we have found that the ratio of negativity and mutual information is one half for pure, maximally entangled two qubit states, and increases to possibly infinity, when correlations between two subsystems decreases. As temperature increases, the ratio drops significantly for pure states which are not maximally entangled, although no pattern, supporting E_I =3₄ was found for these states.

It is remarkable that negativity for states which are maximally entangled in the groundstate are bounded by 3₄ I. There is however, no reason to believe that this result translates to a quantum field, with high degrees of freedom. This makes it inconclusive as to why the ratio E_I = 3₄ arises from holographic theory.

Furthermore, when considering multipartite systems with high number of qubits, the ratio for pure states are seemingly always one half. The reason for is presumably that bipartite systems are more likely to have no or less correlations then multipartite systems. The ratio decreases instantly at higher temperature; thus, multipartite systems are seemingly bounded by one half.

The physical interpretation of the ration, and why it is solely found for two qubit states, which are maximally entangled in the groundstate, remains an open question. Besides the ratio of three-quarter, it is also of interest to study the ratio of one-half, which is more likely the upper bound for quantum fields. Although before continuing with further research, these numerical result need undeniably proof, and is therefore also subject for future research.

Acknowledgement

I would like to dedicate a few words to my supervisor dhr.dr. B.W. Freivogel who guided me through this thesis. His critical questions, advise, and feedback on my performance during this project was much appreciated. I would also like to thank dhr.dr. J.P. van der Schaar for taking on the role examiner and my partner Nic for his support and peer reviewing my report.

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REFERENCES 25

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[20] Rangamani, M., and Takayanagi, T. Holographic Entanglement Entropy, 1 ed., vol. 931. Springer International Publishing, 2017.

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27

Appendices

A

Mathematica code

Function “Partial Trace of a Multiqubit System” by Mark S. Tame, with explanation can be found in wolfram library archive: http://library.wolfram.com/infocenter/MathSource/5571/

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Holographic vs. non-holographic inequalities.