E M E R G E N C E O F C L A S S I C A L R E A L I T Y F R O M Q U A N T U M T H E O R I E S

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1 E M E R G E N C E O F C L A S S I C A L R E A L I T Y F R O M Q U A N T U M T H E O R I E S V O N FA B I A N R. L U X Bachelorarbeit in Physik vorgelegt der Fakultät für Mathematik, Informatik und Naturwissenschaften der RWTH Aachen im Juli 2014 angefertigt im Institut für Theoretische Festkörperphysik bei Prof. Dr. Manuel Schmidt

2 D E C L A R AT I O N Ich versichere, dass ich die Arbeit selbstständig verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel benutzt, sowie Zitate kenntlich gemacht habe. Aachen, Juli 2014 Fabian R. Lux Fabian R. Lux: Emergence of Classical Reality from Quantum Theories, Bachelorarbeit in Physik, Juli ii

3 A B S T R A C T This bachelor thesis deals with the problematic nature of describing classical physics as an emergent phenomenon of quantum physics. Starting from the most common interpretations of quantum mechanics and their shortcomings, some of the recent approaches to this problem are presented. The focus lies on the contributions of Wojciech Hubert Zurek, who suggested that decoherence processes together with a selective proliferation of information could lead to a classical behavior. In the main part of this thesis, numerical studies on spin systems elucidate Zurek s ideas, test its limits and show a way beyond. iii

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5 C O N T E N T S i emergence of classical reality from quantum theories 1 1 introduction 3 2 interpretations of quantum mechanics The Copenhagen Interpretation General Von Neumann Measurements The Many-Worlds Interpretation 11 3 decoherence, einselection and quantum darwinism The Concept of Partial Traces Decoherence and Einselection Decoherence of a Single Qubit Entropy and Information Application of Mutual Information Quantum Darwinism 24 4 numerical studies Implementation Notes The Spin Star-Environment Decoherence in the Bloch Sphere Correlation Amplitudes Mutual Information Partial Information Random Walk Experiments 41 5 summary and outlook 47 ii appendix 49 a self-information 51 b the partial trace 53 b.1 The Uniqueness of the Partial Trace 53 b.2 The Partial Trace in Composite Two-State Systems 53 bibliography 55 v

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7 Part I E M E R G E N C E O F C L A S S I C A L R E A L I T Y F R O M Q U A N T U M T H E O R I E S

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9 I N T R O D U C T I O N 1 Everything we call real is made of things that cannot be regarded as real. Niels Bohr This well-known quotation of Niels Bohr 1 gets to the heart of a problem that gave generations of physicists sleepless nights. Experiments in the early 20th century refused all classical approaches and led to a new kind of physics. The theory of quantum mechanics. Physicists motivated and elaborated the axioms of this quantum theory, which had the ability to explain a wide range of experimental results. They are sometimes referred to as Dirac-von Neumann axioms 2, which state the Hilbert space structure of the space of quantum states and Born s rule, which connects this structure with a probability interpretation. Despite the great success of quantum theory, the interpretation of the mathematical foundations kept controversial and is still an open question. In 1935 Einstein, Podolski and Rosen published their paper Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? where they questioned the physical reality of quantum states. Their objection concerns essentially two statements of quantum mechanics. The first statement is that two non-commutating observables cannot be measured simultaneously 3 with infinite accuracy. As an example, position ˆx and momentum ˆp with the canonical commutation rule [ ] ˆxi, ˆp j = i h δi,j i, j {1, 2, 3} (1.1) The foundations and principles of quantum mechanics. Einstein, Podolski, Rosen The Heisenberg uncertainty principle obey the Heisenberg uncertainty relation σˆxi σˆpj 1 2 [ ˆx i, ˆp j ] = h 2 δ i,j, (1.2) where σ denotes the standard deviation with respect to some state ψ in the corresponding Hilbert space. As the deviation of ˆp i decreases, the deviation of the corresponding position observable ˆx i will have to increase in order to keep up the uncertainty relation and vice versa. In the limit σˆpi 0 the position observable ˆx i will be completely delocalized. Thus, a state vector ψ can only provide precise information about one of these conjugate observables, but not about both. 1 Which ends with the famous phrase If quantum mechanics hasn t profoundly shocked you, you haven t understood it yet. 2 Dirac and von Neumann were the first to publish a textbook on the mathematical foundations of quantum mechanics in 1930 and 1932 respectively ([3],[15]). 3 Where the word simultaneously should be handled carefully. Quantum Entanglement 3

10 4 introduction The EPR paradox Bell s theorem, the confidence in quantum mechanics The second statement concerns the tensorial structure of Hilbert spaces. A prominent example is the product space of two equivalent two-state systems H = H 1 H 2, where H i is isomorphic to C 2. The natural basis of the resulting space is given by the tensor product of the basis kets, which is the intuitive procedure of constructing all possible (classical) configurations of the composite system. And because the composite system needs to be a Hilbert space in agreement with the Dirac-von Neumann axioms, all superpositions of states are also legal states of the systems. In general, such states cannot be written as a tensor product of states of the single systems, which is obvious from a mathematical point of view, but astonishing in its consequences for the nature of quantum systems. In fact, this quantum entanglement makes it possible to determine the state of one subsystem by measuring an entangled partner system. Einstein, Podolski and Rosen connect quantum entanglement and Heisenberg uncertainty to construct a paradox, which is known as EPR paradox 4. Their whole argumentation is based on two assumptions, which are then connected to the previous two statements of quantum mechanics. The first is a criterion of completeness, saying that it is a necessary requirement for a complete theory that every element of the physical reality must have a counterpart in the physical theory (see [4] p. 777). The second assumption is a criterion of reality: If one can predict the value of a physical quantity with certainty without perturbing the physical system, then there exists an element of physical reality corresponding to this physical quantity (ibidem). Einstein et al. construct an entangled system of two particles with the possibility to determine (with certainty) either the position or the momentum of the first particle by measuring the second entangled partner without disturbing the first. Therefore, by the criterion of reality, momentum and position should be an element of physical reality. But, in contradiction to the criterion of completeness, the physical state vector can only describe one of these two conjugate observables with certainty. In the opinion of Einstein et al., this is a sign for the incompleteness of quantum mechanics. Theories that are trying to add variables in order to complete quantum mechanics in this sense are called hidden variable theories. Over the past 50 years a lot of experiments enhanced the confidence in quantum mechanics. Especially the strong experimental evidence of a violation of Bells inequality indicates that local hidden variable theories are impossible. Furthermore it has been accentuated by Fuchs and Peres in 2000 (see [6]) what was already part of Bohr s reply to the EPR paper, namely that a quantum state itself is purely epistemic and cannot be viewed as ontological entity 5. In 4 Named after Einstein, Podolski and Rosen. 5 Ontology and epistemology are the questions to what can be said to exist? and what can be known?.

11 introduction 5 other words, Fuchs and Peres say that a quantum state is formulated operationally in such a way, that the application of Born s rule determines the correct probabilities to describe the experiment. With this definition it is possible to construct wave functions for macroscopic objects as well as for microscopic systems 6, which led Fuchs and Peres to their statement: Quantum theory needs no interpretation. However, the acceptance of quantum mechanics as an adequate description of the microscopical domain ultimately raises the question of how classical reality - a collection of things, to which we attribute the status of an ontological entity with objective existence - could possibly emerge from quantum states, which cannot be viewed as real in this sense. This closes the circle to the opening quotation. Theories which aim to describe classical reality as an emergent phenomenon of quantum mechanics have to give an answer to this question. Wojciech Hubert Zurek is a key figure in this research. Since the early 1980 s, Zurek has been investigating decoherence processes in composite quantum systems where he recognized that decoherence leads to an effective selection of quantum states, called einselection. Furthermore, information about these einselected states can be redundantly imprinted in the environment, which is constantly monitoring the system. This leads to a selective proliferation of information, a phenomenon Zurek named quantum darwinism in analogy to evolutionary processes. Together, these concepts can be used to show a way from quantum physics to classicality, which will be the focus of the following chapters. A few remarks on the outline of this work: Chapter 2 will give an overview on the most common interpretations of quantum mechanics, their shortcomings and the role of the measurement process. Based on this discussion chapter 3 introduces the concepts of decoherence, einselection and quantum darwinism together with the necessary mathematical tools and the basic concepts of information theory. The main part of this work, chapter 4, presents a numerical study of spin systems elucidating Zurek s ideas and testing its limits. The final conclusions and a short outlook can be found in the last chapter. Decoherence, einselection, and quantum darwinism The outline of this work 6 A famous example is Schrödinger s cat.

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13 I N T E R P R E TAT I O N S O F Q U A N T U M M E C H A N I C S 2 In the first half of the 20th century, physicist succeeded to find a mathematical framework that is able to describe experiments whose results are inexplicable from a classical point of view. But the framework alone is unsatisfying for the human curiosity which keeps asking deeper questions, like the question of the ontological status of quantum mechanics or the nature of the measurement process. Interpretations of quantum mechanics want to give a guideline how to deal with these questions. However, there is no known way to decide whether an interpretation is correct or not. What is an interpretation of quantum mechanics? 2.1 the copenhagen interpretation For practical purposes the usual way of thinking about quantum mechanics is saying that a state vector ψ can be used to derive the correct probabilites to describe the outcomes of all experiments that could be performed on the corresponding physical system S. This is the essence of the so called Copenhagen Interpretation of quantum mechanics (short: CI), which was elaborated by Niels Bohr and Werner Heisenberg in The interpretation is based on the following axioms 1 The fundamental postulates i) For each closed quantum system S there is an associated separable Hilbert Space H over C, called the space of states. ii) Every physical measurable quantity A can be described by a linear self-adjoint operator A on H, called observable. iii) Possible measurement outcomes are eigenvalues a n of A with the corresponding eigenkets a n, ν H, where ν indicates a possible discrete (or continuous) degeneracy. iv) If the system is in the (normalized) state ψ, a measurement of A will yield the value a n with the probability w an ( ψ ) = a n, ν ψ 2. v) The Projection Postulate. If the measurement of A yields the eigenvalue a n, the state of the system can be found in the normalized projection of the initial state onto the corresponding eigenspace of a n. 1 As already mentioned in the introduction, axioms i)-iii) are referred to as Dirac-von Neumann axioms, whereas axiom iv) is called Born s Rule. ν 7

14 8 interpretations of quantum mechanics The border between the quantum and the classical vi) The time evolution of quantum system is given by the Schrödinger equation: i h t ψ(t) = H ψ(t), (2.1) where H is a Hermitian operator, connected to the total energy of the system. Bohr and Heisenberg insisted that measuring devices should be objects of a classical description. Thus a superposition of states in the classical domain was not allowed by decree (compare to [18] p. 716). This dualism between the quantum system and the classical measuring device caused many troubles with the Copenhagen Interpretation, because it is difficult (or even impossible) to draw a fixed border between these two ways of describing reality. [...] the CI universe would be governed by two sets of laws, with poorly defined domains or jurisdiction. This fact has kept many students, not to mention their teachers, awake at night (from Zurek in [18] p. 717). 2.2 general von neumann measurements The nature of quantum measurements When looking for this border, it is a natural question to ask how measurement devices work in quantum mechanics 2, because measurements are our interface to the quantum world. The first who investigated measurement schemes in quantum mechanics was von Neumann in 1932 (see [15]). In the following I would like to present the scheme von Neumann suggested, by following the discussion of the problem given by Jacobs in chapter one of [7]. Consider the composite system of a quantum system S interacting with a measurement apparatus A. The resulting tensor product Hilbert space should be of the form H SA = span { s i a j }i,j N, (2.2) where { s i } i N and { a j represent orthonormal basis systems }j N in the Hilbert spaces of S and A respectively. System and apparatus interact unitarily (according to the Schrödinger equation) through an operator U defined on H SA : U = u i,j,i,j a i s i a j sj i,j,i,j (2.3) = A i,j a i a j, i,j where A i,j := i,j u i,j,i,j s i s j. The apparatus should be used to measure the state of the system. In order to do this, consider the apparatus beeing in the initial state a 0 a 0, which leads to the following density matrix before the interaction occurs: ρ = ρ S a 0 a 0. (2.4) 2 Which is a little bit tricky, because it is not possible to do this within the CI framework.

15 2.2 general von neumann measurements 9 A time evolution of the system, followed by a projection on one apparatus pointer a n transforms this density matrix to ρ (id a n a n ) U ρ U (id a n a n ) = A n,0 ρ A n,0 a n a n (2.5) Therefore, the reduced density matrix of the system 3 after the measurement is given by ρ S = A n ρ A n Tr A n ρ A, (2.6) n where A n A n,0. This is the form one would expect from the postulates of quantum mechanics. From unitarity follows: id = a i U U n a j = k A k,i A k,j A na n = id. (2.7) Any sequence of operators {A n } n fulfilling this condition describes a possible measurement of the quantum system. Apart from this result, it is instructive to look at the time evolution of an uncorrelated initial state: ( ) ψ = α i s i i a 0 ) (2.8) U ψ = k β i,k s i a k. i,k A k ( α i s i i a k (2.9) For the special case of orthogonal measurements, i.e. A i A j = δ i,j A j, the system basis can be rewritten in an appropriate way such that ( ) ψ = α i s i i a 0 β i s i a i. (2.10) i This step is called premeasurement, describing the dynamical generation of entanglement between the system and the apparatus before the measurement of the apparatus pointer is performed by an external observer. The interesting feature of von Neumann s scheme is the amplification of superpositions to a macroscopical level, represented by the quantum entanglement of system and environment (compare Schlosshauer [13] p. 52). This is clearly incompatible with the Copenhagen Interpretation, that forbids a quantum description of the measurement apparatus. Thus the suggested scheme shows a way beyond the Copenhagen Interpretation. Von Neumann premeasurement 3 A more detailed justification for this will follow with the concept of partial traces, introduced in the next chapter.

16 10 interpretations of quantum mechanics Preferred basis problem Interestingly enough, it is not yet clear how von Neumann s scheme can describe a completed measurement. This is essentially due to two problems. The first problem is known as the problem of the preferred basis. It is possible to rewrite the right-hand side of equation (2.10) in infinitely many ways, using a different basis for the Hilbert spaces of the apparatus and the system. Consider for example a spin singlet state in the z-basis: ψ = 1 2 ( ). (2.11) Each z-ket can equally well be expressed in terms of x-kets by applying a transformation to the conjugate basis by ± = ( ± )/ 2. (2.12) This leads to a similar expression for the state ψ : ψ = 1 2 ( + + ). (2.13) Who measured whom? The quantum c-not Astonishingly from a physical point of view, the apparatus and the system show correlations in infinitely many ways. Therefore, the von Neumann scheme alone cannot determine which observable has been measured by the apparatus (compare [12] p. 1272). Furthermore, it is not clear who measured whom. Using quantum controlled-not gates, it can be shown that information flow in two coupled two state systems also depends on the choice of basis (see Zurek in [18] p. 720). A quantum gate is a quantum mechanical realization of a logic gate, that uses unitary operations on (usually) finite systems (see Nielsen [9] for an introduction). The controlled-not, or c-not, is a generalization of the logic XOR gate that acts on the composition of two subsystems with basis states { 1, 0 } (see figure 1). It has a matrix representation of the form U CN = , (2.14) with respect to span { 11, 10, 01, 00 }. (2.15) Unitary evolutions like this can be generated by appropriate interaction Hamiltonians. The first system can be interpreted as a control, switching the state of the second system - the target - if its state is 1 and leaving it unaffected if its state is 0. A c-not is therefore a simplified model of a premeasurement process 4. Since the state of the 4 With the identifications control = system and target = apparatus.

17 2.3 the many-worlds interpretation 11 x x y x x + y x y XOR x + y Figure 1: A quantum c-not as a generalization of the logic XOR gate. The operation can be expressed by an addition modulo two. Thus, the control x switches the state of the target y if x = 1, and leaves it unaffected if x = 0. target is the only one that changes, the direction of information flow seems to be obvious, namely from the first system to the second. However, a transformation to the conjugate basis, ± = ( 1 ± 0 )/ 2, yields Ũ CN = B T U CN B = , (2.16) where B is the matrix which transforms U CN to the basis B = { ++, +, +, }. (2.17) As a result of basis ambiguity, the matrix representation shows that system one and two changed their roles of control and target: If the state of the second system is it switches the state of the first system and leaves it unaffected otherwise. Thus, the direction of information flow depends on the choice of basis, or rather, on the initial states of system and apparatus (compare [18] p. 720). The second problem of quantum measurements is the problem of definite outcomes. Von Neumann s scheme does not address the question why we cannot observe a superposition of pointer states. Theories which aim to describe classical reality as an emergent phenomenon of quantum physics have to give a solution to both of the presented problems. The problem of definite outcomes 2.3 the many-worlds interpretation In his famous paper Relative State Formulation of Quantum Mechanics from 1957 (see [5]), Hugh Everett III took a next step from what was already suggested by von Neumann s premeasurement. In order to find a reformulation of quantum mechanics that would be better suited for an application to general relativity, Everett claimed that in principle, there would be no reason to restrict the validity of quantum mechanics to the description of isolated microscopical systems. In analogy to von Neumann s approach, Everett investigated Relative State Formalism

18 12 interpretations of quantum mechanics The splitting universe Shortcomings and ontological implications a compound of many interacting systems, whose time-evolution is given by Schrödinger s equation. Consider for example the following unitary process with N coupled subsystems: ( ) N ψ = α i j αi j j i=1 c j1,...,j N α (1) j 1 α (N) j N ψ T. j 1,...,j N (2.18) Everett recognized, that the resulting state could be rewritten in terms of relative states: ψ T = N i α (1) i ψ rel. ; α (1) i, i where ψ rel. ; α (1) i = 1 N i c i,j2,...,j N α (2) j 2 j 2,...,j N α (N) j N, (2.19) with some normalization constant N i. Accordingly, each quantum state of a system which is part of a composite system could only be interpreted as being relative to the other parts of the composite system. Another way of interpreting this result was given by Bryce DeWitt in 1970 (see [2]). If one extends Everett s formulation to the whole universe, which is an isolated system by definition 5, states of quantum systems would became relative to the states of the rest of the universe (compare to [12] p. 1288). Different quantum states were then realized in different branches of the universe. This is the origin of the term many-worlds interpretation. A (projective) measurement performed on a two state system like Schrödinger s cat system would lead to a splitting of the universe into different worlds. One for each possible measurement outcome (the cat is alive or the cat is dead). Unfortunately, the problems discussed in this chapter still remain unsolved. There is no explanation for the selection of a preferred basis, and no explanation for the definite outcomes of experiments. Nevertheless, the many-worlds interpration was a crucial conceptual breakthrough (Zurek in [18] on page 717), because quantum mechanics got reinstated as a basic tool in the search for its interpretation (ibidem). This tool will be used implicitly throughout the following chapters, and it will be used without discussing its ontological implications. Tegmark summarized in [14] that, confronted with the question of what is physically real, one has to choose between two paradigms. Either the outside view, the universal wave function with its superposition of a tremendous number of possible states is physically real, or the inside view, the perception of one branch. However, this work will be more pragmatic and does not address this question. Instead, the relevant question will be: Why do we seem to perceive things with objective existence? 5 There is no outside.

19 D E C O H E R E N C E, E I N S E L E C T I O N A N D Q U A N T U M D A RW I N I S M 3 A first step towards a solution to the problems of the previous chapter was achieved by the so called decoherence program. Decoherence in quantum mechanics is the destruction of phase relations between parts of a superposition state, and hence a destruction of quantum correlations between those parts. In this sense, the focus on decoherence is the focus on the question what is the effect of environmental monitoring on quantum systems. From a classical point of view, the focus has always been on isolated systems 1, which revealed fundamental physical properties, whereas interactions with the environment was usually viewed as a kind of disturbance (compare [12] p. 1273). In this chapter I want to give an example for an exactly solvable model of decoherence to illustrate the effectiveness of decoherence and how the concept of decoherence leads to the so called einselection, a concept that will be motivated in the second section. This will lead to a spin environment toy model which will be investigated numerically throughout this document. The decoherence program 3.1 the concept of partial traces The discussions in this chapter are all based on physical problems with interacting quantum systems. As the Hilbert space is a tensor product space in these cases, the standard textbook quantum statistics will also be defined on this space. Sometimes however, it can be interesting to ask for a more compact description. For example, consider the composition of a system S and its environment E. Do we really need to know everything about E in order to describe the statistics for measurements on S? Luckily, this question can be answered with a no. The appropriate concept is called the partial trace of density matrices. The idea behind partial traces is, that tracing has something to do with averaging. This is familiar from the computation of expectation values of operators. In this case, the expectation value of an operator M given the corresponding density matrix ρ is determined by A compact description of measurement statistics Tracing as averaging process M = Tr Mρ. (3.1) 1 Or systems with a well-defined and controllable interaction with the environment. 13

20 14 decoherence, einselection and quantum darwinism Now suppose the operator M = M id B acts on the tensor product of the systems A and B with the density matrix ρ = ρ AB. The statement is now, that M = Tr Mρ AB = Tr Mρ A = M, (3.2) with ρ A Tr B ρ AB. (3.3) The right-hand side of (3.3) is the partial trace, as mentioned earlier. Because M has no effect on the subsystem B, it doesn t matter when the trace with respect to B is applied. This is the intuitive justification. A proof that the partial trace is the unique operation fulfilling equation (3.2) can be found in appendix A. If { a i } i IA and { b i } i IB are orthonormal basis systems of A and B, the partial trace can be calculated as follows: a i ρ A a j = k I B b k ρ AB b k. (3.4) 3.2 decoherence and einselection Environmentinduced superselection The term einselection was coined by Zurek in 1982 (see [17]) and is the abbreviation for environment-induced superselection. Einselection describes the decoherence-imposed selection of a preferred set of pointer states (see [18] p.717), and hence, a possible solution to the preferred basis problem. For a simple example consider three interacting two state system with the role of the system S, the apparatus A and the environment E. A general initial state of the system will be the object of a premeasurement induced by A: (α + β ) A 0 α A 1 + β A 0 φ, (3.5) with orthogonal pointer states A 0 and A 1. As discussed in the previous chapter, this state can be rewritten in many different choices of basis vectors. In the next step, φ will be the object of a premeasurement induced by E: The process of decoherence ( φ ) ɛ 0 α A 1 ɛ 1 + β A 0 ɛ 0 Φ. (3.6) After the environment correlated with system and apparatus, it is not possible to introduce a new basis to the system such that the apparatus state stays pure, and there is no preferred basis problem between system and apparatus. Besides that, the reduced density matrix of the system and the apparatus shows an interesting evolution in the second premeasurement process: ρ P SA = φ φ = ( α 2 ) αβ α β ( β 2 ) α β 2 (3.7) = Tr E Φ Φ = ρsa D,

21 3.2 decoherence and einselection 15 where the matrices are shown with respect to the subspace given by span { A 1, A 0 }. In the calculation of the partial trace after the premeasurement it has been assumed that ɛ 0 ɛ 1 = 0. (3.8) This is a non-trivial assumption, which will be justified in following section. If this assumption turns out to be true, the reduced density matrix of system and apparatus would only contain classical correlations, i.e. the off-diagonal terms vanish. This is the process of decoherence. Environmental monitoring destroys phase relations. Notice that the premeasurement step in (3.6) together with the requirement ɛ 0 ɛ 1 = 0 implies a special form for the Hamiltonian of the apparatus-environment-interaction H AE. To illustrate this, consider an initial preparation of the system in the states The process of einselection The SAE-interaction leads to ± = 1 2 ( ± ). (3.9) (( ± ) A 0 ) ɛ ( A 1 ɛ 1 ± A 0 ɛ 0 ). (3.10) The goal of a measurement process is usually to distinguish possible states of a quantum system. But as one can see from this example, it is impossible for the given interaction to distinguish + and using a measurement performed on the environment in the orthogonal basis { ɛ 0, ɛ 1 }. In contrast, an initial preparation of the system in or can be distinguished perfectly. Moreover, this result is true for any orthogonal basis of the environment, because of the basis ambiguity between environment states ɛ and composite system-apparatus states ψ (compare Schlosshauer [13] p. 72): 1 2 ( ψ 1 ɛ 1 ± ψ 0 ɛ 0 ) = 1 2 ( ψ 1 ɛ 1 ± ψ 0 ɛ 0 ) =... (3.11) Thus, due to the special form of interaction, there is no information encoded in the environment, that could be used to distinguish + and. Another way of looking at this process argues in terms of entanglement. The state Ψ = 1 2 ( + + ) = (3.12) will not generate any entanglement with the environment, because, as defined above, it is unaffected by the interaction (compare [13] p. 71 ff.): ( Ψ ) ɛ 0 Ψ ɛ 1. (3.13) An interaction like this is sometimes referred to as non-demolition measurement (ibidem). The phase relations between + and keep intact, not affected by decoherence. On the other hand the initial states

22 16 decoherence, einselection and quantum darwinism Definition of einselection ± develop strong entanglement with the environment, resulting in indistinguishability between the two possible initial configurations. To summarize these results, the interaction of a quantum system (and an apparatus) with the environment leads to a selection of states, which are least affected by the environmental monitoring. In Zurek s words: Einselection is this decoherence-imposed selection of the preferred set of pointer states that remain stable in the presence of the environment [18]. In the model discussed above, the pointer states can be characterized in terms of a commutation relation. To leave the pointer states unaffected, the pointer observable A = a 0 A 0 A 0 + a 1 A 1 A 1 should commute with the Hamiltonian H AE, that describes the interaction between the apparatus and the environment (see [18] p. 722): [H AE, A] = 0. (3.14) This relation can be used in the search for pointer states. For more realistic models however, one has to look for different criteria to determine the least affected states. This will be this discussed later in this document. 3.3 decoherence of a single qubit A decoherence toy-model An exactly solvable model for decoherence was given by Zurek and Blume-Kohout in [18] and [1]. A quantum system S represented by a single qubit interacts with a spin environment E, whose self-hamiltonian vanishes. The interaction of system and environment has the form H SE = S z N k=1 g k S z k (3.15) with spin operator S z subspace operators = h 2 σz and the coupling constants g k to the Sk z = id 2 id }{{} 2 S z id 2 id 2. (3.16) }{{} (k 1) times (N k) times S z is diagonal in the natural basis {, }. Tensor products of diagonal matrices are diagonal and hence H SE is diagonal in the natural basis of the tensor product space: H SE = h2 N 4 ( ) g k ( k k k k ). (3.17) k=1 The matrix exponential is then given by Ht i e h = N k=1 [ e i g k h 4 t ( k k + k k ) + e +i g k h 4 t ( k k + k k ) ]. (3.18)

23 3.3 decoherence of a single qubit 17 Therefore, one obtains the following simple form for the time evolution operator: Time evolution operator where U (t; t 0 ) = ˆɛ (t t 0 ) + ˆɛ ( (t t 0 )), (3.19) ˆɛ (t) = N k=1 ( e i g k h 4 t k k + e i g k h 4 t k k ). (3.20) A general initial state is given by the following equation: Φ (0) = (a + b ) Time evolution of this state results with N k=1 (α k k + β k k ). (3.21) Φ (t) = U (t) Φ (0) = a E(t) + b E( t), (3.22) E(t) = ˆɛ (t) = N k=1 N k=1 (α k k + β k k ) ( α k e i g k h 4 t k + β k e i g k h 4 t k ). (3.23) Φ (t) has the expected relative state form. This leads to the density Density matrices matrix of system and environment: ρ SE = Φ (t) Φ (t) ( a 2 E(t) E(t) = {, } a b E( t) E(t) ) a b E(t) E( t) b 2. E( t) E( t) (3.24) Then, the reduced density matrix for system can be computed by tracing out the environment: ρ S = Tr E Φ (t) Φ (t) ( a 2 a b rs (t) a b r S (t) b 2 = {, } ), (3.25) with the off-diagonal correlation amplitude r S (t) = E(t) E( t). (3.26) Correlation amplitude r S (t) can be expanded, using the definitions above: r S (t) = = N k=1 N k=1 ( α k 2 e i g k h 2 t + β k 2 e i g k h 2 t ) ( ) gk h ( cos 2 t + i α k 2 β k 2) ( ) gk h sin 2 t, (3.27)

24 18 decoherence, einselection and quantum darwinism with the absolute value r S (t) 2 = = N k=1 N k=1 cos 2 ( gk h 2 t ) ( + α k 2 β k 2) ( ) 2 sin 2 gk h 2 t ). 1 4 α k 2 β k 2 sin 2 ( gk h 2 t (3.28) Because α k 2 + β k 2 = 1 the correlation amplitude is always r S (t) 2 1. (3.29) Recurrence times and exponential efficiency Each trigonometric function in r S (t) 2 has the period T k = 4π g h. There- fore the whole system returns to its initial state after the recurrence time, which is - if existent - the least common multiple of the periods T k. If the periods T k are not commensurable, r S (t) 2 becomes almost periodic with the time for almost recurrence τ. For very large systems τ approaches infinity: τ for N. (3.30) Averaging r S (t) 2 over the recurrence time for large systems results 1 τ lim r(t) 2 dt 2 N τ τ N 1 + ( α k 2 β k 2) 2 0 k=1 (3.31) It has been made use of the fact that r S (τ) 2 is close the initial configuration, so that the trigonometric integrals contribute each a factor of 1 2. Decoherence in this sense is therefore exponentially effective (see [18] p. 731). 3.4 entropy and information Entropy production in decoherence processes The previous section demonstrated how environmental monitoring completes a von Neumann measurement process and justified the assumption ɛ 0 ɛ 1 0 (3.32) for the orthogonality of the relative states of the environment, that was made in section 3.2. Thus, the environment selects preferred states by decohering superpositions of this special pointer states 2. In the example of the previous section, these pointer states were eigenstates of the spin z-direction of the system. If we go back to the SAE triple of section 3.2, in the transition from ρ P SA ρd SA, (3.33) a general pure state will decohere into a mixture, unless the pure state coincides with one of the pointer states. This can be quantified 2 If the interaction between system and environment has an appropriate form.

25 3.4 entropy and information 19 in terms of an increasing entropy. A quantum mechanical analog to the classical formulation is the von Neumann entropy of the density matrix, defined by S(ρ) Tr (ρ log 2 ρ). (3.34) The trace is invariant under similarity transformations and hence, S(ρ) can be calculated in the eigenbasis of ρ (with N eigenvalues p 1,..., p N ): S(ρ) = N i=1 p i log 2 p i. (3.35) An interesting result, because it connects the quantum entropy formulation with concepts of classical information theory. As derived in appendix B, the amount of information gained in a probabilistic experiment by obtaining an event E which occurs with probability p, can be quantified with A connection to classical information theory I(p) = log 2 p. (3.36) If the logarithm uses base 2, the unit of information content is the bit. So, if one considers an experiment where N distinct events occur with probabilities p 1,..., p N, the average amount gained in one experiment is simply H = N i=1 p i log 2 p i. (3.37) In classical information theory, this quantity is known as Shannon entropy. The higher the Shannon entropy, the higher the ignorance about the outcome of the experiment. H is limited by the value for highest ignorance, i.e. p i = 1/N: H N i=1 1 N log 2 1 N = log 2 N. (3.38) Application to The connection between entropy and information offers a deeper insight to the process of einselection. In the SAE triple from section 3.2, the pure premeasured density matrix ρsa P will keep a pure density matrix ρsa D despite of environmental monitoring, if the system was initially in one of the einselected states. For example, the state of the system will transform to einselection - a selective loss of information A 0 ɛ 0 A 1 ɛ 1, (3.39) which has a pure ρ D SA. From the definition of von Neumann entropy follows that S(ρ) = 0 for all pure ρ. In other words, the premeasurement performed by the environment will not change the state of knowledge about the system in this scenario. However, superposition states of and result in mixed density matrices and are characterized by an increase of entropy. With the developed justification, an increasing entropy can be interpreted as rising ignorance about

26 20 decoherence, einselection and quantum darwinism Figure 2: Mutual information Venn diagram (Graphic by KonradVoelkel via Wikimedia Commons, released into the Public Domain). This graphic can be used to derive useful relations between the mutual information content of X and Y, the conditional entropies and the joined entropy. Mutual Information the state of the system and hence, einselection is the selective loss of information everywhere except in the pointer states (see Zurek [18] p.733). It is attainable and practical to further generalize the information interpretation. In an experiment, the usual situation is to ask the measurement apparatus for information about the system. All situations in this chapter were constructed this way. Once more it is appropriate to borrow from classical information theory and ask the question: What is the amount of information, that can be obtained about the system by measuring the pointer state of the apparatus? In the classical case, this question can be answered by means of the Shannon entropy. Let X and Y be two random variables. The mutual information content of X and Y is a quantity, that measures the amount of information X and Y (compare [9] p.506). If one adds the information contents of X and Y, H(X) and H(Y), one has to subtract the intersection of information that is common to both, X and Y (ibidem): I(X : Y) H(X) + H(Y) H(X, Y), (3.40) where H(X, Y) = p(x, y) log 2 p(x, y) (3.41) x,y

27 3.4 entropy and information 21 is the joined entropy of X and Y. Correspondingly, p(x, y) is the probability to obtain the instantiations x of X and y of Y. The mutual information can be reformulated using conditional entropies: H(X Y) = y = y = y p(y)h(x Y = y) p(y) x p(y) x =H(X, Y) + y p(x y) log 2 p(x y) p(x y) log 2 p(x, y) p(y) p(y) log 2 p(y) p(x y) x } {{ } =1 (3.42) H(X Y) = H(X, Y) H(Y). With this result, the mutual information can be expressed by I(X : Y) = H(X) H(X Y). (3.43) This new way of writing I(X : Y) elucidates the meaning of mutal information as it is the amount of information the ignorance about X has reduced given Y. An illustration of the connection between joined entropy, conditional entropy and mutual information is shown in figure 2. It can be proven (see for example Nielsen [9] p.507) that The classical limit of mutual information 0 H(X Y) H(X) and hence, I(X : Y) 0. (3.44) Mutual information is therefore limited to I(X : Y) min {H(X), H(Y)}. (3.45) In the quantum mechanical analog, mutual information between to subsystems A and B can be defined by Quantum mutual information I(A : B) = S(ρ A ) + S(ρ B ) S(ρ AB ), (3.46) with the reduced density matrices ρ A, ρ B and ρ AB. In contrast to the classical result, quantum mutual information (QMI) violates the inequality (3.45). Consider for example two spin qubits A and B in a singlet state ψ = 1 2 ( A B A B ). (3.47) This leads to the density matrices ( ) ρ A = ρ B = {, } 2 0 1, ρ AB pure, (3.48) and the mutual information between A and B: { } I(A : B) = 2S(ρ A ) min S(ρ A ), S(ρ B ). (3.49)

28 22 decoherence, einselection and quantum darwinism Asymmetric mutual information A clear violation of (3.45), that has further consequences for the formulation of QMI. The inequality has to be extended to the level { } I(A : B) 2 min S(ρ A ), S(ρ B ). (3.50) Especially, I is in general not equal to the quantities J A (A : B) = S(ρ B ) S(ρ B A ) (3.51) and J B (A : B) = S(ρ A ) S(ρ A B ), (3.52) which is the case for Shannon mutual information. J A and J B are sometimes referred to as asymmetric mutual information If one demands that I(A : B)! = J (A : B), (3.53) it follows that the conditional entropies for the qubit singlet have to be negative, i.e. S(ρ A B ) < 0 and S(ρ B A ) < 0, (3.54) Conditional density matrices which is a very strange consequence. Yet, the definition of ρ A B is indistinct. In 2001, Ollivier and Zurek investigated a possible redefinition of J using complete measurements 3 (see [10]): J (A : Π B ) S(A) S(A Π B ), (3.55) where Π B is a complete set of one-dimensional measurement projectors on subsystem B, i.e. π Π B π = id B. (3.56) The conditional density matrix is then determined by 4 ρ A π = π ρab π Tr AB π ρ AB π Π B, (3.57) in analogy to the measurement procedure derived in section 2.2. This gives a reasonable definition for the conditional entropy S(A Π B ) = π Π B p π S(A π), (3.58) Quantum discord with probabilities p π = Tr AB π ρ AB. Ollivier and Zurek proposed, that the difference between the classical equivalent quantities I and J could be used as a measure of classicality called quantum discord: D(A Π B ) =I(A : B) J (A Π B ) =S(B) S(AB) + S(A Π B ). (3.59) 3 Now and in the following the abbreviation S(X ) S(ρ X ) will be used. 4 Implicitly, π acts now via id π on ρ AB.

29 3.5 application of mutual information 23 Once again, consider the SAE triplet of section 3.2. After the environment completed the decoherence process, the reduced density matrix of system S and apparatus A was given by ( ) ρsa D = α β 2, (3.60) where the matrix is shown with respect to the subspace given by span { A 1, A 0 }. It follows that S(S : A) = S(SA) = S(S) = S(A). (3.61) Let Π A be the set of orthogonal projections onto the apparatus pointer basis. Then by definition ( S(S Π A ) = π ρsa D π log π ρsa D π ) 2 = Tr =0, π Π A Tr {( α 2 log p π 0 β 2 log 2 1 )} (3.62) which is in agreement with the classical conclusion that I = J, i.e. the quantum discord vanishes. In other words, decoherence reduced the amount of mutual information to a level, which is classically allowed from inequality (3.45). At the same time, the conditional entropy, which was initially negative by classical assumptions, was raised to non-negative values (compare Zurek [18] p.734). 3.5 application of mutual information To get a first impression of how mutual information behaves in spin environments, consider again the exactly solvable model from section 3.3. The interesting quantity is the mutual information between one spin of the environment and the system spin qubit. Several density matrices have to be computed. Let E m be the reduced universe consisting of one environmental spin. The corresponding density matrix is given by Mutual information in spin environments with ρ Em = Tr E\Em Tr S ρ SE E(t) E(t) = = Tr E\Em ( a 2 E(t) E(t) + b 2 E( t) E( t) ) k=1 ( N E( t) E( t) = ( ) N α k 2 c.c. e i g k h 2 t α k β k β k 2 k=1 α k 2 c.c. e i g k h 2 t α k β k β k 2 k ) k (3.63) (3.64) (3.65)

30 24 decoherence, einselection and quantum darwinism and one obtains α ρ Em = ( m 2 ) c.c. α mβ m a 2 gm h ei 2 t + b 2 gm h e i 2 t β m 2 (3.66) with eigenvalues λ ± E m = 1 2 ± 1 2 and the correlation amplitude ( α m 2 β m 2 ) α m 2 β m 2 r Em (t) 2 (3.67) The N = 1 case r Em (t) 2 = 1 4 a 2 b 2 sin 2 ( g m h t) (3.68) 2 The result for the reduced universe of system S can be seen as a generalization of this: λ ± S = 1 2 ± 1 ( a 2 b 2 ) a 2 b 2 r S (t) 2. (3.69) For the N = 1 case 5 the symmetry between both amplitudes is obvious and the eigenvalues of system and environment show the same time dependend behaviour. Due to this synchronicity the von Neumann entropy of the system and the entropy of the environment vanish simultaneously at times t = n 2π g h with n Z and so does the mutual information between both, which is determined by I(S : E) = S(S) + S(E) S(SE). (3.70) }{{} =0 (pure) The mutual information reaches its maximum when the correlation amplitudes are minimal, which is the case at times t = (2n + 1) π g h with n Z. It is instructive to investigate the mutual information at this specific times as a function of p 1 = a 2 and p 2 = α 2. Figure 3 shows that I(S : E) has a global maximum for p 1 = p 2 = 0.5, which indicates that maximal ignorance about the initial eigenvalue of S z leads to a build up of perfect correlations. If one of the two spins would initially be polarized in S z direction, one won t be able to obtain information about one by investigating the other. A result which is intuitively clear, because the interaction Hamiltonian generates rotations about the z-axis. 3.6 quantum darwinism Environment as a witness Since the first section of this chapter the focus changed more and more from the system to the environment. So far, two important observations were made. The first observation is, that the environment decoheres the system. But not all system states are influenced in the 5 One spin in the system, one in the environment.

31 3.6 quantum darwinism 25 Figure 3: Mutual information as a function of p 1 = a 2 and p 2 = α 2. The global maximum at p 1 = p 2 = 0.5 indicates that maximal ignorance about the initial eigenvalue of S z leads to perfect correlations. same way. A small set of pointer states remains pure, despite of decoherence. This is einselection. The second observation is related to the flow of information. During the process of einselection and decoherence, the environment learns something about the system. It gathers information in the form of correlations with the system, that can be quantified with the tool of mutual information. Single fragments of the environment contain a record of the systems state and the observer can ask the environment for information about it. This is know as environment as a witness approach (see Zurek [18] p.758). But there is still one important property of classicality missing. And that is objectivity. Ollivier, Poulin and Zurek tried to give a physical definition of objectivity inspired by the EPR criterion of reality (see chapter 1). In [11], they write: A property of a physical system is objective when it is Redundancy and mutual information 1. simultaneously accessible to many observers, 2. who are able to find out what it is without prior knowledge about the system of interest, and 3. who can arrive at consensus about without prior agreement. If different observers have access to different fragments of the environment and all these fragments contain enough information to determine the pointer state of the system, criteria 1-3 are all fulfilled. In other words, the same information has to be redundantly imprinted in the environment in order to achieve objectivity. This can again be quantified using mutual information. Redundancy can be defined as the number of disjoint fragments of the environment, that contain nearly full information about the state of the system (cf. [11] p. 6): R δ max {F j} {#F : I(S : F j ) 12 (1 δ)i(s : E) }, (3.71)

32 26 decoherence, einselection and quantum darwinism where δ represents some error tolerance. The maximization has to be carried out over all disjoint partitions { } F j of the environment E, such that E = #F i=1 F i. (3.72) Usually, measurements on nearly the whole E are required to obtain the maximal allowed quantum information: I(S : F) 2 S(S) for F E. (3.73) Partial Information On the other hand, the classical allowed level of I(S : F) S(S) can be reached more easily. The reason for this is the fact, that the observer has only access to parts of E. Complex quantum correlations between the system and various parts of the environment are hidden to him, unless he can measure all of E. Redundancy is therefore measured at the classical level. The factor 1 2 in equation 3.71 takes care of this. A basic tool in analyzing redundancy is the investigation of partial information plots. I will follow the introduction to partial information plots given by Blume-Kohout and Zurek in [1]. Consider an environment E that can be separated into N subsystems. Let E {m} be the set of all fragments of size m, i.e. these fragments contain m subsystems, and E {N m} the set of all fragments of size N m. For every element of the one set there exists a unique element of the other, such that the union of both elements is equal to E: E {m} = E {N m}. (3.74) For E {m}, the partial information I(m) should represent the average mutual information of all fragments of size m: I(m) ( ) N 1 m I(S : F). (3.75) F E {m} Now consider, that SE is in a pure state and E = E 1 E 2. For this case, mutual information has a nice property. It can be derived from the triangle inequality for entropies, sometimes called Araki-Lieb inequality (a proof can be found in [9] p.516). If ρ A and ρ B are reduced density matrices of ρ AB, then S(ρ AB ) S(ρ A ) S(ρ A ). (3.76) Application of this inequality to the pure SE yields S(SE) = 0 S(S) S(E) (3.77) S(SE) = 0 S(SE 1 ) S(E 2 ) (3.78) S(SE) = 0 S(SE 2 ) S(E 1 ). (3.79)

33 3.6 quantum darwinism 27 Therefore, the mutual information of S and E is given by Further, I(S : E) = 2 S(S) = 2 S(E). (3.80) I(S : E 1 ) + I(S : E 2 ) = 2S(S) + S(E 2 ) S(SE 1 ) + S(E 1 ) S(SE 2 ). }{{}}{{} =0 =0 This means, mutual information is additive, i.e. I(S : E 1 ) + I(S : E 2 ) = I(S : E). (3.81) Additivity of mutual information The additivity of mutual information has consequences for the shape of the I(m) graph. It follows, that ( ) N 1 I(m) + I(N m) = m I(S : F) F E {m} ( ) N 1 + N m I(S : F) F E {N m} ( ) N 1 = m I(S : F) + I(S : E \ F) F E {m} ( ) N 1 = m I(S : E) F E {m} =I(S : E). (3.82) Besides the properties that were proved above, it has been used that ( N m ) = ( N N m ). The relation implies that I(m) has to be antisymmetric around m = N/2 with ( ) N I = I(S : E) = S(S). (3.83) Redundancy in The antisymmetry restricts the possible shapes of partial information graphs. As mentioned above, the quantum sector partial information plots I(m) > S(S) (3.84) will usually be reached only when the observer has access to a large part of the environment. From antisymmetry follows, that this will usually be the case in the region with m > N/2. Furthermore, antisymmetry gives a feeling of how redundancy will look like. If a system state has a highly redundant imprint in the environment, a small fragment of the environment will be sufficient to obtain I(m) (1 δ)s(s). (3.85) Thus, I(m) will increase quickly for small m, reach a plateau at I(m) S(S) and will stay there until m approaches N and the quantum sector gets accessible.

34 28 decoherence, einselection and quantum darwinism Quantum darwinism If the interaction between a system S and its environment E leads to redundancy, there are many copies of the same information about S available in E. But only pointer states can survive the monitoring by the environment and can produce descendants in E. Zurek understands this process in analogy to natural selection, and he coined the term quantum darwinism to accentuate this connection.

35 N U M E R I C A L S T U D I E S 4 The main emphasis of this work is the investigation of emergence of classicality in spin 1 2 systems. A special focus is put on the transition from simple system apparatus interactions like Ising Interaction to more complex situations with anisotropic Heisenberg Hamiltonians. The standard model used throughout this work is based on the triple of a system S, an environment E and observers O, which have access to E. If O interacts with E, the interaction is modeled from O s perception (his branch of the universe) using projective measurements. On the other hand, S-E- and E-E-interaction for a total number of N spin- 1 2 particles should be given by the Hamiltonian The standard model H int = N i,j=1 J x i,j Sx i Sx j + J y i,j Sy i Sy j + J z i,j Sz i Sz j, (4.1) which acts on the Hilbert space H = ( C 2) N with coupling constants J i,j. S x i, Sy i and S z i represent the spin operators for the x, y and z direction on the subspace H i = id 2 id }{{} 2 C 2 id 2 id 2. (4.2) }{{} (i 1) times (N i) times H int is the fully anisotropic Heisenberg Hamiltonian. Special cases are 1. Ji,j x = J y i,j = Ji,j z, and the the coefficients are only nonzero for nearest-neighbors. Then, H int is the usual (isotropic) Heisenberg Hamiltonian. 2. J x i,j = J y i,j = 0, and the the coefficients J z i,j are only nonzero for nearest-neighbors. Then, H int is the usual Ising-ZZ Hamiltonian. Unless specified differently, the self-hamiltonians H S and H E are neglected in the so-called quantum measurement limit, where the interaction Hamiltonian dominates the evolution of the quantum system (cf. [13] p.76) and the total Hamiltonian is approximately H H int. (4.3) 4.1 implementation notes All numerical calculations in this chapter were carried out in Mathematica 9. For the code and its documentation please contact Fabian.Lux@rwth-aachen.de 29

36 30 numerical studies E 7 E 1 E 6 S E 2 E 5 E 3 E 4 Figure 4: The Spin Star-Environment. In this numerical study, the central system S and all fragments of the enviroment E consist of single spin qubits. Various interactions between S E and E E can be activated. Interactions with observers O are modeled as projective measurements from O s perspective. 4.2 the spin star-environment A generalization of Zurek s spin model The first object of research is a generalization of Zurek s model from section 3.3, where the system S consists only of a single qubit. For a reduction of numerical complexity, H int is assumed to be isotropic in x and y direction, i.e. J x i,j = Jy i,j Jxy i,j i, j N. (4.4) The interaction Hamiltonian can then be rewritten in the form H int = N i,j=1 Ji,j z Sz i Sz j + Jxy i,j 2 ( S + i S j + S i S + j ), (4.5) with the spin creation and annihilation operators S + i = S x i + is y i (4.6) S i = S x i is y i. (4.7) Because of its x-y-isotropy, the simplified Hamiltonian is invariant under rotations in the x-y-plane, meaning that the total z-spin is conserved under these transformations, i.e. S z = Si z (4.8) i [H int, S z ] = 0. (4.9)

37 4.2 the spin star-environment 31 Figure 5: The left matrix plot shows the characterstic self-similiar structure of H int for N = 10 in the natural tensor product basis, which is sorted by S z eigenvalues. The right plot is the corresponding time evolution operator, whose entries are shown as absolute values. proof. The fundamental commutation relation is [ S α, S β] = i h ɛ αβγ S γ. (4.10) It follows directly that [H int, S z ] i,j k [ S x i Sx j + S y i Sy j, Sz k = [Si x, Sz k ] Sx j + Si x k = i h = 0, ] [ ] S x j, Sz k + [ S y i, ] [ ] Sz y k S j + S y i S y j, Sz k ( ) S y i Sx j + Si x Sy j S y i Sx j Si x Sy j which completes the proof. This S z conservation law can be used to block diagonalize H int, i.e. H int = H 1 H k. (4.11) One can take advantage of this structure in the computation of the time evolution matrix exponential: ( exp ī ) ( h H intt = exp ī ) ( h H 1t exp ī ) h H kt. (4.12) Block diagonlization of H int and the choice of coupling constants The block diagonal structure of H int and the associated time evolution is shown in figure 5. In the following simulations, all non-zero coupling constants J are chosen randomly from a Gaussian distribution J N ( µ, σ 2), (4.13)

38 32 numerical studies General remarks on implementing spin Hamiltonians with mean value µ = 1 and standard deviation σ = 0.1. Some coefficients are fixed to zero in order to describe the geometrical situation shown in figure 4. I will not explain all details of the implementation in Mathematica (for further information please see the documentation of the code) and confine myself to a few remarks on the crucial points. The implementation of the Hamiltonian is the most sophisticated part and therefore the one I should spend a few words on. In a first step, the basis states are created as all N-tuples consisting of 1 s and 0 s (spin configurations up and down in z-direction). All these states are numbered in a well-defined order. For example, state 1 is chosen to be the state, were all spins show up in z-direction: (4.14) To obtain a matrix representation of the Hamiltonian H, the easiest way is to look at the action of H on the basis states. This scheme can be broken down to the level of tensor subspace operators, for example: S x i Sx j = 1 4 (S+ i + S i )(S + j + S j ) = 1 4 (S+ i S + j + S + i S j + S i S + j + S i S j ) = , (4.15) where the matrix is shown with respect to the subspace } span { i j, i j, i j, i j. (4.16) All other relevant operators can be derived in a similar way and the action of H is well-defined. In general, the action of H on a basisket j will lead to a superposition of these kets: H j = N H i,j i. (4.17) i=1 Then, a readout of the coefficients H i,j gives the column j of H with respect to the chosen basis. All other steps are more or less straightforward to implement Decoherence in the Bloch Sphere A closer look at decoherence If the system S only interacts via Ising-ZZ, i.e. J xy ij = 0, the S z expectation value of S will be a conserved quantity. Thus, time evolution

39 4.2 the spin star-environment 33 Figure 6: Decoherence in the Bloch Sphere. This histogram shows the effect of decoherence on the expectation values S x and S y for N = 8. The z-axis represents the relative number of counts per bin. of the systems state will be restricted to a slice of the Bloch sphere 1 where S z is constant. But the interaction of S with E will also decohere superpositions of z-kets. If decoherence is complete: S x = S y = 0. (4.18) This process can be visualized with a numerical calculation: Initially, S is polarized in x-direction, whereas each E spin is randomly initialized in a pure state. Subsystems of E don t interact. Then, the time evolution of SE is calculated with a step size of δt = 1 10 h J ij, (4.19) whereas the total simulation time is sufficiently larger than the characteristic time of the system 1/( h J ij ). At every instance of time, the values of S x and S y are stored in a two-dimensional array. Figure 6 shows the results as a histogram for 2 S x / h and 2 S y / h. The z-axis represents the relative number of counts per bin. Additionally, the whole simulation can be monitored in Mathematica by plotting the trajectory of S x and S y tuples in the S z -plane. While monitoring the process, one observes that the initial position ( S x = h/2, S y = 0) rapidly decoheres to the region close to the origin. Furthermore, the dynamic of the system seems to slow down when the expectation values approach (0, 0). The reason for this phenomenon is related to the time evolution of the systems density matrix: i h t ρ S (t) = [ ] H eff (t), ρ S (t). (4.20) The effective Hamiltonian H eff (computed from H int ) will commute with z-spin of the system. If the off-diagonal terms of ρ S are of order O(ɛ). This will also be true for the off-diagonal elements of the The setup Freezing dynamics 1 For an introduction to the concept of the Bloch sphere see for example [12] p.295.

40 34 numerical studies Figure 7: The quantum halo. This density-histogram shows the effect of decoherence on the expectation values S x and S y for N = 8. The classical point (0, 0) is enveloped by a quantum halo. A brighter color represents a higher probabilty of catching the system at this point. The quantum halo. commutator, whose diagonal elements are equal to zero. In the limit ɛ 0, the dynamic of the system freezes because of equation 4.20 and thus, it will be more likely to find the system in the neighborhood of (0, 0). This is the essence of figure 6. Smoothing the histogram and generating its density results in figure 7, which has a colorful interpretation. The system can be found in a pointer state, when S x = S y = 0, but this point can never be reached due to the freezing dynamics. Thus, the classical pointer states are enveloped by a quantum halo (cf. [18] p.731), nicely represented by figure 7. Catching the system in a Schrödinger-cat state far away from (0, 0) is very unlikely, because this states are very fragile Correlation Amplitudes Correlation functions In analogy to chapter 3, the focus will now change from the system to the environment. Based on the discussions of the previous chapters, the first question that raises is the question how, and if classicality emerges in these special spin system. But this question ultimately leads to the next: How do we measure, what can we measure? A first intuitive answer uses correlation functions. The idea is that these measure could, in principle, be tested in an experiment. The correlation function is given in the form C α,β (S E) = S α S Sβ E, (4.21) where SS α is the α-component of the system spin and Sβ E is the β- component of one environment spin. Figure 8 shows the three functions C zx, C zy and C zz for two different types of interaction. The first interaction is a simple Ising zz-interaction between the system and

41 4.2 the spin star-environment 35 Figure 8: Correlations in the Spin Star-Environment. For same initial configurations, the left plot shows the case with zz-ising being the only SE-interaction, whereas the right plot allows all sort of interactions between S E and E E. the environment. These interaction is well understood, because it is the analytical solvable model of chapter 3. There is a periodic rise and fall of correlations with a periodicity that depends on the chosen coupling constant. This plot is just an illustration for the von Neumann premeasurement, performed by the environment, while the observer does not interact with the composite system of S and E. Also in accordance with previous considerations is the fact, that there is no correlation between the z-components of system- and environment-spin. That is simply because the interaction commutes with these observables and leaves them unaffected, i.e. the premeasurement cannot establish correlations. Since there were no correlations existent initially, no correlation C zz can be seen in the first plot of figure 8. From the type of interaction it is clear that the interesting pointer observable is the spin in z-direction. Thus, these results are no surprise. The second plot of figure 8 is far more complicated. In contrast to the first plot, anisotropic Heisenberg interaction between S E and E E is allowed, while the coupling within the environment is a factor of 10 weaker than the dominant S E coupling. As a first observation, the characteristic time scale of the oscillatory graph is much shorter now, while the correlation amplitude itself is nearly a factor of 10 smaller than before. The reason for the different time scale is given by the complex interplay of competitive interactions (and not only due to incommensurable coupling constants). As one can see from the first plot of figure 8, the zz-interactions led to a rise of C zx with a simultaneous decrease of C 2 zy. Complementary interactions will lead to complementary behaviour, adding up to a faster change of correlations and preventing the correlation to raise up to a level of the previously discussed Ising interaction. Von Neumann premeasurement Correlations for anisotropic interactions 2 In fact, the simultaneous decrease could as well be a simultaneous increase for different initial conditions. Important is that both components change.

42 36 numerical studies Figure 9: Characteristic damping of the time dependent mean value for N = 8. For same initial configurations, the left plot shows the case with zz-ising being the only SE-interaction, whereas the right plot allows all sort of interactions between S E and E E. Time dependent correlation mean One general remark can be made about the oscillatory form of the graph, by looking at the mean value C(t) = 1 t t 0 C(t )dt. (4.22) Shortcomings of correlation functions The results for C zx are shown in figure 9. Again on the left, the situation for the simple Ising interaction and on the right the complicated interaction in comparison. Despite the different dimension of amplitudes (again a factor of 10), both plots show a qualitative similar behaviour: lim C(t) = 0. (4.23) t The discussion of correlation functions has a few shortcomings. At first, it is difficult to give an interpretation to the actual value of C. And second, mutual information is better suited for a detailed investigation, because the basic properties were already discussed in chapter Mutual Information Environment as a witness approaches Since a typical observer will use the environment as an information channel to acquire knowledge about the state of the system, the natural motivation behind the choice to investigate mutual information is the question what the environment knows about the system. As pointed out in chapter 3, there are basically two different approaches to this concept. The first is the notion of classical Shannon mutual information which depends on a certain choice of measurement basis. This quantity is equal to the asymmetric mutual information J (S : Π E ) discussed in section 3.4 for a given set of measurement operators Π E acting on E. In the following, this quantity will be referred to as classical mutual information (CMI). The second approach

43 4.2 the spin star-environment 37 Figure 10: Comparison of classical and quantum mutual information for N = 8. The quantum discord vanishes when the pointer basis has developed. is the quantum mutual information (QMI), using von Neumann entropy to generalize from classical information theory to the quantum realm. Both ways of looking at the information content fulfill complementary roles in characterizing the information that is available to the observer. QMI indicates whether there is information available or not. It doesn t make any classification of this information, because it is a basis independent measure. The indicated information can still be encoded in a non-trivial manner, such that the observer cannot access this information with the observables that are available to him. This is were classical mutual information comes into play. In order to compute it, one has to choose some observable of interest. The comparison of both quantities (CMI and QMI) can be used to classify the chosen information channel by the amount of information that can be encoded using this specific observable. One example takes a look at the simple scenario of Ising zz - interaction between system and environment as discussed above. The system spin and one spin of the environment should be initially polarized in x+ direction. Figure 10 illustrates a comparison between the QMI and the CMI of the two subsystems. The CMI is calculated for the z-direction spin observable of the system and the x-direction observable of the environment spin. At the moment the QMI reaches its maximum, the decoherence process is completed and the pointer basis has developed. At the same moment, CMI and QMI coincide, i.e. S x is an optimal environment observable to encode information about the S z state of the system. The difference of QMI and CMI is the quantum discord D, already discussed in section 3.4. So the task of finding a way to decode accessible information led to the same quantity Zurek proposed as a CMI vs. QMI Encoding information The quantum discord

44 38 numerical studies Figure 11: Mutual information density for Ising-ZZ E E interaction for N = 8. The y-axis represents the number of the spin in the environment, the x-axis represents time, while the color symbolizes the mutual information in the unit of bits. Intra-environment interactions measure of classicality. This can also explain why QMI is limited to the classical allowed level below one bit of mutual information, which can usually be exceeded in the quantum realm as impressively demonstrated by the singlet state. In the moment of completed decoherence the quantum discord necessarily vanishes in the pointer basis as seen in section 3.4, which forces the quantum mutual information to respect the classical bound of one bit. So far, the environment spins didn t couple to each other. What happens if they do? Generally speaking, the observer will lose a bit of his ability to obtain information about the system if he is restricted to measurements on single E spins. The information about S will still be somewhere in the environment, but it will be delocalized in correlations that involve a large part of the environment. This will be the object of a further numerical study: All E spins get randomly initialized, the S spin is polarized in + x-direction. Then the quantum mutual information between every single spin in E and the spin S is computed after time intervals δt = Information Recurrence 1, 3h h Jij i (4.24) where h Jij i is the mean coupling parameter between system and environment3. For the first part of this study, E E coupling should be Ising-ZZ, just like the interaction between the system and the environment. But the value of coupling constants should be a factor of ten smaller. There are essentially two observations that can be made. On the one hand, the initial rise of mutual information will decay within the char3 This parameter was chosen to be one in the units of Jij.

45 4.2 the spin star-environment 39 Figure 12: Mutual information density for anisotropic E E interaction for N = 8. The y-axis represents the number of the spin in the environment, the x-axis represents time, while the color symbolizes the mutual information in the unit of bits. acteristic time of of the intra-environment action. On the the other hand, this information can come back to the subenvironments within a time approximately five to six times larger than the characteristic time of the intra-environment action. This recurrence of information is most impressive when all E spins have S z = 0 4. An exemplary result for this case is shown in figure 11. In the second part of the study, anisotropic J xy ij couplings between E spins are allowed as well, while the corresponding coupling constants are still a factor of ten smaller than the ones for S E interaction. This time, there will be no recurrence of information. Figure 12 illustrates the results, again for S z = 0 in E. The reason for this phenomenon is not fully understood. Obviously it is related to the way the subenvironments talk to each other. In order to narrow down the possible explanations, it could be interesting to investigate linear spin chain models which allow a better control of the information flow. Its furthermore important to state that the mutual information of subenvironments might decay, but computations show that the mutual information of the whole environment and the system always tends to be at the level of 2 bit, meaning that the information still has to be out there in the environment, but somehow encrypted in correlations that involve large parts of E. For anisotropic interactions within E there are more possible ways of establishing these correlations which might be an explanation for the phenomenon of suppressed recurrence. Suppressed recurrence 4 For this case, the information transfer from S to E is the most effective.

46 40 numerical studies Figure 13: This partial information plot for N = 8 spins reveals the characteristic shape of redundancy. It was computed for the time when decoherence completed the creation of pointer states Partial Information Redundancy of information Implementation notes Loss of Redundancy In section 3.6, redundancy was presented as a main concept in Zurek s theory of quantum darwinism. A basic tool to investigate redundancy are partial information plots. Using this tool, this part of chapter 4 will investigate the influence of environment interactions on redundancy. In order to compute the partial information, all environment spins are labeled: E = {2, 3, 4,...}, (4.25) where number one is reserved for the spin in S. For this set, the powerset P(E) is computed and the mutual information is calculated for the spin in S and every element of P(E) at a given instance of time. Elements of P(E) are the needed fragments F of E. Then, the mutual information gets averaged for all elements of P(E) with the same cardinality, i.e. all fragments of the same size. For this computation γ represents the average relative coupling strength between S E and E E interaction: γ J(E E) J(S E). (4.26) Any anisotropic interaction is allowed inside the environment, whereas the interaction between system and environment is still Ising-ZZ. In the γ = 0 case, the partial information shown in figure 13 reveals the characteristic shape of redundancy discussed in section 3.6. In figure 13, partial information was calculated for the time where decoherence completed the creation of pointer states.

47 4.2 the spin star-environment 41 Figure 14: The time dependence of partial information for N = 8 spins. The relative coupling strength between S E and E E is γ = 0.1. As usual, the time is chosen with respect to S E coupling units. Redundancy can be found in the range between the characteristic times, i.e < t < For γ > 0, the time dependence of partial information gets more important: 1. γ 1. The dynamics of the S E interaction is much slower than the intra-environmental dynamics and redundancy can be found in the range between the characteristic times of these interactions. An illustration for γ = 0.1 is shown in figure 14. This rise and fall of redundancy was also observed by Jess Riedel, Zurek and Zwolak in [8]. 2. γ 1. Intra-environmental interaction is much faster now and hinders S E from creating redundancy. This is in accordance with the results of the previous section, which revealed a rapid decay of mutual information in the subsystems. Furthermore, the information will be encrypted in large parts of the environment within the characteristic time of intra-environment interactions, which can be seen from the shape of figure 14. The cases discussed in this section were two extreme cases. One case, were all spins in the environment are independent and one case where all spins interact. Between these cases there has to be an intermediary sector, which shows a transition from the redundant shape of partial information plots to the contrary one. In this intermediary sector, the environment decomposes into independent groups of spins. Redundancy is directly related to the number of independent groups and will raise when this number raises. The intermediary regime Random Walk Experiments Until now, observers didn t play a major role in the numerical studies. The abstract information content somewhere in the environment and the selection of pointer states is everything that was examined. This is The observer s role

48 42 numerical studies Figure 15: This graph shows two runs of successively measuring all E spins in x-direction for N = 8. On the y-axis: the probability of finding S in in z-direction. After the first run, the available information in this channel is extracted and the state of knowledge converges. The setup going to change in this section. The aim is to model the measurement process from the observer s perspective 5, and to catch the system on its way to classicality. The simulation begins with a random initialization of E and a +xpolarization of S, both described by the initial state vector Ψ 0. The observer O can perform a measurement on E after the application of a time evolution with an increment δt, where δt is randomly chosen from a normal distribution δt N (µ, σ 2 ), (4.27) Simulation of a measurement process with mean value µ = 10 and standard deviation σ = 0.1, with respect to the characteristic time scale given by the S E interaction. From the observer s perspective, measurements seem to be nonunitary jumps, as described by the projection postulate. An appropriate way of simulating this, is to calculate all relevant probabilities for possible measurement outcomes of the observable which O has chosen. A random number generator decides the outcome with respect to these probabilities. The state Ψ 0 is then updated with respect to the projection postulate, using the projector P that corresponds to the random outcome: Ψ 0 1 N P U(δt) Ψ 0 Ψ 1. (4.28) Ising-Interaction For the case of Ising-ZZ interaction, S z of S has the interesting property of being unaffected by time evolutions U, but with the ability to change via projections of Ψ. A sequence of measurements on 5 From his branch of the universe.

49 4.2 the spin star-environment 43 Figure 16: This histogram shows the result of n = 3500 simulated measurement processes with O successively measuring the whole E in one fixed direction. In every run, the knowledge about S converges to a specific probability p ( ). E will therefore lead to a random walk-like behavior of S z. Every measurement on E will extract some piece of information about S z, eventually leading to a system state that is perfectly predictable. However, if O decides to measure E spins in one fixed direction his ability to extract information is limited to one specific channel. It can be seen from subsection that there are at least two spin directions that decode information about S z. In fact, if O measures the whole environment in a fixed direction of his choice, the state of knowledge about S z (as described by the state vector Ψ ) will converge to a constant value after he completed a first run. If he repeats this process, the probability of finding S in the state or won t change. One exemplary case is shown in figure 15. A statistical analysis of the convergence of knowledge shows that the systems already tends to be in one of its pointer states. In the analysis, the measurement process is simulated n = 3500 times with the same initial conditions (which are randomly chosen in the first step). For every step n, the converged probability of finding S in in z-direction is stored. Figure 16 shows the corresponding histogram with an increased probability of finding S in either or. If O is allowed to measure in the two complementary directions x and y, there exists a measurement protocol that will always 6 lead to a perfect predictable system state. Furthermore, it suffices for O to look only at one subenvironment in order to achieve this goal. The idea is to switch the information channel after every measurement, from one observable to the complementary one. This will successively extract information about S until its state is perfectly predictable, i.e. Measurement of one spin component Statistical analysis Measurements in complementary directions 6 Unless the spins O looks at are all initially polarized in z-direction and develop no correlations with the system.