Algorithmic Cooling and Quantumthermodynamic Machines

Diplomarbeit von Florian Rempp

July 20, 2007

Hauptberichter: Prof. Dr. Gunther¨ Mahler Mitberichter: Prof. Dr. Alejandro Muramatsu

1. Institut fur¨ Theoretische Physik Universit¨at Stuttgart Pfaﬀenwaldring 57, 70550 Stuttgart

Ehrenw¨ortliche Erkl¨arung

Ich erkl¨are, daß ich diese Arbeit selbst¨andig verfaßt und keine anderen als die angegebe- nen Quellen und Hilfsmittel benutzt habe.

Stuttgart, July 20, 2007 Florian Rempp

Contents

1. Abstract 1 1.1. Motivation ...... 1 1.2. Outline ...... 1

2. Quantum Mechanics 3 2.1. Operator Representations ...... 3 2.1.1. Transition Operator Representation ...... 3 2.1.2. Pauli Operators ...... 4 2.1.3. Density Operator Representation ...... 5 2.1.4. Dynamics ...... 5 2.2. The Liouville Space ...... 8 2.3. Quantum Thermodynamics ...... 9 2.3.1. Thermodynamic Variables ...... 9 2.3.2. Further Thermodynamic Variables ...... 11 2.3.3. Work and Heat ...... 11 2.3.4. Engine eﬃciencies ...... 12 2.4. Thermodynamic processes ...... 13 2.4.1. Quantum Thermodynamic Machines ...... 14 2.5. The Quantum Otto-cycle ...... 15 2.6. Quantum Computing ...... 17 2.6.1. Quantum Gates ...... 18

3. Heat and Work 21 3.1. The Local Eﬀective Measurement Basis Method (LEMBAS) [42] ..... 22 3.2. Derivation of dW and dQ ...... 23 3.2.1. Derivation of Hˆ eﬀ ...... 23 3.2.2. Temperature ...... 24 3.3. Examples ...... 25 3.3.1. Laser with Detuning ...... 25 3.3.2. Quantum Gates ...... 25

4. Algorithmic cooling 31 4.1. Introduction ...... 31 4.2. The Original Algorithm ...... 32

v Contents

4.2.1. The Basic, Closed Proposal ...... 32 4.2.2. Heat Bath Algorithmic Cooling ...... 33 4.3. Cyclic Extended Algorithm ...... 33 4.3.1. Detailed Investigation of the Algorithm ...... 35 4.3.2. Final Temperatures of the Cyclic Algorithm ...... 37 4.3.3. Eﬃciency ...... 40 4.4. An Ideal Algorithm ...... 41

5. Summary and Perspectives 45 5.1. Summary ...... 45 5.2. Perspectives ...... 46

A. Appendix 47 A.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 of quantum system theory [28]. It describes the emergence of thermodynamic behavior from quantum mechanics [12]. Most efforts so far concerned the definition of the thermodynamic 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 increase the polarisation of a Spin by unitarily moving entropy to an other spin system, do not easily fit into the framework of quantum thermodynamics. Although one has the feeling it should, because it reminds us strongly of a refrigerator. We will have to extend the concept of heat and work employing the principle of local energy measurement basis (LEMBAS) [42] to compute the efficiency of a cooling algorithm. LEMBAS implies an extension of the work and heat definition itself, which holds 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 in cycles as a real refrigerator. The possible algorithms so far include further steps only by increasing 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 of quantum thermodynamics along with a brief introduction to quantum information processing. Then the LEMBAS concept is introduced with the extension of the deﬁnition of heat and work in quantum thermodynamics. In the third chapter we will take a closer look on algorithmic cooling, especially the algorithm by Boykin et al. and its cyclic extension. This is completed by the introduction of an ideal algorithm (with respect to heat 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 formula for black body radiation and with the idea of fixed, finite energy quanta in which radiation is emitted from the black body, quantum mechanics has become one of the most successful theories of physics. Quantum mechanics have originally been invented to describe thermodynamic properties of nature. It is a valid question whether thermodynamics somehow emerge from quantum mechanics. Research in this section, the quantum thermodynamics developed in recent years. A good summary of the results so far is given in [12]. Of course it is far from complete, as the development concentrated on a quasi classic limt of quantum mechanics, by trying to conserve classical thermodynamic properties as the extensivity of energy, entropy etc.. Some amazing effects have been found here: quantum thermodynamic heat engines [19,20] and Fourier’s law in non equilibrium 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 thus say goodbye to extensivity, at least to some degree. In this Chapter basic concepts of quantum mechanics are presented as well as some quantum thermodynamic concepts in their classical definition. In case of work and heat the extension of the definition to arbitrary, bipartite quantum systems is given in Chap. 3 along with the LEMBAS concept.

2.1. Operator Representations

In order to describe a quantum mechanic system one can not rely on the classical phase space but has to use the eigenvalues of Hermitian operators. Their representation requires parameters, deﬁned according to an appropriate reference frame. These reference frames 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 described in the complete orthogonal state basis |ii with

hi| ji = δij with i, j =1, 2,...,k. (2.1)

3 2. Quantum Mechanics we can now deﬁne k2 transition operators

Pˆij = |iihj| (2.2) which in general are non-Hermitian but nevertheless orthonormal

† ˆ ˆ ′ ′ Tr PijPi′j′ = δii δjj (2.3) n o 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 Oˆ can be expanded into this basis

Oˆ = OijPˆij, (2.4) ij X where the Oij are the expansion coeﬃcients

ˆ ˆ† Oij = Tr OPij = hi| A |ji (2.5) n o 2 This makes a total of 2k parameters due to the fact, that coeﬃcients Oij in general are complex. In case of a Hermitian operator all Oij are real because

ˆ ˆ† ˆ ˆ† with O = O and Pij = Pij

Oˆ = OijPˆij (2.6) ij X ˆ† ∗ ˆ† O = OijPij (2.7) ij X∗ ⇒ Oij = Oij. (2.8)

2 So we need k 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 the SU(k) as a complete orthogonal basis of the Hilbert space. For k = 2 these are the Pauli operators σi with i = x, y, z, 0. In terms of transition operators the Pauli operators are

x σ =Pˆ12 − Pˆ21, (2.9) y σ =i (Pˆ21 − Pˆ12), (2.10) z σ =Pˆ11 − Pˆ22, (2.11) 0 σ =½2. (2.12)

4 2.1. Operator Representations

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

i 2 (σ ) = ½2, (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 its density matrix ρij based on the transition operators:

ρˆ = ρij Pˆij. (2.17) i,j=1 X In order to describe a real quantum system,ρ ˆ is subject to the condition

Tr {ρˆ} = ρii =1 (2.18) i X and has to be a positive deﬁnite Hermitian operator. The matrix elements can be projected out again with the aid of the statevectors

ρij = hi| ρˆ|ji . (2.19)

The expectation value of an arbitrary operator Oˆ for the stateρ ˆ is given by

Oˆ = Tr Oˆ ρˆ . (2.20) D E n o 2.1.4. Dynamics The unitary dynamics of a closed quantum system are given by the Schr¨odinger equation ∂ i ~ |ψ(t)i = Hˆ (t) |ψ(t)i . (2.21) ∂t In the so called Schr¨odinger picture the dynamics of the system are exclusively represented by the time dependence of the state vectors |ψ(t)i. Nevertheless, there may

5 2. Quantum Mechanics be explicitly time dependent external potentials resulting in time dependence of the Hamiltonian Hˆ (t). In order to ﬁnd the dynamics of the density operator we take its time derivative

∂ i ρˆ(t)= − Hˆ (t) ρ |ψ(t)ihψ(t)| − ρ |ψ(t)ihψ(t)| Hˆ (t) . (2.23) ∂t ~ i i i i X X ρˆ(t) ρˆ(t)

This leaves us with the Liouville| von{z Neumann} equation| {z }

∂ i ρˆ(t)= − Hˆ (t), ρˆ(t) . (2.24) ∂t ~ h i Alternatively, one may introduce a superoperator D(ˆρ) to describe the dynamics of open quantum systems ∂ i ρˆ(t)= − Hˆ (t), ρˆ(t) + D(ˆρ). (2.25) ∂t ~ h i Eq. (2.24) and (2.25) can also be written in superoperator representation

∂ ρˆ(t)= L ρˆ (2.26) ∂t 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

ρˆ(t)= Uˆ(t)ˆρ(0) Uˆ †(t), (2.27) where Uˆ obeys the Schr¨odinger equation

∂ i ~ Uˆ(t)= Hˆ (t) Uˆ(t). (2.28) ∂t

For ∂tHˆ = 0 the time evolution operator Uˆ(t) has the formal solution

Uˆ(t) = e−i Hˆ t/~ (2.29)

6 2.1. Operator Representations

Eq. (2.29) can be used also to define a so called model Hamiltonian Hîndmodel for the transformation generated by the quantum gate Uˆgate (see Sec. 2.6.1 on page 18) i ~ Hˆ = ln Uˆ . (2.30) model t gate n o These model Hamiltonians, however, do not represent a realistic quantum system, because they do not include local Zeemann splittings of the qbits, something that cannot easily 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 always be influenced by some kind of greatly uncontrollable environment, at least slightly. This means that every description of a real quantum system is necessarily incomplete and closed 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 so called 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 system and can therefore not be included into the Hamiltonian. A dissipator D(ˆρ) has to be added to the Liouville von Neumann equation (2.24) resulting in the so called quantum master equation (2.25). To compute the dissipator we first consider the whole closed system described by the density operatorρ ˆ, which is partitioned into

ρˆs = Trb {ρˆ} (2.31)

ρˆb = Trs {ρˆ} (2.32) with the indices s and b denoting system and bath, respectively. For the entire system the Liouville von Neumann equation (2.24) ∂ i ~ ρˆ(t)= Hˆ (t), ρˆ(t) ∂t h i holds. In analogy to the derivation of the classical master equation one assumes that the 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 ~ ρˆ (t)= fˆ(ˆρ , ρˆ ). (2.33) ∂t s s b Such an equation is usually not solvable. One has to create an autonomous form by applying specialisations, approximations and assumptions, neglecting most of the degrees of freedom. This has been attempted by many authors [5,35,38]. Standard ingredients are

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 are considered week and thus the two systems factorize. This approximation may lead to problems because for longer time periods even weak interaction may lead to strong 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 may lead to problems evaluating the resulting equation for longer time periods. The most general form, conserving the properties of the density operator, namely Hermiticity, positivity and Tr {ρˆ} = 1, was introduced by Lindblad [25]

(k )2−1 ∂ i 1 s ρˆ = − Hˆ , ρˆ + A 2 Lˆ ρˆ Lˆ† − ρˆ Lˆ† Lˆ − Lˆ† Lˆ ρˆ . (2.34) ∂t s ~ s s 2 ij i s j s j i j i s i,j=1 h i X The first term represents coherent dynamics, as (2.24). The second part is the incoherent influence of the environment on the system, defined by the environmental operators Lî. 2 These operators form, together with the ½ , a complete orthogonal basis of the (ks) dimensional Liouville space of the system. Aij is an Hermitian positive definite matrix of parameters. However, not all current master equations (e.g. [35]) are in Lindblad form and 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 derivation of 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 rotating wave approximation, the dissipator reads W D(ˆρ )= 1→0 (2 σ− ρˆ σ+ − ρˆ σ+ σ− − σ+ σ− ρˆ ) s 2 s s s W + 0→1 (2 σ+ ρˆ σ− − ρˆ σ− σ+ − σ− σ+ ρˆ ) (2.35) 2 s s s with the rates W1→0 and W0→1 chosen such that the system will end up in a thermal state with the same temperature as the bath. Because in this thesis only single spins are to be thermalised, this rather intuitive dissipator is feasible.

2.2. The Liouville Space

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

8 2.3. Quantum Thermodynamics deﬁned as (Aˆ|Bˆ) = Tr Aˆ†Bˆ , the trace norm of operators in H. Operators acting on states |Oˆ) in the Liouvillen spaceo are deﬁned as [12]

2.3. Quantum Thermodynamics

2.3.1. Thermodynamic Variables If one wants to investigate thermodynamic properties of quantum systems one has to come up with new “quantum” definitions of the pertinent thermodynamic quantities, such as entropy, temperature, energy, work and heat etc.. These quantities should approach their classical definitions, at least within some limits (normally weak coupling and an equilibrium situation are required). Whether they are still thermodynamic, if they are not restricted by these limitations, e. g. for nonequilibrium situations, is open for discussion. Let us assume that they are, at least near the accounting limits. In this section the definitions of thermodynamic variables will be given following [12]. Some of them, 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 detailed treatise can be found in [12]. Typically, when dealing with quantum systems the entropy used is the von Neumann entropy S given by

S = −Tr {ρˆln {ρˆ}} , (2.37) whereρ ˆ is the density operator of the considered system. It is invariant under unitary transformations. 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 to have 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 they will vanish in a time averaged density operator [10]. It can be shown that the time averaged entropy St rises if the oﬀ diagonal elements vanish due to the dynamics of the system. For small energy spacing (typically for large systems) the necessary averaging time rises and diverges for degenerate systems. The required averaging time 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) as the thermodynamic entropy. The temperature T of a system is given as the derivation of the entropy with respect to the mean energy E (for constant mechanical control V )

dS 1 T = = . (2.38) dE k β V B For a TLS in a diagonal state with P N 1 0 = e−β (E1−E0) (2.39) P0 N1 it follows that E − E T = − 1 0 . (2.40) (ln {P1} − ln {P0}) − (ln {N1} − ln {N0}) th Here Ei is the energy eigenvalue of the i level, Pi the respective occupation probability and Ni its degeneracy. This deﬁnition can easily be extended to the inverse spectral temperature of a multi level system by assigning temperatures to adjacent pairs of energy levels, over which an average is taken:

W + W −1 β := − 1 − 0 M s 2 M W + W − 1 ln {P } − lnP − (ln {N } − ln {N }) i i i i−1 i i−1 , (2.41) 2 E − E i=1 i i−1 X where Wi is the probability of ﬁnding the system at the energy Ei. This βs depends only on the spectra and the occupation probabilities. Thus it is always deﬁned, even for systems, which don’t show thermodynamic behaviour. For equilibrium situations however, the spectral temperature corresponds to the thermodynamic temperature. This can be proven by showing that in equilibrium the relation ∂S ∂ 1 ∂D(E) β = = ln {D(E)} = (2.42) s ∂E ∂E D(E) ∂E holds, with D(E) being the state density, using extensivity of the entropy and the fact that it approaches its maximum in equilibrium.

10 2.3. Quantum Thermodynamics

∆E(V ) ⇒ ∆E(V − ∆V )

∆V

Figure 2.1.: The volume of a quantum well, depicted here as the area of the 1D well, can directly 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 variable to V , the volume. The chemical potential µ and its conjugate variable the particle number N are left aside because they are not needed in this thesis. For a TLS, like a spin with the Hamiltonian Hˆ =σ ˆz ∆E/2, the following correspondences can be veriﬁed by considering the TLS as two energy levels in a quantum well (see Fig. 2.1)

∆E = ∆E(V ) (2.43) dU d ∆E(V ) P = − = σˆ . (2.44) dV dV 2 z S V can be associated with ∆E. We are thus able to call a bath contact without change of the Zeeman splitting isochoric. The justiﬁcation and a simple example is given in Sec. 2.4.1 on page 14. For a cyclic process the energy at the beginning and the end of the process should be the same, dU = 0. The path in the ST plane for one cycle is a closed one and therefore (2.49) leads to

∆W = −∆Q = − T dS. (2.45) I Knowing the heat current Jα, ﬂowing from or into a bath α during the cycle, it is possible to determine the exchanged heat over one period τ τ ∆Qα = Jαdt. (2.46) Z0 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 with the help of the Gibbs fundamental relation [6]

dU(S,V )=dW ¯ +dQ ¯ = −P dV + T dS, (2.47)

11 2. Quantum Mechanics by knowing the inner energy U. P , the pressure, is conjugate to the volume V . The temperature T is conjugate to S, the entropy. The inner energy U of quantum systems is given by the energy expectation value

hEi = Tr Hˆ ρˆ , (2.48) n o with the system Hamiltonian Hˆ and its stateρ ˆ. The change of the energy is given by d d d hEi = Tr Hˆ ρˆ + Tr Hˆ ρˆ . (2.49) dt dt dt dW/¯ dt dQ/¯ dt As indicated by the underbraces, the| ﬁrst{z term,} where| {z only th}e spectrum is changing, is associated with work d dW¯ = Tr Hˆ ρˆ dt = E˙ i pidt. (2.50) dt i X The second term of (2.49), where the state and therefore the entropy is changing, is interpreted as the exchanged heat d dQ¯ = Tr Hˆ ρˆ dt = Ei p˙idt. (2.51) dt i X 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 the absorbed heat from the hot bath, ∆Qh −∆W ηe = . (2.52) ∆Qh For a heat pump the efficiency reads −∆Q ηp = h . (2.53) ∆W For a Carnot engine (2.52) leads to

e Th ηCar =1 − , (2.54) Tc for the classical Otto engine one gets V γ ηe =1 − 1 , (2.55) Otto V 2

12 2.4. Thermodynamic processes

a) b) c)

work Th W Tc Qh Qc ∆Eh ∆Eh ∆Ec ∆Ec

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, while the state does not change. c) Isochoric step: As in a) the TLS exchanges heat 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 called the compression ration). γ is the polytropic exponent which depends on the considered gas. The respective eﬃciencies for the heat pump are

p Th ηCar = (2.56) Th − Tc for the Carnot-cycle and V γ ηp = 1 . (2.57) Otto V − V 1 2 for the Otto-cycle.

2.4. Thermodynamic processes

We will discuss only the relevant process types for this thesis, the adiabatic and the isochoric process. These are the steps which seem easiest to be realised in a quantum engine. We assume that the system is in a canonical state all the time, which is feasible because of the bath contact, which dephases much faster then it thermalises. Both steps are depicted in Fig. 2.2. During an adiabatic step [cf. Fig. 2.2 b)] the von Neumann entropy of the considered system has to stay constant. For a TLS the occupation probabilities remain constant and thus the density matrix does not change. No heat is exchanged. Only the volume V , which is directly mapped onto the Zeemann splitting ∆E of the system (2.43) will be increased or decreased. The work done or gained during this process is simply given by

13 2. Quantum Mechanics the diﬀerence of the energy expectation value at the beginning and the end of the step

∆E ∆E ∆E ∆W = hE i−hE i = tanh i i − f . (2.58) ↑/↓ f i 2T 2 2 i Because the state during an adiabatic step does not change and starts canonically, it is possible to determine the temperature of the TLS after an adiabatic step. This would not be possible for a multi level system, unless the ratio of the frequencies between the diﬀerent energy levels stays constant during the step.

1 e∆Eh/(2Th) 0 ρˆ = . (2.59) Z 0 e−∆Eh/(2Th) Then the adiabatic step as in b) is performed and therefore the temperature of the TLS is decreasing to

e−∆Eh/(2Th) e−∆Ec/(2Tf) = e∆Eh/(2Th) e∆Ec/(2Tf) ∆Ec Tf = Th . (2.60) ∆Eh

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 this step (see (2.49)) and only heat is exchanged with a heat bath. The entire change of the energy expectation value has to be heat Q and can be calculated as

∆E ∆E ∆E ∆Q = hE i−hE i = h tanh h − tanh h . (2.61) h f i 2 2T 2T h

If Tf = Tc at the end of the adiabatic step, no heat with the bath will be exchanged during the isochoric step. For Tf < Tc heat is extracted from the cold bath, the cycle can not work as a heat pump anymore. It is possible to deﬁne a critical temperature diﬀerence ∆E ∆T = T − T = T 1 − c (2.62) crit h c h ∆E h via (2.60), where the process will switch its function from heat pump (∆T < ∆Tcrit) to a heat engine (∆T > ∆Tcrit) and vise versa (see [19]). This behaviour is always valid for a 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 thermodynamic machine become?”.

14 2.5. The Quantum Otto-cycle

It is obvious that on such a small scale, now reachable for experiments, quantum mechanics plays an important role. Answering the question above is therefore linked with the possibility of finding thermodynamic cycles within the quantum regime. First attempts 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 realisation seems to be a machine constructed of two-level systems (TLS), like spins or qbits, as investigated for quantum information purposes. There are many different scenarios possible to control structures built with a few TLS. Examples are quantum optical systems [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 thermodynamic machine was introduced in [19, 20]. Two TLS act as a filter which control the bath contact of the TLS in the middle, working as a heat pump or heat engine, depending on the temperature gradient induced on the TLS-chain. In principle, this setup should pave the way for an experimental realization. Now that we know that thermodynamic machines are realisable, one has to think about what kind of thermodynamic process could possibly be established within the considered system. A promising candidate seems to be the Otto-type cycle, where the gas system is controlled by two adiabatic and two isochoric steps. In Sec. 2.4 on page 13 both steps have been be discussed. In Sec. 2.5 a complete Otto-cycle of a TLS is studied. The corresponding TS diagram as well as the PV diagram are presented. The efficiency is derived and we will discuss what will happen when this efficiency equals the Carnot efficiency — a point which cannot 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 remains constant and therefore no heat will be exchanged. At the isochoric steps the system is in 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 and the left section of Fig. 2.3 with the respective TS diagram we want to show that a TLS can perform Otto-cycles:

1. Isochoric step (A → B): The TLS is in contact with the hot bath at Th. The occupation probabilities change at ∆EA = ∆EB = ∆Eh until the temperature of the TLS equals Th. The exchanged heat Qh is deﬁned by (2.61) and reads ∆E ∆E ∆E ∆Q = hE i−hE i = h tanh c − tanh h , (2.63) h B A 2 2 T 2 T c h

15 2. Quantum Mechanics

P T D A 4 B 0.16

A 0.15 3 C 0.14 C B D 1.7 2 2.3 0.64 0.65 V S

Figure 2.3.: Left: PV diagram for a spin 1/2 performing an Otto-cycle. Right: TS diagram for a spin 1/2 performing an Otto-cycle. The following parameters have been chosen as an example: ∆Eh = 2.25, ∆Ec = 1.75, Th = 4 and Tc =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)

∆E ∆E − ∆E ∆W = hE i−hE i = tanh h h c . (2.64) BC C B 2 T 2 h is released.

3. Isochoric step (C → D): The TLS is in contact with the cold bath at Tc. The temperature of the spin at point C is T = T ∆Eh . The heat ∆Q (cf. (2.61)) C h ∆Ec c

∆E ∆E ∆E ∆Q = hE i−hE i = c tanh h − tanh c , (2.65) c D C 2 2 T 2 T h c

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)

∆E ∆E ∆E ∆W = hE i−hE i = tanh c c − h . (2.66) DA A D 2T 2 2 c

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

WBC + WDA ∆Eh ηqm = − =1 − . (2.67) Qh ∆Ec

16 2.6. Quantum Computing

p e η 50 0.5 η 0.08 pump engine pump engine p e η ∆Qc ηCar Car 0.04 ∆W 30 0.3 0

qp2 e ηqm ∆Qh p -0.04 10 ηqm 0.1 -0.08 0.1 0.5 0.9 1.3 0.1 0.5 0.9 1.3 tcrit2 ∆Tcrit ∆T ∆Tcrit ∆T

p e Figure 2.4.: Left: Carnot-efficiency ηCar for the heat pump and the heat engine ηCar as a function of temperature difference ∆T , while Tc = 2.5, ∆Eh = 2.25 and p e ∆Ec =1.75. ηqm and ηqm are the efficiencies of quantum Otto pump/engine. p ˜ η˜id = 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 and heat ∆Qc from/to the cold bath for the ideal machine as a function of p p temperature difference ∆T . At ∆T = ∆Tcrit, ηqm = ηCar 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 Otto cycle.

Finally we want to discuss the critical temperature ∆Tcrit [cf. (2.62)]. The left side of Fig. 2.4 shows the efficiency of a TLS Otto machine as a function of the difference of the bath temperatures ∆T and the respective Carnot efficiency. As can be seen, the efficiency of the TLS-machine is always less than the Carnot efficiency, except when ∆Tcrit is reached. At this point we get ηqm = ηCar. The right part 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 is running 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 artificially changing the moving direction in the ST -Diagram.

2.6. Quantum Computing

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

17 2. Quantum Mechanics

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

2.6.1. Quantum Gates In classical combinational logic a logical circuit is built of logical gates which can perform the operations AND, OR, XOR and NOT, where AND, OR and XOR combine the states of two bits to one and NOT inverts the state of a single bit. Obviously, all logical expressions can be composed of these gates. They may be translated into a combination of 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 combination of the universal gate. To transfer the concept of logical gates to quantum information processing, one has to take the quantum nature of those systems into account. Because operations on a closed quantum system (see Sec. 2.1.4 on Page 6) have to be unitary, they can be represented by a unitary transformation matrix U, acting on the density matrix ρi in a way that

−1 ρf = UρU (2.68) is the density matrix after the operation has been performed. This description is independent of the exact realization which would be described by a Liouville von Neumann equation (c.f. 2.1.4 page 5). In order to ﬁnd the corresponding transformation matrix for a certain quantum operation one starts with its truth table. Writing the input states and the output states as two vectors, their relationship can be formulated via a matrix As an easy example one ﬁnds for the CNOT gate

|11i|10i|01i|00i |11i→|10i |11i 0100 |10i→|11i |10i 1000 . (2.69) =⇒ |01i→|01i |01i 0010  |00i→|00i |00i 0001      If several of those operations Ui are carried out one after another, the whole sequence can be written as a single transformation U = Un ...U2U1. The separation in substeps is 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 gates one is able to depict an algorithm as a logical circuit, just like in digital electronics. For better comprehensibility the description of these gates is made by the pure states of the

18 2.6. Quantum Computing qbits and thus in binary logic. Of course all superpositions of the binary operations are carried 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 the state |1i, the other qbit is inverted, else nothing is done at all.

qbit 2 0100 1000 UCNOT =   0010 qbit 1  0001      Figure 2.5.: The symbol for the CNOT gate in a circuit and the corresponding transformation matrix UCNOT .

SWAP: The SWAP gate is a two qbit quantum gate, too. It swaps the states of two qbits 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.

1000 qbit 2 0010 U = SWAP  0100  qbit 1  0001      Figure 2.6.: The symbol for the SWAP gate in a circuit and the corresponding transformation matrix USWAP.

19 2. Quantum Mechanics

CSWAP: The CSWAP gate is a three qbit quantum gate where the central qbit controls a swap process of the outer ones. If qbit 2 is in the state |0i qbit 1 is swapped with qbit 3, else no operation is performed on the system. 10000000 qbit 3 01000000  00100000   00000010  U =   qbit 2 CSWAP  00001000     00000100     00010000  qbit 1    00000001      Figure 2.7.: The symbol for the CSWAP gate in a circuit and the corresponding transformation matrix UCSWAP .

There are many more quatum gates, e.g. the Toﬀoli gate, which will not be used in this thesis and therefore are not introduced here. Almost every quantum gate with two or 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 thermodynamics, dU =dW ¯ +dQ, ¯ (3.1) combined with definitions for the infinitesimal change in workdW ¯ and heatdQ ¯ is together with the second law sufficient for computing important quantities like the efficiency of a process. In the quantum realm, the classification of work and heat is less clear. So far, it has mainly been based on the change of the total energy expectation value

dU = dTr Hˆρˆ = Tr ρˆdHˆ + Hˆdˆρ , (3.2) n o n o defining the first term asdW ¯ and the second asdQ ¯ [1, 20, 21, 24]. However, such a classification is problematic, as an external driving, described by a time-independent Hamiltonian, may not lead to work done on the system, although such a driving may even induce inversion in the system. While in some cases this may be fixed by regarding only processes, in which the driving fields are switched on and off, the microscopic interpretation of Eq. (3.2) remains unsatisfactory. Nevertheless, thermodynamic behavior may occur even in small (closed) quantum systems [12]. In principle it should be possible to obtaindW ¯ anddQ ¯ even there. In the following, 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. Based on what would be observed in experiment, we give a definition for the local energy. We then show that the change in local energy can always be split into one part that is associated with a change of entropy and in one part, which is not. in analogy with classical 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, but on properties of the whole system. We explicitly give formulas to calculate the non-local quantities, once the time evolution of the full system is known. Finally, some examples will 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

Hˆ = HÂ + HˆB + HÂB, (3.3) where HÂ acts only on the subsystem A and HˆB only on B, respectively. Contrary to typical open system scenatios there is not necessarily any asymmetry between A and B (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 classical thermodynamics we define the infinitesimal workdW ¯ performed on A as the change in its 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 heatdQ ¯ . Let the dynamics of the subsystem A be given by the Liouville-von Neumann equation ∂ ρˆ = −i[Hˆ + Hˆ eff, ρˆ ]+ L (ˆρ), (3.5) ∂t A A A inc eff whereρ Â is the reduced density operator of A, Hˆ is an effective Hamiltonian describing the unitary dynamics induced by B and Linc is a superoperator describing incoherent processes. Since Linc is a function of the density operator of the full systemρ ˆ, Eqn. (3.5) is not necessarily a closed differential equation. We now consider a hypothetical measurement of the local effective energy in A. One could imagine an experimentalist sweeping a laser over the whole spectrum and recording the absorption profile. However, depending on the angle and the polarisation of the laser beam, a different spectrum will be observed. Thus the experiment introduces a local effective measurement basis (LEMBAS). In the following we study how this concept can be incorporated into the derivation of work and heat. eff Therefore, let Hˆ be expandend on an operator basis {Qî}, which contains the Hamil- eff ton operator of subsystem A as one of the basis operators, say Qˆ1. Hˆ is then uniquely partitioned as k n ˆ eff ˆ† ˆ eff ˆ ˆ† ˆ eff ˆ H = Tr Qi H Qi + Tr Qi H Qi, (3.6) i=1 X n o i=Xk+1 n o ˆ eff ˆ eff Hcom Hncom where we assume the operators| Qî{z, i = 1 ...k} are| all Qî {zwhich commute} with HÂ (k is the Hilbert space dimension). Therefore, we have ˆ eff ˆ ˆ eff ˆ [Hcom, HA]=0, [Hncom, HA] =06 (3.7) ˆ eff except for the case where Hncom = 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 HÂ, the corresponding operator is ˆ ′ ˆ ˆ eff H = HA + Hcom. (3.8) Therefore the change in internal energy within A is given by d ˙ dU = Tr Hˆ ′ ρˆ dt = Tr Hˆ ′ ρˆ + Hˆ ′ ρˆ˙ dt. (3.9) A dt A A A n o n o Using (3.5) and HÂ being time-independent leads to

ˆ˙ eff ˆ ′ ˆ eff ˆ ′ L dUA = Tr Hcom ρÂ − i[H , Hncom]ˆρA + H inc(ˆρ) dt, (3.10) n o where the cyclicity of the trace has been used. Observing that the dynamics induced by the first two terms are unitary, we arrive at

ˆ˙ eff ˆ′ ˆ eff dW¯ A = Tr Hcom ρÂ − i[H , Hncom]ˆρA dt (3.11)

n ′ o dQ¯ A = Tr Hˆ Linc(ˆρ) dt. (3.12) n o 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 the state of the total bipartide systemρ ˆ, not justρ Â. Rather than considering A we might also study B under the influence of A. These two perspectives, though, are mutually exclusive, as they refer to different partitions (The interaction is considered to be part of the observed system in each case).

3.2.1. Derivation of Hˆ eff In order to actually computedW ¯ anddQ ¯ the effective Hamiltonian Hˆ eff is required. By starting with the Liouville-von Neumann equation for the full system ∂ ρˆ = −i[H,ˆ ρˆ] (3.13) ∂t and taking the partial trace over B (see e.g. [5]) yields ∂ ρˆ = Tr [Hˆ + Hˆ + Hˆ , ρˆ] . (3.14) ∂t A B A B AB n o Using some theorems on partial traces, given in appendix A.1 on page 47, shows that terms involving HˆB vanish and HÂ generates the local dynamics in A. For dealing with the terms involving HÂB we first split the density operator

ρˆ =ρ ˆA ⊗ ρˆB + CˆAB, (3.15)

23 3. Heat and Work

whereρ Â,B are the reduced density matrices for A and B respectively, and CÂB is the operator describing the correlations between both subsystems. Since the first term represents a factorizing density matrix the factorization approximation is fulfilled by definition and we can write (cf. [11])

eff TrB [HÂB, ρˆ ⊗ ρˆB] = [Hˆ , ρˆ], (3.16) n o where Hˆ eff is given by ˆ eff ˆ H = TrB HAB (½A ⊗ ρˆB) . (3.17) n o We will now show that the processes generated by [HÂB, CÂB] cannot result in unitary dynamics, but will always change the local von Neumann entropy SA. In order to prove this, we compute its time derivative

S˙A = −Tr [HˆAB, CˆAB] logˆρA ⊗ ½B . (3.18) Therefore, any dynamics generatedn by this term cannot be unio tary, but will result in a contribution to Linc. If the dynamics of the full system are unitary, we get

Linc = −iTrB [HÂB, CÂB] . (3.19) n o 3.2.2. Temperature An open question remains how these new definitions of heat and work are linked with the common usage e.g. [1,20,21,24]. It can easily be seen that in the limit when the interaction 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 corresponds to the definition given in (3.2). A more interesting point is the local temperature of, say, system A. From Gibb’s fundamental relation it is known that 1 dS = dQ.¯ (3.20) T Using now the definition given for the heat in (3.12) one gets 1 dS = Tr Hˆ ′ L (ˆρ) dt. (3.21) A T ∗ A inc T ∗ should indicate a parameter associatedn with the localo temperature. Note that up to now this has not to be equivalent to the thermodynamic temperature. On the other hand we know the derivation of the entropy SA from (3.18), combined with (3.21) gives 1 −Tr {L (ˆρ) logˆρ } = Tr Hˆ ′ L (ˆρ) inc A T ∗ A inc n o Tr {L (ˆρ) logˆρ } T ∗ = inc A . (3.22) ˆ ′ L Tr HA inc(ˆρ) n o

24 3.3. Examples

When dealing with canonical states, T ∗ could explicitly be determined by (3.22) because ˆ ′ ∗ ˆ ′ then HA commutes withρ ˆA. T is given in the energy basis of HA, the local measurement basis. T ∗ can deviate from the global temperature of the complete system due to interaction 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 concrete physical scenarios. First we consider a two-level atom with a local Hamiltonian HÂ driven by a laser field Vˆ . In the semiclassical treatment of the radiation field emitted by a laser the total Hamiltonian is given by ∆E Hˆ = Hˆ + Vˆ = σˆ + g sin ω t σˆ , (3.23) A 2 z x where g is the coupling strength and ω the laser frequency. In the rotating wave approximation the Hamiltonian can be made time-independent. We investigate the situation where the atom is initially in a thermal state described by the density operator

ˆ ρˆ(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¯ anddQ ¯ resulting in

∆Eg2 β ∆E dW¯ = tanh sin Ω t (3.25) 2 Ω 2 dQ¯ = 0, (3.26) where Ω = g2 + δ2 is the Rabi frequency and δ = ω − ∆E/~ is the detuning from the resonance frequency. For comparison, using (3.2) leads to p (∆E + δ) g2 β ∆E dW¯ = tanh sin Ω t. (3.27) 2 Ω 2 Since the maximum of this expression is not at the resonance frequency (i.e. δ = 0), this result is unphysical.

3.3.2. Quantum Gates In this Section we discuss the eﬃciency of the SWAP and CNOT gate as an example for the application of the LEMBAS principle.

25 3. Heat and Work

|11i |1i |1i |10i ⇒ |01i |0i |0i |00i 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 speciﬁc realisation of the quantum gate. One is thus led, e.g. to a series of π-pulses between the pertinent levels of the system, in case of a SWAP gate the |01i and the |10i level (Fig.3.1). The Hamiltonian in the rotating wave approximation reads

4 1 Hˆ (t)= Hˆ + Hˆ (t)= E Pˆ + g ei (EB −E3) tPˆ + e−i (EB −E3) tPˆ (3.28) l int i ii 2 23 32 i=1 X 4 ˆ z with i=1 Ei Pii = µ=1,2 ∆Eµ σµ representing the spectrum of the two uncoupled spins. To solve the equations of motion of this Hamiltonian we use the unitary transformation P P

i (EB−E3) t Pˆ22 Uˆrot = e (3.29) in the rotating basis to get rid of the time dependence of the Hamiltonian [28]

ˆ ˆ ˆ ˆ −1 Hrot = Urot(t) H(t) Urot (t). (3.30)

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

βi ∆EA βi ∆EB g e − e ∆EA − ∆EB dW¯ = sin (g t). (3.31) βi ∆EA βi ∆EB 2 (1 + e )(1+e ) To get the work ∆W done for a SWAP we have to integrate over half a period of the Rabi oscillation

π g 1 1 ∆W = dW¯ dt = − (∆EA − ∆EB) (3.32) 1 + eβi ∆EB 1 + eβi ∆EA Z0

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

26 3.3. Examples

Hamiltonian

βi (∆EA+∆EB) eﬀ −1 + e Hˆ = − σˆ0 ∆EB (1+eβi ∆EA )(1+eβi ∆EB ) eβi ∆EB − eβi ∆EA + ∆EB cos(gt) βi ∆EA βi ∆EB (1+e )(1+e ) #

+ˆσz ∆EA. (3.33)

Due to the fact that σ0 commutes with σz we ﬁnd Hˆ eﬀ = Hˆ ′ and d d ρˆ (t)= Tr {ρˆ(t)} = L , (3.34) dt A dt B inc from (3.12) follows

βi ∆EA βi ∆EB g e − e ∆EA dQ¯ = − sin (g t) (3.35) βi ∆EA βi ∆EB 2(1+e )(1+e ) and thus for the whole heat transferred

π g 1 1 ∆Q = dQ¯ dt = − (∆EA) (3.36) 1 + eβi ∆EB 1 + eβi ∆EA Z0 Putting all terms together the SWAP gate has the engine eﬃciency

∆Q ∆EA ηSWAP = − = , (3.37) ∆W ∆EA − ∆EB 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 change of the energy expectation value of qbit 2 represents the heat

∆Q = ∆hEi2 = Tr Hˆ2 ρˆf − Tr Hˆ2 ρî (3.38) n o n o transferred by the algorithm, with Hˆ2 = ∆E2 σˆ0 ⊗ σˆz,ρ î andρ ˆf representing the density operator of the system before and after the application of the gate. To compute the work W in the system, one has to look at qbit 1. The energy difference is given by

∆hEi1 = Tr Hˆ1 ρˆf − Tr Hˆ1 ρˆi (3.39) n o n o with Hˆ1 = ∆E1 σˆz ⊗ σˆ0. If no work was done on the system, the change of the energy expectation value of this subsystem, given by Eq. (3.39), would be equal to the heat

27 3. Heat and Work

|11i |1i |1i |10i controlled ⇒ |01i |0i |0i |00i qbit A qbit B 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 |∆hEi1|−|∆hEi2| was less than zero, work would be extracted from the system. If it was greater than zero work would be done on the system. In case that ∆hEi1 and ∆hEi2 had diﬀerent signs, |∆hEi1|−|∆hE(n)i2| would be equivalent to the change of the energy expectation value of the entire system,

∆W =∆hEi = Tr Hˆ ρˆf − Tr Hˆ ρî n o n o =Tr (Hˆ1 + Hˆ2)ˆρf − Tr (Hˆ1 + Hˆ2)ˆρi n o n o = Tr Hˆ1 ρˆf − Tr Hˆ1 ρî + Tr Hˆ2 ρˆf − Tr Hˆ2 ρî . (3.40) n o n o n o n o ∆hEi1 ∆hEi2

Now we are able to calculate| the{z efficiency η}of the| SWAP gate{z by η = }−∆Q/∆W and obtain the same result as in (3.37). We will treat the CNOT gate inanalogy with the SWAP gate. The necessary π-pulse is depicted in Fig. 3.2, but the energy difference between |10i and |11i is always equivalent to the |10i-|11i difference. That brings up the problem to apply the pulse selectively to this single transition. In practice the so called minimal pulse is substituted by a series of real pulses, that has the same effect on the system as the selected pulse depicted above. Finding those pulse sequences is not unique and can not be more efficient than the 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 the same temperature, and the Hamiltonian

Hˆ = ∆Eµ σˆz(µ). (3.41) µ=1,2 X Now we use (3.39)and (3.40) to calculate the work and heat transferred by the gate and get β ∆E1 β ∆E2 e e − 1 ∆E2 ∆W = ∆Q = . (3.42) β ∆E1 β ∆E1 2 (1+e ) (1+e )

28 3.3. Examples

This leads to the efficiency ∆Q η = − = −1. (3.43) CNOT ∆W This negative value is due to the fact that the CNOT gate leads to negative temperatures of qbit 2, for finite initial temperatures. In terms of quantum computing just the increased polarisation is relevant, therefor we can neglect the sign of ηCNOT when comparing to other gates as the SWAP gate. The efficiencies of the quantum gates can not easily be combined to the efficiency of a complete algorithm because, in general, they depend on the initial occupation probabilities 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 of their environment. This concept emerged from the backround of quantum information processing and was developed to prepare the initial state of a set of qbits by applying a series of quantum gates. The idea is that the computer could prepare its highly polarised initial state by its own means without an external cooling device. One has to distinguish here 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 system is conserved and so the algorithm is basically only capable of moving the entropy around 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 anymore and thus signiﬁcantly lower temperatures are reachable. The problem is to experimentally achieve the heat bath contact in such a way that there is no eﬀect on the already cooled down qbits.

One possible algorithm was proposed by Boykin et al. in 2002 [4] (see section 4.2 on page 32) and experimentally realised in 2005 by Baugh [3]. Here we ﬁrst discuss this algorithm with emphasis on its thermodynamic properties. Then an extension of the algorithm 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 4. Algorithmic cooling

qbit 3

qbit 2

qbit 1 CNOT CSWAP

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 gates on a set of tree qbits of equal inverse temperature β(0) in order to cool qbit 3 down by transferring energy to the two other qbits. First a CNOT gate is performed between qbit 1 and 2, followed by a CSWAP, as depicted in Fig. 4.1 (For a description of the gates see 2.6.1 on page 18 or [31]). It is not intuitively clear how this procedure could achieve the goal of cooling. One has to take a closer look at the eﬀects the gates have on the system. To study this we start with a product state of the 3 qbits with the same inverse temperature β(0) and same Zeemann splitting, i.e. with the Hamiltonian

Hˆ = ∆Eσˆz(µ). (4.1) µ=1 X and

1 ε (0) ρ (0) = σ + i σ (4.2) µ 2 0 2 z 1 − e−β(0) ∆E with ε (0) = i 1 + e−β(0) ∆E ρ(0) =ρ1(0) ⊗ ρ2(0) ⊗ ρ3(0) (4.3)

Now the transformation

Uˆ = UˆCSWAP σˆ0 ⊗ UˆCNOT (4.4) representing the algorithm is applied to the system

ρˆ(1) = Uˆ ρˆ(0) Uˆ −1. (4.5)

32 4.3. Cyclic Extended Algorithm

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

1+3eβ(0) ∆E (1+eβ(0) ∆E )3 0 ρˆ (1) = 2 β(0) ∆E β(0) ∆E , (4.6) 1  e (3+e )  0 (1+eβ(0) ∆E )3   a canonical state of diﬀerent temperature

8 ln 3 − 3+eβ(0) ∆E β(1) = 2 β(0) − . (4.7) n ∆E o Measured in units of β(0) within the limit of small β(0) follows β(1) lim =1.5. (4.8) β(0)→0 β(0) 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 applications of the algorithm. However, by cooling down two other qbits by applying the same algorithm as described above to two additional sets of 3 qbits, allows a second application of the algorithm to the cooled qbit triple with reduced initial inverse temperature β(1). One qbit could be cooled down to the total of 9/4 β(0). By adding additional qbits arbitrary low temperatures are in principle achievable. One needs 3N qbits to perform N 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 introduced an improvement to the original algorithm by using rapidly thermalising qbits. These qbits could be brought in contact with a heat bath and thus, after an inﬁnite waiting time, be reset to their initial state. The algorithm is implemented in the same way as the basic one described above. Then the qbits 1 and 2 are reset to their initial state and 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 have seen now, heat bath algorithmic cooling needs only 1/3 of the qbits needed for the basic algorithm, but nevertheless it has an exponential growth in qbits. At the current state of quantum computing that seems not viable.

4.3. Cyclic Extended Algorithm

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

33 4. Algorithmic cooling

qbit 3

β(0) qbit 2

β(0) qbit 1

SWAPCNOT CSWAP bath

Figure 4.2.: Cyclic cooling algorithm: 3 quantum gates are applied. First a SWAP gate then a CNOT gate and finally a CSWAP gate. Bath contact at inverse temperature 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 CSWAP gate is applied to the system in the same manner as in Boykin’s algorithm. Now one has to wait until the auxiliary qbits 1 and 2 are relaxed back to the initial temperature by the coupling to a heat bath (see Fig. 4.2). This can intuitively be understood by unfolding the algorithm. Removing the bathcontact and SWAP gate and instead adding two additional auxiliary qbits, the previous step of the algorithm is performed on cooling qbit 1 and so on. This way one generates a chain in which each application of the algorithm precooles the “qbit 1” of the next step. For the bath contact step two classes have 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 not totally 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 set of qbits. We call this algorithm cyclical, despite the fact that after one application the system does not return to its initial state, but is further cooled down. In principle, the complete 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 most interesting question, as in case of a refrigerator, is of course the final inverse temperature β(n) of the first qbit after the nth application of the algorithm in dependence of system parameters and relaxation times between subsequent application steps. On the bottom line a SWAP gate is added between the cooled qbit and one of the auxiliary qbits at the beginning of the algorithm. Starting with one auxiliary at lower temperature 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 4.3. Cyclic Extended Algorithm applying it in two diﬀerent ways on sets of four qbits: First, qbit 1 is cooled using qbits 2 and 3, then qbit 4 is cooled as well with the aid of qbits 2 and 3. In case of a NMR-type quantum computer or something similar, the two stages could each be applied to the whole system simultaneously, thus achieving a rather fast cooling of the system. If one was capable of running quantum gates on any combination of qbits, all but two qbits of the 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 by Ryan [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 product state

ρ(0) =ρ1(0) ⊗ ρ2(0) ⊗ ρ3(0) (4.9) 1 ε (0) ρ (0) = σ + i σ (4.10) µ 2 0 2 z 1 − e−β(0) ∆Eµ with εi(0) = , 1 + e−β(0) ∆Eµ where β(0) denotes the inverse bath temperature and ∆Eµ the Zeemann splitting of the respective spin. Note that we label all quantities belonging to the nth application of the algorithm by (n), so the initial state and the temperature of the bath are labeled (0). The system Hamiltonian is 3 3

Hˆ = Hˆµ = ∆Eµσˆz(µ). (4.11) µ=1 µ=1 X X There is no interaction between the qbits, except the interaction introduced by the application of the quantum gates. The quantum gates can be treated as before

Uˆalg =UˆCSWAP UˆCNOT UˆSWAP, (4.12) ˆ ˆ † ρˆf(n)=Ualg ρˆi(n) Ualg. (4.13) The bath contact cannot be represented by a unitary transformation. A quantum master equation in Lindblad form [5, 20, 38, 43] has to be solved, instead (see also “2.1.4 The Quantum Master Equation” page 7). Because the qbits are not coupled, they must be thermalised separately. The respective quantum master equation for qbit µ reads

ρˆ˙µ = − i[Hˆµ, ρˆµ]

+ W1→0(2σ ˆ−ρˆµσˆ+ − ρˆµσˆ+σˆ− − σˆ+σˆ−ρˆµ)

+ W0→1(2σ ˆ+ρˆµσˆ− − ρˆµσˆ−σˆ+ − σˆ−σˆ+ρˆµ) (4.14)

=L ρˆµ, (4.15)

35 4. Algorithmic cooling

with the rates W1→0 = λ/(1+(1/ε)) and W0→1 = λ/(1+ ε), ε = exp {β(0)∆E} and the bath coupling strength λ. In order to compute the density matrix ρ(n) for an arbitrary number of repetitions of the algorithm, it would be desirable to apply a simple matrix transformation for the bath 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 operator projecting an arbitrary state on a solution of (4.15). Just like in Hilbert space the time evolution operator, the formal solution of (4.14), the solution of (4.15), is given by

Lt ρ(t, n +1)=e ρf(n)= T(t)ρf(n) (4.16) with the limit lim T(t)= T (4.17) t→∞ defining the complete thermalisation super operator. Because the quantum gates merely interchange entries ρii of the density operator, they do not lead to coherences in the system. We start with a thermal state and thus can restrict ourselves to density operators as the initial states of the thermalisationsuperoperator. With this restriction (4.16) leads to −2τλ −2τλ εi(0) − 1 T(τ)= |σˆ0)(ˆσ0| + e |σˆz)(ˆσz| + e − 1 |σˆ0)(ˆσz|. (4.18) εi(0)+1 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 a Liouville product space with the basis |σî) ⊗|σˆj ) ⊗|σˆk)= |σîjk) with i, j, k ∈{0, x, y, z}. There the superoperator thermalising qbit 1 and 2 reads

T12(τ)= T(τ) ⊗ T(τ) ⊗ ½. (4.19) The superoperator of the algorithm U has to be generated in the same product basis as T(τ). Therefore we insert a general density operatorρ ˆ with coefficients ρijk in the basisσ î ⊗ σˆj ⊗ σˆk =σ îjk into its expansion

ρˆ = ρijk σîjk (4.20) Xijk to get a general density operatorρ ˆ in Hilbert space H. Now the algorithm is applied to the system ρˆ′ = Uˆ ρÛˆ −1. (4.21) Thisρ ˆ′ can be interpreted as the left hand side of a system of linear equations in the Liouville space L ′ ρˆ ⇒ U |ρîjk) (4.22) th One whole application of the algorithm now reads B = T12(τ) U. To compute the n application of the algorithm B has to be taken to its nth power |ρˆ(n)) = Bn(τ)|ρˆ(0)), (4.23)

36 4.3. Cyclic Extended Algorithm

β(n) 2.1 β(0)

2.0

1.9

1.8

1.7

1.6

1.5 0 10 20 n

Figure 4.3.: β(n) as a function of cycles n for inﬁnite bath contact time and constant Zeemann splitting ∆E1 = ∆E2 = ∆E3 = ∆E. The asymptotic temperature is approximately reached after seven steps. using the inital state of the algorithm |ρˆ(0)). This result is easily transformed from Liouville space L back into Hilbert space H by inserting |ρˆ(n)) into the expansion 4.20

ρˆ = |ρˆijk)ˆσijk. (4.24) Xijk

4.3.2. Final Temperatures of the Cyclic Algorithm For infinite bath contact time (T(τ) ⇒ T) we are able to diagonalise B analytically and thus calculateρ ˆ(n). We find the density operator of qbit 3 by calculating the partial trace over qbit 1 and 2 and accordingly the final inverse temperature β(∞) of spin 3 (depicted in Fig. 4.3) ∆E + ∆E β(∞)= 2 3 β(0). (4.25) ∆E1

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 complete analytic 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 shown in Fig. 4.4. For long bath contact it evolves like Fig. 4.5. Apparently, in Fig. 4.5 the

37 4. Algorithmic cooling

β(n) 1.6 β(0) 1.5

1.4

1.3

1.2

1.1

1 0 100 200 300 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 up again.

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.

β(n) 2 β(0)

1.9

1.8

1.7

1.6

1.5 0 10 20 n

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

38 4.3. Cyclic Extended Algorithm

For the chosen set of parameters the application of the quantum gate itself needs a single time-unit. Let us measure the bath contact time τ in this time-unit as well. After each application of the quantum gate we wait exactly the same time τ to let qbit 2 and 3 relax towards the equilibrium temperature of the bath. In Fig. 4.6 we show the final inverse temperature of qbit 1 after n = 300 applications of the full algorithm (gates plus bath coupling) in dependence of the bath coupling time τ. Note that even if we have computed 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 is especially the case for τ ≫ 32 as can be seen in Fig. 4.5.

β(300) β(0)

1.5

1.3 0 100 200 τ

Figure 4.6.: The final inverse temperature β(300) as a function of different bath contact times τ. The line at β(300)/β(0) = 2 represents the upper limit for infinite bath contact, the line at β(300)/β(0)= 1, 53 marks the inverse temperature of 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 Boykin algorithm. Waiting time τ ≈ 32 means that qbit 2 and 3 are cooled to 0.77 times the inverse 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 a single cycle. However, multiple application of the algorithm does not always lead to a reduction of the final temperature of the cooled spin, at least if τ < 32. This can be seen in Fig. 4.4 as well, where the temperature after the first application is already lower than after 300 cycles.

39 4. Algorithmic cooling

4.3.3. Eﬃciency

As mentioned earlier, algorithmic cooling can be seen in the context of quantum thermodynamic 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 qbits 1 and 2) into a heat bath of higher temperature (realised by the bath contact). For a thermodynamic machine the efficiency η is of high interest to determine its capabilities. For the heat pump it is defined as

−∆Q η = (4.26) ∆W where ∆Q is the heat transfered into the hot bath and ∆W the work consumed by the process (also see Sec. 3 on page 21). We will have to determine work and heat of the process. Following the definition via the total differential of the energy expectation value (2.49) d d d hEi = Tr Hˆ ρˆ + Tr Hˆ ρˆ dt dt dt dW¯ dQ¯ | {z } | {z } We acquire the same work as by applying the extended definition (3.12) and (3.11)

d ˙ hEi = Tr Hˆ eﬀρˆ − i[Hˆ ′, Hˆ eﬀ]ˆρ dt + Tr Hˆ ′L (ˆρ) dt. . dt 1 2 inc n dW¯ o n dQ¯ o | {z } | {z } To calculate the transfered heat Q the whole pulse sequence has to be modeled in order to obtain the results. This is done for the single gates in Sec. 3.3.2 page 25. We make use 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 hE3(n)i of the cooled qbit 3

∆Q = hE3(n)i−hE3(n − 1)i c1 n+1 eβ(0) ∆E2 + eβ(0) ∆E1 β(0) ∆E3 β(0) (∆E2+∆E1) β(0) ∆E β(0) ∆E e − e z(1+e 2 )(1+e}| 1 ){! = ∆E3. (4.27) (1+eβ(0) ∆E3 ) (eβ(0) ∆E2 + eβ(0) ∆E1 )

The work ∆W is the change of the energy expectation value hE(n)i of the entire

40 4.4. An Ideal Algorithm system with a rather lengthy result ∆W = hE(n)i−hE(n − 1)i c2 eβ(0) (∆E2+∆E1) 1 + eβ(0) ∆E3 −1 + eβ(0) ∆E2 1 + eβ(0) ∆E1 = ∆E2 (1+ez β(0) ∆E3 )(1+e β(0) ∆E2 )(1+e}| β(0) ∆E1 )(1+e β(0) (∆E2+∆E1{)) c3 k 1 − tanh β ∆E2 tanh β ∆E1 2 2 eβ ∆E3 − eβ (∆E2+∆E1) z }|2 {! + (1+eβ(0) ∆E3 )(1+eβ(0) ∆E2 )(1+eβ(0) ∆E1 )(1+eβ(0) (∆E2+∆E1))

β ∆E2 β ∆E1 × 1 + e e (∆E3 − ∆E1) h β ∆E2 β ∆E1 β ∆E2 β ∆E1−1 + e − e + e e ∆E2 . (4.28)

Inserting (4.27) and (4.28) into (4.26) results in i P (n) ∆E η = 1 3 (4.29) P13(n) (∆E3 − ∆E1)+ P2(n) ∆E2 with

n β(0) ∆E3 β(0) (∆E2+∆E1) β(0) (∆E2+∆E1) P1(n)= c1 e − e 1 + e

n β(0) ∆E3 β(0) (∆E2+∆E1) β ∆E2 β ∆E1 P13(n)= c3 e − e 1 + e e n β(0) ∆E3 β(0) (∆E2+∆E1) β ∆E2 β ∆E1 β ∆E2 β ∆E1 P2(n)= c2 + c3 e − e e − e + e e − 1 . To further understand this result one has to look at the transitions which are needed to operate 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 denominator are the energies of the applied transitions. The Pi(n) are the differences of the occupation probabilities of the involved niveaus before the accounting gate is applied. The structure of (4.29) reminds us of the efficiency of the quantum Otto process [20]. Numerically it is always below the accounting Otto efficiency.

4.4. An Ideal Algorithm

To achieve a better understanding of the cooling process we take a look at a simpler system: 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 ˆ z H = ∆Eµ σµ (4.30) µ=1,2 X

41 4. Algorithmic cooling

|11i |1i

|1i |10i ⇒ |01i |0i

|0i |00i

qbit 1 qbit 2 entire system

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

and the initial density operatorρ ˆi. Evidently, the lowest temperature qbit 1 can obtain without coupling to a heat bath, results by applying a SWAP gate (see Sec. 2.6.1 on page 19). If coherences would remain in the system the local temperature of spin 1 would be higher because this situation could be represented by a partial SWAP gate. The application of the SWAP transformation to the initial state

ˆ ˆ −1 ρˆf = USWAP ρˆi USWAP (4.31) results in the ﬁnal inverse temperature of qbit 1

∆E2 βf = βi. (4.32) ∆E1 Calculating the work and heat done on the system (as 3.3.2) we ﬁnd

1 1 ∆W = − (∆E1 − ∆E2) (4.33) 1 + eβi ∆E2 1 + eβi ∆E1 1 1 ∆Q = − ∆E1 (4.34) 1 + eβi ∆E2 1 + eβi ∆E1 This results in an engine eﬃciency of

∆Q ∆E η = − = 1 . (4.35) ∆W ∆E2 − ∆E1 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 of the quantum Otto process. If one adds two isochoric bath coupling steps, the repeated application of SWAP gates would fully resemble an Otto cycle. Although, if qbit 2 was now coupled to a heat bath again and thus set back to its initial temperature, the occupation probabilities of the two qbits would be identical. So the repeated application

42 4.4. An Ideal Algorithm

|111i |11i |110i |101i |1i |10i |011i ⇒ |0i |01i |100i |010i |001i |00i |000i qbit 3 axillary system entire system

Figure 4.8.: Portation of the swapping concept to a three spin system. One has to swap the highest and the lowest energy level of the auxillary system with the qbit to be cooled (i.e. qbit 3). This results in a π-pulse on the levels of the whole system, depicted on the righthand side. of the SWAP gate would not result in further cooling of qbit 1, i.e. the ﬁnal temperature will be reached after one cycle. It is most likely that more heat could be transferred by aprocedure resembling a Carnot cycle. But realising the isothermic steps requires very precise control over the Zeemann splitting of the spins, because a TLS does not seem to perform isothermal expansion or compression 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 probabilities of the hightest and lowest niveau of the auxiliary system have to be swapped with the two niveaus of qbit 3. As depicted in Fig. 4.8, this results in a π-pulse between the |001i and |110i niveau. Now we compute the inverse temperature βf of qbit 3 after the SWAP operation and get ∆E1 + ∆E2 βf = βi. (4.36) ∆E3 This is the same temperature as β(∞) (Eq. (4.25)) of the cyclic algorithm. Again, bath contact 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 an other qbit in the sense of Sec. 4.2.2 (page 33). This scheme can easily be extended to any number of qbits. Altough a mathematical proof was not accessible, calculations for up to 6 qbits suggest that

n−1 ∆Ei i=1 βf = βi (4.37) P∆En

43 4. Algorithmic cooling should be the general formula for the final temperature. It always results in a lower temperature than the heat bath cooling algorithm with the same number of qbits. In order to show this we first take the number n of qbits necessary to perform N cooling N−1 steps with the heat bath algorithm n =2+3 . Assuming that ∆Ei = ∆E ∀ ∆Ei, SWAP as in Boykins algorithm, this results in a final inverse temperature βf = n − 1 βi for Boykin 3 N the swapping algorithm while heat bath algorithmic cooling results in βf ≈ 2 βi. Subtracting the final inverse temperatures of the two algorithms from one another gives 3 N 3 n − 1 − β =1+ 3N−1 1 − β > 0. (4.38) 2 1 2N 1 ! 1 8 − 2 for N =1 =< | > 0{z for N >}1 : The swapping type algorithm would always be better than the heat bath algorithm. The efficiency of all checked systems with the swapping algorithm is of the form ∆E η = n (4.39) n−1 ∆Ei − ∆En i=1 P 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 thermodynamic properties. 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 information processing.

Heat bath algorithmic cooling, where some qbits are coupled to a heat bath and thus 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 working algorithm for a better comparison with a refrigerator or the quantum thermodynamic heat pump proposed by Henrich [20]. We, indeed, found a possibility to run Boykins algorithm in a cyclic manner [33] and calculated its efficiency and lowest reachable temperure analytically for infinite bath contact time and numerically for finite contact times. It became clear that there is a rather easy scheme to “cyclify” any given heat bath cooling algorithm by adding a SWAP gate to it. I further investigated the limitations of algorithmic cooling by considering a simplier algorithm 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 maximum heat 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 qbit and may even be the best possibility to algorithmically cool this qbit. Swapping has the Otto efficiency. It may be possible to further improve the cooling by modelling the bath contact as an isothermic step. This would lead to Carnot efficiency, but would be beyond the means of algorithmic cooling. During the process of investigation we came up with the local effective measurement basis principle (LEMBAS), a method to calculate local properties as an experimentalist would probably measure them. This led straight to an extention of the definition thermodynamic properties of work and heat for arbitrary bipartite quantum systems, which led 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 quantum thermodynamics. What remains, is to do some numerics for quantum gates with permanent bath coupling and maybe expanding the concept to multi level systems. On the other hand, there is plenty of work to be done in conceptional quantum thermodynamics, which is still defined only for quasi classical systems with weak coupling and therefore the energy as an extensive value. In real quantum systems with coherences and finite interaction energies, the energy is not extensive any more, in inhomogeneous systems 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 extend the definitions of quantum thermodynamic variables for coherent systems, as we did here for work and heat.

46 A. Appendix

A.1. Trace theorems

Let H = HA ⊗ HB be a bipartite Hilbert space, with operators Aˆ and Bˆ acting on HA and HB, respectively, and an operator Cˆ acting on the full Hilbert space.

Theorem A.1.

ˆ ˆ ˆ ˆ ½ TrA [A ⊗ ½, C] = TrB [ ⊗ B, C] =0. n o n o Proof. Due to symmetry it suﬃces to prove

ˆ ˆ TrA [A ⊗ ½, C] =0. (A.1) n o

Using an orthonormal basis {Qˆi} with each Qˆi acting only on HA or HB we can write ½ TrA [Aˆ ⊗ ½, Cˆ] = cjk TrA [Aˆ ⊗ , Qˆj ⊗ Qˆk] n o Xjk n o

= cjk TrA [Aˆ ⊗ ½, Qˆj] ⊗ Qˆk Xjk n o = cjk TrA [A,ˆ Qˆj] Qˆk Xjk n o cycl = 0. (A.2)

Theorem A.2.

TrA (Aˆ ⊗ Bˆ)Cˆ = BˆTrA (Aˆ ⊗ ½)Cˆ (A.3) n o n o TrB (Aˆ ⊗ Bˆ)Cˆ = AˆTrB (½ ⊗ Bˆ)Cˆ . (A.4) n o n o 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 (Aˆ ⊗ Bˆ)Cˆ = cjk TrA (Aˆ ⊗ Bˆ)(Qˆj ⊗ Qˆk) n o Xjk n o = cjk TrA (AˆQˆj) ⊗ (BˆQˆk) Xjk n o = Bˆ cjk TrA (AˆQˆj) ⊗ Qˆk Xjk n o

= Bˆ TrA (Aˆ ⊗ ½)Cˆ . (A.5) n o

ˆ Corollary A.3. For B = ½ it immediately follows from theorem A.2

TrB (Aˆ ⊗ ½)Cˆ = Aˆ TrB Cˆ . (A.6) n o n o Corollary A.4. Another immediate consequence of theorem A.2 is

TrB C,ˆ Aˆ ⊗ Bˆ = TrB Cˆ(½ ⊗ Bˆ) , Aˆ . (A.7) nh io h n o i

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52 Index

algorithmic cooling, 31 heat, new definition, 23, 40 heat, old definition, 11, 12, 40 Born approximation, 8 Boykin, 31 ideal algorithm, 41 closed algorithmic cooling, 31 laser, 25 closed Boykin algorithm, 32 Lindblad form, 8, 35 closed quantum system, 5 Liouville space, 8, 36 CNOT gate, 19 Liouville von Neumann equation, 6 control bit, 19 lowering operator, 5 CSWAP gate, 20 cyclic algorithm, 33 Markov assumption, 8 cyclic algorithm analytic results, 37 master equation, 35 model Hamiltonian, 7 density operator, 5 density operator, time averaged, 10 NMR-type quantum computer, 35 detuning, 25 operator representation, 3 discrete Hilbert space, 3 Otto-type cycle, 15 dissipator, 8 Pauli operators, 4 effective Hamiltonian, 23 pressure, 11 efficiency, classical Carnot engine, 12 efficiency, classical Carnot heat pump, 13 quantum computing, 17 efficiency, classical Otto engine, 12 quantum gates, 18 efficiency, classical Otto heat pump, 13 quantum master equation, 7 efficiency, CNOT, 29 quantum Otto cycle, 15 efficiency, cyclic algorithm, 40 quantum Otto process, 42 efficiency, quantum Otto engine, 16 efficiency, SWAP, 27 raising operator, 5 entropy, 9 rotating wave approximation, 8

ﬁnal temperature, cyclic algorithm, 37 Schr¨odinger equation, 5 spectral temperature, 10 Gibbs fundamental relation, 11 SWAP gate, 19, 42 heat bath algorithmic cooling, 31, 33 temperature, 9, 10

53 Index thermalisation superoperator, 36 thermodynamic variables, 9 time evolution operator, 6, 36 tracenorm, 4 transition operator, 3 truth table, 18 universal gate, 18 volume, 11 von Neumann entropy, 9, 13 work, new deﬁnition, 23, 40 work, 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 and always 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 and always 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 a hopefully 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 my LATEX 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 important being there for me.

Special thanks to . . .

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

. . . Florian Hard the most amusing subjects at the coﬀee table and the best Cthulhu game 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!