Algorithmic Cooling and Quantumthermodynamic Machines

Algorithmic Cooling andQuantumthermodynamic Machines

Diplomarbeit von

Florian Rempp

July 20, 2007

Hauptberichter: Prof. Dr. Gunther Mahler

Mitberichter: Prof. Dr. Alejandro Muramatsu

1. Institut fur Theoretische Physik

Universitat Stuttgart

Pfaffenwaldring 57, 70550 Stuttgart

http://itp1.uni-stuttgart.de

http://www.uni-stuttgart.de

Ehrenwortliche Erklarung

Ich erklare, daß ich diese Arbeit selbstandig verfaßt und keine anderen als die angegebe-nen Quellen und Hilfsmittel benutzt habe.

Stuttgart, July 20, 2007 Florian Rempp

Contents

1. Abstract 11.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2. Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. Quantum Mechanics 32.1. Operator Representations . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1. Transition Operator Representation . . . . . . . . . . . . . . . . . 32.1.2. Pauli Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.3. Density Operator Representation . . . . . . . . . . . . . . . . . . 52.1.4. Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2. The Liouville Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3. Quantum Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1. Thermodynamic Variables . . . . . . . . . . . . . . . . . . . . . . 92.3.2. Further Thermodynamic Variables . . . . . . . . . . . . . . . . . 112.3.3. Work and Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.4. Engine efficiencies . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4. Thermodynamic processes . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4.1. Quantum Thermodynamic Machines . . . . . . . . . . . . . . . . 14

2.5. The Quantum Otto-cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.6. Quantum Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6.1. Quantum Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3. Heat and Work 213.1. The Local Effective Measurement Basis Method (LEMBAS) [42] . . . . . 223.2. Derivation of dW and dQ . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.1. Derivation of Heff . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.2. Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3.1. Laser with Detuning . . . . . . . . . . . . . . . . . . . . . . . . . 253.3.2. Quantum Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4. Algorithmic cooling 314.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2. The Original Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

v

Contents

4.2.1. The Basic, Closed Proposal . . . . . . . . . . . . . . . . . . . . . 324.2.2. Heat Bath Algorithmic Cooling . . . . . . . . . . . . . . . . . . . 33

4.3. Cyclic Extended Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 334.3.1. Detailed Investigation of the Algorithm . . . . . . . . . . . . . . . 354.3.2. Final Temperatures of the Cyclic Algorithm . . . . . . . . . . . . 374.3.3. Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.4. An Ideal Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5. Summary and Perspectives 455.1. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.2. Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

A. Appendix 47A.1. Trace theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Bibliography 49

acknowledgments 55

vi

1. Abstract

1.1. Motivation

In recent years the field of quantum thermodynamics has been developed as a part ofquantum system theory [28]. It describes the emergence of thermodynamic behaviorfrom quantum mechanics [12]. Most efforts so far concerned the definition of the ther-modynamic values with all their classic properties, the quasi classic limes so to speak.Some, more explicit “quantum” concepts as algorithmic cooling [4], a method to increasethe polarisation of a Spin by unitarily moving entropy to an other spin system, do noteasily fit into the framework of quantum thermodynamics. Although one has the feelingit should, because it reminds us strongly of a refrigerator.

We will have to extend the concept of heat and work employing the principle oflocal energy measurement basis (LEMBAS) [42] to compute the efficiency of a coolingalgorithm. LEMBAS implies an extension of the work and heat definition itself, whichholds for every quantum system and provides more physical solutions than the old one.

Another question on algorithmic cooling in this context is, whether it can work incycles as a real refrigerator. The possible algorithms so far include further steps only byincreasing the system size, even if contact to a heat bath is part of the algorithm [4].

One can also ask, if an algorithm is ideal in the sense that maximum heat is transferred(per cycle for cyclic algorithms) and if there is an upper limit as in the classical realm,namely the Carnot cycle.

1.2. Outline

First we will recall some basic concepts of quantum mechanics and give a sketch ofquantum thermodynamics along with a brief introduction to quantum information pro-cessing. Then the LEMBAS concept is introduced with the extension of the definition ofheat and work in quantum thermodynamics. In the third chapter we will take a closerlook on algorithmic cooling, especially the algorithm by Boykin et al. and its cyclic ex-tension. This is completed by the introduction of an ideal algorithm (with respect toheat transport) and its connection to the quantum Otto cycle.

1

1. Abstract

2

2. Quantum Mechanics

Since the beginning of the 20th century when Planck came up with his famous for-mula for black body radiation and with the idea of fixed, finite energy quanta in whichradiation is emitted from the black body, quantum mechanics has become one of themost successful theories of physics. Quantum mechanics have originally been inventedto describe thermodynamic properties of nature. It is a valid question whether ther-modynamics somehow emerge from quantum mechanics. Research in this section, thequantum thermodynamics developed in recent years. A good summary of the results sofar is given in [12]. Of course it is far from complete, as the development concentratedon a quasi classic limt of quantum mechanics, by trying to conserve classical thermody-namic properties as the extensivity of energy, entropy etc.. Some amazing effects havebeen found here: quantum thermodynamic heat engines [19,20] and Fourier’s law in nonequilibrium systems [30]. But to incorporate other concepts, e.g. algorithmic cooling,one has to extend the definitions, so to speak put more “quantum” into them and thussay goodbye to extensivity, at least to some degree.

In this Chapter basic concepts of quantum mechanics are presented as well as somequantum thermodynamic concepts in their classical definition. In case of work and heatthe extension of the definition to arbitrary, bipartite quantum systems is given in Chap. 3along with the LEMBAS concept.

2.1. Operator Representations

In order to describe a quantum mechanic system one can not rely on the classical phasespace but has to use the eigenvalues of Hermitian operators. Their representation re-quires parameters, defined according to an appropriate reference frame. These referenceframes are called operator representations. Some of the more common representations,relevant for this thesis are presented in the following section.

2.1.1. Transition Operator Representation

If one is interested in a discrete finite k-dimensional Hilbert space H it can be describedin the complete orthogonal state basis |i〉 with

〈i| j〉 = δij with i, j = 1, 2, . . . , k. (2.1)

3


we can now define k2 transition operators

Pij = |i〉〈j| (2.2)

which in general are non-Hermitian but nevertheless orthonormal

Tr

PijP†i′j′

= δii′δjj′ (2.3)

if the trace operation Tr . . . is used to calculate the norm, this is called tracenorm.The transition operators form a complete basis in the accounting Liouville space L

(see Sec. 2.2 page 8). All Operators O can be expanded into this basis

O =∑

ij

OijPij, (2.4)

where the Oij are the expansion coefficients

Oij = Tr

OP †ij

= 〈i|A |j〉 (2.5)

This makes a total of 2k2 parameters due to the fact, that coefficients Oij in general arecomplex. In case of a Hermitian operator all Oij are real because

with O = O† and Pij = P †ij

O =∑

ij

OijPij (2.6)

O† =∑

ij

O∗ijP

†ij (2.7)

⇒ Oij = O∗ij. (2.8)

So we need k2 coefficients Oij to define a Hermitian operator uniquely.

2.1.2. Pauli Operators

It is convenient, especially for small quantum systems, to use the generators of theSU(k) as a complete orthogonal basis of the Hilbert space. For k = 2 these are the Paulioperators σi with i = x, y, z, 0. In terms of transition operators the Pauli operatorsare

σx =P12 − P21, (2.9)

σy =i (P21 − P12), (2.10)

σz =P11 − P22, (2.11)

σ0 =12. (2.12)

4


They are Hermitian and, expect σ0, traceless. Some important relations for the Paulioperators are

(σi)2 = 12, (2.13)

[σx, σy] = 2iσz, (2.14)

and cyclic permutations of (2.14). Additionally, raising and lowering operators

σ+ = σx + iσy (2.15)

σ− = σx − iσy (2.16)

can be introduced.

2.1.3. Density Operator Representation

In this thesis the most utilised representation of the state of a quantum system is by itsdensity matrix ρij based on the transition operators:

ρ =k∑

i,j=1

ρij Pij . (2.17)

In order to describe a real quantum system, ρ is subject to the condition

Tr ρ =∑

i

ρii = 1 (2.18)

and has to be a positive definite Hermitian operator. The matrix elements can beprojected out again with the aid of the statevectors

ρij = 〈i| ρ |j〉 . (2.19)

The expectation value of an arbitrary operator O for the state ρ is given by

⟨

O⟩

= Tr

O ρ

. (2.20)

2.1.4. Dynamics

The unitary dynamics of a closed quantum system are given by the Schrodinger equation

i ~∂

∂t|ψ(t)〉 = H(t) |ψ(t)〉 . (2.21)

In the so called Schrodinger picture the dynamics of the system are exclusively rep-resented by the time dependence of the state vectors |ψ(t)〉. Nevertheless, there may

5


be explicitly time dependent external potentials resulting in time dependence of theHamiltonian H(t).

In order to find the dynamics of the density operator we take its time derivative

∂

∂tρ(t) =

∑

i

ρi

(

|ψ(t)〉〈ψ(t)| + |ψ(t)〉〈ψ(t)|)

(2.22)

and insert (2.21) for |ψ(t)〉 and 〈ψ(t)|

∂

∂tρ(t) = −

i

~

(

H(t)∑

i

ρi |ψ(t)〉〈ψ(t)|

︸︷︷︸

ρ(t)

−∑

i

ρi |ψ(t)〉〈ψ(t)|

︸︷︷︸

ρ(t)

H(t))

. (2.23)

This leaves us with the Liouville von Neumann equation

∂

∂tρ(t) = −

i

~

[

H(t), ρ(t)]

. (2.24)

Alternatively, one may introduce a superoperator D(ρ) to describe the dynamics of openquantum systems

∂

∂tρ(t) = −

i

~

[

H(t), ρ(t)]

+ D(ρ). (2.25)

Eq. (2.24) and (2.25) can also be written in superoperator representation

∂

∂tρ(t) = L ρ (2.26)

where, of course, L is not identical in both cases.

Closed Systems

For closed systems Eq. (2.24) can formally be solved by a unitary transformation

ρ(t) = U(t) ρ(0) U †(t), (2.27)

where U obeys the Schrodinger equation

i ~∂

∂tU(t) = H(t) U(t). (2.28)

For ∂tH = 0 the time evolution operator U(t) has the formal solution

U(t) = e−i H t/~ (2.29)

6


Eq. (2.29) can be used also to define a so called model Hamiltonian Hindmodel for thetransformation generated by the quantum gate Ugate (see Sec. 2.6.1 on page 18)

Hmodel =i ~

tln

Ugate

. (2.30)

These model Hamiltonians, however, do not represent a realistic quantum system, be-cause they do not include local Zeemann splittings of the qbits, something that cannoteasily be added. Therefore one has to find the accounting π-pulse sequence of the gates(see Sec. 3.3.2 on page 25).

Open systems

In reality it is hardly possible to set up a closed system as described above. It will alwaysbe influenced by some kind of greatly uncontrollable environment, at least slightly. Thismeans that every description of a real quantum system is necessarily incomplete andclosed quantum systems can only be realised approximately. Some environmental effects,e.g. a Laser pulse, result in unitary dynamics of the system. They can be modeled as a socalled classical driver by an explicit time dependence of the Hamiltonian. Other effects,like coupling to a heat bath, are able to change the entropy S of the observed systemand can therefore not be included into the Hamiltonian. A dissipator D(ρ) has to beadded to the Liouville von Neumann equation (2.24) resulting in the so called quantummaster equation (2.25).

To compute the dissipator we first consider the whole closed system described by thedensity operator ρ, which is partitioned into

ρs = Trb ρ (2.31)

ρb = Trs ρ (2.32)

with the indices s and b denoting system and bath, respectively. For the entire systemthe Liouville von Neumann equation (2.24)

i ~∂

∂tρ(t) =

[

H(t), ρ(t)]

holds. In analogy to the derivation of the classical master equation one assumes thatthe interesting part of the dynamics (here the system) can be projected out [44]. Thus(2.24) has to be replaced by the non autonomous form

i ~∂

∂tρs(t) = f(ρs, ρb). (2.33)

Such an equation is usually not solvable. One has to create an autonomous form byapplying specialisations, approximations and assumptions, neglecting most of the degreesof freedom. This has been attempted by many authors [5, 35, 38]. Standard ingredientsare

7

2. Quantum Mechanics the Markov assumption (dynamics of the system do not affect the environment). the Born approximation (the interactions between system and environment areconsidered week and thus the two systems factorize. This approximation may leadto problems because for longer time periods even weak interaction may lead tostrong quantum correlations). the assumption of special interaction type (e.g. rotating wave approximation).

Some authors [41] construct a Dyson series and truncate at some order. But this maylead to problems evaluating the resulting equation for longer time periods. The mostgeneral form, conserving the properties of the density operator, namely Hermiticity,positivity and Tr ρ = 1, was introduced by Lindblad [25]

∂

∂tρs = −

i

~

[

Hs, ρs

]

+1

2

(ks)2−1∑

i,j=1

Aij

(

2 Li ρs L†j − ρs L

†j Li − L†

j Li ρs

)

. (2.34)

The first term represents coherent dynamics, as (2.24). The second part is the incoherentinfluence of the environment on the system, defined by the environmental operators Li.These operators form, together with the 1, a complete orthogonal basis of the (ks)

2

dimensional Liouville space of the system. Aij is an Hermitian positive definite matrixof parameters. However, not all current master equations (e.g. [35]) are in Lindblad formand thus do not conserve positivity of the density operator for the whole parameter range.But within reasonable parameter windows they do, of course. For a detailed derivationof the master equation see [19, 26, 28].

Considering the environment as a set of decoupled harmonic oscillators, with an eigen-frequency density that makes the Markov assumption feasible, and within the rotatingwave approximation, the dissipator reads

D(ρs) =W1→0

2(2 σ− ρs σ

+ − ρs σ+ σ− − σ+ σ− ρs)

+W0→1

2(2 σ+ ρs σ

− − ρs σ− σ+ − σ− σ+ ρs) (2.35)

with the rates W1→0 and W0→1 chosen such that the system will end up in a thermalstate with the same temperature as the bath. Because in this thesis only single spinsare to be thermalised, this rather intuitive dissipator is feasible.

2.2. The Liouville Space

Sorting the entries of an operator O on a discrete finite k-dimensional Hilbert spaceH into a k2 dimensional vector, we define ”ket” and ”bra” like vectors O → |O) andO† → (O| in this super space, the so called Liouville space L. Their inner product is

8

2.3. Quantum Thermodynamics

defined as (A|B) = Tr

A†B

, the trace norm of operators in H. Operators acting on

states |O) in the Liouville space are defined as [12]

O|O) = |A)(B|O) = Tr

B†O

A, (2.36)

where the superoperator O = |A)(B| represents a k2 × k2 dimensional matrix.


2.3.1. Thermodynamic Variables

If one wants to investigate thermodynamic properties of quantum systems one has tocome up with new “quantum” definitions of the pertinent thermodynamic quantities,such as entropy, temperature, energy, work and heat etc.. These quantities shouldapproach their classical definitions, at least within some limits (normally weak couplingand an equilibrium situation are required). Whether they are still thermodynamic, ifthey are not restricted by these limitations, e. g. for nonequilibrium situations, is openfor discussion. Let us assume that they are, at least near the accounting limits. In thissection the definitions of thermodynamic variables will be given following [12]. Some ofthem, namely those for work and heat, will be extended in chapter 3 (page 21).

Entropy and Temperature

The following concept of entropy and temperature is only a rough sketch. A detailedtreatise can be found in [12]. Typically, when dealing with quantum systems the entropyused is the von Neumann entropy S given by

S = −Tr ρ ln ρ , (2.37)

where ρ is the density operator of the considered system. It is invariant under unitarytransformations. That has two consequences for closed systems: S is independent of the chosen basis. S is time independent.

This corresponds to the Gibbs entropy in the ensemble approach. Since one wants tohave an entropy that may possibly change in time, two approaches are viable: Transit to an open system approach as described in Sec. 2.1.4 on page 7.

9

2. Quantum Mechanics Time averaged density operator: Since all off diagonal elements are oscillating theywill vanish in a time averaged density operator [10]. It can be shown that the timeaveraged entropy St rises if the off diagonal elements vanish due to the dynamicsof the system. For small energy spacing (typically for large systems) the necessaryaveraging time rises and diverges for degenerate systems. The required averagingtime may exceed the thermodynamic relaxation time of such a system.

If the state is diagonal in the energy eigenbasis of the system one can consider (2.37) asthe thermodynamic entropy.

The temperature T of a system is given as the derivation of the entropy with respectto the mean energy E (for constant mechanical control V )

T =

(dS

dE

)

V

=1

kB β. (2.38)

For a TLS in a diagonal state with

P1N0

P0N1= e−β (E1−E0) (2.39)

it follows that

T = −E1 − E0

(ln P1 − ln P0) − (ln N1 − ln N0). (2.40)

Here Ei is the energy eigenvalue of the ith level, Pi the respective occupation probabilityand Ni its degeneracy. This definition can easily be extended to the inverse spectraltemperature of a multi level system by assigning temperatures to adjacent pairs of energylevels, over which an average is taken:

βs := −

(

1 −W0 +WM

2

)−1

M∑

i=1

(Wi +Wi − 1

2

)ln Pi − lnPi−1 − (ln Ni − ln Ni−1)

Ei −Ei−1, (2.41)

where Wi is the probability of finding the system at the energy Ei. This βs dependsonly on the spectra and the occupation probabilities. Thus it is always defined, evenfor systems, which don’t show thermodynamic behaviour. For equilibrium situationshowever, the spectral temperature corresponds to the thermodynamic temperature. Thiscan be proven by showing that in equilibrium the relation

βs =∂S

∂E=

∂

∂Eln D(E) =

1

D(E)

∂D(E)

∂E(2.42)

holds, with D(E) being the state density, using extensivity of the entropy and the factthat it approaches its maximum in equilibrium.

10


∆V

⇒∆E(V ) ∆E(V − ∆V )

Figure 2.1.: The volume of a quantum well, depicted here as the area of the 1D well, candirectly be mapped to the Zeemann splitting of the TLS.

2.3.2. Further Thermodynamic Variables

After temperature and entropy we may introduce the pressure P as the conjugate variableto V , the volume. The chemical potential µ and its conjugate variable the particlenumber N are left aside because they are not needed in this thesis. For a TLS, like aspin with the Hamiltonian H = σz ∆E/2, the following correspondences can be verifiedby considering the TLS as two energy levels in a quantum well (see Fig. 2.1)

∆E = ∆E(V ) (2.43)

P = −

(dU

dV

)

S

=d

dV

∆E(V )

2σz. (2.44)

V can be associated with ∆E. We are thus able to call a bath contact without changeof the Zeeman splitting isochoric. The justification and a simple example is given inSec. 2.4.1 on page 14. For a cyclic process the energy at the beginning and the end ofthe process should be the same, dU = 0. The path in the ST plane for one cycle is aclosed one and therefore (2.49) leads to

∆W = −∆Q = −

∮

TdS. (2.45)

Knowing the heat current Jα, flowing from or into a bath α during the cycle, it is possibleto determine the exchanged heat over one period τ

∆Qα =

∫ τ

0

Jαdt. (2.46)

In the following α = h or c indicating a hot or cold bath.

2.3.3. Work and Heat

In classical thermodynamics all thermodynamic information of the system is given withthe help of the Gibbs fundamental relation [6]

dU(S, V ) = dW + dQ = −PdV + TdS, (2.47)

11


by knowing the inner energy U . P , the pressure, is conjugate to the volume V . Thetemperature T is conjugate to S, the entropy.

The inner energy U of quantum systems is given by the energy expectation value

〈E〉 = Tr

H ρ

, (2.48)

with the system Hamiltonian H and its state ρ. The change of the energy is given by

d

dt〈E〉 = Tr

d

dtH ρ

︸︷︷︸

dW/dt

+ Tr

Hd

dtρ

︸︷︷︸

dQ/dt

. (2.49)

As indicated by the underbraces, the first term, where only the spectrum is changing, isassociated with work

dW = Tr

d

dtH ρ

dt =∑

i

Ei pidt. (2.50)

The second term of (2.49), where the state and therefore the entropy is changing, isinterpreted as the exchanged heat

dQ = Tr

Hd

dtρ

dt =∑

i

Ei pidt. (2.51)

Thus (2.49) is again the fundamental relation (2.47).

2.3.4. Engine efficiencies

The efficiency of a heat engine is given by the ratio of the released work ∆W and theabsorbed heat from the hot bath, ∆Qh

ηe =−∆W

∆Qh. (2.52)

For a heat pump the efficiency reads

ηp =−∆Qh

∆W. (2.53)

For a Carnot engine (2.52) leads to

ηeCar = 1 −

Th

Tc, (2.54)

for the classical Otto engine one gets

ηeOtto = 1 −

(V1

V2

)γ

, (2.55)

12

2.4. Thermodynamic processes

a) b) c)

Th Tc

Qh Qc

∆Ec∆Ec∆Eh ∆Eh

W

work

Figure 2.2.: a) Isochoric step: TLS in contact with a heat bath at temperature Th ex-changing heat Qh. The energy splitting ∆Eh is constant during this step. b)Adiabatic step: TLS is driven by a work reservoir from ∆Eh to ∆Ec, whilethe state does not change. c) Isochoric step: As in a) the TLS exchangesheat with a bath at Tc and the energy splitting stays constant at ∆Ec.

where V1 and V2 are the volumes between which the gas system is driven (V 1/V 2 is calledthe compression ration). γ is the polytropic exponent which depends on the consideredgas. The respective efficiencies for the heat pump are

ηpCar =

Th

Th − Tc

(2.56)

for the Carnot-cycle and

ηpOtto =

(V1

V1 − V2

)γ

. (2.57)

for the Otto-cycle.

2.4. Thermodynamic processes

We will discuss only the relevant process types for this thesis, the adiabatic and theisochoric process. These are the steps which seem easiest to be realised in a quantumengine. We assume that the system is in a canonical state all the time, which is feasiblebecause of the bath contact, which dephases much faster then it thermalises. Both stepsare depicted in Fig. 2.2.

During an adiabatic step [cf. Fig. 2.2 b)] the von Neumann entropy of the consideredsystem has to stay constant. For a TLS the occupation probabilities remain constantand thus the density matrix does not change. No heat is exchanged. Only the volumeV , which is directly mapped onto the Zeemann splitting ∆E of the system (2.43) will beincreased or decreased. The work done or gained during this process is simply given by

13


the difference of the energy expectation value at the beginning and the end of the step

∆W↑/↓ = 〈Ef〉 − 〈Ei〉 = tanh

(∆Ei

2Ti

)[∆Ei

2−

∆Ef

2

]

. (2.58)

Because the state during an adiabatic step does not change and starts canonically, it ispossible to determine the temperature of the TLS after an adiabatic step. This wouldnot be possible for a multi level system, unless the ratio of the frequencies between thedifferent energy levels stays constant during the step.

ρ =1

Z

(e∆Eh/(2Th) 0

0 e−∆Eh/(2Th)

)

. (2.59)

Then the adiabatic step as in b) is performed and therefore the temperature of the TLSis decreasing to

e−∆Eh/(2Th)

e∆Eh/(2Th)=

e−∆Ec/(2Tf)

e∆Ec/(2Tf)

Tf = Th∆Ec

∆Eh. (2.60)

For an isochoric step [cf. Fig. 2.2 a) and c)] the volume and thus the energy splitting∆E stays constant. Therefore, the Hamiltonian is constant. No work is done in thisstep (see (2.49)) and only heat is exchanged with a heat bath. The entire change of theenergy expectation value has to be heat Q and can be calculated as

∆Qh = 〈Ef〉 − 〈Ei〉 =∆Eh

2

[

tanh

(∆Eh

2T

)

− tanh

(∆Eh

2Th

)]

. (2.61)

If Tf = Tc at the end of the adiabatic step, no heat with the bath will be exchangedduring the isochoric step. For Tf < Tc heat is extracted from the cold bath, the cyclecan not work as a heat pump anymore. It is possible to define a critical temperaturedifference

∆Tcrit = Th − Tc = Th

(

1 −∆Ec

∆Eh

)

(2.62)

via (2.60), where the process will switch its function from heat pump (∆T < ∆Tcrit) toa heat engine (∆T > ∆Tcrit) and vise versa (see [19]). This behaviour is always valid fora TLS performing cycles which include adiabatic steps [18].

2.4.1. Quantum Thermodynamic Machines

One major questions of quantum thermodynamics is: “How small can a thermodynamicmachine become?”.

14

2.5. The Quantum Otto-cycle

It is obvious that on such a small scale, now reachable for experiments, quantummechanics plays an important role. Answering the question above is therefore linkedwith the possibility of finding thermodynamic cycles within the quantum regime. Firstattempts have been made in [14,37] and many more approaches are now being studied,beginning with harmonic oscillators, uncoupled spins, three-level systems etc. [2,8,9,22,23, 32, 39].

Despite the growing theoretical knowledge about quantum thermodynamic machines,experiments have not yet been realised. A good candidate for experimental realisationseems to be a machine constructed of two-level systems (TLS), like spins or qbits, asinvestigated for quantum information purposes. There are many different scenariospossible to control structures built with a few TLS. Examples are quantum opticalsystems [7], nuclear magnetic resonance [13] and solid state systems [29].

A setup where at least three TLS arranged in a chain can work as a thermodynamicmachine was introduced in [19, 20]. Two TLS act as a filter which control the bathcontact of the TLS in the middle, working as a heat pump or heat engine, dependingon the temperature gradient induced on the TLS-chain. In principle, this setup shouldpave the way for an experimental realization.

Now that we know that thermodynamic machines are realisable, one has to thinkabout what kind of thermodynamic process could possibly be established within theconsidered system. A promising candidate seems to be the Otto-type cycle, where thegas system is controlled by two adiabatic and two isochoric steps. In Sec. 2.4 on page13 both steps have been be discussed.

In Sec. 2.5 a complete Otto-cycle of a TLS is studied. The corresponding TS diagramas well as the PV diagram are presented. The efficiency is derived and we will discusswhat will happen when this efficiency equals the Carnot efficiency — a point whichcannot be studied easily in classical thermodynamics.

2.5. The Quantum Otto-cycle

As already mentioned, the Otto-cycle consists of two adiabatic and two isochoric steps.On the adiabatic steps work is done or released from the system. The entropy remainsconstant and therefore no heat will be exchanged. At the isochoric steps the system isin contact with a heat bath without performing work (see Fig. 2.2).

With the help of the right section of Fig. 2.3, showing the PV diagram of TLS andthe left section of Fig. 2.3 with the respective TS diagram we want to show that a TLScan perform Otto-cycles:

1. Isochoric step (A → B): The TLS is in contact with the hot bath at Th. Theoccupation probabilities change at ∆EA = ∆EB = ∆Eh until the temperature ofthe TLS equals Th. The exchanged heat Qh is defined by (2.61) and reads

∆Qh = 〈EB〉 − 〈EA〉 =∆Eh

2

[

tanh

(∆Ec

2 Tc

)

− tanh

(∆Eh

2 Th

)]

, (2.63)

15


A

BC

D

0.16

0.15

0.14

1.7 2 2.3V

P

A

B

C

D

3

4

0.64 0.65

T

S

Figure 2.3.: Left: PV diagram for a spin 1/2 performing an Otto-cycle. Right: TSdiagram for a spin 1/2 performing an Otto-cycle. The following parametershave been chosen as an example: ∆Eh = 2.25, ∆Ec = 1.75, Th = 4 andTc = 2.5.

2. Adiabatic step (B → C): By changing the local energy splitting from ∆EB = ∆Eh

to ∆EC = ∆Ec without transitions in the TLS, the work ∆WBC given by (2.58)

∆WBC = 〈EC〉 − 〈EB〉 = tanh

(∆Eh

2 Th

)[∆Eh − ∆Ec

2

]

. (2.64)

is released.

3. Isochoric step (C → D): The TLS is in contact with the cold bath at Tc. Thetemperature of the spin at point C is TC = Th

∆Eh

∆Ec. The heat ∆Qc (cf. (2.61))

∆Qc = 〈ED〉 − 〈EC〉 =∆Ec

2

[

tanh

(∆Eh

2 Th

)

− tanh

(∆Ec

2 Tc

)]

, (2.65)

will have been exchanged when the temperature of the spin equals Tc.

4. Adiabatic step (C → D): The system is driven back from ∆ED = ∆Ec to ∆EA =∆Ec as in step 1. The work done in this step is given by (2.58)

∆WDA = 〈EA〉 − 〈ED〉 = tanh

(∆Ec

2Tc

)[∆Ec

2−

∆Eh

2

]

. (2.66)

The efficiency for this Otto-type engine can then be calculated via (2.52) and theresults obtained above ((2.63)-(2.66))

ηqm = −WBC +WDA

Qh= 1 −

∆Eh

∆Ec. (2.67)

16

2.6. Quantum Computing

qp2

tcrit2

pump engine

∆Tcrit ∆T0.1

0.1

0.5

0.5

0.9 1.3

10

30

50

0.3

ηpCar

ηpqm

ηeCar

ηeqm

ηp ηe

pump engine∆Qc∆W

∆Qh

∆Tcrit ∆T

-0.08

-0.04

0

0.04

0.08

0.1 0.5 0.9 1.3

Figure 2.4.: Left: Carnot-efficiency ηpCar for the heat pump and the heat engine ηe

Car asa function of temperature difference ∆T , while Tc = 2.5,∆Eh = 2.25 and∆Ec = 1.75. ηp

qm and ηeqm are the efficiencies of quantum Otto pump/engine.

ηpid = 12.36 and ∆Tcrit = 0.22 can be realized for ∆Eh = 1.904, ∆Ec = 1.75

and Tc = 2.5. Right: Work ∆W , heat ∆Qh from/to the hot bath andheat ∆Qc from/to the cold bath for the ideal machine as a function oftemperature difference ∆T . At ∆T = ∆Tcrit, η

pqm = ηp

Car and therefore∆W = 0, ∆Qh = 0 and ∆Qc = 0.

Eq. (2.67) is equivalent to (2.55), therefore the quantum Otto cycle is indeed an Ottocycle.

Finally we want to discuss the critical temperature ∆Tcrit [cf. (2.62)]. The left side ofFig. 2.4 shows the efficiency of a TLS Otto machine as a function of the difference ofthe bath temperatures ∆T and the respective Carnot efficiency.

As can be seen, the efficiency of the TLS-machine is always less than the Carnotefficiency, except when ∆Tcrit is reached. At this point we get ηqm = ηCar. The rightpart of Fig. 2.4 (showing ∆W , ∆Qh and ∆Qc as function of ∆T ) shows that at ∆Tcrit

no heat will be pumped or transported and no work is released or done. The machine isrunning idle, literally speaking. If ∆T < ∆Tcrit the machine is acting as a heat pump.At ∆Tcrit the machine switches from a heat pump to a heat engine without artificiallychanging the moving direction in the ST -Diagram.


From the point of view of a computer scientist quantum computing is most similar tothe technical realization of a probabilistic Turing machine which uses superposed statesbetween 0 and 1 to calculate results with the highest possible probability to be correct. Inrecent years some quantum algorithms have been published [15, 40] which theoreticallyexceed the possibilities of classical computers. Besides the many unsolved technicalproblems an actual quantum computer is facing, there is the need of a highly polarised

17


TLS to be used as a qbit. This can be achived by working at low temperatures. Butthat will not be feasible for large quantum computers. Another way of preparing such apolarised state is via a refrigeration process, called algorithmic cooling. The small initialpolarisation of some of the qbits is amplified by further decreasing the polarisation ofthe others.

2.6.1. Quantum Gates

In classical combinational logic a logical circuit is built of logical gates which can performthe operations AND, OR, XOR and NOT, where AND, OR and XOR combine thestates of two bits to one and NOT inverts the state of a single bit. Obviously, all logicalexpressions can be composed of these gates. They may be translated into a combinationof NAND (NOT AND) gates for example. NAND gates are called “universal”.

Definition: A gate is called universal if all possible gates can be realized by a combina-tion of the universal gate.

To transfer the concept of logical gates to quantum information processing, one has totake the quantum nature of those systems into account. Because operations on a closedquantum system (see Sec. 2.1.4 on Page 6) have to be unitary, they can be representedby a unitary transformation matrix U , acting on the density matrix ρi in a way that

ρf = UρU−1 (2.68)

is the density matrix after the operation has been performed. This description is inde-pendent of the exact realization which would be described by a Liouville von Neumannequation (c.f. 2.1.4 page 5).

In order to find the corresponding transformation matrix for a certain quantum oper-ation one starts with its truth table. Writing the input states and the output states astwo vectors, their relationship can be formulated via a matrix As an easy example onefinds for the CNOT gate

|11〉 |10〉 |01〉 |00〉|11〉 → |10〉|10〉 → |11〉|01〉 → |01〉|00〉 → |00〉

=⇒

|11〉|10〉|01〉|00〉

0 1 0 01 0 0 00 0 1 00 0 0 1

. (2.69)

If several of those operations Ui are carried out one after another, the whole sequencecan be written as a single transformation U = Un . . . U2U1. The separation in substepsis rather arbitrary but for the sake of an intuitive access some elementary operations,the so called quantum gates, have been defined. With the aid of those quantum gatesone is able to depict an algorithm as a logical circuit, just like in digital electronics. Forbetter comprehensibility the description of these gates is made by the pure states of the

18


qbits and thus in binary logic. Of course all superpositions of the binary operations arecarried out. Quantum gates are:

CNOT: The controlled not gate is a two qbit quantum gate which represents the coun-terpart of a classical XOR gate. If the controlling qbit (or control qbit) is in thestate |1〉, the other qbit is inverted, else nothing is done at all.

qbit 1

qbit 2

UCNOT =

0 1 0 01 0 0 00 0 1 00 0 0 1

Figure 2.5.: The symbol for the CNOT gate in a circuit and the corresponding transfor-mation matrix UCNOT .

SWAP: The SWAP gate is a two qbit quantum gate, too. It swaps the states of twoqbits with each other. It is the closest thing to a copy operation you can get,because copy would violate the reversibility of a quantum circuit.

qbit 1

qbit 2USWAP =

1 0 0 00 0 1 00 1 0 00 0 0 1

Figure 2.6.: The symbol for the SWAP gate in a circuit and the corresponding trans-formation matrix USWAP.

19


CSWAP: The CSWAP gate is a three qbit quantum gate where the central qbit controlsa swap process of the outer ones. If qbit 2 is in the state |0〉 qbit 1 is swappedwith qbit 3, else no operation is performed on the system.

qbit 3

qbit 2

qbit 1

UCSWAP =

1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 0 0 0 1 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 1 0 0 0 00 0 0 0 0 0 0 1

Figure 2.7.: The symbol for the CSWAP gate in a circuit and the corresponding trans-formation matrix UCSWAP .

There are many more quatum gates, e.g. the Toffoli gate, which will not be used inthis thesis and therefore are not introduced here. Almost every quantum gate with twoor more inputs can be called universal [27].

20

3. Heat and Work

The formulation of classical thermodynamics was one of the most important achieve-ments of the 19th century, as it allowed to investigate a large variety of phenomena,including the operation of thermodynamical machines. The first law of thermodynam-ics,

dU = dW + dQ, (3.1)

combined with definitions for the infinitesimal change in work dW and heat dQ is to-gether with the second law sufficient for computing important quantities like the effi-ciency of a process.

In the quantum realm, the classification of work and heat is less clear. So far, it hasmainly been based on the change of the total energy expectation value

dU = dTr

Hρ

= Tr

ρdH + Hdρ

, (3.2)

defining the first term as dW and the second as dQ [1, 20, 21, 24]. However, such aclassification is problematic, as an external driving, described by a time-independentHamiltonian, may not lead to work done on the system, although such a driving mayeven induce inversion in the system. While in some cases this may be fixed by regardingonly processes, in which the driving fields are switched on and off, the microscopicinterpretation of Eq. (3.2) remains unsatisfactory.

Nevertheless, thermodynamic behavior may occur even in small (closed) quantumsystems [12]. In principle it should be possible to obtain dW and dQ even there. In thefollowing, we will present a definition that does not suffer from the problems above.

We will first discuss the effective local dynamics of a bipartite quantum system. Basedon what would be observed in experiment, we give a definition for the local energy. Wethen show that the change in local energy can always be split into one part that isassociated with a change of entropy and in one part, which is not. in analogy withclassical thermodynamics, the former is called “heat” and the latter is called “work”.Our definitions for the local heat and work will not only depend on local properties, buton properties of the whole system. We explicitly give formulas to calculate the non-localquantities, once the time evolution of the full system is known. Finally, some exampleswill be given.

21

3. Heat and Work

3.1. The Local Effective Measurement Basis Method

(LEMBAS) [42]

We consider an autonomous bipartite system described by the Hamiltonian

H = HA + HB + HAB, (3.3)

where HA acts only on the subsystem A and HB only on B, respectively. Contrary totypical open system scenatios there is not necessarily any asymmetry between A andB (in terms of size etc.). As a consequence we cannot expect a closed (if approximate)dynamical description of A or B separatly In agreement with the results from classicalthermodynamics we define the infinitesimal work dW performed on A as the change inits internal energy dUA that does not change its local von Neumann entropy, i.e.

dS = 0 ⇔ dW = dUA. (3.4)

The remainder is defined as the infinitesimal heat dQ.Let the dynamics of the subsystem A be given by the Liouville-von Neumann equation

∂

∂tρA = −i[HA + Heff, ρA] + Linc(ρ), (3.5)

where ρA is the reduced density operator of A, Heff is an effective Hamiltonian describingthe unitary dynamics induced by B and Linc is a superoperator describing incoherentprocesses. Since Linc is a function of the density operator of the full system ρ, Eqn. (3.5)is not necessarily a closed differential equation.

We now consider a hypothetical measurement of the local effective energy in A. Onecould imagine an experimentalist sweeping a laser over the whole spectrum and recordingthe absorption profile. However, depending on the angle and the polarisation of the laserbeam, a different spectrum will be observed. Thus the experiment introduces a localeffective measurement basis (LEMBAS). In the following we study how this concept canbe incorporated into the derivation of work and heat.

Therefore, let Heff be expandend on an operator basis Qi, which contains the Hamil-ton operator of subsystem A as one of the basis operators, say Q1. H

eff is then uniquelypartitioned as

Heff =

k∑

i=1

Tr

Q†iH

eff

Qi

︸︷︷︸

Heffcom

+

n∑

i=k+1

Tr

Q†iH

eff

Qi

︸︷︷︸

Heffncom

, (3.6)

where we assume the operators Qi, i = 1 . . . k are all Qi which commute with HA (k isthe Hilbert space dimension). Therefore, we have

[Heffcom, HA] = 0, [Heff

ncom, HA] 6= 0 (3.7)

except for the case where Heffncom = 0.

22

3.2. Derivation of dW and dQ

3.2. Derivation of dW and dQ

If a measurement of the local energy is performed in the energy Eigenbasis of HA, thecorresponding operator is

H ′ = HA + Heffcom. (3.8)

Therefore the change in internal energy within A is given by

dUA =d

dtTr

H ′ ρA

dt = Tr

˙H ′ ρA + H ′ ˙ρA

dt. (3.9)

Using (3.5) and HA being time-independent leads to

dUA = Tr

˙Heff

com ρA − i[H ′, Heffncom] ρA + H ′

Linc(ρ)

dt, (3.10)

where the cyclicity of the trace has been used. Observing that the dynamics induced bythe first two terms are unitary, we arrive at

dWA = Tr

˙Heff

com ρA − i[H ′, Heffncom] ρA

dt (3.11)

dQA = Tr

H ′Linc(ρ)

dt. (3.12)

Therefore, it is possible to define heat and work for any quantum mechanical process,regardless of the type of dynamics or the states involved. Needed, however, is here thestate of the total bipartide system ρ, not just ρA.

Rather than considering A we might also study B under the influence of A. These twoperspectives, though, are mutually exclusive, as they refer to different partitions (Theinteraction is considered to be part of the observed system in each case).

3.2.1. Derivation of Heff

In order to actually compute dW and dQ the effective Hamiltonian Heff is required. Bystarting with the Liouville-von Neumann equation for the full system

∂

∂tρ = −i[H, ρ] (3.13)

and taking the partial trace over B (see e.g. [5]) yields

∂

∂tρA = TrB

[HA + HB + HAB, ρ]

. (3.14)

Using some theorems on partial traces, given in appendix A.1 on page 47, shows thatterms involving HB vanish and HA generates the local dynamics in A. For dealing withthe terms involving HAB we first split the density operator

ρ = ρA ⊗ ρB + CAB, (3.15)

23

3. Heat and Work

where ρA,B are the reduced density matrices for A and B respectively, and CAB isthe operator describing the correlations between both subsystems. Since the first termrepresents a factorizing density matrix the factorization approximation is fulfilled bydefinition and we can write (cf. [11])

TrB

[HAB, ρ⊗ ρB]

= [Heff, ρ], (3.16)

where Heff is given by

Heff = TrB

HAB (1A ⊗ ρB)

. (3.17)

We will now show that the processes generated by [HAB, CAB] cannot result in unitarydynamics, but will always change the local von Neumann entropy SA. In order to provethis, we compute its time derivative

SA = −Tr

[HAB, CAB] log ρA ⊗ 1B

. (3.18)

Therefore, any dynamics generated by this term cannot be unitary, but will result in acontribution to Linc. If the dynamics of the full system are unitary, we get

Linc = −iTrB

[HAB, CAB]

. (3.19)

3.2.2. Temperature

An open question remains how these new definitions of heat and work are linked withthe common usage e.g. [1,20,21,24]. It can easily be seen that in the limit when the in-teraction strength between both systems goes to zero, the effective dynamics will vanish.Only for a time dependent Hamiltonian the system is able to do work which correspondsto the definition given in (3.2).

A more interesting point is the local temperature of, say, system A. From Gibb’sfundamental relation it is known that

dS =1

TdQ. (3.20)

Using now the definition given for the heat in (3.12) one gets

dSA =1

T ∗Tr

H ′A Linc(ρ)

dt. (3.21)

T ∗ should indicate a parameter associated with the local temperature. Note that upto now this has not to be equivalent to the thermodynamic temperature. On the otherhand we know the derivation of the entropy SA from (3.18), combined with (3.21) gives

−Tr Linc(ρ) log ρA =1

T ∗Tr

H ′A Linc(ρ)

T ∗ =Tr Linc(ρ) log ρA

Tr

H ′A Linc(ρ)

. (3.22)

24

3.3. Examples

When dealing with canonical states, T ∗ could explicitly be determined by (3.22) becausethen H ′

A commutes with ρA. T ∗ is given in the energy basis of H ′A, the local measure-

ment basis. T ∗ can deviate from the global temperature of the complete system due tointeraction between the individual systems, inducing correlations [16, 17].

3.3. Examples

3.3.1. Laser with Detuning

Using the LEMBAS principle it is now possible to investigate work and heat in concretephysical scenarios. First we consider a two-level atom with a local Hamiltonian HA

driven by a laser field V . In the semiclassical treatment of the radiation field emittedby a laser the total Hamiltonian is given by

H = HA + V =∆E

2σz + g sinω t σx, (3.23)

where g is the coupling strength and ω the laser frequency. In the rotating wave approx-imation the Hamiltonian can be made time-independent. We investigate the situationwhere the atom is initially in a thermal state described by the density operator

ρ(0) = Z−1e−β HA, (3.24)

with Z being the partition function and β being the inverse temperature.Since (3.23) is already an effective description we can directly compute dW and dQ

resulting in

dW =∆E g2

2 Ωtanh

β∆E

2sin Ω t (3.25)

dQ = 0, (3.26)

where Ω =√

g2 + δ2 is the Rabi frequency and δ = ω−∆E/~ is the detuning from theresonance frequency. For comparison, using (3.2) leads to

dW =(∆E + δ) g2

2 Ωtanh

β∆E

2sin Ω t. (3.27)

Since the maximum of this expression is not at the resonance frequency (i.e. δ = 0), thisresult is unphysical.

3.3.2. Quantum Gates

In this Section we discuss the efficiency of the SWAP and CNOT gate as an example forthe application of the LEMBAS principle.

25

3. Heat and Work

|0〉

|0〉

|1〉

|1〉

|00〉

|01〉

|10〉

|11〉

⇒

qbit A qbit B entire system

Figure 3.1.: Scheme of the SWAP gate and the accounting π-pulse.

To compute work W and heat Q of a quantum Gate one has to refer to a specificrealisation of the quantum gate. One is thus led, e.g. to a series of π-pulses betweenthe pertinent levels of the system, in case of a SWAP gate the |01〉 and the |10〉 level(Fig.3.1). The Hamiltonian in the rotating wave approximation reads

H(t) = Hl + Hint(t) =4∑

i=1

Ei Pii +1

2g(

ei (EB−E3) tP23 + e−i (EB−E3) tP32

)

(3.28)

with∑4

i=1Ei Pii =∑

µ=1,2 ∆Eµ σzµ representing the spectrum of the two uncoupled spins.

To solve the equations of motion of this Hamiltonian we use the unitary transformation

Urot = ei (EB−E3) t P22 (3.29)

in the rotating basis to get rid of the time dependence of the Hamiltonian [28]

Hrot = Urot(t) H(t) U−1rot (t). (3.30)

Thus we are able to calculate the infinitesimal change of work dW delivered to thesystem by (3.11) and get

dW =g(eβi ∆EA − eβi ∆EB

) (∆EA − ∆EB

)

2 (1 + eβi ∆EA) (1 + eβi ∆EB)sin (g t). (3.31)

To get the work ∆W done for a SWAP we have to integrate over half a period of theRabi oscillation

∆W =

πg∫

0

dW dt =

(1

1 + eβi ∆EB−

1

1 + eβi ∆EA

)

(∆EA − ∆EB) (3.32)

This is the same result as if calculated via (3.2) because there is no detuning. In order tocompute the heat ∆Q transferred from spin 1 to spin 2 we use (3.17) to find the effective

26

3.3. Examples

Hamiltonian

Heff = − σ0

[−1 + eβi (∆EA+∆EB)

(1 + eβi ∆EA) (1 + eβi ∆EB)∆EB

+

(eβi ∆EB − eβi ∆EA

)

(1 + eβi ∆EA) (1 + eβi ∆EB)∆EB cos (gt)

]

+ σz ∆EA. (3.33)

Due to the fact that σ0 commutes with σz we find Heff = H ′ and

d

dtρA(t) =

d

dtTrB ρ(t) = Linc, (3.34)

from (3.12) follows

dQ = −g(eβi ∆EA − eβi ∆EB

)∆EA

2 (1 + eβi ∆EA) (1 + eβi ∆EB)sin (g t) (3.35)

and thus for the whole heat transferred

∆Q =

πg∫

0

dQ dt =

(1

1 + eβi ∆EB−

1

1 + eβi ∆EA

)

(∆EA) (3.36)

Putting all terms together the SWAP gate has the engine efficiency

ηSWAP = −∆Q

∆W=

∆EA

∆EA − ∆EB

, (3.37)

which is the same as the quantum Otto process in the heat pump mode [20].Due to the fact that there are no complications from the LEMBAS principle, we are

able to calculate work and heat for the quantum gates in a much easier way. The changeof the energy expectation value of qbit 2 represents the heat

∆Q = ∆〈E〉2 = Tr

H2 ρf

− Tr

H2 ρi

(3.38)

transferred by the algorithm, with H2 = ∆E2 σ0 ⊗ σz, ρi and ρf representing the densityoperator of the system before and after the application of the gate. To compute thework W in the system, one has to look at qbit 1. The energy difference is given by

∆〈E〉1 = Tr

H1 ρf

− Tr

H1 ρi

(3.39)

with H1 = ∆E1 σz ⊗ σ0. If no work was done on the system, the change of the energyexpectation value of this subsystem, given by Eq. (3.39), would be equal to the heat

27

3. Heat and Work

|0〉

|0〉

|1〉

|1〉

|00〉

|01〉

|10〉

|11〉

⇒

qbit A qbit B

controlled

entire system

Figure 3.2.: Scheme of the CNOT gate and the accounting π-pulse.

(3.38) with opposite sign. Energy would only be moved around within the system.Supposing that |∆〈E〉1|− |∆〈E〉2| was less than zero, work would be extracted from thesystem. If it was greater than zero work would be done on the system. In case that∆〈E〉1 and ∆〈E〉2 had different signs, |∆〈E〉1| − |∆〈E(n)〉2| would be equivalent to thechange of the energy expectation value of the entire system,

∆W =∆〈E〉 = Tr

H ρf

− Tr

H ρi

=Tr

(H1 + H2) ρf

− Tr

(H1 + H2) ρi

=Tr

H1 ρf

− Tr

H1 ρi

︸︷︷︸

∆〈E〉1

+ Tr

H2 ρf

− Tr

H2 ρi

︸︷︷︸

∆〈E〉2

. (3.40)

Now we are able to calculate the efficiency η of the SWAP gate by η = −∆Q/∆W andobtain the same result as in (3.37).

We will treat the CNOT gate inanalogy with the SWAP gate. The necessary π-pulse isdepicted in Fig. 3.2, but the energy difference between |10〉 and |11〉 is always equivalentto the |10〉-|11〉 difference. That brings up the problem to apply the pulse selectively tothis single transition. In practice the so called minimal pulse is substituted by a seriesof real pulses, that has the same effect on the system as the selected pulse depictedabove. Finding those pulse sequences is not unique and can not be more efficient thanthe minimal pulse. Therefore we use the latter to calculate the efficiency.

We start again with two qbits A and B with different Zeemann splittings but with thesame temperature, and the Hamiltonian

H =∑

µ=1,2

∆Eµ σz(µ). (3.41)

Now we use (3.39)and (3.40) to calculate the work and heat transferred by the gate andget

∆W = ∆Q =eβ ∆E1

(eβ ∆E2 − 1

)∆E2

2 (1 + eβ ∆E1) (1 + eβ ∆E1). (3.42)

28

3.3. Examples

This leads to the efficiency

ηCNOT = −∆Q

∆W= −1. (3.43)

This negative value is due to the fact that the CNOT gate leads to negative temper-atures of qbit 2, for finite initial temperatures. In terms of quantum computing justthe increased polarisation is relevant, therefor we can neglect the sign of ηCNOT whencomparing to other gates as the SWAP gate.

The efficiencies of the quantum gates can not easily be combined to the efficiency ofa complete algorithm because, in general, they depend on the initial occupation proba-bilities of the qbits.

29

3. Heat and Work

30

4. Algorithmic cooling

4.1. Introduction

Algorithmic cooling is a method to achieve highly polarised spins without cooling oftheir environment. This concept emerged from the backround of quantum informationprocessing and was developed to prepare the initial state of a set of qbits by applying aseries of quantum gates. The idea is that the computer could prepare its highly polarisedinitial state by its own means without an external cooling device. One has to distinguishhere between two kinds of algorithmic cooling: First, the so called closed algorithm, which makes use only of a closed set of qbits.

Due to the unitarity of the dynamics of closed systems the entropy S of the systemis conserved and so the algorithm is basically only capable of moving the entropyaround in the system and into coherences between the qbits. Second, the so called heat bath algorithmic cooling, which uses, at least temporally,local coupling to a heat bath. The algorithm is not entropy-conserving anymoreand thus significantly lower temperatures are reachable. The problem is to exper-imentally achieve the heat bath contact in such a way that there is no effect onthe already cooled down qbits.

One possible algorithm was proposed by Boykin et al. in 2002 [4] (see section 4.2 onpage 32) and experimentally realised in 2005 by Baugh [3]. Here we first discuss thisalgorithm with emphasis on its thermodynamic properties. Then an extension of thealgorithm is presented, which reduces the number of required qbits by making it cyclical.Finally we will examine of the best possible algorithm within this framework.

31


qbit 3

qbit 2

qbit 1

CSWAPCNOT

Figure 4.1.: Quantum circuit of the original closed algorithm by Boykin et al..

4.2. The Original Algorithm

4.2.1. The Basic, Closed Proposal

In 2002 Boykin et al. [4] proposed a cooling algorithm which applies two quantum gateson a set of tree qbits of equal inverse temperature β(0) in order to cool qbit 3 down bytransferring energy to the two other qbits. First a CNOT gate is performed betweenqbit 1 and 2, followed by a CSWAP, as depicted in Fig. 4.1 (For a description of thegates see 2.6.1 on page 18 or [31]). It is not intuitively clear how this procedure couldachieve the goal of cooling. One has to take a closer look at the effects the gates haveon the system. To study this we start with a product state of the 3 qbits with the sameinverse temperature β(0) and same Zeemann splitting, i.e. with the Hamiltonian

H =

3∑

µ=1

∆Eσz(µ). (4.1)

and

ρµ(0) =1

2σ0 +

εi(0)

2σz (4.2)

with εi(0) =1 − e−β(0) ∆E

1 + e−β(0) ∆E

ρ(0) =ρ1(0) ⊗ ρ2(0) ⊗ ρ3(0) (4.3)

Now the transformation

U = UCSWAP σ0 ⊗ UCNOT (4.4)

representing the algorithm is applied to the system

ρ(1) = U ρ(0) U−1. (4.5)

32

4.3. Cyclic Extended Algorithm

Then the qbits 2 and 3 are traced out. This leaves us with

ρ1(1) =

1+3eβ(0)∆E

(1+eβ(0)∆E)30

0e2 β(0) ∆E(3+eβ(0) ∆E)

(1+eβ(0)∆E)3

, (4.6)

a canonical state of different temperature

β(1) = 2 β(0) −ln

3 − 83+eβ(0) ∆E

∆E. (4.7)

Measured in units of β(0) within the limit of small β(0) follows

limβ(0)→0

β(1)

β(0)= 1.5. (4.8)

The application of the algorithm thus results in the inverse temperature β(1) ≈ 3/2 β(0)of spin 3 within the limit of high temperatures.

Having applied the algorithm once, the initial state is recovered by two further appli-cations of the algorithm. However, by cooling down two other qbits by applying the samealgorithm as described above to two additional sets of 3 qbits, allows a second applica-tion of the algorithm to the cooled qbit triple with reduced initial inverse temperatureβ(1). One qbit could be cooled down to the total of 9/4 β(0). By adding additional qbitsarbitrary low temperatures are in principle achievable. One needs 3N qbits to performN cooling steps.

4.2.2. Heat Bath Algorithmic Cooling

In the same paper [4] as the basic algorithm (see Sec. 4.2 on page 32) Boykin introducedan improvement to the original algorithm by using rapidly thermalising qbits. Theseqbits could be brought in contact with a heat bath and thus, after an infinite waitingtime, be reset to their initial state. The algorithm is implemented in the same way asthe basic one described above. Then the qbits 1 and 2 are reset to their initial stateand thus used to cool a second and, after another bath contact, a third qbit and so on.This way one needs 2 + 3N−1 qbits to perform N cooling steps as above. As we haveseen now, heat bath algorithmic cooling needs only 1/3 of the qbits needed for the basicalgorithm, but nevertheless it has an exponential growth in qbits. At the current stateof quantum computing that seems not viable.


Instead of using an infinite number of qbits to reach an arbitrary low temperature itwould be highly desirable to have an algorithm which reaches at least some lower tem-perature without using more qbits. As will be shown below, this is obtained by starting

33


qbit 3

qbit 2

qbit 1

SWAP CSWAPCNOT bath

β(0)

β(0)

Figure 4.2.: Cyclic cooling algorithm: 3 quantum gates are applied. First a SWAP gatethen a CNOT gate and finally a CSWAP gate. Bath contact at inversetemperature β(0) is symbolized by the boxes on qbit 2 and 3.

with a SWAP gate between qbit 1 and 3. After the SWAP again a CNOT and CSWAPgate is applied to the system in the same manner as in Boykin’s algorithm. Now onehas to wait until the auxiliary qbits 1 and 2 are relaxed back to the initial temperatureby the coupling to a heat bath (see Fig. 4.2). This can intuitively be understood byunfolding the algorithm. Removing the bathcontact and SWAP gate and instead addingtwo additional auxiliary qbits, the previous step of the algorithm is performed on cool-ing qbit 1 and so on. This way one generates a chain in which each application of thealgorithm precooles the “qbit 1” of the next step. For the bath contact step two classeshave been investigated: Total relaxation to the bath temperature (infinite contact time)and a finite coupling time τ to the bath. That means coherences in the system are nottotally damped and the initial occupation probabilities are not entirely regained.

This new algorithm can be applied for an arbitrary number of times to the same setof qbits. We call this algorithm cyclical, despite the fact that after one application thesystem does not return to its initial state, but is further cooled down. In principle, thecomplete process could be seen in the context of thermodynamical machines, i.e. re-frigerators, cooling down a finite “environment” (here only the single spin 1). The mostinteresting question, as in case of a refrigerator, is of course the final inverse temperatureβ(n) of the first qbit after the nth application of the algorithm in dependence of systemparameters and relaxation times between subsequent application steps.

On the bottom line a SWAP gate is added between the cooled qbit and one of theauxiliary qbits at the beginning of the algorithm. Starting with one auxiliary at lowertemperature makes it possible to apply the algorithm cyclical. This scheme of“cyclifying”an algorithm can be applied to any heat bath cooling algorithm.

With this cyclic algorithm it is possible to cool down one half of the accessible qbits by

34


applying it in two different ways on sets of four qbits: First, qbit 1 is cooled using qbits 2and 3, then qbit 4 is cooled as well with the aid of qbits 2 and 3. In case of a NMR-typequantum computer or something similar, the two stages could each be applied to thewhole system simultaneously, thus achieving a rather fast cooling of the system. If onewas capable of running quantum gates on any combination of qbits, all but two qbits ofthe system could be cooled down, which leaves more of the register for actual computing.In 2007 a similar algorithm introduced by Schulman [36] was experimentally realised byRyan [34].

4.3.1. Detailed Investigation of the Algorithm

As with the original algorithm (see Sec. 4.2 on page 32), we start in a canonical productstate

ρ(0) =ρ1(0) ⊗ ρ2(0) ⊗ ρ3(0) (4.9)

ρµ(0) =1

2σ0 +

εi(0)

2σz (4.10)

with εi(0) =1 − e−β(0) ∆Eµ

1 + e−β(0) ∆Eµ,

where β(0) denotes the inverse bath temperature and ∆Eµ the Zeemann splitting of therespective spin. Note that we label all quantities belonging to the nth application of thealgorithm by (n), so the initial state and the temperature of the bath are labeled (0).The system Hamiltonian is

H =3∑

µ=1

Hµ =3∑

µ=1

∆Eµσz(µ). (4.11)

There is no interaction between the qbits, except the interaction introduced by theapplication of the quantum gates.

The quantum gates can be treated as before

Ualg =UCSWAP UCNOT USWAP, (4.12)

ρf(n) =Ualg ρi(n) U †alg. (4.13)

The bath contact cannot be represented by a unitary transformation. A quantum masterequation in Lindblad form [5, 20, 38, 43] has to be solved, instead (see also “2.1.4 TheQuantum Master Equation” page 7). Because the qbits are not coupled, they must bethermalised separately. The respective quantum master equation for qbit µ reads

˙ρµ = − i[Hµ, ρµ]

+W1→0(2 σ−ρµσ+ − ρµσ+σ− − σ+σ−ρµ)

+W0→1(2 σ+ρµσ− − ρµσ−σ+ − σ−σ+ρµ) (4.14)

=L ρµ, (4.15)

35


with the rates W1→0 = λ/(1+ (1/ε)) and W0→1 = λ/(1+ ε), ε = exp β(0) ∆E and thebath coupling strength λ.

In order to compute the density matrix ρ(n) for an arbitrary number of repetitionsof the algorithm, it would be desirable to apply a simple matrix transformation for thebath contact as well. Therefore one has to compute the algorithm in the Liouville Space(see “2.2 The Liouville Space” page 8). There we are able to write down a super operatorprojecting an arbitrary state on a solution of (4.15). Just like in Hilbert space the timeevolution operator, the formal solution of (4.14), the solution of (4.15), is given by

ρ(t, n+ 1) = eLtρf(n) = T(t)ρf(n) (4.16)

with the limitlimt→∞

T(t) = T (4.17)

defining the complete thermalisation super operator. Because the quantum gates merelyinterchange entries ρii of the density operator, they do not lead to coherences in thesystem. We start with a thermal state and thus can restrict ourselves to density operatorsas the initial states of the thermalisationsuperoperator. With this restriction (4.16) leadsto

T(τ) = |σ0)(σ0| + e−2τλ |σz)(σz| +(e−2τλ − 1

) εi(0) − 1

εi(0) + 1|σ0)(σz|. (4.18)

This superoperator represents the thermalisation process truncated after a time step τ .In order to write down the thermalising super operator for the whole system we utilize aLiouville product space with the basis |σi)⊗|σj)⊗|σk) = |σijk) with i, j, k ∈ 0, x, y, z.There the superoperator thermalising qbit 1 and 2 reads

T12(τ) = T(τ) ⊗ T(τ) ⊗ 1. (4.19)

The superoperator of the algorithm U has to be generated in the same product basisas T(τ). Therefore we insert a general density operator ρ with coefficients ρijk in thebasis σi ⊗ σj ⊗ σk = σijk into its expansion

ρ =∑

ijk

ρijk σijk (4.20)

to get a general density operator ρ in Hilbert space H. Now the algorithm is applied tothe system

ρ′ = U ρ U−1. (4.21)

This ρ′ can be interpreted as the left hand side of a system of linear equations in theLiouville space L

ρ′ ⇒ U |ρijk) (4.22)

One whole application of the algorithm now reads B = T12(τ) U. To compute the nth

application of the algorithm B has to be taken to its nth power

|ρ(n)) = Bn(τ)|ρ(0)), (4.23)

36


2.0

1.5

1.6

1.7

1.8

1.9

2.1

0 10 20

β(n)β(0)

n

Figure 4.3.: β(n) as a function of cycles n for infinite bath contact time and constantZeemann splitting ∆E1 = ∆E2 = ∆E3 = ∆E. The asymptotic temperatureis approximately reached after seven steps.

using the inital state of the algorithm |ρ(0)). This result is easily transformed fromLiouville space L back into Hilbert space H by inserting |ρ(n)) into the expansion 4.20

ρ =∑

ijk

|ρijk) σijk. (4.24)

4.3.2. Final Temperatures of the Cyclic Algorithm

For infinite bath contact time (T(τ) ⇒ T) we are able to diagonalise B analytically andthus calculate ρ(n). We find the density operator of qbit 3 by calculating the partialtrace over qbit 1 and 2 and accordingly the final inverse temperature β(∞) of spin 3(depicted in Fig. 4.3)

β(∞) =∆E2 + ∆E3

∆E1

β(0). (4.25)

For constant Zeemann splitting ∆E1 = ∆E2 = ∆E3 = ∆E this results into β(∞) =2 β(0), significantly lower than (4.8) with 3 spins to its disposal.

Unfortunately, for the truncated relaxation, i.e. a finite relaxation time τ , a completeanalytic solution is not available, hence we investigate the algorithm numerically (with∆E1 = ∆E2 = ∆E3 = 1 and β(0) = 1).

For short bath contact time τ the inverse temperature β(n) of the cooled spin is shownin Fig. 4.4. For long bath contact it evolves like Fig. 4.5. Apparently, in Fig. 4.5 the

37


1

1.1

1.2

1.3

1.4

1.5

1.6

0 100 200 300

β(n)β(0)

n

Figure 4.4.: Trend of β(n) as a function of cycles n for short bath contact time (τ = 1).The three curves together describe the temperature evolution of qbit 1: β(n)jumps from the upper curve to the lowest, then to the middle one and upagain.

final stationary state is reached quickly after less then ten applications of the algorithm.This is not the case for very short relaxation times, as can be seen in Fig. 4.4.

1.5

1.6

1.7

1.8

1.9

2

0 10 20

β(n)β(0)

n

Figure 4.5.: β(n) as a function of cycles n for long bath contact time (τ = 200). Herethe final temperature is reached approximately after a few steps.

38


For the chosen set of parameters the application of the quantum gate itself needs asingle time-unit. Let us measure the bath contact time τ in this time-unit as well. Aftereach application of the quantum gate we wait exactly the same time τ to let qbit 2 and3 relax towards the equilibrium temperature of the bath. In Fig. 4.6 we show the finalinverse temperature of qbit 1 after n = 300 applications of the full algorithm (gates plusbath coupling) in dependence of the bath coupling time τ . Note that even if we havecomputed three hundred applications of the algorithm for each different waiting time τ ,the final temperature could already be reached after a couple of applications. This isespecially the case for τ ≫ 32 as can be seen in Fig. 4.5.

1.3

1.5

2

0 100 200

β(300)β(0)

τ

Figure 4.6.: The final inverse temperature β(300) as a function of different bath contacttimes τ . The line at β(300)/β(0) = 2 represents the upper limit for infinitebath contact, the line at β(300)/β(0) = 1, 53 marks the inverse temperatureof qbit 1 after one application of the algorithm.

At τ ≈ 32 the final temperature is as low as in the closed Boykin algorithm. Note,that a single cycle of the new algorithm leads to the same temperature as the Boykinalgorithm. Waiting time τ ≈ 32 means that qbit 2 and 3 are cooled to 0.77 times theinverse bath temperature β(0) after the first step.

For τ > 32 the final temperature is lower than it would be with the Boykin algorithm(line in Fig. 4.6 at ≈ 3/2). The cooling of the algorithm is improved compared to asingle cycle. However, multiple application of the algorithm does not always lead to areduction of the final temperature of the cooled spin, at least if τ < 32. This can beseen in Fig. 4.4 as well, where the temperature after the first application is already lowerthan after 300 cycles.

39


4.3.3. Efficiency

As mentioned earlier, algorithmic cooling can be seen in the context of quantum thermo-dynamic machines, more precisely, as some kind of refrigerator cooling a finite reservoir(here only one spin, qbit 3) by pumping heat via a working gas (here the auxiliary qbits1 and 2) into a heat bath of higher temperature (realised by the bath contact). For athermodynamic machine the efficiency η is of high interest to determine its capabilities.For the heat pump it is defined as

η =−∆Q

∆W(4.26)

where ∆Q is the heat transfered into the hot bath and ∆W the work consumed by theprocess (also see Sec. 3 on page 21). We will have to determine work and heat of theprocess. Following the definition via the total differential of the energy expectationvalue (2.49)

d

dt〈E〉 = Tr

d

dtHρ

︸︷︷︸

dW

+ Tr

Hd

dtρ

︸︷︷︸

dQ

We acquire the same work as by applying the extended definition (3.12) and (3.11)

d

dt〈E〉 = Tr

˙Heff

1 ρ− i[H ′, Heff2 ]ρ

dt︸︷︷︸

dW

+ Tr

H ′Linc(ρ)

dt.︸︷︷︸

dQ

.

To calculate the transfered heat Q the whole pulse sequence has to be modeled in orderto obtain the results. This is done for the single gates in Sec. 3.3.2 page 25. We makeuse of the results presented there and take the intuitive approach. The transported heat∆Q is represented by the change of the local energy expectation value 〈E3(n)〉 of thecooled qbit 3

∆Q = 〈E3(n)〉 − 〈E3(n− 1)〉

=

(c1

︷︸︸︷

eβ(0) ∆E2 + eβ(0) ∆E1

(1 + eβ(0) ∆E2) (1 + eβ(0) ∆E1)

)n+1(eβ(0) ∆E3 − eβ(0) (∆E2+∆E1)

)

(1 + eβ(0) ∆E3) (eβ(0) ∆E2 + eβ(0) ∆E1)∆E3. (4.27)

The work ∆W is the change of the energy expectation value 〈E(n)〉 of the entire

40

4.4. An Ideal Algorithm

system with a rather lengthy result

∆W = 〈E(n)〉 − 〈E(n− 1)〉

=

(c2

︷︸︸︷

eβ(0) (∆E2+∆E1)(1 + eβ(0) ∆E3

) (−1 + eβ(0) ∆E2

) (1 + eβ(0) ∆E1

) )

(1 + eβ(0) ∆E3) (1 + eβ(0) ∆E2) (1 + eβ(0) ∆E1) (1 + eβ(0) (∆E2+∆E1))∆E2

+

(c3

︷︸︸︷

1 − tanh

β ∆E2

2

tanh

β ∆E1

2

2

)k(eβ ∆E3 − eβ (∆E2+∆E1)

)

(1 + eβ(0) ∆E3) (1 + eβ(0) ∆E2) (1 + eβ(0) ∆E1) (1 + eβ(0) (∆E2+∆E1))

×[(

1 + eβ ∆E2 eβ ∆E1)(∆E3 − ∆E1)

+(eβ ∆E2 − eβ ∆E1 + eβ ∆E2 eβ ∆E1−1

)∆E2

]

. (4.28)

Inserting (4.27) and (4.28) into (4.26) results in

η =P1(n) ∆E3

P13(n) (∆E3 − ∆E1) + P2(n) ∆E2(4.29)

with

P1(n) = cn1(eβ(0) ∆E3 − eβ(0) (∆E2+∆E1)

) (1 + eβ(0) (∆E2+∆E1)

)

P13(n) = cn3(eβ(0) ∆E3 − eβ(0) (∆E2+∆E1)

) (1 + eβ ∆E2 eβ ∆E1

)

P2(n) = c2 + cn3(eβ(0) ∆E3 − eβ(0) (∆E2+∆E1)

) (eβ ∆E2 − eβ ∆E1 + eβ ∆E2 eβ ∆E1 − 1

).

To further understand this result one has to look at the transitions which are needed tooperate the quantum gates (see Sec. 3.3.2 on page 25). The ∆E3 in the numerator of(4.29) is the Zeemann splitting of the cooled qbit while the energies in the denominatorare the energies of the applied transitions. The Pi(n) are the differences of the occupationprobabilities of the involved niveaus before the accounting gate is applied. The structureof (4.29) reminds us of the efficiency of the quantum Otto process [20]. Numerically itis always below the accounting Otto efficiency.


To achieve a better understanding of the cooling process we take a look at a simplersystem: Two uncoupled qbits with different Zeemann splitting (∆E1 < ∆E2 w.l.o.g.),as depicted in Fig. 4.7, at the same initial inverse temperature βi, with the Hamiltonian

H =∑

µ=1,2

∆Eµ σzµ (4.30)

41


|0〉

|0〉

|1〉

|1〉

|00〉

|01〉

|10〉

|11〉

⇒

qbit 1 qbit 2 entire system

Figure 4.7.: Scheme of the SWAP gate and the accounting π-pulse.

and the initial density operator ρi. Evidently, the lowest temperature qbit 1 can obtainwithout coupling to a heat bath, results by applying a SWAP gate (see Sec. 2.6.1 on page19). If coherences would remain in the system the local temperature of spin 1 would behigher because this situation could be represented by a partial SWAP gate.

The application of the SWAP transformation to the initial state

ρf = USWAP ρi U−1SWAP (4.31)

results in the final inverse temperature of qbit 1

βf =∆E2

∆E1

βi. (4.32)

Calculating the work and heat done on the system (as 3.3.2) we find

∆W =

(1

1 + eβi ∆E2−

1

1 + eβi ∆E1

)

(∆E1 − ∆E2) (4.33)

∆Q =

(1

1 + eβi ∆E2−

1

1 + eβi ∆E1

)

∆E1 (4.34)

This results in an engine efficiency of

η = −∆Q

∆W=

∆E1

∆E2 − ∆E1. (4.35)

This is exactly the efficiency of the quantum Otto process ( [19] or Sec. 2.4.1 on page 14).One can state that the application of a SWAP gate is equivalent to the adiabatic step ofthe quantum Otto process. If one adds two isochoric bath coupling steps, the repeatedapplication of SWAP gates would fully resemble an Otto cycle. Although, if qbit 2was now coupled to a heat bath again and thus set back to its initial temperature, theoccupation probabilities of the two qbits would be identical. So the repeated application

42


|0〉

|1〉

|00〉

|01〉

|10〉

|11〉

|000〉

|001〉|010〉

|011〉

|100〉

|101〉|110〉

|111〉

⇒

qbit 3 axillary system entire system

Figure 4.8.: Portation of the swapping concept to a three spin system. One has to swapthe highest and the lowest energy level of the auxillary system with the qbitto be cooled (i.e. qbit 3). This results in a π-pulse on the levels of the wholesystem, depicted on the righthand side.

of the SWAP gate would not result in further cooling of qbit 1, i.e. the final temperaturewill be reached after one cycle.

It is most likely that more heat could be transferred by aprocedure resembling a Carnotcycle. But realising the isothermic steps requires very precise control over the Zeemannsplitting of the spins, because a TLS does not seem to perform isothermal expansion orcompression on its own. Then work has to be performed on the isothermal step.

We are now going to translate this idea of swapping to the original three qbit system.Therefore we consider it as a bipartite system, the cooled qbit 3 and the auxiliary system(qbits 1 and 2). In order to achive the best swapping effect the occupation probabilitiesof the hightest and lowest niveau of the auxiliary system have to be swapped with thetwo niveaus of qbit 3. As depicted in Fig. 4.8, this results in a π-pulse between the |001〉and |110〉 niveau. Now we compute the inverse temperature βf of qbit 3 after the SWAPoperation and get

βf =∆E1 + ∆E2

∆E3βi. (4.36)

This is the same temperature as β(∞) (Eq. (4.25)) of the cyclic algorithm. Again, bathcontact of the auxiliary system does not lead to further cooling of qbit 1. Nevertheless,adding the bath contact allows us to use the same set of auxiliary qbits again to cool another qbit in the sense of Sec. 4.2.2 (page 33).

This scheme can easily be extended to any number of qbits. Altough a mathematicalproof was not accessible, calculations for up to 6 qbits suggest that

βf =

n−1∑

i=1

∆Ei

∆Enβi (4.37)

43


should be the general formula for the final temperature. It always results in a lowertemperature than the heat bath cooling algorithm with the same number of qbits. Inorder to show this we first take the number n of qbits necessary to perform N coolingsteps with the heat bath algorithm n = 2 + 3N−1. Assuming that ∆Ei = ∆E ∀ ∆Ei,as in Boykins algorithm, this results in a final inverse temperature βSWAP

f = n− 1 βi for

the swapping algorithm while heat bath algorithmic cooling results in βBoykinf ≈

(32

)Nβi.

Subtracting the final inverse temperatures of the two algorithms from one another gives

(

n− 1 −

(3

2

)N)

β1 = 1 + 3N−1

(

1 −3

2N

)

︸︷︷︸

=

8

<

:

−12

for N = 1> 0 for N > 1

β1 > 0. (4.38)

The swapping type algorithm would always be better than the heat bath algorithm. Theefficiency of all checked systems with the swapping algorithm is of the form

η =∆En

n−1∑

i=1

∆Ei − ∆En

(4.39)

Again, this is the Otto efficiency as in the two spin case (4.35).

44

5. Summary and Perspectives

5.1. Summary

In this thesis algorithmic cooling was investigated with respect to its thermodynamicproperties. There are two different types of cooling algorithms: Closed algorithms, which only apply quantum gates and thus are limited by Shan-

non’s bound, the pendant tho the second law of thermodynamic in informationprocessing. Heat bath algorithmic cooling, where some qbits are coupled to a heat bath andthus the Entropy is moved out of the system.

We have started by studing the cooling algorithm proposed by Boykin et al. in 2002 [4],because both types of algorithms are featured in this proposal.

One of the first questions was, whether it would be possible to build a cyclic workingalgorithm for a better comparison with a refrigerator or the quantum thermodynamicheat pump proposed by Henrich [20]. We, indeed, found a possibility to run Boykinsalgorithm in a cyclic manner [33] and calculated its efficiency and lowest reachabletemperure analytically for infinite bath contact time and numerically for finite contacttimes. It became clear that there is a rather easy scheme to “cyclify” any given heatbath cooling algorithm by adding a SWAP gate to it.

I further investigated the limitations of algorithmic cooling by considering a simplieralgorithm than Boykin’s, i.e. by just swapping the occupation probabilities of two TLS.It became evident that this SWAP operation is the algorithm transporting maximumheat between the two qbits and that it cannot be further improved by cyclifying it.This swapping scheme can be extended to a multi qbit system, cooling a single qbitand may even be the best possibility to algorithmically cool this qbit. Swapping hasthe Otto efficiency. It may be possible to further improve the cooling by modelling thebath contact as an isothermic step. This would lead to Carnot efficiency, but would bebeyond the means of algorithmic cooling.

During the process of investigation we came up with the local effective measurementbasis principle (LEMBAS), a method to calculate local properties as an experimentalistwould probably measure them. This led straight to an extention of the definition ther-modynamic properties of work and heat for arbitrary bipartite quantum systems, whichled to more physically convincing results than the old one, e.g. for a Laser driving a TLS

45

5. Summary and Perspectives

with detuning. As we regard it now, LEMBAS has many other interesting implications,for example on heat transport phenomena, and so on.

5.2. Perspectives

Conceptionally, algorithmic cooling has been shown to be completely embedded in quan-tum thermodynamics. What remains, is to do some numerics for quantum gates withpermanent bath coupling and maybe expanding the concept to multi level systems. Onthe other hand, there is plenty of work to be done in conceptional quantum thermody-namics, which is still defined only for quasi classical systems with weak coupling andtherefore the energy as an extensive value. In real quantum systems with coherencesand finite interaction energies, the energy is not extensive any more, in inhomogeneoussystems it even does not follow a continuity equation. But as experiments show, they,nevertheless, obey thermodynamic rules. Therefore it would be the next step to extendthe definitions of quantum thermodynamic variables for coherent systems, as we did herefor work and heat.

46

A. Appendix

A.1. Trace theorems

Let H = HA ⊗HB be a bipartite Hilbert space, with operators A and B acting on HA

and HB, respectively, and an operator C acting on the full Hilbert space.

Theorem A.1.

TrA

[A⊗ 1, C]

= TrB

[1⊗ B, C]

= 0.

Proof. Due to symmetry it suffices to prove

TrA

[A⊗ 1, C]

= 0. (A.1)

Using an orthonormal basis Qi with each Qi acting only on HA or HB we can write

TrA

[A⊗ 1, C]

=∑

jk

cjk TrA

[A⊗ 1, Qj ⊗ Qk]

=∑

jk

cjk TrA

[A⊗ 1, Qj] ⊗ Qk

=∑

jk

cjk TrA

[A, Qj]

Qk

cycl= 0. (A.2)

Theorem A.2.

TrA

(A⊗ B)C

= BTrA

(A⊗ 1)C

(A.3)

TrB

(A⊗ B)C

= ATrB

(1⊗ B)C

. (A.4)

Proof. Again, for symmetry reasons we only need to prove the first part. Using the same

47

A. Appendix

basis operators as in the proof of theorem A.1 we have

TrA

(A⊗ B)C

=∑

jk

cjk TrA

(A⊗ B)(Qj ⊗ Qk)

=∑

jk

cjk TrA

(AQj) ⊗ (BQk)

= B∑

jk

cjk TrA

(AQj) ⊗ Qk

= B TrA

(A⊗ 1)C

. (A.5)

Corollary A.3. For B = 1 it immediately follows from theorem A.2

TrB

(A⊗ 1)C

= ATrB

C

. (A.6)

Corollary A.4. Another immediate consequence of theorem A.2 is

TrB

[

C, A⊗ B]

=[

TrB

C(1⊗ B)

, A]

. (A.7)

48

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52

Index

algorithmic cooling, 31

Born approximation, 8Boykin, 31

closed algorithmic cooling, 31closed Boykin algorithm, 32closed quantum system, 5CNOT gate, 19control bit, 19CSWAP gate, 20cyclic algorithm, 33cyclic algorithm analytic results, 37

density operator, 5density operator, time averaged, 10detuning, 25discrete Hilbert space, 3dissipator, 8

effective Hamiltonian, 23efficiency, classical Carnot engine, 12efficiency, classical Carnot heat pump, 13efficiency, classical Otto engine, 12efficiency, classical Otto heat pump, 13efficiency, CNOT, 29efficiency, cyclic algorithm, 40efficiency, quantum Otto engine, 16efficiency, SWAP, 27entropy, 9

final temperature, cyclic algorithm, 37

Gibbs fundamental relation, 11

heat bath algorithmic cooling, 31, 33

heat, new definition, 23, 40heat, old definition, 11, 12, 40

ideal algorithm, 41

laser, 25Lindblad form, 8, 35Liouville space, 8, 36Liouville von Neumann equation, 6lowering operator, 5

Markov assumption, 8master equation, 35model Hamiltonian, 7

NMR-type quantum computer, 35

operator representation, 3Otto-type cycle, 15

Pauli operators, 4pressure, 11

quantum computing, 17quantum gates, 18quantum master equation, 7quantum Otto cycle, 15quantum Otto process, 42

raising operator, 5rotating wave approximation, 8

Schrodinger equation, 5spectral temperature, 10SWAP gate, 19, 42

temperature, 9, 10

53

Index

thermalisation superoperator, 36thermodynamic variables, 9time evolution operator, 6, 36tracenorm, 4transition operator, 3truth table, 18

universal gate, 18

volume, 11von Neumann entropy, 9, 13

work, new definition, 23, 40work, old definition, 11, 12, 40

54

Acknowledgments

I would like to thank. . .

. . . Professor Mahler for letting me do my thesis in his group, having an open ear andalways asking the right questions.

. . . Professor Muramatsu for accepting the second report of this thesis.

. . . Professor Wunner for providing good working conditions at his institute.

. . .Mathias Michel for being a great roommate, teaching me how science works andalways taking time for my stupid questions, even when he had no time to spare.

. . . Valerio Scarani for encouraging me in what i am doing and letting me join in ahopefully fruitful collaboration.

. . .Markus Henrich, Heiko Schroeder, Hendrik Weimer, Alexander Kettler, Harry Schmidt,Petro Vidal, Thomas Jahnke, Georg Reuther and Jens Teifel from the Group of Mr.Mahler, especially the first four for really great discussions and Harry for solving all myLATEX and Mathematica problems.

. . . Ingo Schaefer and Alexander Deiss for spending lots of hours proofreading this thesis.

. . .My parents for making my studies possible in the first place and more importantbeing there for me.

Special thanks to . . .

. . .my roommates Alexander Deiss and Ralf Henschel for tolerating my bubbling aboutphysics, the parties with and without a reason and simply good friendship.

. . . Florian Hard the most amusing subjects at the coffee table and the best Cthulhugame i ever played.

. . . Hendrik Weimer, Heiko Schroeder and Alexander Kettler for fun at our fragging

55

Index

hours.

. . .my fencing crew for training with me and thus keeping me balanced.

. . . everybody i forgot to thank — sorry!

56

Algorithmic Cooling and Quantumthermodynamic Machines

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