On multipartite symmetric states in Quantum Information Theory

This dissertation describes the application of symmetry to multipartite quantum systems from the viewpoint of quantum information theory. It is written in the framework of abstract quantum information theory, i.e. without distinguishing the physical nature of the d-level systems considered. The results are mathematically rigorous and aim at investigating entanglement properties as well as operational properties of states of multipartite systems. The first chapter gathers the mathematical tools and concepts needed in the following chapters and can be omitted by the experienced reader. In chapter two we apply the concept of symmetry to multipartite systems and show how to construct families of symmetric states that can be described by few parameters only regardless of the size of single systems. Chapter three is devoted to the states of tripartite systems having Werner symmetry. We characterize their separability properties and analyze the strength of the known separability criteria. Furthermore we investigate their entanglement properties in terms of the relative entropy of entanglement, the entanglement monotone induced by the trace norm distance and the maximal violation of Bell inequalities. As a third aspect we analyze the possibility of embedding bipartite Werner states and explore the inner geometry of the manifold given by the statistical distance of two neighbouring tripartite Werner states. In the fourth chapter we introduce multipartite quantum data hiding as an application of multipartite Werner states. We prove the security of this protocol for any coalition and give various examples. The last chapter is concerned with the shared fidelity, its frustration and how it can be maximized in certain multipartite systems.

[1]  R F Werner,et al.  Hiding classical data in multipartite quantum states. , 2002, Physical review letters.

[2]  David P. DiVincenzo,et al.  Hiding Quantum Data , 2002 .

[3]  M. Horodecki,et al.  Reduction criterion of separability and limits for a class of distillation protocols , 1999 .

[4]  I. Chuang,et al.  Quantum Computation and Quantum Information: Introduction to the Tenth Anniversary Edition , 2010 .

[5]  Oliver Rudolph.  A separability criterion for density operators , 2000, quant-ph/0002026.

[6]  M. Nielsen,et al.  Separable states are more disordered globally than locally. , 2000, Physical review letters.

[7]  P. Grangier,et al.  Experimental Tests of Realistic Local Theories via Bell's Theorem , 1981 .

[8]  R. Ellis,et al.  Entropy, large deviations, and statistical mechanics , 1985 .

[9]  V. Buzek,et al.  Quantum secret sharing , 1998, quant-ph/9806063.

[10]  W. Wootters,et al.  Almost every pure state of three qubits is completely determined by its two-particle reduced density matrices. , 2002, Physical review letters.

[11]  G. Lindblad.  On the generators of quantum dynamical semigroups , 1976 .

[12]  C. Fuchs.  Quantum Mechanics as Quantum Information (and only a little more) , 2002, quant-ph/0205039.

[13]  M. Nielsen.  Conditions for a Class of Entanglement Transformations , 1998, quant-ph/9811053.

[14]  W. Wootters.  Statistical distance and Hilbert space , 1981 .

[15]  W. Wootters,et al.  Entanglement sharing among quantum particles with more than two orthogonal states , 2001 .

[16]  G. Pedersen.  C-Algebras and Their Automorphism Groups , 1979 .

[17]  J. Cirac,et al.  Irreversibility in asymptotic manipulations of entanglement. , 2001, Physical review letters.

[18]  D. Petz,et al.  Quantum Entropy and Its Use , 1993 .

[19]  V. Paulsen.  Completely bounded maps and dilations , 1987 .

[20]  Lov K. Grover.  A fast quantum mechanical algorithm for database search , 1996, STOC '96.

[21]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[22]  J. J. Pollitt,et al.  The Ancient View of Greek Art , 1974 .

[23]  M. Wolf,et al.  Distillability via protocols respecting the positivity of partial transpose. , 2001, Physical review letters.

[24]  E. Schmidt.  Zur Theorie der linearen und nichtlinearen Integralgleichungen , 1989 .

[25]  W. Dür.  Multipartite entanglement that is robust against disposal of particles , 2001 .

[26]  R. Werner,et al.  Optimal cloning of pure states, testing single clones , 1998, quant-ph/9807010.

[27]  R. Werner,et al.  Separability properties of tripartite states with U ⊗ U ⊗ U symmetry , 2000, quant-ph/0003008.

[28]  M. Horodecki,et al.  Separability of mixed quantum states: linear contractions approach , 2002, quant-ph/0206008.

[29]  J. Eisert,et al.  Remarks on entanglement measures and non-local state distinguishability , 2003 .

[30]  David Avis,et al.  A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra , 1991, SCG '91.

[31]  W. Wootters,et al.  A single quantum cannot be cloned , 1982, Nature.

[32]  Debbie W. Leung,et al.  Quantum data hiding , 2002, IEEE Trans. Inf. Theory.

[33]  R. Werner.  All teleportation and dense coding schemes , 2000, quant-ph/0003070.

[34]  M. Horodecki,et al.  Separability of mixed states: necessary and sufficient conditions , 1996, quant-ph/9605038.

[35]  P. Horodecki.  Separability criterion and inseparable mixed states with positive partial transposition , 1997, quant-ph/9703004.

[36]  Nicolas Hadjisavvas.  Metrics on the set of states of a W∗-algebra , 1986 .

[37]  R. Feynman.  Quantum mechanical computers , 1986 .

[38]  A. Fine.  Hidden Variables, Joint Probability, and the Bell Inequalities , 1982 .

[39]  C. H. Bennett,et al.  Unextendible product bases and bound entanglement , 1998, quant-ph/9808030.

[40]  A. Shimony,et al.  Proposed Experiment to Test Local Hidden Variable Theories. , 1969 .

[41]  M. Wolf,et al.  All-multipartite Bell-correlation inequalities for two dichotomic observables per site , 2001, quant-ph/0102024.

[42]  R. Werner,et al.  Semicausal operations are semilocalizable , 2001, quant-ph/0104027.

[43]  Slovakia,et al.  Entangled webs: Tight bound for symmetric sharing of entanglement , 2000, quant-ph/0007086.

[44]  D. Deutsch.  Quantum theory, the Church–Turing principle and the universal quantum computer , 1985, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[45]  J. Bell.  On the Einstein-Podolsky-Rosen paradox , 1964 .

[46]  H. Fan.  A note on separability criteria for multipartite state , 2002, quant-ph/0210168.

[47]  Albert Einstein,et al.  Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? , 1935 .

[48]  M. Hübner.  Explicit computation of the Bures distance for density matrices , 1992 .

[49]  A. Jamiołkowski.  Linear transformations which preserve trace and positive semidefiniteness of operators , 1972 .

[50]  A. Uhlmann.  The "transition probability" in the state space of a ∗-algebra , 1976 .

[51]  Peter W. Shor,et al.  Algorithms for quantum computation: discrete logarithms and factoring , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[52]  M. Horodecki,et al.  The Uniqueness Theorem for Entanglement Measures , 2001, quant-ph/0105017.

[53]  Bruno Nachtergaele,et al.  Finitely Correlated Pure States , 1994 .

[54]  Ryszard Horodecki,et al.  Informationally coherent quantum systems , 1994 .

[55]  N Gisin,et al.  No-signaling condition and quantum dynamics. , 2001, Physical review letters.

[56]  Pérès.  Separability Criterion for Density Matrices. , 1996, Physical review letters.

[57]  Charles H. Bennett,et al.  Mixed-state entanglement and quantum error correction. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[58]  E. Rains.  Bound on distillable entanglement , 1998, quant-ph/9809082.

[59]  A. Cabello.  N-particle N-level singlet States: some properties and applications. , 2002, Physical review letters.

[60]  E. Jaynes.  Information Theory and Statistical Mechanics , 1957 .

[61]  D. Leung,et al.  Hiding bits in bell states. , 2000, Physical Review Letters.

[62]  M. Wolf,et al.  Conditional entropies and their relation to entanglement criteria , 2002, quant-ph/0202058.

[63]  Adi Shamir,et al.  How to share a secret , 1979, CACM.

[64]  S. Braunstein,et al.  Statistical distance and the geometry of quantum states. , 1994, Physical review letters.

[65]  Elby,et al.  Triorthogonal uniqueness theorem and its relevance to the interpretation of quantum mechanics. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[66]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[67]  R. Werner.  OPTIMAL CLONING OF PURE STATES , 1998, quant-ph/9804001.

[68]  W. Wootters,et al.  The parts determine the whole in a generic pure quantum state. , 2002, Physical review letters.

[69]  P. Benioff.  The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines , 1980 .

[70]  Charles H. Bennett,et al.  Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. , 1993, Physical review letters.

[71]  R. Werner,et al.  Entanglement measures under symmetry , 2000, quant-ph/0010095.

[72]  E. Schrödinger.  Die gegenwärtige Situation in der Quantenmechanik , 1935, Naturwissenschaften.