Entanglement in Open Quantum Systems

Transcription

1 SEKTION PHYSIK DER LUDWIG-MAXIMILIANS-UNIVERSITÄT MÜNCHEN Entanglement in Open Quantum Systems Diplomarbeit von Marc Busse Mai 2006

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3 SEKTION PHYSIK DER LUDWIG-MAXIMILIANS-UNIVERSITÄT MÜNCHEN Entanglement in Open Quantum Systems Diplomarbeit von Marc Busse MAX-PLANCK-INSTITUT Angefertigt am FÜR PHYSIK KOMPLEXER SYSTEME Dresden Aufgabensteller: Zweitgutachter: Priv. Doz. Dr. Andreas Buchleitner Prof. Dr. Harald Weinfurter Eingereicht am 15. Mai 2006

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5 Acknowledgments First of all it s very important for me to thank Andreas Buchleitner, who was not only a chief, but also a friend. I m grateful for his great overview of physics, for his ability to hold the group together, for the inspiring discussions about science and about life, and most of all, for taking care about us. I also enjoyed working together with André, Olivier and Carlos about unraveling entanglement. Special thanks to André for his optimism, that motivated us to go on with our project, and Olivier, Artem, Alexej and Alexey for being good companies beyond physics. For introducing me to entanglement measures, I would like to thank Florian Mintert. Very inspiring was my trip to Warsaw, funded by the project Entanglement measures and the influence of noise by the VW-foundation. I benefited a lot from the constructive collaboration with Marek Kuś and Rafal Demkowicz- Dobrzański. It made the second part of this thesis possible. Very exciting was also my trip to Krakow. I would like to thank Karol Życzkowski for this experience. For being pretty cool colleagues and a wonderful time in Dresden thanks to the whole group. In particular I would like to thank my parents for supporting my efforts over the last 26 years.

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7 Contents 1 Introduction 1 I Dynamics of Entanglement 5 2 Entanglement and its Measures Schmidt Decomposition Entanglement and Nonlocality Mixed State Entanglement Entanglement Measures from Basic Principles LOCC Operations and Entanglement Monotones The Convex Roof Construction Explicit Evaluation of Concurrence and of Entanglement of Formation Quantum State Tomography Open Quantum Systems Quantum Operations Master Equation System-Bath Interaction Models vi

8 CONTENTS vii Thermal Bath Dephasing Two Qubits and Private Baths Solution of the Master Equation Quantum Trajectories Spontaneous Emission Quantum Monte Carlo Method - General Scheme Freedom in the Choice of Jump Operators and the Realization of Different Pure State Decompositions Dynamics of Entanglement Master Equations for Entanglement Perturbation of Entanglement Quantum Trajectories and Entanglement General Method Simulations Outlook I 73 II Detection of Genuine Multipartite Entanglement 75 6 Multipartite Entanglement Generalized Multipartite Concurrence Genuine Multipartite Entanglement (GME) Detection of GME 83

9 viii CONTENTS 7.1 Detection of GME with a Function f(ρ) Detection of GME with Concurrence What Kind of States can be Detected? Maximization of Concurrence Graph States Graph States Maximize Concurrence Outlook II 89

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11 Chapter 1 Introduction Everything flows Virtually the entire state of the art theory of mixed state entanglement is based on static measures. For instance, the spectrum of the partially transposed density matrix is analyzed, or optimizations over the decomposition space of the density matrix are performed. The degree of entanglement at a given moment can then be evaluated. These intricate mathematical notions are not only opposed to our intuition, but they also fail to give direct access to the dynamics of entanglement. When it comes to describe the time evolution of entanglement [1 3] essentially all approaches published so far employ the above static measures, point-wise in time, after propagating the density matrix of the state itself. However, this gives little physical insight in the cause of the observed entanglement evolution. In the first part of the thesis, we present a characterization of entanglement, that is dynamic already by its construction: Mixed state entanglement is given by the average entanglement in a suitably chosen continuous measurement setup. A continuously observed quantum system is forced into a stochastic pure state evolution, a so-called quantum trajectory. The entanglement along a single trajectory can be evaluated in terms of pure state measures. We show that there exists an optimal measurement strategy, which guarantees that the mixed state entanglement is given by the pure state entanglement, averaged over many realizations of the stochastic process. The entanglement at a given time is then identical to the result of the commonly accepted static measures. Since the quantum trajectories can be inferred from a complete detection record, this method is also suggestive of possible measurement setups which might allow to monitor entanglement dynamics in real time. We want to stress that a framework for entanglement dynamics is not 1

12 2 CHAPTER 1. INTRODUCTION only of fundamental interest, it is also highly relevant for quantum applications: Entanglement forms the basis for a variety of protocols, including quantum cryptography [4], quantum teleportation [5], and the one way quantum computer [6]. However, it is experimentally not possible to keep a system completely isolated from its environment. Due to this environment coupling, one has to consider that initially pure states evolve into mixed states. As a consequence of this decoherence process, entanglement is very easily lost! An example of a decay process of entanglement is shown in figure 1.1. This measurement was performed with two trapped ions [7], initially prepared in a Bell state. At different times, the density matrix was measured and an entanglement measure (see Chapter 2) was evaluated. As the offdiagonal entries of the density matrix are redistributed, entanglement fades away. This above measurement was done only with two qubits, and all state of the art experiments in quantum information are performed just with few qubits. However, a commercial application, for instance of quantum computing, would require a coherent manipulation of thousands of qubits. E n ta n g lemen t o f fo rma tio n (a ) 8 ms 2 ms Time (ms) Figure 1.1: Decay of entanglement. Two trapped ions are initially prepared in a Bell state and evolve under spontaneous emission [7]. Circles show the time evolution the state s entanglement of formation. The insets show the entries of the density matrix at short and long times. (Courtesy of Christian Roos.) With such increasing numbers of subsystems, another challenge moves into focus: We have to decide whether the complete system is entangled, or whether some subsystems can be separated from an entangled remainder. This leads to the problem of identifying genuine multipartite entanglement (GME). In the second part of the thesis, we analyze the potential of entanglement measures for the detection of GME. As a result, we obtain a general detection scheme for mixed state GME, based on the generalized multipartite concurrence [8]. Concurrence of a bipartite state was recently obtained experimentally with a single measurement [9]. Therefore, we have the justified hope that GME can be experimentally obtained with few measurements, using our detection scheme.

13 3 We start out with a recollection of entanglement measures (chapter 2), as far as it is of relevance for our specific purposes. Then we switch to a discussion of the dynamics of open quantum systems (chapter 3), with emphasis on the quantum trajectory approach. Combining these two concepts will lead to a theory of entanglement dynamics (chapter 4). This first part of the thesis ends with a conclusion and outlook in chapter 5. The potential of entanglement measures for the detection of genuine multipartite entanglement is exploited in the second part. The discussion starts with an analysis of the generalized multipartite concurrence, and some general remarks on multipartite entanglement (chapter 6). In chapter 7, we draw a general detection scheme of GME, based on concurrence. The conclusion of the second part of the thesis is given in chapter 8.

14 4 CHAPTER 1. INTRODUCTION

15 Part I Dynamics of Entanglement 5

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17 Chapter 2 Entanglement and its Measures The appearance of entanglement is a consequence of the mathematical structure of quantum mechanics. A pure state ψ of a composite quantum system lives in a linear space, constructed by a tensor product of vector spaces referring to its subsystems. Such tensor vector spaces contain states that cannot be factorized into pure states of their individual components. These states are called entangled. To be more precise: Definition 2.1: A pure state ψ H A H B... H N is called entangled, if and only if: φ A H A, φ B H B,..., φ N H N : (2.1) Otherwise the state is separable. ψ = φ A φ B... φ N. (2.2) In the presence of classical statistical uncertainty, the quantum state is described by its density matrix ρ. However, infinitely many ensembles of states are leading to the same density operator. The mixture ρ is said to be entangled, if and only if all those ensembles contain at least one pure entangled state. Definition 2.2: A mixed state ρ L(H A H B... H N ) is entangled, if and only if any pure state decomposition ρ = j p j ψ j ψ j, p j > 0, (2.3) contains at least one non-separable ψ j. Since entanglement is a crucial resource for many applications in quantum information, it is important to quantify the amount of entanglement in L(H) denotes the space of linear operators on H. For finite dimensional spaces, this is equivalent to the more commonly used space of bounded operators B(H), [10]. 7

18 8 CHAPTER 2. ENTANGLEMENT AND ITS MEASURES a given system. However, there is a diversity of possible correlations, if the system has more than two subsystems. So, in general, one will need a family of maps: M α : L(H A H B... H N ) ρ M( α) R + 0, (2.4) to completely describe the amount of entanglement in a given system. Though, our following discussion will be limited to the bipartite case, i.e., to systems that are composed of two subsystem A and B, living in Hilbert spaces H A and H B, respectively. Multipartite entanglement will be discussed in chapter 6. In the bipartite scenario, entanglement can be measured by a single map M(ρ), that ascribes a real positive number to each state ρ. However, once again, there is a diversity of possible maps that can be used for this purpose. The aim of this chapter is not to give an exhaustive overview on entanglement measures, but, instead, to motivate the measures which will be used in subsequent chapters, as a basis for the analysis of entanglement dynamics. 2.1 Schmidt Decomposition The quantification of pure state entanglement relies on a special basis representation of bipartite states ψ, the so-called Schmidt decomposition. Surprisingly, the coordinates in this representation - the Schmidt coefficients - contain the complete information about the entanglement of the state ψ. Any measure of entanglement is then a function of these Schmidt coefficients. To perform the Schmidt decomposition, we need a linear algebra method called singular value decomposition, which allows to diagonalize an arbitrary, even rectangular matrix [11]. Singular value decomposition (SVD): Let A C m n be an arbitrary matrix with rank k. There exist unitary matrices V C m m, U C n n, and a diagonal matrix D = [d ij ] C m n, such that: A = V DU. (2.5) The only non vanishing entries of D are the so-called singular values s i : d 11 = s 1, d 22 = s 2,..., d kk = s k, (2.6) which are given by the square root of the eigenvalues of AA : s i = Spec{AA } i. (2.7) The connection between entanglement and quantum correlations will be outlined in section 2.2. The space α parameterizes the set of measures {M α }, which is necessary to characterize the correlations in the multipartite system. The size of α will be investigated in Chapter 6. The k th eigenvalue of a matrix M will be abbreviated by Spec{M} k

19 2.2. ENTANGLEMENT AND NONLOCALITY 9 The columns of V and U are eigenvectors of AA and A A, respectively. Schmidt decomposition: Let ψ H A H B be the state of a bipartite d d dimensional system, and let { i A } and { j B } be an orthonormal basis set of H A and H B, respectively. Then { i A j B } forms a basis in H A H B, so we can represent ψ using its coordinates [A] i,j : ψ = i,j [A] ij i A j B. (2.8) Now we apply SVD to the matrix A: ψ = i,j [V DU ] ij i A j B = i,j,l V i,l D l,l U lj i A j B (2.9) = l = l s l ( V il i A ) ( Ujl j B ) i j }{{}}{{} k A k B (2.10) λl l A l B (2.11) and finally obtain the Schmidt decomposition of ψ. The transformed basis k A k B is called Schmidt basis, and the coordinates {λ k } are said to be the Schmidt coefficients. They are given by the singular values s i = λ i of the original coordinate matrix A, and can also be obtained, more efficiently, by the spectrum of the reduced density matrix ρ A of ψ : This can be easily verified: λ k = Spec{T r B ψ ψ } k. (2.12) T r B ψ ψ = l B λ m λ n m A m B n A n B l B (2.13) = l l,m,n λ l l A l A (2.14) where we used the orthonormality of the Schmidt basis. Armed with the tool of Schmidt decompositions, we can derive pure state entanglement measures. 2.2 Entanglement and Nonlocality Probably the most surprising feature of entangled states is the appearance of quantum correlations between local measurements. Since this property lies at the heart of entanglement, it also serves as a path to pure state

20 10 CHAPTER 2. ENTANGLEMENT AND ITS MEASURES entanglement measures. The approach presented in this section is close to a recent paper, [13], by M.A. Cirone. Suppose we let two particles interact in order to create entanglement. We keep the particles very well isolated from the environment, so that the entanglement - which relies on coherences between multipartite states - is preserved and the system is well described by a pure state ψ. However, we separate the two subsystems and bring them to two different laboratories, so that the interaction is completely shut off. Let ψ be the pure state of the total (bipartite) system, with Schmidt decomposition ψ = d λl l A l B, l=1 λ l = 1. (2.15) l Now a measurement in the Schmidt basis is performed in each of the labs. The results of the measurements are random variables, and will be denoted X A and X B for the outcomes in lab A and lab B, respectively. In the limit of many repeated measurements, the relative weight of the outcomes will converge to the following probability distributions: p(x A = k) = k A (T r B ψ ψ ) k A (2.16) = λ k (2.17) p(x B = k) = k B (T r A ψ ψ ) k B (2.18) = λ k (2.19) Now suppose the two experimentalists in lab A and B compare their results and evaluate the joint probability p(x A, X B ). This will lead to the following result: p(x A, X B = i, j) = i A j B ( ψ ψ ) i A j B = δ ij λ i λ i λ j = p(x A = i)p(x B = j), where the inequality in the last line holds, if at least two of the Schmidt coefficients are larger than zero. In this case, we see from the inequality between the joint probability distribution p(x A, X B ) and the product of the individual distributions, that the two measurement results X A and X B are not independent, although the two particles do not interact anymore! This phenomenon is called nonlocality. Since the difference between p(x A, X B ) and p(x A )p(x B ) is a signature of nonlocality - therefore also for entanglement - it is suggestive to take the distance between p(x A, X B ) and p(x A )p(x B ) as a measure for entanglement: M( ψ ) D(p(X A, X B ), p(x A )p(x B )). (2.20) There is a variety of different distances for probability distributions, hence there

21 2.2. ENTANGLEMENT AND NONLOCALITY 11 is also a diversity of induced measures. We will focus on the L 1 -distance, and, in addition, on the relative entropy: M( ψ ) 1 = i,j = i,j p(i, j) p(i)p(j) δ ij λ j λ i λ j = i { j δ ij λ i λ j } = i {(1 λ i )λ i + i j λ i λ j } = 1 i λ 2 i + i λ i ( j λ j λ i ) = 2(1 i λ 2 i ) c 2 M( ψ ) H = i,j = i,j p(i, j) ln p(i, j) p(i)p(j) δ ij λ j ln δ ijλ j λ i λ j = i λ i ln λ i E The induced measures are the so-called concurrence c - for the choice of the L 1 -norm - and the entanglement of formation (EOF) - in the case of the relative entropy. These measures form the basis for our discussion in the subsequent chapters. With the help of equation (2.12), we can rewrite the definitions of concurrence and EOF, in terms of the reduced density matrix ρ A = T r B ψ ψ : c = 2(1 i λ 2 i ) = 2(1 i (Spec{ρ A } i ) 2 ) c = 2(1 T rρ 2 A ) (2.21) E = i λ i ln λ i = i Spec{ρ A } i ln Spec{ρ A } i E = T rρ A ln ρ A (2.22) An open question is what happens if we choose another distance, e.g., a general L p - distance [15], or the L -distance [15], or the fidelity F (p ij, q ij ) P ij pij q ij. Do all of them yield a measure, i.e, a monotone? The definitions of monotones will be given in the next section. Can all or at least some of the other entanglement measures - like the negativity - be derived by this method, from a corresponding distance of probability distributions?

22 12 CHAPTER 2. ENTANGLEMENT AND ITS MEASURES These expressions have nice, intuitive interpretations: Suppose the state of the total system is known exactly, i.e., it is in a pure state ψ. Then the state of one of the subsystems is given by the reduced density matrix ρ A = T r B ψ ψ. If entanglement is present in ψ, then this reduced density matrix is mixed, i.e., its state is not precisely known! So the subsystems alone do not contain the complete information about the entire system. This is one of the strange properties of entanglement, and was first recognized by Schrödinger, [18]. The degree of ignorance is given by the entropy, so it is suggestive to measure entanglement by the entropy of the reduced density matrix. The von Neumann entropy of the reduced density matrix is precisely the entanglement of formation (2.22), and the square root of its linear entropy gives the concurrence (2.21). The linear entropy is obtained from the von Neumann entropy, by a first order expansion i λ i ln λ 1 i λ i (λ i 1) (2.23) = i λ i i λ 2 i = 1 i λ 2 i. (2.24) Due to this linear expansion, concurrence is in general easier to evaluate, [8]. The reduced density matrix of a separable state ψ = φ A ϕ B is pure: ρ A = T r B ( ψ ψ ) = T r B ( φ A φ A ϕ B ϕ B ) = φ A φ A. Hence, its entropy - and therefore also its entanglement measured by (2.21) or (2.22) - vanishes. Entanglement is maximal, E max = ln d, c max = 2(1 1/d), for states with a maximally mixed reduced density operator maximally mixed state: ρ A = 1 d I, where d is the dimension of one of the subsystems. Examples for two qubits are Bell states Φ ± = ( 00 ± 11 )/ 2, (2.25) Ψ ± = ( 01 ± 10 )/ 2, (2.26) and any state χ which is related to a Bell state via a local unitary transformation { χ = U 1 U 2 Bell : U 1, U 2 unitary}. (2.27) We finish this section with a list of equivalent expressions for concurrence,

23 2.3. MIXED STATE ENTANGLEMENT 13 equation (2.21), that can be useful for later calculations c 2(1 T rρ 2 A ) (2.28) ψ σ y σ y ψ (2.29) 2(ψ 00 ψ 11 ψ 01 ψ 10 ) (2.30) The first expression (2.28) is valid for bipartite systems of arbitrary dimension, while the later two formulas (2.29), (2.30) are restricted to 2 by 2 systems. ψ ij are the coordinates of ψ in the computational basis, i.e., σ y is one of the Pauli matrices ψ = ψ ψ ψ ψ σ y = ( ) 0 i, (2.31) i 0 thus the tensor product σ y σ y has the explicit matrix form σ y σ y = (2.32) Finally, the expression ψ denotes the complex conjugate of the bra ψ, or simply the transpose of the ket ψ, that is: ψ = (ψ 00, ψ 01, ψ 10, ψ 11 ). (2.33) 2.3 Mixed State Entanglement Entanglement of pure bipartite (finite dimensional) systems is very well understood. There is just one type of correlations: quantum correlations between subsystems A and B. Therefore, any ambiguity in the description of entanglement comes only from the way how these correlations are quantified. The unambiguous case of dealing solely with quantum correlations is lost in the presence of classical statistical uncertainty. Under these circumstances, also classical correlations appear, which do not have their origin in quantum entanglement. The following example tries to clarify this complication: Classical Correlations: Suppose we have two spin-1/2 particles, that are prepared with probability 1/2 in the state 00 (i.e., both spins point downwards along the quantization axis) and with probability 1/2 in the state

24 14 CHAPTER 2. ENTANGLEMENT AND ITS MEASURES 11. The ensemble generated by infinitely many repetitions of this preparation step is described by the density matrix 1/ ρ = 1/2( ) = (2.34) /2 Let X A and X B be the random variables which represent the measurement outcomes of spin A and spin B - in the quantization direction - respectively. The probability distributions for X A and X B can be deduced from the reduced density matrices: p(x A ) = p(x B ) = x A (T r B ρ) x A (2.35) ( ) 1/2 0 = x A x 0 1/2 A (2.36) The joint distribution p(x A, x B ) is given by: = 1/2 x A, x B = 0, 1. (2.37) p(x A, x B = i, j) = ij ρ ij = δ ij 1/2 (2.38) p(x A = i) p(x B = j) = 1/4. (2.39) Hence, with a measurement in the chosen basis, we observe the same correlations as we would expect from the Bell state ψ = 1/ 2( ). (2.40) Yet, the origin of the above correlation (2.38), (2.39) is not a coherence between two-particle states! Therefore, we are here dealing with classical correlations rather then with a signature of entanglement! Consequently, the entanglement inscribed in a mixed state cannot simply be estimated by the correlations, quantified e.g., by the distance between p(x A )p(x B ) and p(x A, X B ), for pure states. Such an approach would lump both - quantum and classical correlations - together. To cure this complication, one might come with up the idea, to use one of the available pure state entanglement measures, and a pure state decomposition of the mixed state ρ under consideration. The mixed state s entanglement would then be given by an average over the pure state s entanglement, ρ = i M(ρ) = i p i ψ i ψ i, (2.41) p i M( ψ i ). (2.42) Though, we have to remember that such decomposition of ρ is not unique! Indeed, there are infinitely many decompositions that constitute the same density matrix. Unlike expectation values of observables, the average pure state

25 2.3. MIXED STATE ENTANGLEMENT 15 entanglement M(ρ) = i p i M( ψ i ) (2.43) is decomposition-dependent, see the proof at the end of this section. Hence the function M : ρ M(ρ) = p i M( ψ i ) i is not well defined, and M(ρ) = i p im( ψ i ) is not a reasonable measure! In this sense, the distinction of classical from quantum correlations is ambivalent. That is what makes the measurement of mixed state entanglement extremely ambitious! To proceed, we will regard the problem from some distance, and adopt a slightly different perspective. We start out with the statement of those properties which any measure should fulfill - and this will give us important hints how to construct a suitable measure. In some sense, this is an axiomatic approach Entanglement Measures from Basic Principles So far, there is no uniquely accepted set of axioms or basic properties that entanglement measures have to fulfill. This gives us some freedom to choose our own list: We seek for a measure M(ρ) that fulfills the following properties: M(ρ) vanishes identically for separable states. If ρ is pure, then M(ρ) reduces to one of the two pure state measures defined in the last section, see equations (2.21) and (2.22). M(ρ) is a monotone, i.e., it is invariant under local operations assisted by classical communication ( LOCC operations). In particular, the last of these properties is absolutely indispensable for entanglement measures. We will devote the next two subsections to explain its relevance LOCC Operations and Entanglement Monotones LOCC operations define that class of system evolutions which do not involve any kind of coherent interactions between the subsystems. The only admissible

26 16 CHAPTER 2. ENTANGLEMENT AND ITS MEASURES interaction between the different parties is the exchange of classical information. Nonlocal applications of quantum information theory - i.e., information processing tasks performed on quantum systems which are spatially separated over sufficiently large distances - are restricted to this kind of operations. For such applications, nonlocal resources must be available from the outset: the protocol involves these resources, but all steps of the protocol are of LOCC type. In such context, an entanglement measure serves to provide quantitive answers to the following questions: How much a priori entanglement is required to create a certain desired (two particle) state, using LOCC? Which kind of nonlocal tasks can be performed through LOCC alone, given a certain entangled initial state? This operational definition of an entanglement measure implies that the measure itself must not increase under LOCC. The following example tries to clarify this statement: Suppose the measure M quantifies the resources that are necessary to create a state using LOCC. Furthermore we assume that M(ρ 1 ) = 0.1 for a given state ρ 1. This means that, for instance, ten copies of ρ 1 can be created with a LOCC operation - say LOCC 1 - from a set of one Bell state and nine separable states. Now assume that there exists a LOCC operation - LOCC 2 - which maps ρ 1 on ρ 2 and simultaneously increases the value of M to M(ρ 2 ) = 0.2. This tells us that it is necessary to have at least two Bell states and eight separable states, in order to create ten copies of ρ 2. However, one can start with one Bell state and nine separable states, apply LOCC 1 in order to create ten copies of ρ 1 and then apply LOCC 2, which gives ten copies of ρ 2. This shows that the measure M does not correctly quantify the resources. The source for the paradox is the assumption that M is not a monotone. The requirement that entanglement measures must not increase under LOCC also follows immediately from the fact that entanglement itself expresses the nonlocal character of a quantum state, which cannot be enhanced by LOCC. This non-increasing character of entanglement under LOCC is summarized under the term monotonicity. Basic constituents of LOCC operations (G. Vidal, [36]) A local operation, assisted by classical communication (LOCC operation), can be decomposed into the following basic steps: L1.1: A local unitary transformation ρ i ρ f = Uρ i U, U = U A I B or I A U B. (2.44)

27 2.3. MIXED STATE ENTANGLEMENT 17 L1.2: A local von Neumann measurement The result of the local projective measurement, ρ k = appears with probability ρ i {p k, ρ k }. (2.45) P kρ i P k T r(p k ρ i ), P k = P A,k I B or I A P B,k, (2.46) p k = T r(p k ρ i ). (2.47) L1.3: The local addition of an uncorrelated ancilla, ρ i ρ f = ρ i ρ ancilla, (2.48) where ρ ancilla is just added to one of the subsystems. L1.4: The dismissal of a local part of the complete system, ρ i ρ f = T r π (ρ i ), (2.49) where the part π, which is traced out, is held by only one of the subsystems. This operation also comprises decoherence processes. L2: The decrease of the available information on the state of the system. Suppose the state ψ k is produced with probability p k, e.g., by a projective measurement. Then, for a given experiment, the state of the system is ψ k. However, the ensemble of many experiments is described by an ensemble of states, denoted with {p k, ψ k }. Now suppose a single experiment is performed, the state ψ k is obtained, but then the information about this state is lost. Then the final state of the system is described by the density matrix ρ f = k p k ψ k ψ k and the transformation reads: ψ k ρ f = k p k ψ k ψ k. (2.50) Upon average over many experiments, the mapping reads: {p k, ψ k } ρ f = k p k ψ k ψ k. (2.51) This process does not include any coherent interactions, and therefore also counts within the class of local operations. Both parties may perform these operations, conditioned on outcomes of previous ones, and are also allowed to communicate the results to the other partys, by means of classical communication. With the above we can now introduce entanglement monotones.

28 18 CHAPTER 2. ENTANGLEMENT AND ITS MEASURES Definition: A monotone µ is a function on the state space µ : L(H A H B ) ρ µ(ρ) R + 0, (2.52) which does - on average over many repetitions of the experiment - not increase under LOCC operations, that is (ρ i LOCC ρ f ) (µ(ρ f ) µ(ρ i )). (2.53) Practical conditions for monotonicity can be inferred from the above decomposition of LOCC operations into basic steps: Conditions for monotonicity (G. Vidal, [36]): A function on the state space is a monotone, if and only if it satisfies the following conditions: C1.1: µ(ρ) = µ(uρu ), for U a local unitary transformation. C1.2: µ(ρ) k p kµ(ρ k ), if {p k, ρ k } is the result of a local projective measurement on ρ. C1.3: µ(ρ) = µ(ρ ρ ancilla ), if ρ ancilla is just locally added to one of the parties. C1.4: µ(ρ) µ(t r π (ρ)), if the part which is traced out is held by just one subsystem. This guarantees that the measure does not increase under decoherence caused by the interaction of the system with private bathes, see section C2: Suppose we have the scenario described in L2. Then the entanglement before the loss of information is given by k p kµ(ψ k ) on average over many repetitions of the experiment. After the information loss, entanglement reads µ(ρ). Monotonicity implies that: µ(ρ) k p k µ(ψ k ). (2.54) Since the ensemble {p k, ψ k } was chosen arbitrarily, the condition (2.54) must hold for any convex decomposition of the density matrix, ρ = k p k ψ k ψ k. This condition guarantees that the measure does not misinterpret the increase of classical correlations as an increase of quantum correlations The Convex Roof Construction We seek for a generalization of a given pure state measure M(ψ) to mixed states, with the restriction that the induced measure M(ρ) be a monotone, and that it reduces to the pure state measure M(ψ) if the state is pure, i.e., ρ = ψ ψ.

29 2.3. MIXED STATE ENTANGLEMENT 19 From the monotonicity condition C2 we conclude: C2 M(ρ) inf p k M(ψ k ), (2.55) {p k,ψ k } where the infimum is performed over all convex pure state decompositions of ρ = k p k ψ k ψ k, p k 0. Suppose ρ = ψ ψ, and M(ρ) < inf p k M(ψ k ). {p k,ψ k } k k Then M(ρ) = M( ψ ψ ) < inf p k M(ψ k ) = M(ψ) {p k,ψ k } M( ψ ψ ) M(ψ), k which is obviously a contradiction. This implies that the mixed state measure is given by the convex-roof construction, [19] M(ρ) = inf p k M(ψ k ). (2.56) {p k,ψ k } However, the conditions C.1.1 to C.1.4 have to be proved in addition, to assure that M(ρ) is a monotone. The explicit evaluation of the convex roof is much simplified by the following trick: Let ρ = i ψ i ψ i be the spectral decomposition of a given density matrix, where ψ i is subnormalised, that is ψ i = p i. Then any possible decomposition ρ = i φ i φ i of the density operator is related to { ψ i } by the linear combination φ i = j U ij ψ j, see [18]. The weights U ij are elements of a left unitary matrix. If M(N ψ) = N 2 M(ψ) - which is the case for concurrence (2.21) - then one can rewrite the optimization (2.56) as M(ρ) = inf U k M( i j U ij ψ j ). (2.57) A solution for this optimization problem is given in the next section, for two qubits. For the pure state measure on the right hand side of (2.57), we can chose entanglement of formation (2.22) or concurrence (2.21). Before finishing this section, we demonstrate that k p km(ψ k ) is indeed

30 20 CHAPTER 2. ENTANGLEMENT AND ITS MEASURES decomposition dependent. Proof that the average pure state expectation value of observables is decomposition-independent, while the average pure state concurrence is not. Suppose A is an hermitian operator, and thus an observable. Then the average pure state expectation value is independent of the decomposition: φ i A φ i = j UijAU i,j,k ψ ik ψ k (2.58) i = ψ j ( U ji U ik) A ψ k (2.59) j,k i }{{} δ jk = i ψ i A ψ i (2.60) Now we try a similar trick with concurrence, for which we choose the expression (2.29), c( ψ i ) ψ i σ y σ y ψ i, φ i σ y σ y φ i = ψj U ij σ y σ y U ik ψ k (2.61) i i,j,k = j,k ( UjiU T ik ) ψj σ y σ y ψ k (2.62) i }{{} δ jk in general Therefore, in general, the pure state ensemble average of concurrence dependents on the decomposition! 2.4 Explicit Evaluation of Concurrence and of Entanglement of Formation Since concurrence, as well as EOF, are for mixed states constructed via the convex roof, they are given in terms of an optimization problem. Therefore an upper bound of these measures is obtained by numerical optimization for any finite dimensional N-partite system. In the case of two qubits there is an exact algebraic solution known, due to Wootters [17]. In this special case, concurrence and EOF happen to be equivalent, in the sense that they are related to each other via an explicit expression, and that the optimum is simultaneously reached for both quantities. Let us first recapitulate Wootters result:

31 2.5. QUANTUM STATE TOMOGRAPHY 21 Analytic expression for concurrence Let ρ C 2 2 be the state of a two-qubit system. The exact expression for the mixed states concurrence, defined as c(ρ) := inf p i c(ψ i ), (2.63) {p i,ψ i } i c(ψ i ) := 1 T r(t r A ψ i ψ i ) 2, (2.64) is given by c(ρ) = max( λ 1 i>1 λi, 0), (2.65) where {λ i } is the spectrum of the auxiliary operator A: A := ρ σ y σ y ρ T σ y σ y. (2.66) The spectrum {λ i } must be given in decreasing order, that is λ 1 must be the largest eigenvalue of H. This method is exact only for two qubits (or for 2 3 systems). Generalizations can be derived, which, after suitable estimates, provide lower bounds for higher dimensional- and multipartite systems, [8]. For two qubits the above result also allows to calculate the value of the entanglement of formation [17], since these measures are related via the following relation. Relation between concurrence and EOF for two qubits Let ρ C 2 2 be the state of a two qubit system. Then the entanglement of formation, defined as E(ρ) := inf p i E(ψ i ), (2.67) {p i,ψ i } i E(ψ i ) := T r{(t r A ψ i ψ i ) ln(t r A ψ i ψ i ), } (2.68) is given by the concurrence c(ρ), according to [17]: E(c) = σ= 1,1 1 2 (1 + σ 1 c 2 1 log 2 2 (1 + σ 1 c 2 ). (2.69) 2.5 Quantum State Tomography Above we explained how entanglement measures can be evaluated from a given density matrix. Here we want to address how the density matrix itself can be The infimum is taken over all possible decompositions ρ = P i p i ψ i ψ i of the density matrix, and T r A denotes the trace over one of the two subsystems.

32 22 CHAPTER 2. ENTANGLEMENT AND ITS MEASURES inferred from experimental data. The standard method of choice is Quantum state tomography [20]. The set {σ 0, σ 1, σ 2, σ 3 } 1 2 {I, σ x, σ y, σ z } (2.70) of Pauli matrices together with the identity forms an orthonormal basis of the hermitian 2 2 matrices, L(C 2 ) {ρ C 2 2 ρ = ρ }, (2.71) with respect to the Hilbert-Schmidt scalar product [12] < ρ, ϱ > Tr(ρϱ ) = Tr(ρϱ). (2.72) Thus any hermitian 2 2 matrix, for instance the density matrix of a qubit, can be expressed as 3 ρ = Tr(ρσ i ) σ i. (2.73) i=0 Consequently, the state of a two-level system can be completely determined by measuring the three spin expectation values (Tr(ρσ 0 ) = 1) which are the elements of the Bloch-vector. σ x ρ, σ y ρ, σ z ρ (2.74) This concept is easily generalized for multipartite two-level systems: An orthonormal basis of the tensor space L(C 2n ) L(C 2 ) L(C 2 )... L(C 2 ) }{{} n times (2.75) is given by the set {σ i1 σ i2... σ in i 1, i 2,..., i n {0, 1, 2, 3}}. (2.76) Hence, any 2 n 2 n hermitian matrix, for instance the density matrix ρ of a n-qubit system, can be expressed as [12] ρ = 3 i 1,..,i n =0 Tr(ρ σ i1 σ i2... σ in ) σ i1 σ i2... σ in. (2.77) This means that the mixed state of a n-qubit system can be recovered from the 4 n 1 expectation values σ i1 σ i2... σ in ρ Tr(ρ σ i1 σ i2... σ in ) (2.78)

33 2.5. QUANTUM STATE TOMOGRAPHY 23 In practice, these are obtained as the average over N measurements of all spins in the chain, what formally reads lim N N k=1 1 N n (measured spin of particle j in direction i j ) k (2.79) j=1 where we attribute the value 1 to the projection on σ 0, for all spins. In the last expression the index k runs over different measurements, and j is the particle index. In order to guarantee convergence of (2.79) to the exact expectation value (2.78), N must be large. Consequently to infer a single expectation value, a large number of measurements is needed. Measurement errors cannot be avoided, and therefore the measurement outcome ρ will deviate from the exact density matrix ρ. However, the result ρ can be improved with computational methods from the theory of inverse problems, for instance the maximum likelihood principle. These methods require prior knowledge about the exact result, which is available, ρ being a density matrix Tr(ρ) = 1, ρ 0. (2.80) This information is useful, since the measured density matrix does in general not fulfill these properties.

34 24 CHAPTER 2. ENTANGLEMENT AND ITS MEASURES

35 Chapter 3 Open Quantum Systems 3.1 Quantum Operations Closed systems exhibit a unitary in time evolution. However, if the system is part of a larger system - i.e., if it interacts with an environment - the dynamics gets more complicated. A possible description of the time evolution is given by a parameterized map, ξ t : H i ρ(0) ρ(t) H f, (3.1) that connects the initial with the final state. The map ξ is called a quantum operation, if it is consistent with the laws of quantum mechanics. This request can be condensed into three axioms [12]: A1. ξ is trace preserving: T r(ξ(ρ)) = 1 A2. ξ is convex-linear: p i 0 : ξ( i p i ρ i ) = i p i ξ(ρ i ) A3. ξ is completely positive, that is, any trivial extensions of ξ, ξ I : H i H E ϱ i ϱ f H f H E must be positive - i.e., map positive operators ϱ i to positive operators ϱ f - for any dimension of the extension H E. Motivation of the axioms (A1) ξ(ρ) is a density operator and hence has trace one. 25

36 26 CHAPTER 3. OPEN QUANTUM SYSTEMS (A2) Suppose we prepare the state ρ i with probability p i. Then the final state will be ξ(ρ i ), with probability p i. Hence, in the ensemble average, the final state is described by the density operator i p iξ(ρ i ). On the other hand, in the ensemble average, the initial state is i p iρ i, so that its image under the map ξ is ξ( i p iρ i ). (A3) One can regard the system S as a subsystem of a larger one, say SE. Then ξ can be extended to SE by multiplication with the identity operator ξ I. The image under this extended map must be a density operator, and hence has to be positive. The axioms A1 to A3 are rather abstract, hence inconvenient for calculations. Fortunately, there is an explicit standard form for quantum operations, the so-called Kraus representation. Kraus Representation A map ξ satisfies A1 to A3 if and only if with the completeness relation ξ(ρ) = N E k ρe k, (3.2) k=1 E k E k = I. (3.3) i The number of Kraus operators E k is limited by the dimension of the state space, N dim(h) 2 [12]. The proof of the equivalence of (3.2) and (3.3) with A1-A3 can be found in [12]. 3.2 Master Equation The unitary evolution of isolated systems ρ(t) = U(t)ρ(0)U (t) (3.4) is described by a differential equation for the density operator, the von Neumann equation: ρ = i [H, ρ]. (3.5) How does the corresponding differential equation look like if the evolution is of the general form (3.2), i.e., for an open system? Formally, we may write: ρ t+dt ρ t ρ t = lim dt 0 dt ξ t+dt (ρ(0)) ξ t (ρ(0)) = lim = F(ρ(0)). (3.6) dt 0 dt

37 3.2. MASTER EQUATION 27 So, in general, the time derivate of the density matrix will depend on the past of the state. This is reasonable, since the system is surrounded by a bath, which may serve as a memory of the state s past. For instance, the bath might store energy. However, if the environment is huge compared to the system size, and if the typical time scale of the system evolution is much larger than the correlation time of the bath, it is a very good approximation to assume that the bath has no memory effects. This is the so-called Markov assumption [21]. Markov assumption Thus we may write: ξ t1 +t 2 (ρ) = ξ t1 (ξ t2 (ρ)), (3.7) ρ t+dt = ξ t+dt (ρ(0)) = ξ dt (ρ(t)). (3.8) Therefore, (3.6) can be written as a closed differential equation: ξ t+dt (ρ(0)) ξ t (ρ(0)) ξ dt (ρ(t)) ρ(t) ρ t = lim = lim = L(ρ(t)) (3.9) dt 0 dt dt 0 dt By including the Kraus representation (3.2) for ξ dt (ρ(t)), one can derive the general form of the superoperator L, the Lindblad form. An explicit demonstration thereof can be found in [21]. The resulting differential equation is called master equation. Lindblad form The time evolution of Markovian open quantum systems has the general structure: ρ = Lρ = i N [H, ρ] + L k ρ, (3.10) } {{} k=1 unitary part }{{} incoherent part L k ρ = (J k ρj k 1 2 J k J kρ 1 2 ρj k J k). (3.12) The first term on the right hand side describes the unitary evolution, generated by the system Hamiltonian H, while the second part is due to the environment coupling. This coupling is characterized by N d 2 1 in general nonhermitian jump operators J k, where d denotes the dimension of the state space. The origin of the name jump operators will become clear in the next chapter on quantum trajectories.

38 28 CHAPTER 3. OPEN QUANTUM SYSTEMS The explicit form of the jump operators for a specific environment coupling is usually derived from a microscopic, Hamiltonian dynamics which includes the environment. Importantly, the choice of the jump operators is not unique. This property will be of crucial importance for our later treatment of entanglement dynamics. Freedom in choice of the jump operators The master equation is invariant under the transformations [26] J k J k = i U ki J i, (3.13) J k J k,± = µi ± J k 2. (3.14) The first amounts to a linear combination of jump operators, with weights given by elements of a left unitary matrix. Due to this left unitarity, the number of jumps may increase, depending on the number of rows of U. The second of the above transformations doubles the number of jumps. Each sign generates a new jump operator. A complex multiple µ of the identity is added. Combining both ((3.13) and (3.14)) leads to the most general form of invariance transformations J k J k,± = µi ± i U kij i (3.15) 2 Proof: We show the invariance of the master equation (3.10) under the transformations (3.13) and (3.14). Its unitary part does not change, thus we only have to show the invariance of k L k. 1) J k J k = i U kij i : k L k = k {( i U ki J i )ρ( j U kj J j ) (3.16) 1 2 ( i U ki J i ) ( j U kj J j )ρ (3.17) 1 2 ρ( i U ki J i ) ( j U kj J j )} (3.18) = ( U jk U ki) J i ρj j 1 ( U 2 ik U kj) J i J jρ (3.19) ij k ij k }{{}}{{} δ ij δ ij 1 ( U 2 ik U kj) ρj i J j = L i (3.20) ij k i }{{} δ ij

39 3.3. SYSTEM-BATH INTERACTION MODELS 29 2) J k J k,± = µi±j k 2 : k,± L k,± ρ = 1 {(µ + J k )ρ(µ + J k ) (µ + J k) (µ + J k )ρ (3.21) k 1 2 ρ(µ + J k) (µ + J k ) + (µ J k )ρ(µ J k ) (3.22) 1 2 (µ J k) (µ J k )ρ 1 2 ρ(µ J k) (µ J k )} (3.23) = k {J k ρj k 1 2 J k J kρ 1 2 ρj k J k} = k L k ρ (3.25) 3.3 System-Bath Interaction Models We will now discuss different types of system-bath interaction models, which enter the master equation via the corresponding jump operators. The models are close to the decoherence mechanisms which frequently dominate in realistic experimental setups, for instance in ion traps or in cavity QED Thermal Bath This model is suited for systems embedded in a thermal environment. Examples are atoms (= system) coupled to the free electromagnetic field in the thermal state of temperature T (= environment) or single mode cavities (= system) in contact with a finite temperature environment. The bath is modeled by a collection of harmonic oscillators, each in a thermal state ρ E = i e H i/k B T T r e H i/k B T, H i = ω i a i a i. (3.26) The reduced dynamics of the system can be described by a master equation with the following jump operators - derived, e.g., in [23]: J 1 = Γ ( n + 1)σ, J 2 = Γ nσ +. (3.27) n is the mean excitation number of the field mode, which is resonant to the atomic transition [23], n = Tr E (ρ E ˆn A ) = e ω A/k B T 1 e ω A/k B T. (3.28)

40 30 CHAPTER 3. OPEN QUANTUM SYSTEMS Γ is the system-bath coupling strength and ( ) 0 1 σ = 0 1 =, (3.29) 0 0 ( ) 0 0 σ + = 1 0 =, (3.30) 1 0 are the atomic annihilation- and creation operators, respectively. In total, the master equation of a two level system coupled to a thermal bath reads: ρ = Γ( n + 1)(σ ρσ σ +σ ρ 1 2 ρσ +σ ) (3.31) + Γ n(σ + ρσ 1 2 σ σ + ρ 1 2 ρσ σ + ) (3.32) The jump operators can be readily interpreted. In the next section we will see that the mixed state dynamics of the system can be interpreted as an average over stochastic pure state evolutions characterized by jumps, which occur upon application of the jump operators J k on the evolving state ψ with probability J k ψ. This means that with a probability the atom absorbs, Pr(absorption) = σ + ψ Γ n (3.33) ψ σ + ψ (3.34) a quantum from the thermal bath, e.g., a photon. And with probability the atom emits a quantum, Pr(emission) = σ ψ Γ( n + 1) (3.35) ψ σ ψ. (3.36) The process has two limits. Either the temperature of the bath is very low (T 0), i.e., n 0, then only (3.35) yields a non-vanishing contribution and the system evolution is exclusively driven by spontaneous emission. Or the temperature of the bath is very high (T ), i.e., n n + 1 and emission and absorption rates are almost equal Pr(emission) Pr(absorption) = n + 1 n T 1 (3.37) This so-called T-infinity case is a noise process, i.e., a process with equal probability for a jump from the ground to the exited state, and vice versa ( bit flips ). Also note that T leads to a well-defined limit only if we simultaneously let Γ 0, i.e., n, Γ 0, Γ n Γ <. (3.38) From now we will switch to the interaction picture, so that the internal dynamics is converted from the state to the operators. Hence the master equation only contains the incoherent part

41 3.3. SYSTEM-BATH INTERACTION MODELS 31 We summarize the two limits. Spontaneous Emission In the limit of small temperatures, the system dynamics is dominated by spontaneous emission of energy quanta into the bath modes. There is only one jump operator J k = Γσ, (3.39) so that the system evolution is described by the master equation ρ = Γ(σ ρσ σ +σ ρ 1 2 ρσ +σ ). (3.40) This decoherence process is the fundamental limit for coherent atomic dynamics. T infinity case In the limit of large bath temperature and small coupling strengths, the jump operators are and the master equation reads T, Γ 0, Γ n Γ, (3.41) J 1 = Γσ, J 2 = Γσ+, (3.42) ρ = Γ(σ ρσ σ +σ ρ 1 2 ρσ +σ ) (3.43) + Γ(σ + ρσ 1 2 σ σ + ρ 1 2 ρσ σ + ). (3.44) The decoherence process described by this master equation is a noise process, that is bit flips occur with equal probabilities Dephasing Dephasing is a phase-destroying decoherence process, that does not affect the populations of the system s eigenstates. The diagonal entries of the density matrix, represented in the energy eigenbasis, remain constant, while the off diagonal elements asymptotically decay in time. Examples for such processes are elastic atomic collisions [23], elastic phonon collisions [23] or phase instabilities of a driving laser beam [22]. These mechanisms can be accounted for by jump operators of the form J k = Γσ + σ (3.45) in the master equation, which for a pure dephasing environment reads ρ = Γ(σ + σ ρσ + σ 1 2 σ +σ ρ 1 2 ρσ +σ ). (3.46)

42 32 CHAPTER 3. OPEN QUANTUM SYSTEMS Two Qubits and Private Baths In later applications of open system theory to entanglement dynamics, we will apply master equations to bipartite systems. Compared to the monopartite case, the number of jump operators for a given bath is doubled, by the substitution J k J k I, I J k. (3.47) For instance, in the case of spontaneous emission and an exited initial state 11, either of the two subsystem can emit a photon. Thus, the possible jumps are 11 σ I 11 = 01, (3.48) 11 I σ 11 = 10. (3.49) The resulting master equation reads (spontaneous emission, two qubits) ρ = Γ(σ I ρ σ + I 1 2 σ + I σ I ρ 1 2 ρ σ + I σ I) (3.50) +Γ(I σ ρ I σ I σ + I σ ρ 1 2 ρ I σ + I σ ).(3.51) The substitution (3.47) is only correct if each subsystem has its own bath. This scenario is called private baths. Imagine two atoms that are coupled to the same set of field modes. A detected photon then can have its origin in the decay of either the first atom or of atom number two, which corresponds to the jump 11 (σ I + I σ ) 11 = , (3.52) different from (3.47). Since the final state is entangled, it is clear that this evolution is not a LOCC operation, described in chapter 2. However, private baths (in the absence of global unitary operations) belong to the class of LOCC operations, and guarantee non-increasing entanglement! 3.4 Solution of the Master Equation A possible strategy to solve the master equation is to choose a special matrix representation. This gives a set of coupled ordinary differential equations (ODE s) that can be solved with standard methods. We will demonstrate this procedure with the simplest example, spontaneous emission of a two-level system. The master equation reads ρ = Γ(σ ρσ σ +σ ρ 1 2 ρσ +σ ) (3.53) = Γ( 0 1 ρ ρ 1 ρ ) (3.54) 2 = Γ( 0 ρ ρ 1 ρ 1 1 ). (3.55) 2 The relation between quantum jumps and photodetection will be discussed in the section on quantum trajectories