Quantum entanglement in finite-dimensional Hilbert spaces by Szil´ardSzalay Dissertation presented to the Doctoral School of Physics of the Budapest University of Technology and Economics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Supervisor: Dr. P´eterP´alL´evay research associate professor Department of Theoretical Physics Budapest University of Technology and Economics arXiv:1302.4654v1 [quant-ph] 19 Feb 2013 2013 To my wife, daughter and son. v Abstract. In the past decades, quantum entanglement has been recognized to be the basic resource in quantum information theory. A fundamental need is then the understanding its qualification and its quantification: Is the quantum state entangled, and if it is, then how much entanglement is carried by that? These questions introduce the topics of separability criteria and entanglement measures, both of which are based on the issue of classification of multipartite entanglement. In this dissertation, after reviewing these three fundamental topics for finite dimensional Hilbert spaces, I present my contribution to knowledge. My main result is the elaboration of the partial separability classification of mixed states of quantum systems composed of arbitrary number of subsystems of Hilbert spaces of arbitrary dimensions. This problem is simple for pure states, however, for mixed states it has not been considered in full detail yet. I give not only the classification but also necessary and sufficient criteria for the classes, which make it possible to determine to which class a mixed state belongs. Moreover, these criteria are given by the vanishing of quantities measuring entanglement. Apart from these, I present some side results related to the entanglement of mixed states. These results are obtained in the learning phase of my studies and give some illustrations and examples. Contents Acknowledgements ix Certifications in hungarian xi List of publications xiii Thesis statements xv Prologue xvii Chapter 1. Quantum entanglement1 1.1. Quantum systems2 1.2. Composite systems and entanglement 13 1.3. Quantifying entanglement 28 1.4. Summary 40 Chapter 2. Two-qubit mixed states with fermionic purifications 43 2.1. The density matrix 43 2.2. Measures of entanglement for the density matrix 46 2.3. Relating different measures of entanglement 52 2.4. Summary and remarks 55 Chapter 3. All degree 6 local unitary invariants of multipartite systems 57 3.1. Local unitary invariant polynomials 58 3.2. Graphs and matrix operations 61 3.3. Pure state invariants 63 3.4. Mixed-state invariants 69 r 3.5. Algorithm for S3=S3 71 3.6. Summary and remarks 73 Chapter 4. Separability criteria for mixed three-qubit states 75 4.1. A symmetric family of mixed three-qubit states 76 4.2. Bipartite separability criteria 78 4.3. Tripartite separability criteria 86 4.4. Tripartite entanglement 96 4.5. Summary and remarks 99 Chapter 5. Partial separability classification 101 5.1. Partial separability of tripartite mixed states 101 5.2. Generalizations: Partial separability of multipartite systems 108 5.3. Summary and remarks 118 Chapter 6. Three-qubit systems and FTS approach 121 vii viii CONTENTS 6.1. State vectors of three qubits 122 6.2. Mixed states of three qubits 127 6.3. Generalizations: Three subsystems 131 6.4. Summary and remarks 133 Epilogue 135 Bibliography 137 Acknowledgements This work would not have been possible without the help of several people, whom I would like to mention here. First and foremost, it is a pleasure to thank my adviser P´eterL´evay for his supervising through the years of learning and research, and for giving me independence to pursue research on the ideas that came across my mind. His helpful discussions together with his insight and passion for research have always been inspiring. I would like to extend my gratitude to some of my other teachers as well, D´enesPetz, Tam´asGeszti and Tam´asMatolcsi, the lectures and books of whom were guides of great value in studying quantum mechanics and mathematical physics. I am grateful to L´aszl´oSzunyogh, the head of the Department of Theoretical Physics, and Gy¨orgy Mih´aly, the head of the Doctoral School of Physics, as well as M´ariaVida, my admin- istrator, for the flexible, effective and helpful attitude for administrative issues, supporting my studies to a large extent. My Ph.D. studies were partially supported by the New Hungary De- velopment Plan (project ID: TAMOP-4.2.1.B-09/1/KMR-2010-0002),´ the New Sz´echenyiPlan of Hungary (project ID: TAMOP-4.2.2.B-10/1{2010-0009)´ and the Strongly correlated systems research group of the \Momentum" program of the Hungarian Academy of Sciences (project ID: 81010-00). I am gerateful to my parents for supproting my studies financially and in principles as well. I would not be succesful without this. Last but not least, I thank my wife, M´arta, for her faithful love and everlasting support, pro- viding the affectionate and peaceful atmosphere which is an essential condition of any absorbed research. I would like to dedicate this piece of work to her and to our children. ix Certifications in hungarian Alul´ırott Szalay Szil´ard kijelentem, hogy ezt a doktori ´ertekez´estmagam k´esz´ıtettem´esabban csak a megadott forr´asokat haszn´altamfel. Minden olyan r´eszt,amelyet sz´oszerint vagy azonos tartalommal, de ´atfogalmazva m´asforr´asb´ol´atvettem, egy´ertelm}uen,a forr´asmegad´as´aval meg- jel¨oltem. Budapest, 2013. februr 14. Szalay Szil´ard Alul´ırott Szalay Szil´ard hozz´aj´aruloka doktori ´ertekez´eseminterneten t¨ort´en}okorl´atoz´asn´elk¨uli nyilv´anoss´agrahozatal´ahoz. Budapest, 2013. februr 14. Szalay Szil´ard xi List of publications The research articles [1], [4], [5] and [6] are covered by this thesis. The research articles [2] and [3] are the results of another research project done in the related field of Black Hole / Qubit correspondence. The publications are listed in chronological order. [1] Szil´ardSzalay, P´eterL´evay, Szilvia Nagy, J´anosPipek, A study of two-qubit density matrices with fermionic purifications, J. Phys. A 41, 505304 (2008)(arXiv: 0807.1804 [quant-ph]) [2] P´eterL´evay, Szil´ardSzalay, Attractor mechanism as a distillation procedure, Phys. Rev. D 82, 026002 (2010)(arXiv: 1004.2346 [hep-th]) [3] P´eterL´evay, Szil´ardSzalay, ST U attractors from vanishing concurrence, Phys. Rev. D 84, 045005 (2011)(arXiv: 1011.4180 [hep-th]) [4] Szil´ardSzalay, Separability criteria for mixed three-qubit states, Phys. Rev. A 83, 062337 (2011)(arXiv: 1101.3256 [quant-ph]) [5] Szil´ardSzalay, All degree 6 local unitary invariants of k qudits, J. Phys. A 45, 065302 (2012)(arXiv: 1105.3086 [quant-ph]) [6] Szil´ardSzalay, Zolt´anK¨ok´enyesi Partial separability revisited: Necessary and sufficient criteria, Phys. Rev. A 86, 032341 (2012)(arXiv: 1206.6253 [quant-ph]) xiii Thesis statements In the past decades, quantum entanglement has been recognized to be the basic resource in quantum information theory. A fundamental need is the understanding of its qualification and its quantification: Is the state entangled, and in this case how much entanglement is carried by it? These questions introduce the topics of separability criteria and entanglement measures, both of which are based on the problem of classification of multipartite entanglement. In the following thesis statements I present my contribution to these three issues. I. I study a 12-parameter family of two-qubit mixed states, arising from a special class of two-fermion systems with four single particle states or alternatively from a four-qubit state vector with amplitudes arranged in an antisymmetric matrix. I obtain a local unitary canonical form for those states. By the use of this I calculate two famous entanglement measures which are the Wooters concurrence and the negativity in a closed form. I obtain bounds on the negativity for given Wootters concurrence, which are strictly stronger than those for general two-qubit states. I show that the relevant entanglement measures satisfy the generalized Coffman-Kundu-Wootters formula of distributed entanglement. I give an explicit formula for the residual tangle as well. The publication belonging to this thesis statement is [1] of the list on page xiii. The main references belonging to this thesis statement are [LNP05, VADM01, CKW00, OV06]. II. Local unitary invariance is a fundamental property of all entanglement measures.I study quantities having this property for general multipartite systems. In particular, I give explicit index-free formulas for all the algebraically independent local unitary invariant polynomials up to degree six, for finite dimensional multipartite pure and mixed quantum states. I carry out this task by the use of graph-technical methods, which provide illustrations for this rather abstract topic. The publication belonging to this thesis statement is [5] of the list on page xiii. The main references belonging to this thesis statement are [HW09, HWW09, Vra11a, Vra11b]. III. I study the noisy GHZ-W mixture and demonstrate some necessary but not sufficient criteria for different classes of separability of these states. I find that the partial transposition criterion of Peres and the criteria of G¨uhneand Seevinck dealing directly with matrix elements are the strongest ones for different separability classes of this xv xvi THESIS STATEMENTS two-parameter state. I determine a set of entangled states of positive partial transpose. I also give constraints on three-qubit entanglement classes related to the pure SLOCC- classes, and I calculate the Wootters concurrences of the two-qubit subsystems. The publication belonging to this thesis statement is [4] of the list on page xiii. The main references belonging to this thesis statement are [Per96, GS10]. IV. I elaborate the partial separability classification of mixed states of quantum systems composed of arbitrary number of subsystems of Hilbert spaces of arbitrary dimensions.