Non-equilibrium quantum entanglement evolution on top of classical many-body systems - two case scenarios

(1)

Non-equilibrium quantum entanglement evolution on top of classical

many-body systems: two case scenarios

Juan David Lizarazo Ferro

Master’s Thesis

∼Advisor∼ Luis Quiroga Puello PhD

Universidad de los Andes

Physics Department Bogot´a-Colombia

(2)

Humpty Dumpty sat on a wall,

Humpty Dumpty had a great fall.

All the king’s horses and all the king’s men

Couldn’t put Humpty together again.

(3)

Introduction

This thesis contains the analysis of two scenarios at the overlap of many-body quantum entanglement, and non-equilibrium physics. We consider sets of qubits which move in classical trajectories, place them in non-non-equilibrium scenarios, and study the entanglement that unfolds. As a first scenario we take two spin gases and mix them. Then, based on the ASEP (asymmetric simple exclusion process), a classical model used in the description of non-equilibrium transport phenomena, we build a quantum hybrid for which we follow the ensuing entanglement both numerically in simulations, and by using analytical results supplied by the solvability of the classical model. First, a spin gas is a model in which the constituent particles describe continuous classical trajectories with an associated two level quantum system for each one. The model was proposed by Calsamiglia et al. [9] in 2005 and is exactly soluble. In this thesis we consider two spin gases initially at different temperatures, allow them to mix, and study the unfolding entropy of entanglement between them. In our analysis we use elementary kinetic theory to calculate the rate of collisions between the groups of qubits at different temperatures, and also use molecular dynamics simulations for a hard disk gas containing sixteen particles. The entropy of entanglement found in the computational experiments is contrasted with analytical expressions showing a good agreement between them.

Secondly we consider the ASEP model, a discrete one-dimensional transport model in which particles move about in random walks with the condition that each lattice site can only contain one particle at a time [46]. Using periodic boundary conditions, and considering the totally asymmetric variant in which particles move only in one direction, we embed a two level quantum system to each mobile particle. As the quantum layer of description we consider an interaction Hamiltonian for the qubits in the spirit of the XY model, with first neighbours interactions only, and with spin-spin coupling strengths as determined by the dynamical rules of the TASEP. In computer simulations we study the unfolding concurrence between pairs of qubits at different lattice distances as a function of time, and also the resulting statistical distributions.

Additionally, in the course of the above simulations/calculations we are led to consider two interesting problems, one in elementary kinetic theory, and another in statistical mechanics. In the mixing of the two spin gases it becomes necessary to study the transfer of heat between the two underlying Boltzmann gases; we find that the analysis of this process lead to Fourier’s heat law. Our analysis of the hybrid TASEP-XY model inspires a simple variation of the one dimensional Ising model, namely a model where the spins are variable in number, we successfully find the model’s corresponding statistics using the transfer matrix method.

We begin in chapter 2 with a short review on the historical and theoretical foundations related to quantum entanglement, non-equilibrium physics, and kinetic theory. Then in chapter 3 we take on the problem of mixing

(6)

spin gases, giving first the theoretical underpinnings of the spin gas model. In chapter 4, after introducing the XY and ASEP models, we give the analysis of our hybrid TASEP-XY model. The appendices at the end complement this thesis, they contain the computer code we used for our molecular dynamics simulation, several worked out problems in kinetic theory, two demonstrations of mathematical relationships used in the body of the thesis, a short postscript on the combinatorial aspects of the TASEP-XY model, and the statistical mechanics of a variation on the Ising 1D model.

(7)

Chapter 2

Theoretical Foundations

In this chapter we give a brief review of the theoretical minima, and of the history, of entanglement, non-equilibrium physics, and kinetic theory. Atomism, and by extension also kinetic theory, stand at opposition to entanglement. On one side a great many phenomena are successfully analysed on the basis of rules obeyed by minute morsels of matter, while on the other side our divisions are laughed upon, and the tiny specks, so ably manipulated on the other side, remain inextricablyentangled, even when taken apart by stellar distances. This clash between atoms and entangled things, delivers many challenges at the doorstep of science, and although we limit ourselves in this thesis to very simple, perhaps childish scenarios, we would like to begin first by explaining what is so classically unpalatable in entanglement, and then continue by describing the arc that atomism drew back from the times of the greeks up to to end of the nineteenth century.

2.1 Entanglement

Given two quantum systems AandB with their corresponding Hilbert spacesHA andHB the tensor product of the two Hilbert spaces HA⊗ HB gives the space into which their common dynamics is contained, an arbitrary pure state having the form

|ψi=X i,j

ai,j|φiiA⊗ |φjiB, with {|φiA,B} basis for their corresponding Hilbert space. (2.1)

If there exists |αi ∈ HA and |βi ∈ HB such that |ψi =|αi ⊗ |βi then the state |ψi is said to be separable, if not, the state is entangled. To measure the amount of entanglement in cases in-between several measures have been proposed, and for our purposes, which deal with pure global bipartite states, or mixed states for two qubits, entropy of entanglement and concurrence will be sufficient. Before giving their definition, a summary of the history of entanglement.

2.1.1 The EPR argument

In 1935 Einstein, Podolsky and Rosen published an article [16] which contented the privilege of quantum mechanics as a complete description of physical reality. They showed how quantum mechanical theory, as applied to a bipartite system, leads one either to assert the possibility of simultaneous reality for two noncommuting quantities, or to

(8)

concede the existence of “spooky actions at a distance”1.

First they proposed a sufficient condition for a physical quantity to be an element of reality as follows: “If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.” Then, they discussed when this criterion is met in quantum mechanics, that is, when the wave function |ψi is an eigenstate of the operator at hand: ˆC|Ψi=c|Ψi. In particular they recall that in quantum mechanics “when the momentum of a particle is known, its coordinate has no physical reality.”

Later they introduce the analysis of a composite system,A⊗B, whose components have interacted for some finite time. If an operator ˆC is then considered forA, then, as quantum mechanics proposes,|ψican be expanded in the form

|ψi=X m

|µmiA⊗ |φmiB, (2.2)

ˆ

D, having eigenstates|νmiA, and eigenvaluesdm: |ψi=P

m|νmiA⊗ |χmiB, and ifdlresults for a measurement in A, thenBis left in|χli; in general|χli 6=|φni. Results of measurements on one part determine the result of another measurement made on the other part,“On the other hand”, they remark,“since at the time of measurement the two systems no longer interact, no real change can take place in the second system in consequence of anything that may be done to the first system.” The former is the so called locality assumption.

Furthermore, they propose an example with two particlesA and B, with respective momenta and positions {x1, x2, p1, p2}for which the wave function on the space representation is

hx1, x2|ψi=

Z ∞

−∞

e~i(x1−x2+x0)pdp, (2.3)

and show that if the momentum of A is measured (p) then also the momentum of B is certain (p), and if the position ofAis determined (x) then the position ofB is fixed at (x+x0). Given the assumption of locality, then

both the position and momentum of B are elements of reality: one thinks that the kind of measurement to be made onAcould be decided at the last moment. Quantum mechanics forbids simultaneous knowledge of position and momentum (along the same direction), they thus conclude it to be incomplete.

Finally, they argue that such a conclusion could not be reached if they insisted“that two or more physical quantities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted”; however, they observe“No reasonable definition of reality could be expected to permit this”.

1_{The origin of this expression is from a letter sent by Einstein to Max Born, in which the former says “I cannot seriously believe in}

it because the theory cannot be reconciled with the idea that physics should represent a reality in time and space, free from spooky actions at a distance. I am, however, not yet firmly convinced that it can really be achieved with a continuous field theory, although I have discovered a possible way of doing this which so far seems quite reasonable.” (Letter sent on March 3, 1947 [15])

In his own context Newton abhored too actions at a distance: “That one body may act upon another at a distance through a vacuum, without the mediation of any thing else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity, that I believe no man, who has in philosophical matters a competent faculty of thinking, can ever fall into it.” (Letter from Newton to Richard Bentley as quoted in Florian Cajori’s historical appendix in Motte’s translation of thePrincipia[38])

(9)

2.1.2 The EPR argument in a system of two spins

In 1957 David Bohm and Yakir Aharonov [6], proposed a simplified version of the EPR argument which considered a molecule composed of two spin1₂particles in the singlet state|ψi= √1

2(| ↑↓ i − | ↓↑ i).

2 _{The molecule is separated}

into its constituent particles, and when the particles are far apart and no longer interact, a spin measurement is made on one of them yielding either ~

2 or −~2. Now, since the total spin is zero, one concludes that if a spin

measurement, along the same axis, is made on the other particle, then its resultmustbe opposite to the one found for the first.

One is led again either to embrace quantum orthodoxy, and allow for nonlocal effects, or to retain locality, and conclude that both particles had definite values for all spin components, in stark contrast with conventional quantum mechanics, which in its most radical proclaims that

“. . . observations not only disturb what has to be measured, they produce it. In a measurement of position, the electron is forced to a decision. We compel it to assume a definite position; previously it was neither here nor there, it had not yet made its decision for a definite position.” Pascual Jordan (1936)3

Furthermore, quantum formalism seems in trouble, for it maintains that knowing the value of angular mo-mentum along an axis automatically precludes knowing with certainty its value along any of the two other axes ([ ˆSi,Sˆj] =i~ijkSˆk). However, since the experiment could have been otherwise4, and another direction could have been measured at one end, it seems as if all spin components for both particles are definite.

They mentioned also that if one admits the correctness of the quantum description, then it is requisite to acknowledge a more intricate role for observation than the one envisaged in the simple thought experiments of early quantum mechanics, in them the quantum of action_~was thought to introduce an inescapable fluctuation, via the measurement instrument, in one of the quantities of a pair of canonically conjugate quantities whenever the other is measured. This effect was understood to be local, and is of a very different tenor than the nonlocal effect produced by measurements in EPR scenarios.

2.1.3 Bell’s inequalities

Hidden variable theories were an attempt to complete5 _{quantum physics by proposing new quantities which in}

conjunction with the wave function would determine precisely the outcome of individual processes, something which quantum mechanics demurs from doing on occasion. This theory construction advanced in parallel with the study of the restrictions such theories ought to obey, John Bell examined the restriction imposed on account of locality, and found that correlations, that is relations between sets of distinct sets of measurements, had to follow a certain inequality.

2_{This state has the peculiarity that it retains its form for any pair of equally oriented coordinate systems.} 3_{As quoted by Max Jammer in [26].}

4_{It is interesting to note, that we are also assuming free will, in the words of Bell:} _{“It has been assumed that the settings of}

instruments are in some sense free variables - say at the whim of experimenters - or in any case not determined in the overlap of the backward light cones. Indeed without such freedom I would not know how to formulate any idea of local causality, even the modest human one.” [4]

5_{In his 1936 philosophical credo,}_{Physics and Reality} _{after praising the success and beauty of quantum physics, Einstein remarks}

that “However, I do not believe that quantum mechanics will be the starting point in the search for this basis, just as, vice versa, one could not go from thermodynamics (resp. statistical mechanics) to the foundations of mechanics.” [14]

(10)

It was argued that an unknown theory could be to quantum mechanics what classical physics is to statistical mechanics, and that if probability came about in quantum theory, it was only because the problem was not sufficiently parametrised. Defining new quantities to differentiate different experimental outcomes had been a fruitful6 practice up till that point in the history of physics, and perhaps still undiscovered properties of matter could free physics from the new kind of probability, anunbeatableprobability, as proposed by quantum mechanics. Having to be compatible with the success of quantum theory, this hidden variable theories had to reproduce the statistical predictions of quantum mechanics when the additional variables λwere averaged over their probability distributionρ(λ).

Bell analysed Bohm’s version of the EPR thought experiment, but instead of just noting what happens when the detectors are along the same direction, he wondered how the results for spin measurements along different directions would relate to one another.

He supposed the existence of hidden variables λwith a probability distribution ρ(λ), the results of the two experiments,AandB, being determined byλ, and by the orientation of the corresponding spin measurements~a and~b

A(~a, λ) =±1, and

B(~b, λ) =±1. (2.4)

The assumption of locality is already introduced above by demanding that the result of the local measurement depend only on the local settings, i.e. one precludes the possibility ofA~a,~b, λ. Now, because when the directions of the detectors are the same, the results are opposite for the singlet state, one must have that

A(~a, λ) =−B(~a, λ). (2.5)

If we know look at the correlation between two directions then

C(~a,~b) =

Z

ρ(λ)A(~a, λ)B(~b, λ) dλ

=−

Z

ρ(λ)A(~a, λ)A(~b, λ) dλ.

(2.6)

And if another direction~cis considered, then

C(~a,~b)−C(~a, ~c) =−

Z

ρ(λ)hA(~a, λ)B(~b, λ)− A(~a, λ)B(~c, λ)idλ ,

and sinceA2_{= 1 , then =}₋

Z

ρ(λ)A(~a, λ)B(~b, λ)h1− A(~b, λ)B(~c, λ)idλ.

(2.7)

Finally, because|A(~b, λ)A(~a, λ)|= 1, h1− A(~b, λ)B(~c, λ)i≥0 andρ(λ)≥0, then7

|C(~a,~b)−C(~a, ~c)|=

Z

ρ(λ)A(~a, λ)B(~b, λ)h1− A(~b, λ)B(~c, λ)idλ

≤1 +C(~b,~c); (2.8)

and since for|ψi= _√1

2(| ↑↓ i − | ↓↑ i), given that the matrix representation of the operator ˆS1·~a⊗Sˆ2·~b in the 6_{Not always of course, remember ether and caloric.}

7_If_f₌Rb

ag(x)dxthen|f| ≤

Rb

(11)

basis{| ↑↑i,| ↑↓i,| ↓↑i,| ↓↓i}is

     

azbz az(bx−iby) (ax−iay)bz (ax−iay) (bx−iby) az(bx+iby) −azbz (ax−iay) (bx+iby) −(ax−iay)bz (ax+iay)bz (ax+iay) (bx−iby) −azbz −az(bx−iby) (ax+iay) (bx+iby) −(ax+iay)bz −az(bx+iby) azbz

     

(2.9)

for|ψi= √1

2(0,1,−1,0) one gets

hS1·~a⊗S2·~bi=−~a·~b, (2.10)

and inequality (2.8) turns out to be

|~a·~b−~a·~c| ≤1−~b·~c, (2.11) which in figure 2.1 is shown to be violated for many pairs of angles, being the conclusion that on theory quantum mechanics infringes the assumption of locality, it was later proved in experiments[2].

0 Π 2Π

0 Π

2Π0 Π 2Π

0 Π 2Π

Θ

Φ

a

Ó

b

Ó

c

Ó

Θ

Φ

Figure 2.1: Bell’s inequality|cos(θ)−cos(θ+φ)| ≤1−cos(φ), angles in the black region satisfty the inequality, while those in the shaded part do not, and maximally not in the four indicated points.

2.1.4 Entanglement measures

Proposed measures of entanglement must satisfy the so-called monotonicity axiom [48]: an entanglement measure, call it ˜E, must never increase under local operations assisted by classical communications (LOCC)8_{. For bipartite}

systems, it can be shown that the monotonicity axiom implies the existence of maximal entanglement, that there exist states from which any other state can be created by LOCC, for instance, Bell states9 are the maximally

8_{Any measurement or unitary evolution, is a quantum operation. The allowance for classical communications means that one}

party can decide to make an operation based on classical information (a phone call) received from the other party. Any LOCC Λ, for bipartite systems, can be expressed in the form Λ( ˆρ) =P

iAî⊗BîρÂˆ†i⊗Bˆ

†

i 9_{Bell states:} _|_ψ±_i₌_√1

2(|01i ± |10i), and|φ

±_i₌_√1

(12)

entangled states for a system of two qubits. This axiom also implies [24] that any entanglement measure ˜E is constant on separable states.

Two examples of entanglement measures are concurrence, and entropy of entanglement. Concurrence is a good measure for two level bipartite systems, with states either mixed or pure. Entropy of entanglement is a good measure for arbitrary bipartite systems, but the global state must be pure.

2.1.4.1 Entropy of entanglement

Given a density matrix ˆρfor a quantum system with a corresponding Hilbert spaceH, one would like to measure the degree to which it represents a mixed state, and by extension a tool by which a pure state can be detected from its density matrix. One such measure is the von Neumann entropyS [36], which assigns a real non negative number to each state ˆρaccording to the equationS( ˆρ) =−Tr [ ˆρlog ( ˆρ)]. It has the following properties

• ∀ˆρ∈ T (H)10_,₀_≤_S_{( ˆ}_ρ₎_≤_{log (}_D₎_, _with_D_{= dim (H),}

• S( ˆρ) = 0 iff ˆρis a pure state, and

• S( ˆρ) = logD iff ˆρ=_D111 (infinite temperature state).

If bipartite systems are considered, with ˆρAB∈ T (HA⊗ HB), then • |S( ˆρA)−S( ˆρB)| ≤S( ˆρAB)≤S( ˆρA) +S( ˆρB),

• if ˆρAB = ˆρA⊗ρˆB, then11 S( ˆρAB) =S( ˆρA) +S( ˆρB), and • if ˆρAB is a pure state then S( ˆρA) =S( ˆρB).

Now, if the quantum system has two parts, so thatH=HA⊗ HB, and if ˆρAB is a pure state, then the von Neumann entropy of the reduced density matrices measures the entanglement between the two parts; the heuristic logic being that if S is a measure of how mixed any given density matrix is, evaluating it after taking the partial trace tells us how much is “lost” when one part is analysed apart form the other, and the bigger this loss the greater the entanglement. For bipartite pure states ˆρAB, one defines itsentropy of entanglementas

E( ˆρAB) :=S( ˆρA) =S( ˆρB). (2.12) As a simple illustration take a pair of qubits in the EPR state |ψi = √1

2(|01i − |10i)

12_{, their corresponding}

density matrix, in the basis{|11i,|10i,|01i,|00i}, being

ˆ ρ=

     

0 0 0 0

0 1₂ −1

2 0

0 −1 2

1

2 0

0 0 0 0

     

, and its entropy S( ˆρ) = 0. (2.13)

10_Let_H_{denote a Hilbert space, and}_T ₍_H_{) the set of trace class operators on}_H_{. A trace class operator is a linear operator on}_H

with a finite trace. Density matrices are trace class operators with unit trace.

11_{Whenever a bipartite system is being discussed, ˆ}_ρ

A= TrB( ˆρ) and ˆρB= TrA( ˆρ)

12_{From this point on we favor the information theory notation}_|₀_i_,_|₁_i_{, over the spin 1/2 representative of two level quantum systems,}

(13)

Now, the reduced density matrix corresponding to the first qubit turns out to be,

ˆ ρA=

1

2 0

0 1₂

!

, (2.14)

namely the infinite temperature state, the entropy of entanglement being E( ˆρ) = log 2. To emphasise, if both qubits are taken into account, then the system is in a state of perfect knowledge, but if only one of them is available, then our knowledge is of perfect ignorance, nothing could be done to guess that the global state is pure from the most exhaustive examination of only one part.

To contrast, suppose instead that the state is separable|ψi = √1

2(|0i+|1i)A⊗

1 √

2(|1i+|0i)B, the density matrix being

ˆ ρ= 1

4      

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

     

, and ˆρA= 1 2

1 1 1 1

!

withE( ˆρ) = 0, (2.15)

result which confirms what could have identified from the beginning, both parts are independent from one another, measurements made on one part have no bearing on the results for measurements made on the other.

Worthwhile to notice thatEhas an operational interpretation, Bennett et al. [5] showed that if one has a pure bipartite state |ψiof two qubits AandB, then E(|ψihψ|) corresponds approximately to the amount of Bell pair states which can be “distilled” by LOCC applied on m copies of |ψi, see fig. 2.2. Therefore Bell states can be defined as units of entanglement, oneebit is defined as the amount of entanglement contained in any of them.

A

B

ÈΨ 1 ÈΨ 2 ÈΨ 3

... ...

ÈΨ

m-2

ÈΨ

m-1

ÈΨ m

L

O

C

1 2

HÈ01-È10L

1

2HÈ01-È10L 2

...

1

2HÈ01-È10L n-1

1

2HÈ01-È10L n

n

m

»

S

H

Ρ

A

L

Figure 2.2: Alice and Bobshare m copies of|ψi, and by using LOCC on them they can approximately generate nBell pair states.

(14)

2.1.4.2 Concurrence

Concurrence is a measure of entanglement defined for mixed or pure states of two qubits as a function of its density matrix ˆρ:

C( ˆρ) := max(0, λ1−(λ2+λ3+λ4)) (2.16)

where the λi are the square roots, in decreasing order, of the eigenvalues of ˆρ˚ρˆwith ˚ρˆ= (ˆσy⊗σˆy) ˆρ∗(ˆσy⊗σˆy); it was proposed by Hill and Wootters in 1997[23].

2.2 Non-equilibrium thermodynamics

Non-equilibrium thermodynamics studies physical phenomena outside of thermal equilibrium, and as such deals with systems changing in time, accompanied by currents of matter and energy, enveloping the description and study of irreversible processes. A few examples of the phenomena dealt by the theory are diffusion, percolation, electrical currents, heat conduction, convection, turbulence, viscosity, and plasticity.

Equilibrium states are determined by intrinsic factors, their properties are independent of their history, and are time independent. They tend to be homogeneous, and are described by a few quantities: internal energy, volume, and mole numbers. In contraposition, non-equilibrium states can have intricate structure, and their mathematical description is more involved, for instance, a few thermodynamic variables might be replaced by a temperature field.

Being flows of all types of capital importance in non-equilibrium processes, transport equations play a central role in the theory [31], the most well known being the following:

~

q=−λ∇T (Fourier’s law), ~

J =−D∇c(Fick’s law), and ~

I=σe∇φe(Ohm’s law). ~

q: heat flux, ~J : diffusion flux13, ~I: electric current density,

T: temperature, c: concentration, φe: electric potential.

Whereλ: thermal conductivity, D: diffusion coefficient, σe: electric conductivity.

From afar two types of starting points can be distinguished in the development of the theory. The top-down, or thermodynamical approach, starts from phenomenological relations (such as the transport equations above) and conservation principles (matter, energy, momentum, time-invariance). The bottom-up, or statistical approach, attempts to describe the macroscopic behaviour of matter from its microscopic rules. Lebon [31] has an excel-lent review on the thermodynamic theories (extended thermodynamics, theories with internal variables, rational thermodynamics, Hamiltonian formulation, and mesoscopic approaches). One statistical approach, and one of the first, is due to Boltzmann14, who in 1872 derived a partial integro-differential equation for the distribution function which determines the number of atoms in each state as a function of time, the equation as given by him at that time was

∂f(x, t)

∂t =

Z ∞

0

Z x+x0

0

"

f(ξ, t) √

ξ

f(x+x0−ξ, t)

p

(x+x0₋_ξ₎ −

f(x, t) √

x

f(x0, t) √

x0

#

√

xx0_ψ₍_{x, x}0_{, ξ}_{) d}_x0_d_ξ, _(2.17)

13_{Amount of moles transported per unit time, per unit area.}

(15)

where x and x0 indicate the kinetic energies of two molecules before a collision, ξ and (x+x0−ξ) their kinetic energies after, and ψ(x, x0, ξ) being a function which depends on the forces between the molecules. After giving the above equation, to which one arrives on the basis of probabilistic and mechanical arguments, he then showed that the Maxwell distributionf(x) =q_π₍_kT4x₎3e−

x

kT satisfies the equation for the time independent case.

In this thesis, when we arrive to the problem of the mixing spin gases, the quantum mechanical state of the qubits will be driven by the non-equilibrium process corresponding to the mixing of the two gases, we will have the opportunity to apply, in a very simple sense, the Maxwell-Boltzmann distribution, which in the field of non-equilibrium thermodynamics, belongs to the kinetic-theory approach, whose exposition we amplify a little further in the next section.

2.2.1 Kinetic theory

In kinetic theory all is reduced to matter in motion. Matter and its different arrangements, which turn by motion into one another, encompass all natural phenomena. In it complexity is obtained form the composition, and dissolution, of simple constituents, one brick at a time. When these moving parts are evident to the senses, as in the beginning of the scientific revolution, kinetic theory, being nothing but sensorial information reified by mathematics, barely needs justification and seems inseparable from an intelligible study of nature. But when the moving parts are of subtle character, such as the atoms discovered by modern science, kinetic theory took time to gain ground on the scientific mindset, and when it did, it did so at the same time than thermodynamics, following the lead of Maxwell, Boltzmann, Clausius, and others. Heat and sound are atoms in motion, thunder and thought are electric charges in motion, matter is the noun and motion is the verb; nature was kind enough to have simple laws governing motion, and few parts as its building blocks.

First proposed by Leucippus, and later elaborated by Democritus, the doctrine of atomism claims that the ultimate constitution of the universe resides in eternal and indivisible particles, of varying shapes and weights, forming ephemeral aggregates, and moving in never ending space. Not only an attempt to give an explanation of the nature of reality, atomism also had a moral objective, namely to banish superstition by replacing the whim of the gods with entities whose essence does not exceed what the senses perceive; materialism gives, so to speak, a solid ground to reality; it can be grasped in full, as a hand holding a rock. In the atomistic mindset, if something occurs, the culprit is a material entity moving through space; thus, for instance, if objects can be seen, atomists held the view that it was because something had traveled from the object to the eye; today we would recognise these entities as photons. With regard to kinetic theory as it flourished in the 19th century, take this fragment from Simplicius of Cilicia (c. 490 - c. 560): “These atoms, which are separated from each other in the infinite void and distinguised from each other in shape, size, position and arrangement, move in the void, overtake each other

and collide. Some of them rebound in random directions, while others interlock because of the symmetry of their

shapes, sizes, positions and arrangements, and remain together. This was how compound bodies were begun.”15

The laws of motion set forth by Isaac Newton propelled science into the quest of exploring how much could be explained on its basis, this not only included the motions of the heavenly bodies, but also the properties of gases. In his Principia, [37] Newton attempted to explain Boyle’s gas law by proposing a model where gases are composed of particles subject to repelling forces between them. If such forces are taken to vanish beyond a certain distance, but are otherwise of magnitude inversely proportional to the distance separating the particles, then pressure rises in proportion with a reduction in the volume occupied by the fluid. Although such forces turned out

(16)

to be inexistent, his attempt clearly shows the ambition of his mechanistic project16, and gives a valuable lesson to be taken to heart, several microscopic rules can give rise to indistinct macroscopic behaviour.

As part of his analysis of hydrodynamics, Daniel Bernoulli showed in 1738 [7] how Boyle’s law again followed if one took gases to be composed of a myriad of small and rapidly moving particles, their collisions with the walls of their container giving rise to pressure, and the pressure being proportional to the square of the speed of the particles. Bernoulli’s proposal was essentially correct, gases adequately described by Boyle’s law are indeed composed of particles which move freely except when colliding with the walls which contain them; however, his model was not readily accepted, among other reasons, because of lack of experimental evidence which could decide between one microscopic model and another.

To give a sense of the sort of calculations involved in kinetic theory, we solve the model proposed by Bernoulli using somewhat newer tools. Suppose our gas is contained in a cube, of sideLand volumeV, that it is composed of N identical particles of massm, moving with random directions, uniformly distributed along the gas volume, and with the same speed ˜v, what is the pressure to which the particle collisions subject any one of its faces?

Without loss of generality, orient the cube parallel to a given set of cartesian axes, make one of the faces lie in the xyplane, and have the cube be in the z >0 region. To find the pressure on the the lower face, we find the total change in momentum produced by the force impressed by this face on the particles which collide with it in a time ∆t. To do so we divide the face into differentials of areadA, count how many particles are to hit it in the available time, and take into account that the change of momentum for each colliding particle is equal to double its momentum perpendicular to the face and posterior to the collision, that is 2˜vcosθ, whereθis as referred in figure 2.3. Over each differentialdAwe must integrate over the entire volume occupied by the gas, taking into account that all particles within a hemisphere of radius ˜v∆t, and within the proper solid angledΩ, will hitdA, being them equal to dN = N_VdV ₄dA_πr2, dΩ =

cosθdA

4πr2 , and the change in their momentum equal todPz = 2˜vcosθNVdV

cosθdA

4πr2 .

Putting it all together

∆Pz=

Z

A

Z

V

2mv˜cos2θN V dV

dA 4πr2

=A

Z v˜∆t

r=0

Z θ=π/2

θ=0

Z φ=2π

φ=0

2m˜vcos2θN V

1 4πr2r

2_sin_θ_d_r_d_θ_d_φ₌_A1

3 N V m˜v

2_∆_t

→ ∆Pz

A∆t =P = 1 3

N V mv˜

2_, _(2.18)

which is exactly the equation of state for a monoatomic gas.

After Bernoulli, one next significant advance in the kinetic theory of gases came in the 19th century when heat was finally recognised as nothing but motion, and probability theory was mature enough to contribute as a mathematical tool. In the beginning of that century the prevailing theory held the view that heat was a fluid, called caloric, which flowed back and forth between hot and cold bodies, and whose amount, given the repulsive force which was ascribed to it, determined whether a body was solid, fluid, or gaseous[29]. A sequence of experiments, including Joule’s experiment on the equivalence between mechanical energy and heat, and Count Rumford’s experiment on the creation of heat by friction[12], finally inclined the balance in favour of the mechanical theory of heat, wherein it is found unnecessary to ascribe an origin for heat different than the motion of the parts which

16_{“. . . and therefore I offer this work as the mathematical principles of philosophy, for the whole burden of philosophy seems to}

consist in this – from the phenomena of motions to investigate the forces of nature, and then form these forces to demonstrate the other phenomena . . . ” (from Newton’s preface to the first edition of thePrincipia)

(17)

ɸ

θ

dV

dA

Figure 2.3: For every area differential collisions with all of the gas particles must be considered, near the borders the walls act like mirrors, so the integral limits remain the same.

constitute a substance. Being heat and mechanics unified, the next step was to find all possible consequences of this unification, and to look for them in experiments. Maxwell [35] elaborated on the model proposed by Bernoulli, and used it to study viscosity, diffusion, evaporation, condensation, and conduction of heat in gases, all of them being phenomena outside of thermal equilibrium. He also inaugurated the use of statistics in kinetic theory, instead of taking all particles to share a common speed, he proposed that a more successful description would refrain from stating definite microscopic states, but would rather consider particles with velocities obeying a probability distribution, and found that this distribution obeyed the functional form of the Gaussian distribution of errors.

(18)

Chapter 3

The Mixing of Two Spin Gases

In this chapter we take the spin gas model, as proposed by Calsamiglia et. al [9], give a summary of its analytical details as found by them, and make the slight modifications required for our purpose. We consider the mixing of two spin gases, and show the results obtained for the entropy of entanglement as found in computational experiments. The appendices which correspond to this chapter contain several kinetic theory calculations, including a novel –at least for this author– derivation of Fourier’s law, and some details of the analytical machinery used in the analysis of entropy.

3.1 The spin gas model

Take a gas composed of particles whose kinematics are adequately described by classical mechanics, incorporate to each of these a two level quantum system, and suppose that the classical dynamics of such particles drives the evolution of the quantum mechanical state but not the other way around, what results is what we will understand as a spin gas.

The interaction Hamiltonian for the qubits is taken to be

ˆ

H(t) =X k<l

g[~rk(t), ~rl(t)] ˆH(kl) (3.1)

where the~rj describe the positions of the particles and the ˆH(kl) are two-body Hamiltonians which all commute in pairs.

The state|ψtiat timetis determined byN(N−1)/2φklphases which in turn depend on the collisions between the gas particles via the time dependent couplingsg[~rk(t), ~rl(t)]

|ψti= ˆUt|ψ0i=

Y

k>l ˆ

U(kl)[φkl(t)]|ψ0i

with ˆU(kl)=e−iφkl(t) ˆH(kl)_{, and}_φ_kl(_t_{) =}

Z t

0

g[~rk(t0), ~rl(t0)] dt0. (3.2)

Suppose in addition that ˆU(kl) = |11iklh11|1 _{is the two-body operator, and that} _|_ψ

0i = |+i⊗N with |+i = 1_{This type of interaction, of a class called Ising type interactions, can be achieved using controlled collisions between atoms in}

(19)

Figure 3.1: The spin gas model: N particles with spin 1/2, enclosed in a vessel of volumeV and temperatureT.

1 √

2(|0i+|1i). Taking this into account

ˆ

Ut|+i⊗N _{= 2}−N/2X

s

ei/2s·Γ(t)·s|si (3.3)

withsranging over all 2N _{possible binary vectors of size}_N _{and Γ(}_t_{) a real symmetric matrix whose elements are} defined as Γ(t)ij =φij(t).

Divide theN gas particles in two setsAandB, so that the reduced density matrix corresponding toAis

ˆ

ρA= TrB(|ψtihψt|) = 1 2NTrB

 

2N₋₁

X

s,s0

ei/2(s·Γ·s−s0·Γ·s0)



. (3.4)

Divide the phase matrix Γ in blocks that contain the interactions between particles within same sets, and between particles in different sets; ΓAB is the off-diagonal block which contains the collisional phases of Aparticles with B particles, ΓAArefers to the block which only makes reference to particles in set A, and ΓBB likewise.

If in eqn. (3.4) ΓAA is set to zero, the resulting density matrix is particularly simple to calculate (the second equality uses a trigonometric identity which permits expressing a sum of 2NB _{cosines as a product of}_N

B terms, see appendix A.6)

ˆ ˜ ρA=

1 2NA

X

sA,s0A

"

1 2NB

X

sB

ei(sA−s0A)·ΓAB·sB

#

|sAihs0_A|

= X

sA,s0A

(

ei/2Pk(sA−s0A)·Γk

NB

Y

k=1

cos

1

2(sA−s

0

A)·Γk

)

|sAihs0_A|with (Γk)i= Γki.

(3.5)

If one defines

CsA,s0A:=e

i/2P

k(sA−s0A).Γk

NB

Y

k=1

cos

₁

2(sA−s

0

a)·Γk

(20)

then ˜ρˆ(t) =CsA,s0Aρ˜ˆ(0). In our computer simulations for mixing spin gases, we used eqn. (3.5) to calculate reduced

density matrices, being the partial trace a generally expensive computational operation, this equation represents a significant reduction in the computational complexity.2

3.1.1 The three particle spin gas

Before going on, to better understand the solution given above, we now show in full the procedure for a spin gas composed of only three particlesa, bandc. Its Hamiltonian will then be

ˆ

whose matrix representation in the ordered basis {|111i,|110i,|101i,|100i,|011i,|010i,|001i,|001i,|000i}is

ˆ H=                

gab+gac+gbc 0 0 0 0 0 0 0

0 gab 0 0 0 0 0 0

0 0 gac 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 gbc 0 0 0

0 0 0 0 0 0 0 0

                , (3.8)

with the initial state given by

|ψ0i=

1

23/2(|1i+|0i)a⊗(|1i+|0i)b⊗(|1i+|0i)c = 1

23/2(|111i+|110i+|101i+|100i+|011i+|010i+|001i+|000i).

(3.9)

Each of the basis states is an eigenstate of ˆH so that|ψ0ievolves after timetto

|ψti= 1 23/2 e

−iβ4_|111i₊_e−iβ1_|110i₊_e−iβ2_|101i₊_|100i₊_e−iβ3_|011i₊_|010i₊_|001i₊_|000i

whereβ1=φab=

Z t

0

gabdt0, β2=φac=

Z t

0

gacdt0, β3=φbc=

Z t

0 gbcdt0,

andβ4=φab+φac+φbc. (3.10)

The corresponding density matrix is

ˆ ρ=1

8                

1 eiβ1−iβ4 _eiβ2−iβ4 _e−iβ4 _eiβ3−iβ4 _e−iβ4 _e−iβ4 _e−iβ4 eiβ4−iβ1 ₁ _eiβ2−iβ1 _e−iβ1 _eiβ3−iβ1 _e−iβ1 _e−iβ1 _e−iβ1 eiβ4−iβ2 _eiβ1−iβ2 ₁ _e−iβ2 _eiβ3−iβ2 _e−iβ2 _e−iβ2 _e−iβ2

eiβ4 _eiβ1 _eiβ2 ₁ _eiβ3 ₁ ₁ ₁

eiβ4−iβ3 _eiβ1−iβ3 _eiβ2−iβ3 _e−iβ3 ₁ _e−iβ3 _e−iβ3 _e−iβ3

eiβ4 _eiβ1 _eiβ2 ₁ _eiβ3 ₁ ₁ ₁

                . (3.11)

2_{The amount of bytes used in}_Mathematica_{to hold the density matrix of a system with}_n_{qubits is}_≈₁₀₀_×₄n_{, which for 16 qubits}

(21)

If set Acontains particlesaandb, then

ˆ ρA=

1 4      

1 1₂e−iβ1₊1 2e

iβ2−iβ4 1 2e

−iβ1₊1 2e

iβ3−iβ4 1 2e

−iβ1₊1 2e

−iβ4

eiβ1

2 +

1 2e

iβ4−iβ2 ₁ 1 2+

1 2e

iβ3−iβ2 1 2+

1 2e

−iβ2

eiβ1

2 +

1 2e

iβ4−iβ3 1 2+

1 2e

iβ2−iβ3 ₁ 1 2+

1 2e

−iβ3

eiβ1

2 +

eiβ4 2

1 2+

eiβ2 2

1 2+

eiβ3

2 1       , (3.12)

and settingβ1= 0

˜ ˆ ρA=

1 4

 

1 1₂

1 +e−iβ3 1 2

1 +e−iβ2 1 2+ 12e

−i(β2 +β3 ) 1

2

1 +eiβ3

1 1₂1 +e−i(β2−β3 ) ₁

2

1 +e−iβ2

1 2

1 +eiβ2 1 2

1 +ei(β2−β3 ) 1 1₂

1 +e−iβ3 1

2+ 12ei(β2 +β3 ) 12

1 +eiβ2 ₁

2

1 +eiβ3

1   =1 4     

1 e−

iβ₃ 2 cos β3 2 e− iβ₂ 2 cos β2 2 e− i

2(β2 +β_{3 ) cos}

β2 +β3 2 e iβ3 2 cos β3 2

1 e−2i(β2−β_{3 ) cos}

β2−β3 2 e− iβ2 2 cos β2 2 e iβ2 2 cos β₂ 2 e 1

2i(β2−β3 ) cos

β₂−β₃

2

1 e−

iβ3 2 cos β₃ 2 e i

2(β2 +β_{3 ) cos}

β2 +β3 2 e iβ₂ 2 cos β2 2 e iβ₃ 2 cos β3 2 1      . (3.13)

From which the entropy of entanglement evaluates to3

S( ˜ρˆA) =−[ζlog₂ζ+ (1−ζ) log₂(1−ζ)] withζ= 1

2

cosβ2 2 cos

β3

2 + 1

.

(3.14)

3.1.2 Rate of entropy creation

Now we would like to review the general features ofSρ˜ˆA

, as given in eqn. (3.5). In order to do so we take our gas to be a dilute gas in thermal equilibrium, composed of N particles with a mean free path similar in size to the characteristic length of the enclosing volume, with a uniform spatial density, and velocities described by the

Boltzmann distribution p(~v) =√1 2πσ

d

e−2σ12|~v|2_{, with} _d_{equal to the dimension of the gas, and} _σ ₌

q

kT m. A hard disk model4 _{is assumed for the collisions [9], and every time two of them collide, the corresponding phase} φkl increases according to the rule ∆ (φkl) = _|_~_vγ_r_|, with~vr their relative velocity at the moment of collision; γ is taken as a parameter which allows us to explore several regimes, and is indicative of the strength of the interaction between two qubits. The hard disk gas occupies an areaA, has temperatureT, and is composed ofN particles of massmand radiusR.

Given our rule for the phases, and considering that our gas is a Boltzmann gas, we would like to find the probability distributionpφ(φ). For this we need the probability distribution for the relative speed between particles

p(˜vr) =

Z

~v1

Z

~ v2

p(~v1)p(~v2)δ(|v~1−v~2| −˜vr)d~v1d~v2

making the change of variables~vr:=v~1−v~2, ~vc := ~ v1+~v2

2

p(˜vr) = 1 (2π)2_σ4

Z

~vr

Z

~ vc

e−

v_r,x2 +v_r,y2 +4(v2_c,x+v2_x,y)

4σ2 δ(|~vr| −vr)d~v_rd~v_c

= v˜r 2σ2e

−v˜2r

4σ2.

(3.15)

3_{From this point onwards we adopt the quantum information theory convention of taking the logarithms in base 2.} 4_{Calsamiglia’s article makes all calculations in 3D, we modify its results to be compatible with our 2D simulations.}

(22)

Havingp(˜vr) we can find that the probability distribution forφis

pφ(φ) = γ

2

2σ2 e−

γ2

4σ2φ2

φ3 . (3.16)

However, taking into account the fact that a change of 2πin φis indifferent for the result, we redefinepφ over the interval [0,2π]5

˜ pφ(φ) =

∞ X n=0 ζ2 2 e− ζ2

4(φ+2πn)2

(φ+ 2πn)3withζ= γ

σ. (3.17)

In the regime of large collisional phases ζ 1, the above expression converges (as found in numerical ap-proximations) to a constant ₂1_π (see fig. 3.2), as such a random phase in the interval [0,2π] is assigned to each collision. The small times regime is determined by the condition rt <1, wherer= 4N

ARhvriis the collision rate (see Appendix A.1), with the average relative velocity hvri =R∞

0

v_r2

2σ2e −v2r

4σ2 dv_r =

√

πσ (see Appendix A.2). In this regime a disk will collide at most with another, the resulting entropy of entanglement, owed to that collision being ₂1_πRS( ˆρ1) dφ, where ˆρ1 is the reduced density matrix of one particle

ˆ ρ1=

1 2

1 eiφ/2cos1₂φ e−iφ/2_cos1

2φ 1

!

, (3.18)

whose eigenvalues are

λ1= cos2 φ

4, λ2= 1−λ1= sin

2φ

4, (3.19)

in consequence the entropy corresponding to one collision is on average

1 2π

Z

S( ˆρ1)dφ=−

1 2π

Z 2π

0

2

cos2φ

4log2cos φ 4 + sin

2φ

4log2sin φ 4

dφ= 2−log₂e. (3.20)

The above has a probabilityrtof occurring, and can occur in a number of ways which depends on the average number of collisions between particles of groupsAandB, withNAandNB particles, which is equal to N_NA₋N₁B, in conclusion

hSAi ≈NANB

N−1rt(2−log2e) = NANB

N−14 N AR

r

πkT

m (2−log2e)t forrt <1 andζ1. (3.21) In the above equation one must take into consideration that in the small times regime the global density matrix remains approximately a direct product of individual density matrices|+ih+|, and therefore one can add the individual entropies to approximate the entropy of entanglement.

3.1.3 Lower bound for the entropy of entanglement

In addition, Calsamiglia et. al show how one can obtain a lower bound for the entropy entanglement valid for arbitrary times, using the sequence of inequalities: hSAi ≥ −hlog2 Tr ˆρ2A

i ≥ −log₂hTrρ˜ˆ2

A

i = −log₂P

sA,s0Ah|CsA,s0A|

2

i/22NA

. The first inequality follows from general considerations regarding the von Neumann entropy (see appendix A.4), we were not able to prove the second inequality, however we checked it in the 3 particle case explained above (in which case it is an equality), and also in a system with 8 particles taking groups of 4 (also an equality).

5_{In the original article of Calsamiglia this is merely stated as a fact, we took it on ourselves to give a bit more clarity to the}

(23)

Π

2 Π 32Π 2Π

0.0 0.5 1.0 1.5 2.0 Φ p Φ Ζ=1 Ζ=10 Ζ=20 Ζ=50

Figure 3.2: Probability distribution for the collisional phases corresponding to several values ofζ.

Now in eqn. (3.6) one can see that CsA,s0A only depends on the difference zA := sA−s

0_{A, and its average}

h|CsA,s0A|

2

i only on the number ZA of nonzero entries of zA. If a particle in B has collided with at least one in A wherezA has nonzero entries, then it will contribute a factor 1/2 to the product in eqn. (3.6), otherwise it contributes a factor of 1. Being the collisions in the gas a Poisson process, the probability that no collision occurs ispzA =e

−rtZA/N_{, therefore each term in the product contributes on average with a factor 1}_/_{2 (1}−_p

zA) +pzA=

1 +e−rtZA/N_/_{2. Taking into consideration combinatoric factors}

hSA(t)i ≥ −log2

(

1 2N

NA

X

ZA=0

_N

A ZA

h

1 +e−rtZA/(N−1)i

NB

)

. (3.22)

For long times the following approximation can be made

1 2N

NA

X

ZA=0

_N

A ZA

1 +e−rtZA/N−1

NB

≈ 1 2N

(

2NB₊_N

A

1 +e−rt/N−1 NB

+ NA

X

ZA=2

_N A ZA ) ≈ 1 2N n

2NB₊_N

A

h

1 +NBe−rt/(N−1)

i

−1−NA+ 2NA

o

= 1 2N

h

2NB₋_{1 +}_N

ANBe−rt/(N−1)+ 2NA

i

= 1 2NA +

1 2NB −

1 2N +

NANB 2N e

−rt/(N−1)

→ hSA(t)i ≥ −log2

₁

2NA +

1 2NB −

1 2N +

NANB 2N e

−rt/(N−1)

.

(3.23)

From the above inequality, and reminding that entropy of entanglement is always bound above by

log₂(min

2NA_,₂NB _{) = min}_{_N

A, NB}, (3.24)

one can see that after a sufficiently long time, the entropy of entanglement is almost equal to its maximum, both parts are tightly entangled.

(24)

3.2 Mixture of two spin gases

We now consider a new scenario were two spin gases initially at different temperatures are allowed to mix. Imagine a container with two chambers, each holding a spin gas of disks of the same nature, equal density but different temperaturesTLandTR, and each in an initial pure quantum state,|ψLiand|ψRi. When the wall that separates

them is removed, how does the entropy of entanglement between the disks evolve?

T

L

,

N

L

,

È

Ψ

L

T

R

,

N

R

,

È

Ψ

R

Figure 3.3: Mixing spin gases: a container is divided in two equal volumes, each containing disks of the same mass and radius, but at different temperatures TL and TR, their corresponding quantum states being |ψLiand |ψRi,

the division is removed and the gases mix.

The analysis of this new situation is the same as the one exposed in section 3.1, and the arguments of section 3.1.2 apply up to the point where the collision rates comes into play, in the appendices [see eqn. (A.3.5)] we find that the collision rate for L particles with R particles is

τ−1=r= 4√πN AR

r

k m

r

TL+TR

2 withN =NL+NR. (3.25)

The entropy of entanglement in the regime of short times and large collisional phases [see eqn. (3.21)] being modified to

hSLi ≈ NRNL

N−1rt(2−log2e) = NRNL N−14

N AR

r

2πk m

r

TL+TR

2 (2−log2e)tforrt <1 andζ1. (3.26) The new rate of entropy of entanglement creation being equal as if the gases had been already at equilibrium at temperature TL+TR

2 .

3.2.1 Computational experiments

We took a hard disk gas with 16 particles in total, 8 of them initially on each side of a container, simulated the collisions between them (see appendix A.5), calculated the corresponding concurrency matrix Γ(t), and from it

(25)

the entropy of entanglement6for the particles initially on the left with those initially on the right. The number of particles we were able to simulate were limited by the exponential growth in the computational cost, corresponding to the manipulation of exponentially increasing Hilbert space sizes. In our simulations the side of the container is equal to the unit of length,k= 1,m= 1 and the radius of the disks taken so that the fraction of area the covered by them was equal to 2%. Initial velocities were sampled according to the Boltzmann distribution corresponding to each temperature, and the disks uniformly distributed in space.

Figures 3.4 and 3.5, show the results of our experiments, in both our modified short time approximation [eqn. (3.26)] is plotted against the data, and are in good agreement; this is indicative that the rate of collisions we calculated, adequately describes the collisions between the gas particles.

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0 2 4 6 8

t

E

TL=1,TR=3

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0 2 4 6 8

E

TL=1,TR=1

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0 2 4 6 8

t

E

TL=1,TR=4

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0 2 4 6 8

E

TL=1,TR=2

Figure 3.4: Entropy of entanglement during the mixing of two spin gases for four pairs of initial temperatures. In light shading the time series for ten different initial conditions, and the dot dashed curve their interpolated average. The dashed red line corresponds to the short time, large collisional phases approximation, as given in eqn. (3.21), with the intersection with the time axis moved arbitrarily to t= 0.15, we interpret this shift as the time required for the particles in either chamber to expand and occupy the whole chamber. In allγ= 1000., and NR=NL= 8.

6_{More precisely we find}_S_ρˆ_˜_A

(26)

0.0 0.2 0.4 0.6 0.8 1.0 0

2 4 6 8

t

E

TL=1,TR=3

0.0 0.2 0.4 0.6 0.8 1.0

0 2 4 6 8

E

TL=1,TR=1

0.0 0.2 0.4 0.6 0.8 1.0

0 2 4 6 8

t

E

TL=1,TR=4

0.0 0.2 0.4 0.6 0.8 1.0

0 2 4 6 8

E

TL=1,TR=2

Figure 3.5: Short time detail of the same scenario shown fig. 3.4, number of initial conditions increased to 40.

3.2.2 Concluding remarks

Our analysis in a mixture of two spin gases has much in common with the model of only one spin gas, but has the additional virtue of specifying the speed in which entanglement grows between two bodies at different temperatures as a function of their temperatures and densities, hS_ti ∝ NR

A √

TL+TR. As such it speaks for the importance of improving the vacuum in an experimental chamber, and of reducing both the temperatures of the system under study and of its surrounding, if one is interested in avoiding the entanglement of one, which might be the object under scrutiny, with another, which might be taken as a surrounding environment. It would be interesting to compare how these two rules of thumb, that the speed of decoherence decreases linearly with an improving vacuum, and that itincreases as the square root of the average temperature, compare to the experimental realities in the design of quantum systems which aim at protecting entanglement. This is not to say we have discovered the importance of low temperatures and densities in the study of quantum phenomena, since these have been for quite some time the daily bread of quantum laboratories, but only to hint at a relationship between important control parameters in the field of the experimental study of entanglement.

One could also hint at a difference between the flow of heat and the “flow” of entanglement, being the first dependent only on the difference of temperatures between two bodies, and the second on their sum. This points to a well known fact in the study of decoherence where a quantum system’s contact with a heat reservoir is known to go in detriment of entanglement. Heat flow stops when two systems have reached the same temperature, but they keep entangling with each other well beyond that point. One would well dream of having an entanglement insulation industry as we already have one of thermal insulation.

(27)

Chapter 4

The hybrid ASEP-XY model

In this chapter we first introduce the XY and ASEP models, later we propose a possible manner to hybridise them, and then study the model in simulations. Our hope in this attempt is to observe the behaviour of entanglement in a non-equilibrium scenario. The study of entanglement in non-equilibrium scenarios is justified by the frailty that it has in equilibrium, when a quantum system has interacted with its environment and has reached equilibrium with it, when we go look, there is little entanglement left for us; this frailty has been studied by Fine et. al [18], Arnesen et. al [1], and Wang [49]. There are already indications of the possibility of having robust entanglement outside of equilibrium, Guerreschi et. al [22] have proposed a molecular motor which ‘generates’ entanglement, Quiroga et. al [42] have shown how heat currents can enhance it, and Castillo et. al [10] have shown how evidence of “quantumness”, via the violation of a Legget-Garg inequality (which is a sort of temporal version of a Bell inequality), can be enhanced, again by the presence of a heat current.

In our proposed model we study the degree of entanglement, via the concurrence, between qubits separated by different distances; we also study the entropy of entanglement between a group of two consecutive qubits and the rest of the chain; and in both cases we search for statistical regularities in their distributions.

4.1 The XY model

Introduced by Lieb et. al in 1961 [32] the XY model is a many-body quantum model for an antiferromagnetic chain of spin 1/2 particles, the model has several variants all of which are exactly soluble, we analyse the isotropic periodic case for which the Hamiltonian is

ˆ H =J

4 N

X

i=1

ˆ

σ_ixσˆx_i₊₁+ ˆσ_iyσˆy_i₊₁

, (4.1)

with N+ 1→1, N the number of particles, ˆσx,y _{Pauli spin operators, and} _{J >}_{0 their coupling strength. The} eigenvalues and eigenvectors of this Hamiltonian can be found by a sequence of transformations which terminate in a Hamiltonian of free fermions.

As a first step ˆH is rewritten in terms of raising and lowering operators{ˆa}and

ˆ

a† , and the energy scale is set by putting J= 1:

(28)

ˆ a†_i = 1

2(ˆσ x i +iσˆ

y

i), (4.2)

ˆ ai =

1 2(ˆσ

x i −iσˆ

y

i), (4.3)

ˆ H = 1

2 N−1

X

i=1

ˆ

a†_iâi+1+ â†i+1âi

. (4.4)

The raising and lowering operators partly obey the anticommuting relationships for fermion creation and annihi-lation operators, and partly the commuting reannihi-lationships for boson operators

n

ˆ aj,aˆ†j

o

= 1,ˆa2_j=ˆa†_j 2

= 0, and fori6=j,hˆa†_i,ˆaj

i

=hâ†_i,â†_ji= [âi,aˆj] = 0. (4.5)

To remedy this ‘defect’, as a second step the following transformation is useful1

ˆ ci:=eπi

Pi−1

j=1ˆa †

jˆaj_ˆ_a

i, and ˆc†_i := ˆa†_ie−πiPjj==1i−1ˆa

†

jˆaj_, _(4.6)

for which

ˆ

c†_iˆci= ˆa†iˆai, (4.7)

and the inverse transformation

ˆ

a†_i =e−πiPij−=11cˆ †

jcˆj_ˆ_c

i ˆ

a†_i = ˆc†_ieπiPij−=11ˆc †

jˆcj_. _(4.8)

The intended result being that the operators{ˆc} obey fermionic anticommuting relationships:

n

ˆ ci,ˆc†j

o

=δij,{ˆci,ˆcj}=

n

ˆ

c†_i,ˆc†_jo= 0. (4.9)

If one takes into account the relationships â†_iaî+1 = ˆci†ˆci+1, and â†iâ

†

i+1= ˆc †

icˆ

†

i+1, and the fact

2 _that _eπiˆc†_jˆcj ₌

e−πicˆ†jcˆj_{, the Hamiltonian takes the form of a quadratic form in fermionic operators}

ˆ H =1

2 N X j=1 ˆ

c†_iˆci+1+ ˆc†i+1cˆi

−1 2

ˆ

c†₁ˆcN + ˆc†Ncˆ1 eπi

PN

j=1cˆ †

jˆcj _{+ 1}

= ˆHc+ ˆHb with ˆHc = 1 2 N X j=1 ˆ

c†_iˆci+1+ ˆc†i+1ˆci

,and ˆHb = ˆH−Hˆc.

(4.10)

The term ˆHb introduces a correction in the energy eigenvalues of the order _N1 and we neglect it ˆH ≈Hˆc =~cˆ†A~ˆc, where Ais ann×nmatrix having a banded structure:

1_{First proposed in 1928 by Pascual Jordan and Eugene Wigner[27].} 2_{The occupation number operators ˆ}_c†

(29)

A= 1 2              

0 1 0 . . . 0 0 1

1 0 1 . . . 0 0 0

0 1 0 . . . 0 0 0

. . . 1 0 0

0 0 0 . . . 0 1 0

0 0 0 . . . 1 0 1

1 0 0 . . . 0 1 0

             

Ai,j = 1

2(δi,j−1+δi,1δj,N+δi−1,j+δi,Nδj,N). (4.11) The matrixAis symmetric, and circulant3, its eigenvalues Λk, and eigenvectorsφ~k are (see [28], p. 167)

Λk =−cosφk,withφk= 2π k

N, andk=− N

2 ,− N

2 + 1, . . . , N

2 −1, (4.12)

the corresponding eigenvectors beingφ~k

j = √1

Ne

−kj_{, j}_{= 0}_,₁_{, ..., N}₋₁_. _(4.13)

Since Ais hermitic there exists a unitary matrixU such thatA=U†ΛU with Λa diagonal matrix containing the eigenvalues ofA, in this way the linear transformation of the operators~cˆ

~_ˆ η=U~c,ˆ

yields ˆHc=~ηˆ†Λ~ηˆ=

X

k

Λkηˆ_k†ηˆk. (4.14)

Some of the eigenvalues Λk are negative, so it is convenient to make the final transformation

ˆ ζk :=

  

ˆ

ηk if Λk≥0 ˆ

η†_k if Λk<0

, (4.15)

in terms of which

ˆ Hc =

X

k |Λk|

ˆ ζ_k†ζˆk−

1 2

. (4.16)

In this final form the problem has been reduced to a problem where the solutions in the occupation number representation are binary vectors~nof sizeN with corresponding energy

E(~n) =X k

|Λk|

nk− 1 2

. (4.17)

EG=− 1 2

X

k

|Λk|=− 1 2cot π N . _(4.18)

In the limit ofN → ∞, the contribution per spin is lim

N→∞

EG

N = limN→∞−

1 2N cot

π

N

=−1

π. (4.19)

Equation (4.13) gives the elemental excitations spectrum, and equation (4.17) the energy of each eigenstate. The energy spectrum for the original Hamiltonian can be then calculated from these, and for large N tends to a normal distribution (see fig. 4.1 ) with an standard deviation σ(N)≈0.354√N, and meanhEi= 0.

(30)

-

6 -

4 -

2

0

2

4

6

0.00

0.05

0.10

0.15

0.20

0.25 E

J

p

H

E

L

-Π -Π

2 0

Π

2 Π

0 1

k

Lk J

Elementary excitations spectrum

Figure 4.1: This histogram shows the distribution energies in the spectrum of theXY model for 20 qubits. The fit shown is for a normal distribution. On the top right corner the elementary excitations spectrum is shown.

Entanglement in theXY4 _{model has been studied by Wang [49], as well as by Franchini [19, 20], and Peschel}

[40]. In theXY model, the ground state is separable, however, if an anisotropy is included between thexandy components the richness of the model’s ground state is significantly enhanced, and is one of the toy models used in the study of quantum phase transitions[39].

4.2 The asymmetric simple exclusion process

Theasymmetric simple exclusion processis a discrete classical transport model where particles undergo a random walk in a lattice with the additional condition that at most one particle can occupy each site at a given time. First proposed in 1968 [33] as a model to describe the formation of biopolymers on nucleic acid templates, the ASEP is nowadays considered as the paradigmatic model for non-equilibrium transport phenomena [11], this partly because of the wealth of analytic results relating to it [13, 17, 21, 30, 34, 47], and partly because of the variety of phenomena to which the model can be mapped into, being a few of them, vehicular traffic [46], molecular motors, and particle deposition [3].

The several variants of the ASEP are distinguished by the type of time dynamics, the dynamical rules, the update procedure, and the boundary conditions. Time dynamics can be either discrete or continuous. The dynamical rules specify the rates at which particles can go forward (q) or backward (p). The update procedure refers to the manner in which the dynamical rules are to be applied, it can be random-sequential, in which case a particle or a site is picked randomly in the lattice, and to it the dynamical rules applied, or it can be ordered-sequential, in which case a particular sequence is always used to apply the rules, or it can be in parallel, in which case the rules are applied at the same time in every lattice site; choosing between these update procedures can lead to significantly different behaviours [43]. Boundary conditions can be either periodic or open, and if open, then particles can enter or leave the lattice at either end.

From the mathematical perspective, the ASEP is an stochastic process governed by the master equation, which

Non-equilibrium quantum entanglement evolution on top of classical many-body systems - two case scenarios