Quantum Entanglement: Detection, Classification, and Quantification Diplomarbeit zur Erlangung des akademischen Grades ,,Magister der Naturwissenschaften'' an der UniversitatÄ Wien eingereicht von Philipp Krammer betreut von Ao. Univ. Prof. Dr. Reinhold A. Bertlmann Wien, Oktober 2005 CONTENTS 1. Introduction ::::::::::::::::::::::::::::: 4 2. Basic Mathematical Description :::::::::::::::: 6 2.1 Spaces, Operators and States in a Finite Dimensional Hilbert Space . 6 2.2 Bipartite Systems . 7 2.2.1 Qubits . 11 2.2.2 Qutrits . 12 2.3 Positive and Completely Positive Maps . 14 3. Detection of Entanglement ::::::::::::::::::: 16 3.1 Introduction . 16 3.2 Pure States . 17 3.3 General States . 17 3.3.1 Nonoperational Separability Criteria . 17 3.3.2 Operational Separability Criteria . 20 4. Classification of Entanglement :::::::::::::::: 32 4.1 Introduction . 32 4.2 Free and Bound Entanglement . 32 4.2.1 Distillation of Entangled States . 32 4.2.2 Bound Entanglement . 36 4.3 Locality vs. Non-locality . 42 4.3.1 EPR and Bell Inequalities . 42 4.3.2 General Bell Inequality . 44 4.3.3 Bell Inequalities and the Entanglement Witness Theorem 49 5. Quantification of Entanglement :::::::::::::::: 52 5.1 Introduction . 52 5.2 Pure States . 52 5.3 General States . 54 5.3.1 Entanglement of Formation . 54 Contents 3 5.3.2 Concurrence and Calculating the Entanglement of For- mation for 2 Qubits . 56 5.3.3 Entanglement of Distillation . 58 5.3.4 Distance Measures . 59 5.3.5 Comparison of Di®erent Entanglement Measures for the 2-Qubit Werner State . 64 6. Hilbert-Schmidt Measure and Entanglement Witness :: 66 6.1 Introduction . 66 6.2 Geometrical Considerations about the Hilbert-Schmidt Distance 66 6.3 The Bertlmann-Narnhofer-Thirring Theorem . 68 6.4 How to Check a Guess of the Nearest Separable State . 70 6.5 Examples . 72 6.5.1 Isotropic Qubit States . 72 6.5.2 Isotropic Qutrit States . 74 6.5.3 Isotropic States in Higher Dimensions . 76 7. Tripartite Systems ::::::::::::::::::::::::: 79 7.1 Introduction . 79 7.2 Basics . 79 7.3 Pure States . 81 7.3.1 Detection of Entangled Pure States . 81 7.3.2 Equivalence Classes of Pure Tripartite States . 81 7.4 General States . 87 7.4.1 Equivalence Classes of General Tripartite States . 87 7.4.2 Tripartite Witnesses . 88 8. Conclusion :::::::::::::::::::::::::::::: 90 1. INTRODUCTION What is Quantum Entanglement? If we look up the word `entanglement' in a dictionary, we ¯nd something like `state of being involved in complicated circumstances', the term also denotes an a®air between two people. Thus in quantum mechanics we could describe quantum entanglement literally as a `complicated a®air' between two or more particles. The ¯rst one to introduce the term was Erwin SchrÄodingerin Ref. [68]. Since this article was published in German, `entanglement' is a later transla- tion of the word `VerschrÄankung'.SchrÄodingerdoes not refer to a mathemat- ical de¯nition of entanglement. He introduces entanglement as a correlation of possible measurement outcomes and states the following: Maximal knowledge of a whole system does not necessarily include knowledge of all of its parts, even if these are totally divided from each other and do not influence each other at the present time. Note that `system' is always a generalized expression for some physical real- ization; in this context a system of two or more particles is meant. Nowadays the de¯nition of entanglement is a mathematical one and rather simple (see Chapter 2) { however, the phenomenological description of en- tanglement is still di±cult. Since J.S. Bell introduced his `Bell Inequality' [3] it has become clear that the correlations related to quantum entanglement can be stronger than merely classical correlations. Classical correlations are those that are explainable by a local realistic theory, and it was propagated by Einstein, Podolsky, and Rosen [31] that quantum mechanics should also be a local realistic theory (see Sec. 4.3). The mysteriousness inherent to quantum entanglement mainly comes from the fact that often in cannot be explained with a classical deterministic model [3, 4] and so underlines the `new physics' that comes with quantum mechanics and distinguishes it from classical physics. Why do we need quantum entanglement? What at ¯rst seemed to be a more philosophical investigation became of practical use in recent years. With the development of quantum information theory a new `quantum' way of informa- tion processing and communication was initiated which makes direct use of 1. Introduction 5 quantum entanglement and takes advantage of it (see, e.g., Refs. [16, 13, 45]). There are various tasks involving quantum entanglement that are an improve- ment to classical information theory, for example quantum teleportation and cryptography (see, e.g., Refs. [5, 17, 15, 32]). In the course of years comput- ing has become and still becomes more and more e±cient, information has to be encoded into less physical material. To be able to keep pace with the technological demand, quantum information theory could serve as the future concept of information processing and communication devices. It is therefore not only of philosophical but also of practical use to deepen and extend the description of quantum entanglement. Aim and Structure of this Work. The aim of this work is to provide a basic mathematical overview of quantum entanglement which includes the fundamental aspects of detecting, classifying, and quantifying entanglement. Several examples should give insight of the explicit application of the given theory. There is no emphasis on detailed proofs. Nevertheless some proofs that are useful to be explicitly mentioned and do not take too long are given, otherwise the reader is refered to other literature. The main part of the work is concerned with bipartite systems, these are systems that consist of two parts (i.e. particles in experimental application). The work is organized as follows: In Chapter 2 we start with basic math- ematical concepts. Next, in Chapter 3, we address the problem of detecting entanglement; in Chapter 4 entanglement is classi¯ed according to certain properties, and in Chapter 5 we discuss several methods to quantify entan- glement. In Chapter 6 we combine the concept of detecting and quantifying entanglement. Finally, in Chapter 7, we briefly take a look on tripartite systems. 2. BASIC MATHEMATICAL DESCRIPTION 2.1 Spaces, Operators and States in a Finite Dimensional Hilbert Space Operators act on the Hilbert space H of a quantum mechanical system, they make up a Hilbert Space themselves, called Hilbert-Schmidt space A. We are only interested in ¯nite dimensional Hilbert spaces, so that in fact A can be regarded as a space of matrices, taking into account that in ¯nite dimensions operators can be written in matrix form. A scalar product de¯ned on A is (A; B 2 A) hA; Bi = TrAyB; (2.1) with the corresponding Hilbert-Schmidt norm p kAk = hA; Ai A 2 A : (2.2) Matrix Notation. Generally, any operator A 2 A can be expressed as a matrix with the elements Aij = heij A jeji ; (2.3) where ei and ej are vectors of an arbitrary basis feig of the Hilbert Space. Of course the same holds for states, since they are operators. De¯nition of a State. An operator ½ is called `state' (or density operator or density matrix) if 1 Tr½ = 1; ½ ¸ 0 ; (2.4) where ½ ¸ 0 means that ½ is a positive operator (more precise: positive semide¯nite), that is, if all its eigenvalues are larger than or equal to zero. Positivity of ½ can be equivalently expressed as Tr½P ¸ 0 8P; (2.5) where P is any projector, de¯ned by P 2 = P . 2 1 Certainly the presented conditions refer to the matrix form of a state ½. 2 Eq. (2.5) follows from the fact that the eigenvalues are nonnegative, since ½ can be written in appropriate matrix notation in which it is diagonal, where the eigenvalues are 2. Basic Mathematical Description 7 Remark. In early quantum mechanics (pure) states are represented as vec- tors jÃi in Hilbert Space. This concept is widened with the introduction of mixed states, so that in general states are viewed as operators. If one is inter- ested in pure states only, either the vector representation jÃi or the operator representation ½pure = jÃi hÃj can be used. Note that Eqs. (2.4) and (2.5) imply Tr½2 · 1.3 In particular we have Tr½2 = 1 ) ½ is a pure state ; Tr½2 < 1 ) ½ is a mixed state : (2.6) 2.2 Bipartite Systems In all chapters but the last we will consider bipartite systems. Following the convention of quantum communication, the two parties are usually referred to as `Alice' and `Bob'. For bipartite systems the Hilbert space is denoted as d1 d2 HA ­ HB , where d1 is the dimension of Alice's subspace and d2 is Bob's, or just HA ­ HB when there need not be a special indication to the dimensions. We may also drop for convenience the indices `A' and `B', e.g. we will often consider the Hilbert Space H2 ­ H2, states on this space are called 2-qubit states. Matrix Notation. In general, we can write a state ½ as a matrix according to Eq. (2.3). However, often we have to use a product basis, to guarantee that certain calculations etc. make sense. In this case for the matrix notation of d1 d2 a state ½ on HA ­ HB we have ½m¹;nº = hem ­ f¹j ½ jen ­ fºi : (2.7) Here feig and ffig are bases of Alice's and Bob's subspaces. Reduced Density Matrices. The notation in a product basis is for example needed to calculate the reduced density matrices of a state ½. These are obtained if Alice neglects Bob's system, or vice versa, which mathematically means she takes a partial trace of the density matrix, she \traces out" Bob's the diagonal elements.