Bachelor Thesis Entanglement or Separability an introduction Lukas Schneiderbauer December 22, 2012 Quantum entanglement is a huge and active research field these days. Not only the philosophical aspects of these ’spooky’ features in quantum mechanics are quite interesting, but also the possibilities to make use of it in our everyday life is thrilling. In the last few years many possible applications, mostly within the ’Quantum Information’ field, have been developed. Of course to make use of this feature one demands tools to control entanglement in a certain sense. How can one define entanglement? How can one identify an entangled quantum system? Can entanglement be measured? These are questions one desires an answer for and indeed many answers have been found. However today entanglement is not yet fully in control by mathematics; many problems are still not solved. This paper aims to provide a theoretical introduction to get a feeling for the mathematical problems concerning entanglement and presents approaches to handle entanglement identification or entanglement measures for simple cases. The reader should be aware of the fact that this paper constitutes in no way the claim to be a summary of all available methods, there exist many more than demonstrated in the following pages. Student ID number 0907633 Degree course Physics assisted by ao. Univ.-Prof. i.R. Dr. Reinhold Bertlmann Contents 1 Introduction 3 2 Preliminary definitions 4 2.1 Composite quantum systems . 4 2.2 Density operator . 4 2.2.1 Reduced density operator for a bipartite quantum system . 4 2.3 Entangled states . 5 2.3.1 A pure correlated composite state . 5 2.4 Hilbert-Schmidt space . 6 2.5 Qudit systems . 6 2.5.1 Qubit systems . 6 3 Pure bipartite qudit states 7 3.1 Schmidt decomposition . 7 3.2 Von-Neumann entropy . 8 4 Mixed bipartite qudit states 11 4.1 A criterion for non-entanglement . 11 4.2 Generalization of the Von-Neumann entropy . 12 4.2.1 Requirements for entanglement measures . 12 4.2.2 Entanglement measures . 13 4.3 Generalization of the Schmidt rank . 14 4.4 Entanglement witnesses . 14 4.4.1 Entanglement Witness Theorem (EWT) . 15 4.4.2 Positive Map Theorem (PMT) . 15 4.4.3 Positive Partial Transpose (PPT) Criterion . 16 4.4.4 Bertlmann-Narnhofer-Thirring Theorem . 17 5 Multipartite states 20 5.1 Some entanglement measures . 20 5.1.1 Geometric measure of entanglement . 20 5.1.2 Measure of entanglement by Barnum . 20 5.2 An entanglement witness . 20 6 The choice of factorization 22 6.1 The choice of factorization for pure states . 22 6.2 The choice of factorization for mixed states . 22 7 Conclusion 24 References 24 1 INTRODUCTION 1 Introduction Quantum entanglement, named by Erwin Schrödinger1, is a quantum mechanics phenomenon which was first outlined by Albert Einstein, Boris Podolski and Nathan Rosen in 1935 and led to the fa- mous EPR paradoxon.[6] Their result was that quantum mechanics can’t be a correct theory since entanglement would lead to a violation of the classical principle of local realism: »We are thus forced to conclude that the quantum-mechanical description of physical reality given by wave functions is not complete.« - Einstein, Podolski, Rosen [6] On the other hand quantum mechanics was succussfully confirmed by experiments. Therefore the idea of ’hidden variables’ arose, meaning quantum theory is not fundamental but a statistical theory which covers the fundamental theory. This makes it possible to deny a violation of local realism as matter of principle and at the same time to ’believe’ in the success of the theory. 1964 John Stewart Bell provided a way to determine experimentally whether hidden variables exist or not (see his Bell inequalities[4]). Surprisingly the experiments turned out to deny the existence of such hidden variables. Although the interpretation of quantum theory may vary, today quantum mechanics is widely ac- cepted as fundamental theory by physicists. This work is an introduction developing criteria to identify entangled quantum systems for specific cases. To start at an uniform level it first provides some fundamental mathematical definitions in section 2. Mathematical entities with physical meaning like the Hilbert space or a density operator are defined. Also a (mathematical) answer to the question »What is entanglement?« is given. The following derivations are based on these definitions. Next it faces one of the simplest non trivial problems in section 3: a pure bipartite qudit state. Two important concepts, the Schmidt-decomposition and the Von-Neumann entropy, are introduced which will prove to be useful in further studies. The Bell states will function as examples. In chapter 4 mixed bipartite qudit systems are discussed. First a simple but important criterion for non-entanglement is derived. Generalization approaches of the Von-Neumann entropy and the Schmidt rank are presented as entanglement measures which will include the entanglement of formation, the relative entropy of entanglement, the entanglement of distillation, entanglement cost and a Hilbert- Schmidt measure. Also the concept of entanglement witnesses with a few applications like the Positive Partial Transpose Criterion is displayed. Furthermore a small outlook for multipartite systems is given in section 5. Two more measures of entanglement are presented and a specific entanglement witness for an N-partite system is given. And last but not least the issue of the possibility to choose different algebra factorizations with respect to the consequences to entanglement is discussed in chapter 6. 1The original German name for this ’spooky’ feature was »Verschränkung«. 3 2 PRELIMINARY DEFINITIONS 2 Preliminary definitions 2.1 Composite quantum systems A quantum system is represented by a Hilbert space H. Let’s consider a number of such systems, denoted by HA, HB and so forth. Definition 2.1. It is postulated that the composite system of these subsystems HA, HB, ... is represented by their Product-Hilbert space HAB::: HAB::: = HA ⊗ HB ⊗ ::: (2.1) An operator O in a composite system SAB::: is denoted by OAB::: . For a product state jΨAi⊗jΨBi⊗::: one also writes jΨA; ΨB; :::i. 2.2 Density operator Definition 2.2. Given an ensemble fj'ii; pig of N possible pure states j'ii with probability pi one defines the density operator ρ N N X X ρ := pij'iih'ij; pi = 1 (2.2) i=1 i=1 Note that the j'ii are not the eigenstates of ρ, therefore not orthogonal in general. This density operator ρ represents a mixed state2 (see e.g. [2]) of a quantum system and has the following important properties: h'jρj'i ≥ 0 8j'i 2 H () ρy = ρ (2.3) tr(ρ) = 1 (2.4) Proof. Eq. (2.3) is obvious and eq. (2.4) can be shown by just using the definition of the trace: P tr(ρ) ≡ ihΨijρjΨii with an arbitrary ON basis fΨig. 2.2.1 Reduced density operator for a bipartite quantum system Definition 2.3. Let ρAB be a density operator on HAB. Then the reduced density operator ρA is defined as A AB ρ := trB(ρ ); (2.5) AB AB P B AB B whereas trB(Z ) is called a partial trace, which is defined by trB(Z ) := nhΨn jZ jΨn i with A arbitrary basis fjΨiig. The result is an operator on H . More to partial traces can be found in [2]. The reduced density operator ρA can be envisioned as the state in the subsystem SA. All probability A predictions for local measurements (the related observable is in the form of A ⊗ I) on system S can be allocated to the reduced density operator. 2A mixed state is a generalization of a pure state. To see this, set N = 1 in eq. (2.2) and one gets the density operator of a pure state, namely ρpure = j'ih'j. 4 2.3 Entangled states 2 PRELIMINARY DEFINITIONS 2.3 Entangled states Definition 2.4. A composite state is called correlated if and only if its density operator ρAB::: can not be written as a product operator, i.e. ρAB::: 6= ρA ⊗ ρB ⊗ ::: (2.6) Since ρ represents a mixed state in general, correlation alone may not necessarily imply a deviation from classical views. To classify a non classical effect we go on with further definitions. Definition 2.5. A state is called separable if and only if the density operator ρAB::: can be written as n AB::: X A B ρ = pr ρr ⊗ ρr ⊗ ::: (2.7) r=1 Note, that for n = 1, ρAB::: is not correlated. The state for n 6= 1, that is a separable correlated state, is called a classical correlated state. From the definition it is clear that the family of separable states is a convex set 3. Definition 2.6. A state is called entangled if and only if the state is correlated and not separable, i.e. n AB::: X A B ρ 6= pr ρr ⊗ ρr ⊗ ::: (2.8) r=1 It may be useful to take a look at a special case: a pure and correlated state. 2.3.1 A pure correlated composite state Claim. A pure correlated composite state is an entangled one. Proof. Let ρ ≡ ρAB::: be the density operator of a pure state jΨi ≡ jΨAB:::i, i.e. ρ = jΨihΨj. Choose a vector jΦi, so that hΨjΦi = 0. First, we make the attempt to decompose ρ to a convex sum4 of other density operators: n X ρ = λrρr r=1 n X =) hΦjρjΦi = 0 = λrhΦjρrjΦi r=1 Since all hΦjρrjΦi are positive (see eq. (2.3)) and λr are positive, the above equation holds, if and only if hΦjρrjΦi = 0. Now complete jΨi to an orthonormal basis fjφkig; jφ1i ≡ jΨi and look at the matrix elements of ρ and ρr in this basis: hφijρrjφji.