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Quantifying Entanglement of Overlapping Indistinguishable Particles Quantifying entanglement of overlapping indistinguishable particles A thesis submitted to University College London for the degree of Doctor of Philosophy in the Department of Physics by Joseph R. Gittings February 2006 UMI Number: U592844 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Dissertation Publishing UMI U592844 Published by ProQuest LLC 2013. Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 I declare that the work presented in this thesis is my own. Joe Gittings A bstract This thesis develops the quantitative study of quantum entanglement in systems of identical particles. Understanding this topic is essential for the construction of quantum information processing devices involving identical particles. A brief overview of necessary concepts and methods, such as the density matrix, the entanglement in pure and mixed states of distinguishable particles, and some common applications of entanglement is given in the introduction. Some competing methods of calculating the entanglement in bipartite pure states of indistinguishable particles are examined. It is shown that only the ’site entropy’ measure introduced by Zanardi satisfies all the criteria for a correct entanglement measure. A teleportation protocol which utilizes all the entan­ glement carried (in both the spin and space degrees of freedom) in a doubly- occupied molecular bonding orbital is presented. The output from an interferometer in a thought experiment described by Omar et al. is studied as an example to see whether entanglement can be separated into space-only, spin-only, and space-spin components. A similar exercise is performed for a doubly-occupied molecular bonding orbital. The relationship between these results and the application of superselection rules (SSRs) to the quantification of useful entanglement is discussed. A numerical method for estimating the entanglement of formation of a mixed state of arbitrary dimension by a conjugate gradient algorithm is described. The results of applying an implementation of the algorithm to both random and isotropic states of 2 qutrits (i.e. two three-dimensional systems) is described. Existing work on calculating entanglement between two sites in various spin systems is outlined. New methods for calculating the entanglement between two sites in various types of degenerate quantum gas - a Fermi gas, a Bose condensate, and a BCS superconductor - are described. The results of numerical studies of the entanglement in a normal metal and a BCS superconductor are reported, both with and without the application of superselection rules for local particle number conservation. Contents 1 Introduction 16 1.1 The density m atrix ................................................................................. 16 1 .1 .1 Pure and impure states and the density operator .............. 16 1.1.2 The density m a trix ................................................................... 18 1.1.3 The density matrix expresses measurement probabilities . 19 1.1.4 The density matrix gives correct measurement averages . 19 1.1.5 Distinguishing pure from mixed states ................................. 20 1 .1 .6 Freedom in the representation of p ........................................ 20 1.1.7 The reduced density m a tr ix .................................................... 21 The partial trace ...................................................................... 21 Justification for using the partial trace ................................ 22 Example of the reduced density operator .......................... 22 1 .2 An introduction to entanglement ....................................................... 23 1.2.1 The qubit and the e b it ............................................................ 24 1.2.2 The Bell s t a t e s ......................................................................... 24 1.2.3 Von Neumann e n tro p y ............................................................ 25 1.2.4 Other properties of the von Neumann entropy .................... 25 1.2.5 Wootters tan g le ........................................................................... 26 1.2.6 Pure state concurrence .............................................................. 26 1.2.7 Entanglement m o n o to n e s ....................................................... 27 1.2.8 Quantum relative e n tro p y ............................................... 28 1.2.9 Entanglement of formation ...................................................... 28 Entanglement of formation for a pure quantum state .. 29 Entanglement of formation for a mixed quantum state . 31 4 Wootters formula for entanglement of formation of two q u b its .......................................................................... 32 Postulated additivity of entanglement of formation .... 32 1.2.10 Entanglement of d istillatio n .................................................. 34 1.3 What can entanglement do? ................................................................. 34 1.3.1 Bell’s theorem .......................................................................... 35 P re a m b le ................................................................................... 35 Nomenclature ............................................................................. 35 Introduce a hidden variable ................................................... 36 Hidden variables are consistent with the results of mea­ surements on a single spin ................................... 37 Hidden variables are not consistent with the results of separate measurements on two spins in a singlet s ta te ............................................................................. 38 A choice of measurement axes which violates Bell’s in­ equality ....................................................................... 40 Bell’s theorem and entanglement .......................................... 40 1.3.2 Quantum teleportation ........................................................... 41 Summary of the teleportation protocol ................................ 41 Description of the teleportation protocol ............................. 41 Significance of the teleportation protocol ............................. 45 1.3.3 Quantum key d istrib u tio n ........................................................ 45 The BB84 QKD protocol ...................................................... 46 The EPR QKD protocol .......................................................... 48 1.3.4 Quantum com puting ................................................................. 52 Overview of quantum computing .......................................... 52 A universal set of quantum gates must have at least one entangling gate .......................................................... 54 Quantum logic gate model of the teleportation protocol . 59 1.4 Conclusion .............................................................................................. 59 2 Entanglement of indistinguishable particles 61 2.1 Introduction .............................................................................................. 61 2.1.1 Why the focus on ferm ions? .................................................... 62 2.1.2 The Kane proposal .................................................................... 62 5 2.1.3 Why is quantifying entanglement between indistinguish­ able particles a problem? ...................................................... 63 2.2 There is no entanglement due to the overall symmetry of the wavefunction .......................................................................................... 64 2.2.1 Definition of a ‘remote’ state for any measurement .... 64 2 .2 .2 Entanglement and overall sym m etry ..................................... 65 2.3 Methods of partitioning Hilbert space of two entangled spinful p a rticle s ................................................................................................... 6 6 2.3.1 Requirements for p artitioning ................................................. 6 6 Tensor product structure ...................................................... 6 6 Delocalization ............................................................................. 67 Indistinguishability ................................................................... 67 2.3.2 Partitioning used by existing entanglement measures . 6 8 2.4 Desirable properties of any entanglement measure ........................... 69 2.4.1 Invariance under local unitary tran sfo rm atio n s ................. 69 2.4.2 Non-in variance under some non-local unitary transforma­ tions is not an undesirable property ................................... 69 2.4.3 Correct behaviour as distinguishability of subsystems A and B is lost ............................................................................. 69 Behaviour of |\I')± Bell sta te s ................................................ 70 Behaviour of 1^)* Bell states ................................................ 71 2.5 Wootters measure for distinguishable particles (tangle) ................
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