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Algorithmic Cooling and Quantumthermodynamic Machines 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 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 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 pro- cessing. 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 ex- tension. 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 for- mula 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 ther- modynamics 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 thermody- namic 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 re- quires 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: k 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 rep- resented 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 ∂ ρˆ(t)= ρ |ψ˙(t)ihψ(t)| + |ψ(t)ihψ˙(t)| (2.22) ∂t i i X and insert (2.21) for |ψ˙(t)i and hψ˙(t)| ∂ 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) .
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