Geometry of quantum states: an introduction to quantum entanglement by Ingemar Bengtsson and Karol Zyczkowski

Preface 1. Convexity, colours and statistics 2. Geometry of probability distributions 3. Much ado about spheres 4. Complex projective spaces 5. Outline of quantum mechanics 6. Coherent states and group actions 7. The stellar representation 8. The space of density matrices 9. Purification of mixed quantum states 10. Quantum operations 11. Duality: maps versus states 12. Density matrices and entropies 13. Distinguishability measures 14. Monotone metrics and measures 15. Quantum entanglement Epilogue Appendices References Index.

[1]  Ingemar Bengtsson,et al.  How to mix a density matrix , 2003 .

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

[3]  H. Weinfurter,et al.  Experimental quantum teleportation , 1997, Nature.

[4]  J. Ginibre Statistical Ensembles of Complex, Quaternion, and Real Matrices , 1965 .

[5]  G. Ewald Combinatorial Convexity and Algebraic Geometry , 1996 .

[6]  G. Bjork,et al.  Entanglement measure for general pure multipartite quantum states , 2004, quant-ph/0401129.

[7]  Amiel Feinstein,et al.  Information and information stability of random variables and processes , 1964 .

[8]  K. Horadam Hadamard Matrices and Their Applications , 2006 .

[9]  M. B. Plenio,et al.  The emergence of typical entanglement in two-party random processes , 2007 .

[10]  P. Lévay The geometry of entanglement: metrics, connections and the geometric phase , 2003, quant-ph/0306115.

[11]  P. Oscar Boykin,et al.  A New Proof for the Existence of Mutually Unbiased Bases , 2002, Algorithmica.

[12]  M. Horodecki,et al.  Quantum α-entropy inequalities: independent condition for local realism? , 1996 .

[13]  A. Uhlmann Entropy and Optimal Decompositions of States Relative to a Maximal Commutative Subalgebra , 1997, quant-ph/9704017.

[14]  Michał Horodecki,et al.  Separability of Mixed Quantum States: Linear Contractions and Permutation Criteria , 2006, Open Syst. Inf. Dyn..

[15]  Michael D. Westmoreland,et al.  Sending classical information via noisy quantum channels , 1997 .

[16]  Ling-An Wu,et al.  A matrix realignment method for recognizing entanglement , 2003, Quantum Inf. Comput..

[17]  N. Gisin Bell's inequality holds for all non-product states , 1991 .

[18]  Fritz Haake,et al.  Quantum chaos of SU3 observables , 2000 .

[19]  S. Fei,et al.  Concurrence of arbitrary dimensional bipartite quantum states. , 2005, Physical review letters.

[20]  Karol Zyczkowski,et al.  Geometry of sets of quantum maps: A generic positive map acting on a high-dimensional system is not completely positive , 2007, 0710.1571.

[21]  Robert R. Tucci All Moments of the Uniform Ensemble of Quantum Density Matrices , 2002, quant-ph/0206193.

[22]  H. Sommers,et al.  Average fidelity between random quantum states , 2003, quant-ph/0311117.

[23]  M. Lewenstein,et al.  Volume of the set of separable states , 1998, quant-ph/9804024.

[24]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[25]  H. Bacry Orbits of the rotation group on spin states , 1974 .

[26]  R. Jozsa,et al.  A Complete Classification of Quantum Ensembles Having a Given Density Matrix , 1993 .

[27]  M. Klimesh,et al.  Mathematical structure of entanglement catalysis , 2001 .

[28]  R. Hudson When is the wigner quasi-probability density non-negative? , 1974 .

[29]  Hong-Jong Kim,et al.  Indecomposable Extreme Positive Linear Maps in Matrix Algebras , 1994 .

[30]  Julio I. De Vicente,et al.  Separability criteria based on the bloch representation of density matrices , 2006, Quantum Inf. Comput..

[31]  K. Audenaert,et al.  Asymptotic relative entropy of entanglement. , 2001, Physical review letters.

[32]  Vittorio Gorini,et al.  Fundamental aspects of quantum theory , 1986 .

[33]  Man-Duen Choi Completely positive linear maps on complex matrices , 1975 .

[34]  David P. DiVincenzo,et al.  Entanglement of Assistance , 1998, QCQC.

[35]  L. J. Landau,et al.  On Birkhoff's theorem for doubly stochastic completely positive maps of matrix algebras , 1993 .

[36]  P. K. Aravind The “twirl”, stella octangula and mixed state entanglement , 1997 .

[37]  Karol Zyczkowski,et al.  Rényi-Wehrl entropies as measures of localization in phase space , 2001 .

[38]  Jianhua Lin,et al.  Divergence measures based on the Shannon entropy , 1991, IEEE Trans. Inf. Theory.

[39]  J. Dittmann The scalar curvature of the Bures metric on the space of density matrices , 1998, quant-ph/9810012.

[40]  R. Delbourgo,et al.  Maximum weight vectors possess minimal uncertainty , 1977 .

[41]  Chi-Kwong Li,et al.  Convexity of the Joint Numerical Range , 1999, SIAM J. Matrix Anal. Appl..

[42]  Marie Paule Malliavin,et al.  Séminaire d'algèbre Paul Dubreil et Marie-Paule Malliavin , 1980 .

[43]  G. Vidal Efficient classical simulation of slightly entangled quantum computations. , 2003, Physical review letters.

[44]  K. Wódkiewicz,et al.  Stochastic decoherence of qubits. , 2001, Optics express.

[45]  A. Wehrl On the relation between classical and quantum-mechanical entropy , 1979 .

[46]  K. Życzkowski,et al.  Composed ensembles of random unitary matrices , 1997, chao-dyn/9707006.

[47]  D. Petz Geometry of canonical correlation on the state space of a quantum system , 1994 .

[48]  D. Brody,et al.  Geometric quantum mechanics , 1999, quant-ph/9906086.

[49]  J. M. Radcliffe Some properties of coherent spin states , 1971 .

[50]  J. Cirac,et al.  Entanglement cost of bipartite mixed states. , 2001, Physical Review Letters.

[51]  D. Varshalovich,et al.  Quantum Theory of Angular Momentum , 1988 .

[52]  Abhay Ashtekar,et al.  Geometrical Formulation of Quantum Mechanics , 1999 .

[53]  W. Wootters Random quantum states , 1990 .

[54]  Gérard G. Emch,et al.  Algebraic methods in statistical mechanics and quantum field theory , 1972 .

[55]  Christopher King,et al.  Minimal entropy of states emerging from noisy quantum channels , 2001, IEEE Trans. Inf. Theory.

[56]  H. Sommers,et al.  Bures volume of the set of mixed quantum states , 2003, quant-ph/0304041.

[57]  T. G. Room,et al.  On the Clifford collineation, transform and similarity groups. II. , 1961 .

[58]  M.A.Jafarizadeh,et al.  Optimal Lewenstein?Sanpera decomposition for some bipartite systems , 2004 .

[59]  Uniwersytet Jagiello,et al.  Hilbert-Schmidt volume of the set of mixed quantum states , 2003 .

[60]  S. Woronowicz Nonextendible positive maps , 1976 .

[61]  P. Schupp On Lieb's Conjecture for the Wehrl Entropy¶of Bloch Coherent States , 1999, math-ph/9902017.

[62]  Aleksandr Yakovlevich Khinchin,et al.  Mathematical foundations of information theory , 1959 .

[63]  A. Robertson AUTOMORPHISMS OF SPIN FACTORS AND THE DECOMPOSITION OF POSITIVE MAPS , 1983 .

[64]  M. Tribus,et al.  Energy and information , 1971 .

[65]  Barry Simon,et al.  Chern numbers, quaternions, and Berry's phases in Fermi systems , 1989 .

[66]  C. Fuchs Distinguishability and Accessible Information in Quantum Theory , 1996, quant-ph/9601020.

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

[68]  A. Orłowski,et al.  Classical entropy of quantum states of light. , 1993, Physical review. A, Atomic, molecular, and optical physics.

[69]  J. Cirac,et al.  Distillability and partial transposition in bipartite systems , 1999, quant-ph/9910022.

[70]  Sung Dahm Oh,et al.  Convex-roof extended negativity as an entanglement measure for bipartite quantum systems , 2003, quant-ph/0310027.

[71]  W. Arveson On subalgebras of $C^*$-algebras , 1969 .

[72]  R. Mcweeny On the Einstein-Podolsky-Rosen Paradox , 2000 .

[73]  I. D. Ivonovic Geometrical description of quantal state determination , 1981 .

[74]  Agarwal,et al.  Wigner distribution of a general angular-momentum state: Applications to a collection of two-level atoms. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[75]  Göran Lindblad,et al.  An entropy inequality for quantum measurements , 1972 .

[76]  A. Perelomov Generalized coherent states and some of their applications , 1977 .

[77]  Joseph M. Renes,et al.  Symmetric informationally complete quantum measurements , 2003, quant-ph/0310075.

[78]  A. Plastino,et al.  Multiqubit systems: highly entangled states and entanglement distribution , 2007, 0803.3979.

[79]  J. Ignacio Cirac,et al.  Equivalence classes of non-local unitary operations , 2002, Quantum Inf. Comput..

[80]  M. Horodecki,et al.  Mixed-State Entanglement and Distillation: Is there a “Bound” Entanglement in Nature? , 1998, quant-ph/9801069.

[81]  K. Farahmand On the Average Number of Real Roots of a Random Algebraic Equation , 1986 .

[82]  S. Chern Complex manifolds without potential theory , 1979 .

[83]  C. Tsallis,et al.  Peres criterion for separability through nonextensive entropy , 2001 .

[84]  M. Keyl Fundamentals of quantum information theory , 2002, quant-ph/0202122.

[85]  L. Landau Das Dämpfungsproblem in der Wellenmechanik , 1927 .

[86]  J. Schwinger UNITARY OPERATOR BASES. , 1960, Proceedings of the National Academy of Sciences of the United States of America.

[87]  M. Tinkham Group Theory and Quantum Mechanics , 1964 .

[88]  L. Da̧browski,et al.  Quantum statistical holonomy , 1989 .

[89]  Bernhard G. Bodmann A Lower Bound for the Wehrl Entropy of Quantum Spin with Sharp High-Spin Asymptotics , 2004 .

[90]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[91]  Feza Gürsey,et al.  Complex and quaternionic analyticity in chiral and gauge theories, I☆ , 1980 .

[92]  Ashish V. Thapliyal,et al.  Evidence for bound entangled states with negative partial transpose , 1999, quant-ph/9910026.

[93]  A. J. Coleman,et al.  Reduced Density Matrices , 2000 .

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

[95]  Sergey Bravyi Requirements for copatibility between local and multipartite quantum states , 2004, Quantum Inf. Comput..

[96]  M. Plenio,et al.  Quantifying Entanglement , 1997, quant-ph/9702027.

[97]  R. Bhatia Matrix Analysis , 1996 .

[98]  E. Saff,et al.  Distributing many points on a sphere , 1997 .

[99]  E. Sudarshan,et al.  Volumes of compact manifolds , 2002, math-ph/0210033.

[100]  F. Benatti,et al.  Quantum dynamical semigroups and non-decomposable positive maps , 2004 .

[101]  Karol Zyczkowski,et al.  Rényi Extrapolation of Shannon Entropy , 2003, Open Syst. Inf. Dyn..

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

[103]  King,et al.  Demonstration of a fundamental quantum logic gate. , 1995, Physical review letters.

[104]  E. Thiele,et al.  A measure of quantum chaos , 1984 .

[105]  S. Golden LOWER BOUNDS FOR THE HELMHOLTZ FUNCTION , 1965 .

[106]  Ling-An Wu,et al.  Test for entanglement using physically observable witness operators and positive maps , 2003, quant-ph/0306041.

[107]  M. Grasselli,et al.  On the Uniqueness of the Chentsov Metric in Quantum Information Geometry , 2000, math-ph/0006030.

[108]  G. Vidal Entanglement of pure states for a single copy , 1999, quant-ph/9902033.

[109]  Kennedy,et al.  Rigorous results on valence-bond ground states in antiferromagnets. , 1987, Physical review letters.

[110]  M. Lewenstein,et al.  Entropic uncertainty relations and entanglement , 2004, quant-ph/0403219.

[111]  Tamara G. Kolda,et al.  Tensor Decompositions and Applications , 2009, SIAM Rev..

[112]  Léon Autonne,et al.  Sur les matrices hypohermitiennes et sur les matrices unitaires , 1915 .

[113]  A. Uhlmann Spheres and hemispheres as quantum state spaces , 1996 .

[114]  R. Simon,et al.  Quantum Kinematic Approach to the Geometric Phase. I. General Formalism , 1993 .

[115]  A. Miyake Classification of multipartite entangled states by multidimensional determinants , 2002, quant-ph/0206111.

[116]  P. Shor Equivalence of Additivity Questions in Quantum Information Theory , 2004 .

[117]  Mervin E. Muller,et al.  A note on a method for generating points uniformly on n-dimensional spheres , 1959, CACM.

[118]  Alan Edelman,et al.  How many zeros of a random polynomial are real , 1995 .

[119]  K. Jones Entropy of random quantum states , 1990 .

[120]  E. Schrödinger Die gegenwärtige Situation in der Quantenmechanik , 2005, Naturwissenschaften.

[121]  T. Erber,et al.  Complex systems: Equilibrium configurations of N equal charges on a sphere (2 <= N <= 112) , 1995 .

[122]  I. Segal Postulates for General Quantum Mechanics , 1947 .

[123]  A. Acín,et al.  Three-qubit pure-state canonical forms , 2000, quant-ph/0009107.

[124]  P. Slater Silver mean conjectures for 15-dimensional volumes and 14-dimensional hyperareas of the separable two-qubit systems , 2003, quant-ph/0308037.

[125]  K. Życzkowski,et al.  Quantum dynamical entropy and decoherence rate , 2003, quant-ph/0309194.

[126]  A. Uhlmann Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory , 1977 .

[127]  Fabio Benatti,et al.  Additivity of the entanglement of formation , 2001 .

[128]  H. Barnum,et al.  Monotones and invariants for multi-particle quantum states , 2001, quant-ph/0103155.

[129]  W. Hoeffding Probability Inequalities for sums of Bounded Random Variables , 1963 .

[130]  Anirvan M. Sengupta,et al.  Distributions of singular values for some random matrices. , 1997, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[131]  K. Hammerer,et al.  Characterization of nonlocal gates , 2002 .

[132]  B. Terhal Bell inequalities and the separability criterion , 1999, quant-ph/9911057.

[133]  Reck,et al.  Experimental realization of any discrete unitary operator. , 1994, Physical review letters.

[134]  Ayumu Sugita LETTER TO THE EDITOR: Proof of the generalized Lieb-Wehrl conjecture for integer indices larger than one , 2002 .

[135]  E. Rains RIGOROUS TREATMENT OF DISTILLABLE ENTANGLEMENT , 1998, quant-ph/9809078.

[136]  Christian Kurtsiefer,et al.  Experimental detection of multipartite entanglement using witness operators. , 2004, Physical review letters.

[137]  A. Sudbery,et al.  How entangled can two couples get , 2000, quant-ph/0005013.

[138]  K. Ball An Elementary Introduction to Modern Convex Geometry , 1997 .

[139]  M. Nielsen Continuity bounds for entanglement , 1999, quant-ph/9908086.

[140]  P. Zanardi Entanglement of Quantum Evolutions , 2001 .

[141]  Belgium,et al.  Maximal entanglement versus entropy for mixed quantum states , 2002, quant-ph/0208138.

[142]  S. Popescu,et al.  Thermodynamics and the measure of entanglement , 1996, quant-ph/9610044.

[143]  J. Mayer,et al.  On the Quantum Correction for Thermodynamic Equilibrium , 1947 .

[144]  A. Cabello,et al.  Bell-Kochen-Specker theorem: A proof with 18 vectors , 1996, quant-ph/9706009.

[145]  A. T. Rezakhani,et al.  On a suggestion relating topological and quantum mechanical entanglements [rapid communication] , 2004 .

[146]  E. Sudarshan,et al.  RELATIVISTIC INVARIANCE AND HAMILTONIAN THEORIES OF INTERACTING PARTICLES , 1963 .

[147]  A. Rajagopal,et al.  Separability and correlations in composite states based on entropy methods , 2002, quant-ph/0203124.

[148]  Victor Guillemin,et al.  Convexity Theorems for Varieties Invariant under a Borel Subgroup , 2005 .

[149]  Joe W. Harris,et al.  Algebraic Geometry: A First Course , 1995 .

[150]  M. Fannes,et al.  Finitely correlated states on quantum spin chains , 1992 .

[151]  Brian S. Hepburn,et al.  Generic incomparability of infinite-dimensional entangled states , 2002, quant-ph/0205063.

[152]  K. Życzkowski,et al.  Geometry of entangled states , 2000, quant-ph/0006068.

[153]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[154]  F. Verstraete,et al.  Lieb-Robinson bounds and the generation of correlations and topological quantum order. , 2006, Physical review letters.

[155]  Rory A. Fisher,et al.  Theory of Statistical Estimation , 1925, Mathematical Proceedings of the Cambridge Philosophical Society.

[156]  Man-Duen Choi,et al.  Extremal positive semidefinite forms , 1977 .

[157]  E. Study Kürzeste Wege im komplexen Gebiet , 1905 .

[158]  Philippe Blanchard,et al.  Measures of entanglement based on decoherence , 2001 .

[159]  W. Wootters A Wigner-function formulation of finite-state quantum mechanics , 1987 .

[160]  J. Neumann Mathematical Foundations of Quantum Mechanics , 1955 .

[161]  C. Lam,et al.  The search for a finite projective plane of order 10 , 1991 .

[162]  N. Gisin Stochastic quantum dynamics and relativity , 1989 .

[163]  Pham Huu Tiep,et al.  Orthogonal decompositions and integral lattices , 1994 .

[164]  A. Sudbery,et al.  On Gell-Mann's λ-matrices,d- andf-tensors, octets, and parametrizations ofS U (3) , 1968 .

[165]  Seung-Hyeok Kye,et al.  DUALITY FOR POSITIVE LINEAR MAPS IN MATRIX ALGEBRAS , 2000 .

[166]  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.

[167]  Max Delbrück,et al.  Statistische Quantenmechanik und Thermodynamik , 1936 .

[168]  C. Beck,et al.  Thermodynamics of chaotic systems , 1993 .

[169]  M. Horodecki,et al.  The asymptotic entanglement cost of preparing a quantum state , 2000, quant-ph/0008134.

[170]  O. Gühne,et al.  Experimental detection of entanglement via witness operators and local measurements , 2002, quant-ph/0210134.

[171]  Karol Życzkowski,et al.  Random quantum operations , 2008, 0804.2361.

[172]  D. Bruß Characterizing Entanglement , 2001, quant-ph/0110078.

[173]  M. Jafarizadeh,et al.  Robustness of entanglement for Bell decomposable states , 2002, quant-ph/0208019.

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

[175]  Paul Erdös,et al.  On an Elementary Proof of Some Asymptotic Formulas in the Theory of Partitions , 1942 .

[176]  L. M. M.-T. Theory of Probability , 1929, Nature.

[177]  Kae Nemoto,et al.  Typical entanglement in multiple-qubit systems , 2001, quant-ph/0106023.

[178]  D. M. Appleby Symmetric informationally complete–positive operator valued measures and the extended Clifford group , 2005 .

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

[180]  L. Gurvits,et al.  Largest separable balls around the maximally mixed bipartite quantum state , 2002, quant-ph/0204159.

[181]  J. Cirac,et al.  Reflections upon separability and distillability , 2001, quant-ph/0110081.

[182]  Barbara M. Terhal Detecting quantum entanglement , 2002, Theor. Comput. Sci..

[183]  V. Varadarajan Geometry of quantum theory , 1968 .

[184]  Wojciech Słomczyński,et al.  The Monge metric on the sphere and geometry of quantum states , 2000, quant-ph/0008016.

[185]  J. V. Corbett,et al.  The geometry of state space , 1993 .

[186]  J. Bell,et al.  Speakable and Unspeakable in Quatum Mechanics , 1988 .

[187]  E. Knill,et al.  Power of One Bit of Quantum Information , 1998, quant-ph/9802037.

[188]  J. Preskill Reliable quantum computers , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[189]  Oliver Rudolph Some properties of the computable cross-norm criterion for separability , 2002, quant-ph/0212047.

[190]  Y. Wong Differential geometry of grassmann manifolds. , 1967, Proceedings of the National Academy of Sciences of the United States of America.

[191]  M. Kus,et al.  Separable approximation for mixed states of composite quantum systems. , 2001, quant-ph/0104098.

[192]  V. Buzek,et al.  Entangled graphs: Bipartite entanglement in multiqubit systems , 2002, quant-ph/0211020.

[193]  Wayne C. Myrvold,et al.  Bell’s Theorem , 2011 .

[194]  Silvio Levy,et al.  Flavors of Geometry , 1997 .

[195]  F. Wilczek,et al.  Geometric Phases in Physics , 1989 .

[196]  J. Dittmann On the Riemannian metric on the space of density matrices , 1995 .

[197]  E. Lieb Some Convexity and Subadditivity Properties of Entropy , 1975 .

[198]  A. Uhlmann,et al.  Stochasticity and Partial Order: Doubly Stochastic Maps and Unitary Mixing , 1982 .

[199]  Wojciech Tadej,et al.  A Concise Guide to Complex Hadamard Matrices , 2006, Open Syst. Inf. Dyn..

[200]  V. Milman,et al.  Asymptotic Theory Of Finite Dimensional Normed Spaces , 1986 .

[201]  L. Mandel,et al.  Optical Coherence and Quantum Optics , 1995 .

[202]  Leboeuf,et al.  Distribution of roots of random polynomials. , 1992, Physical review letters.

[203]  P. Zanardi,et al.  Entangling power of quantum evolutions , 2000, quant-ph/0005031.

[204]  N. J. A. Sloane,et al.  McLaren’s improved snub cube and other new spherical designs in three dimensions , 1996, Discret. Comput. Geom..

[205]  M. Donald Further results on the relative entropy , 1987, Mathematical Proceedings of the Cambridge Philosophical Society.

[206]  R. D. Hill,et al.  On the matrix reorderings Γ and ψ , 1985 .

[207]  A. Plastino,et al.  Inclusion relations among separability criteria , 2003, quant-ph/0307167.

[208]  J. Cirac,et al.  Optimization of entanglement witnesses , 2000, quant-ph/0005014.

[209]  A. O. Pittenger Unextendible product bases and the construction of inseparable states , 2002, quant-ph/0208028.

[210]  A. Montanaro,et al.  On the dimension of subspaces with bounded Schmidt rank , 2007, 0706.0705.

[211]  Kanno,et al.  Proof of Page's conjecture on the average entropy of a subsystem. , 1994, Physical review letters.

[212]  P. Busch,et al.  The quantum theory of measurement , 1991 .

[213]  Karol Zyczkowski,et al.  Mean Dynamical Entropy of Quantum Maps on the Sphere Diverges in the Semiclassical Limit , 1997, chao-dyn/9707008.

[214]  R. Alicki,et al.  Comment on "Reduced dynamics need not be completely positive" , 1995, Physical review letters.

[215]  E. Sudarshan Structure and Parametrization of Stochastic Maps of Density Matrices , 2001, quant-ph/0109158.

[216]  A. Miranowicz,et al.  Inseparability criteria based on matrices of moments , 2009 .

[217]  J. Klauder,et al.  COHERENT STATES: APPLICATIONS IN PHYSICS AND MATHEMATICAL PHYSICS , 1985 .

[218]  S. Luo A simple proof of Wehrl's conjecture on entropy , 2000 .

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

[220]  A. Uhlmann Density operators as an arena for differential geometry , 1993 .

[221]  A. Calderbank,et al.  Quantum Error Correction and Orthogonal Geometry , 1996, quant-ph/9605005.

[222]  G. Roger,et al.  Experimental Test of Bell's Inequalities Using Time- Varying Analyzers , 1982 .

[223]  V. Vedral,et al.  Information, relative entropy of entanglement, and irreversibility. , 1999, Physical review letters.

[224]  Emil Wolf,et al.  Principles of Optics: Contents , 1999 .

[225]  R. Gilmore,et al.  Coherent states: Theory and some Applications , 1990 .

[226]  M. Lewenstein,et al.  Quantum Entanglement , 2020, Quantum Mechanics.

[227]  S. G. Hoggar,et al.  t-Designs in Projective Spaces , 1982, Eur. J. Comb..

[228]  Wai-Shing Tang On positive linear maps between matrix algebras , 1986 .

[229]  H. Weinfurter,et al.  Witnessing multipartite entanglement , 2003, quant-ph/0309043.

[230]  Leonid Gurvits,et al.  Classical complexity and quantum entanglement , 2004, J. Comput. Syst. Sci..

[231]  T. Kibble,et al.  Geometrization of quantum mechanics , 1979 .

[232]  D. Gross,et al.  Evenly distributed unitaries: On the structure of unitary designs , 2006, quant-ph/0611002.

[233]  D. Petz Monotone metrics on matrix spaces , 1996 .

[234]  Michael Atiyah,et al.  Convexity and Commuting Hamiltonians , 1982 .

[235]  R. Jozsa Fidelity for Mixed Quantum States , 1994 .

[236]  F. Hausdorff Der Wertvorrat einer Bilinearform , 1919 .

[237]  K. B. Whaley,et al.  Geometric theory of nonlocal two-qubit operations , 2002, quant-ph/0209120.

[238]  A. Sugita Moments of generalized Husimi distributions and complexity of many-body quantum states , 2001, nlin/0112042.

[239]  Östlund,et al.  Thermodynamic limit of density matrix renormalization. , 1995, Physical review letters.

[240]  E. Prugovec̆ki Information-theoretical aspects of quantum measurement , 1977 .

[241]  K. Kraus General state changes in quantum theory , 1971 .

[242]  E. Schrödinger Discussion of Probability Relations between Separated Systems , 1935, Mathematical Proceedings of the Cambridge Philosophical Society.

[243]  Andreas Klappenecker,et al.  On the monomiality of nice error bases , 2005, IEEE Transactions on Information Theory.

[244]  Man-Duen Choi Positive semidefinite biquadratic forms , 1975 .

[245]  R. Penrose The road to reality. , 1984, Nursing times.

[246]  A. Holevo Statistical structure of quantum theory , 2001 .

[247]  A. R. Davidson Theory of Probabilities. , 1953 .

[248]  C. A. Coulson,et al.  Present State of Molecular Structure Calculations , 1960 .

[249]  N. Linden,et al.  Parts of quantum states , 2004, quant-ph/0407117.

[250]  B. De Moor,et al.  Variational characterizations of separability and entanglement of formation , 2001 .

[251]  C. Bendjaballah,et al.  Quantum Aspects of Optical Communications , 1991 .

[252]  Mark Fannes,et al.  Quantum Dynamical Systems , 2001 .

[253]  Jagat Narain Kapur,et al.  Measures of information and their applications , 1994 .

[254]  E. Specker,et al.  The Problem of Hidden Variables in Quantum Mechanics , 1967 .

[255]  Ingram Olkin,et al.  Inequalities: Theory of Majorization and Its Application , 1979 .

[256]  T. Prosen Exact statistics of complex zeros for Gaussian random polynomials with real coefficients , 1996, chao-dyn/9612005.

[257]  William O. Alltop,et al.  Complex sequences with low periodic correlations (Corresp.) , 1980, IEEE Trans. Inf. Theory.

[258]  A. Sudbery On local invariants of pure three-qubit states , 2000, quant-ph/0001116.

[259]  T. Andô Majorization, doubly stochastic matrices, and comparison of eigenvalues , 1989 .

[260]  J. Hadamard,et al.  Leçons D'Analyse Fonctionnelle , 1934, The Mathematical Gazette.

[261]  E. Kashefi,et al.  Uniqueness of the entanglement measure for bipartite pure states and thermodynamics. , 2002, Physical review letters.

[262]  R. Baxter,et al.  Dimers on a Rectangular Lattice , 1968 .

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

[264]  P. Slater Hall normalization constants for the Bures volumes of the n-state quantum systems , 1999, quant-ph/9904101.

[265]  J. Eisert,et al.  Complete hierarchies of efficient approximations to problems in entanglement theory , 2004 .

[266]  K. Gustafson,et al.  Numerical Range: The Field Of Values Of Linear Operators And Matrices , 1996 .

[267]  F. Verstraete,et al.  Matrix product states represent ground states faithfully , 2005, cond-mat/0505140.

[268]  A. Sudbery Computer-friendly d-tensor identities for SU(n) , 1990 .

[269]  A. Uhlmann Concurrence and Foliations Induced by Some 1-Qubit Channels , 2003, quant-ph/0301088.

[270]  M. Berry Quantal phase factors accompanying adiabatic changes , 1984, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[271]  A. Winter,et al.  Communications in Mathematical Physics Structure of States Which Satisfy Strong Subadditivity of Quantum Entropy with Equality , 2022 .

[272]  Alexander S Shumovsky,et al.  General entanglement , 2005 .

[273]  K. Życzkowski Volume of the set of separable states . II , 1999 .

[274]  M. Nowak,et al.  Wishart and anti-Wishart random matrices , 2001, math-ph/0112017.

[275]  T. Prosen Parametric statistics of zeros of Husimi representations of quantum chaotic eigenstates and random polynomials , 1996, chao-dyn/9612006.

[276]  G. Vidal On the characterization of entanglement , 1998 .

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

[278]  Robert Alicki,et al.  Quantum Dynamical Semigroups , 2006 .

[279]  J. Cirac,et al.  Three qubits can be entangled in two inequivalent ways , 2000, quant-ph/0005115.

[280]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[281]  E. Jaynes Probability theory : the logic of science , 2003 .

[282]  M. M. Miller,et al.  Fundamentals of Quantum Optics , 1968 .

[283]  J. Eisert,et al.  Schmidt measure as a tool for quantifying multiparticle entanglement , 2000, quant-ph/0007081.

[284]  O. Klein Zur quantenmechanischen Begründung des zweiten Hauptsatzes der Wärmelehre , 1931 .

[285]  K. Lendi,et al.  Quantum Dynamical Semigroups and Applications , 1987 .

[286]  Schumacher,et al.  Noncommuting mixed states cannot be broadcast. , 1995, Physical review letters.

[287]  Barry C. Sanders,et al.  Bounds on general entropy measures , 2003 .

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

[289]  W. Majewski Transformations between quantum states , 1975 .

[290]  A. Sudbery,et al.  One-qubit reduced states of a pure many-qubit state: polygon inequalities. , 2002, Physical review letters.

[291]  Gilad Gour Family of concurrence monotones and its applications , 2005 .

[292]  Lieven Clarisse The distillability problem revisited , 2006, Quantum Inf. Comput..

[293]  V. P. Belavkin,et al.  C*-algebraic generalization of relative entropy and entropy , 1982 .

[294]  P. Shor,et al.  Nonadditivity of bipartite distillable entanglement follows from a conjecture on bound entangled Werner states. , 2000, Physical review letters.

[295]  L. Hua Harmonic analysis of functions of several complex variables in the classical domains , 1963 .

[296]  R. A. Leibler,et al.  On Information and Sufficiency , 1951 .

[297]  D. Petz,et al.  Geometries of quantum states , 1996 .

[298]  D. Gitman,et al.  Coherent states of SU(N) groups , 1992 .

[299]  A. Zeilinger,et al.  Going Beyond Bell’s Theorem , 2007, 0712.0921.

[300]  E. Davies,et al.  PROBABILISTIC AND STATISTICAL ASPECTS OF QUANTUM THEORY (North‐Holland Series in Statistics and Probability, 1) , 1984 .

[301]  Lorenza Viola,et al.  NMR quantum information processing and entanglement , 2002, Quantum Inf. Comput..

[302]  A. Wehrl General properties of entropy , 1978 .

[303]  H. Osaka Indecomposable positive maps in low dimensional matrix algebras , 1991 .

[304]  Atsushi Higuchi,et al.  On maximal entanglement between two pairs in four-qubit pure states , 2007, 0704.2961.

[305]  Daniel R. Terno,et al.  Convex probability domain of generalized quantum measurements , 1998, quant-ph/9806024.

[306]  M. Ruskai Inequalities for quantum entropy: A review with conditions for equality , 2002, quant-ph/0205064.

[307]  K. Życzkowski,et al.  Entanglement of 2×K quantum systems , 2003, quant-ph/0302144.

[308]  Jan Havrda,et al.  Quantification method of classification processes. Concept of structural a-entropy , 1967, Kybernetika.

[309]  I. Chuang,et al.  Simulating quantum operations with mixed environments , 1998, quant-ph/9806095.

[310]  S. Fei,et al.  Generalized reduction criterion for separability of quantum states (7 pages) , 2003, quant-ph/0312185.

[311]  B. Moor,et al.  Maximally entangled mixed states of two qubits , 2000, quant-ph/0011110.

[312]  Nesa L'abbe Wu,et al.  Linear programming and extensions , 1981 .

[313]  F. Verstraete,et al.  Computational complexity of projected entangled pair states. , 2007, Physical review letters.

[314]  F. Strocchi,et al.  COMPLEX COORDINATES AND QUANTUM MECHANICS , 1966 .

[315]  Timothy F. Havel Robust procedures for converting among Lindblad, Kraus and matrix representations of quantum dynamical semigroups , 2002, quant-ph/0201127.

[316]  E. Gerjuoy Lower bound on entanglement of formation for the qubit-qudit system , 2003, quant-ph/0301015.

[317]  Asher Peres,et al.  Quantum Theory: Concepts and Methods , 1994 .

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

[319]  Andrew M. Childs,et al.  The limitations of nice mutually unbiased bases , 2004, quant-ph/0412066.

[320]  J. Hannay,et al.  LETTER TO THE EDITOR: The Berry phase for spin in the Majorana representation , 1998 .

[321]  Aidan Roy,et al.  Equiangular lines, mutually unbiased bases, and spin models , 2009, Eur. J. Comb..

[322]  A. Uhlmann GEOMETRIC PHASES AND RELATED STRUCTURES , 1995 .

[323]  Kil-Chan Ha,et al.  Atomic positive linear maps in matrix algebras , 1998 .

[324]  B. Mielnik Theory of filters , 1969 .

[325]  Mahdad Khatirinejad,et al.  On Weyl-Heisenberg orbits of equiangular lines , 2008 .

[326]  L. Accardi Topics in quantum probability , 1981 .

[327]  L. Jakóbczyk,et al.  Geometry of Bloch vectors in two-qubit system , 2001 .

[328]  J. Dittmann Explicit formulae for the Bures metric , 1998, quant-ph/9808044.

[329]  A. O. Pittenger,et al.  Separability and Fourier representations of density matrices , 2000, quant-ph/0001014.

[330]  Francesco Petruccione,et al.  The Theory of Open Quantum Systems , 2002 .

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

[332]  I. Olkin,et al.  Inequalities: Theory of Majorization and Its Applications , 1980 .

[333]  C. King Additivity for unital qubit channels , 2001, quant-ph/0103156.

[334]  P. Bertrand,et al.  A tomographic approach to Wigner's function , 1987 .

[335]  W. Wootters,et al.  Distributed Entanglement , 1999, quant-ph/9907047.

[336]  V. Bargmann On a Hilbert space of analytic functions and an associated integral transform part I , 1961 .

[337]  P. Zanardi,et al.  Purity and state fidelity of quantum channels , 2004, quant-ph/0403074.

[338]  R. Gilmore,et al.  Lie Groups, Lie Algebras, and Some of Their Applications , 1974 .

[339]  Sen Average Entropy of a Quantum Subsystem. , 1996, Physical review letters.

[340]  M. Lewenstein,et al.  Detection of entanglement with few local measurements , 2002, quant-ph/0205089.

[341]  N. Mermin Hidden variables and the two theorems of John Bell , 1993, 1802.10119.

[342]  P Shuklayzx Universal fluctuations of zeros of chaotic wavefunctions , 1996 .

[343]  M. Ruskai,et al.  Entanglement Breaking Channels , 2003, quant-ph/0302031.

[344]  Yong-Cheng Ou,et al.  Violation of monogamy inequality for higher-dimensional objects , 2006, quant-ph/0612127.

[345]  Leonid Gurvits Classical deterministic complexity of Edmonds' Problem and quantum entanglement , 2003, STOC '03.

[346]  J. Neumann,et al.  On an Algebraic generalization of the quantum mechanical formalism , 1934 .

[347]  A. Gleason Measures on the Closed Subspaces of a Hilbert Space , 1957 .

[348]  B. Mirbach,et al.  A GENERALIZED ENTROPY MEASURING QUANTUM LOCALIZATION , 1998 .

[349]  S. Schirmer,et al.  Orbits of quantum states and geometry of Bloch vectors for N-level systems , 2003, quant-ph/0308004.

[350]  Luigi Accardi,et al.  Nonrelativistic Quantum Mechanics as a Noncommutative Markof Process , 1976 .

[351]  J. Neumann Thermodynamik quantenmechanischer Gesamtheiten , 1927 .

[352]  S. Szarek Volume of separable states is super-doubly-exponentially small in the number of qubits , 2005 .

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

[354]  M. Nielsen A simple formula for the average gate fidelity of a quantum dynamical operation [rapid communication] , 2002, quant-ph/0205035.

[355]  E. Knill,et al.  Generalizations of entanglement based on coherent states and convex sets , 2002, quant-ph/0207149.

[356]  M. Horodecki,et al.  The entanglement of purification , 2002, quant-ph/0202044.

[357]  G. Gibbons Typical states and density matrices , 1992 .

[358]  Aleksandr I︠A︡kovlevich Khinchin Arbeiten zur Informationstheorie , 1957 .

[359]  A. Winter,et al.  Aspects of Generic Entanglement , 2004, quant-ph/0407049.

[360]  M. Plenio,et al.  MINIMAL CONDITIONS FOR LOCAL PURE-STATE ENTANGLEMENT MANIPULATION , 1999, quant-ph/9903054.

[361]  M. Hall RANDOM QUANTUM CORRELATIONS AND DENSITY OPERATOR DISTRIBUTIONS , 1998, quant-ph/9802052.

[362]  Ettore Majorana Atomi orientati in campo magnetico variabile , 1932 .

[363]  C. Fuchs,et al.  Entanglement of Formation of an Arbitrary State of Two Rebits , 2000, quant-ph/0009063.

[364]  David W. Lewis,et al.  Matrix theory , 1991 .

[365]  J. Oppenheim,et al.  Secure key from bound entanglement. , 2003, Physical Review Letters.

[366]  E. Sudarshan,et al.  Completely Positive Dynamical Semigroups of N Level Systems , 1976 .

[367]  T. Brun,et al.  Parametrization and distillability of three-qubit entanglement , 2000, quant-ph/0005124.

[368]  T. Isola,et al.  Wigner–Yanase information on quantum state space: The geometric approach , 2003 .

[369]  John M. Howie,et al.  Fields and Galois Theory , 2006 .

[370]  M. Horodecki,et al.  Local environment can enhance fidelity of quantum teleportation , 1999, quant-ph/9912098.

[371]  Chi-Kwong Li A simple proof of the elliptical range theorem , 1996 .

[372]  R. A. Fisher,et al.  Design of Experiments , 1936 .

[373]  H. Lo,et al.  Concentrating entanglement by local actions: Beyond mean values , 1997, quant-ph/9707038.

[374]  R. Ingarden,et al.  Information Dynamics and Open Systems: Classical and Quantum Approach , 1997 .

[375]  Martin B. Plenio,et al.  An introduction to entanglement measures , 2005, Quantum Inf. Comput..

[376]  G. Lüders Über die Zustandsänderung durch den Meßprozeß , 1950 .

[377]  U. Brehm The shape invariant of triangles and trigonometry in two-point homogeneous spaces , 1990 .

[378]  A. Kitaev,et al.  Universal quantum computation with ideal Clifford gates and noisy ancillas (14 pages) , 2004, quant-ph/0403025.

[379]  A. J. Coleman THE STRUCTURE OF FERMION DENSITY MATRICES , 1963 .

[380]  Werner,et al.  Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. , 1989, Physical review. A, General physics.

[381]  S. Sherman,et al.  Monotone and convex operator functions , 1955 .

[382]  Freeman J. Dyson,et al.  The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics , 1962 .

[383]  G. Lindblad Expectations and entropy inequalities for finite quantum systems , 1974 .

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

[385]  A. Winter,et al.  Distillation of secret key and entanglement from quantum states , 2003, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[386]  Ling-An Wu,et al.  The generalized partial transposition criterion for separability of multipartite quantum states , 2002, quant-ph/0208058.

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

[388]  G. Hardy,et al.  Asymptotic formulae in combinatory analysis , 1918 .

[389]  Seung-Hyeok Kye Facial structures for unital positive linear maps in the two-dimensional matrix algebra , 2003 .

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

[391]  E. Wigner,et al.  Book Reviews: Group Theory. And Its Application to the Quantum Mechanics of Atomic Spectra , 1959 .

[392]  Timothy F. Havel,et al.  Ensemble quantum computing by NMR spectroscopy , 1997, Proc. Natl. Acad. Sci. USA.

[393]  L. Hardy METHOD OF AREAS FOR MANIPULATING THE ENTANGLEMENT PROPERTIES OF ONE COPY OF A TWO-PARTICLE PURE ENTANGLED STATE , 1999, quant-ph/9903001.

[394]  O. Gühne Characterizing entanglement via uncertainty relations. , 2003, Physical review letters.

[395]  E. Sudarshan Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams , 1963 .

[396]  P. Arrighi,et al.  On quantum operations as quantum states , 2003, quant-ph/0307024.

[397]  R. Delbourgo,et al.  Minimal uncertainty states for the rotation and allied groups , 1977 .

[398]  Jun Tomiyama,et al.  Indecomposable Positive Maps in Matrix Algebras , 1988, Canadian Mathematical Bulletin.

[399]  F. Hiai,et al.  Sufficiency, KMS condition and relative entropy in von Neumann algebras. , 1981 .

[400]  A. Calderbank,et al.  Z4‐Kerdock Codes, Orthogonal Spreads, and Extremal Euclidean Line‐Sets , 1997 .

[401]  Kenneth S. Williams,et al.  Introductory Algebraic Number Theory , 2003 .

[402]  C. H. Bennett,et al.  Quantum nonlocality without entanglement , 1998, quant-ph/9804053.

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

[404]  R. Morrow,et al.  Foundations of Quantum Mechanics , 1968 .

[405]  Steane,et al.  Error Correcting Codes in Quantum Theory. , 1996, Physical review letters.

[406]  Page,et al.  Average entropy of a subsystem. , 1993, Physical review letters.

[407]  C. Aragone,et al.  Intelligent spin states , 1974 .

[408]  Funabashi,et al.  Nonadditive conditional entropy and its significance for local realism , 2000, quant-ph/0001085.

[409]  M. Lewenstein,et al.  Separable approximations of density matrices of composite quantum systems , 2000, quant-ph/0011066.

[410]  Nick Herbert FLASH—A superluminal communicator based upon a new kind of quantum measurement , 1982 .

[411]  I. Pitowsky Infinite and finite Gleason’s theorems and the logic of indeterminacy , 1998 .

[412]  Howard Barnum,et al.  Better bound on the exponent of the radius of the multipartite separable ball (12 pages) , 2005 .

[413]  G. Hardy,et al.  Asymptotic Formulaæ in Combinatory Analysis , 1918 .

[414]  A. Robert Calderbank,et al.  The Finite Heisenberg-Weyl Groups in Radar and Communications , 2006, EURASIP J. Adv. Signal Process..

[415]  Andreas Klappenecker,et al.  Constructions of Mutually Unbiased Bases , 2003, International Conference on Finite Fields and Applications.

[416]  J. Luque,et al.  Polynomial invariants of four qubits , 2002, quant-ph/0212069.

[417]  A. Sudbery,et al.  Non-local properties of multi-particle density matrices , 1998, quant-ph/9801076.

[418]  P. Goldbart,et al.  Geometric measure of entanglement and applications to bipartite and multipartite quantum states , 2003, quant-ph/0307219.

[419]  H. Hofmann,et al.  Violation of local uncertainty relations as a signature of entanglement , 2002, quant-ph/0212090.

[420]  G. W. Gibbons,et al.  Complex numbers, quantum mechanics and the beginning of time , 1993 .

[421]  Pechukas,et al.  Reduced dynamics need not be completely positive. , 1994, Physical review letters.

[422]  Wojciech Słlomczyński,et al.  Subadditivity of Entropy for Stochastic Matrices , 2002 .

[423]  William Hall,et al.  A new criterion for indecomposability of positive maps , 2006 .

[424]  Martin Rötteler,et al.  Computing local invariants of qubit systems , 1998 .

[425]  T. Andô Cones and norms in the tensor product of matrix spaces , 2004 .

[426]  V. Pless Introduction to the Theory of Error-Correcting Codes , 1991 .

[427]  R. Brockett,et al.  Time optimal control in spin systems , 2000, quant-ph/0006114.

[428]  R. Delbourgo,et al.  Covariance, correlation and entanglement , 2000, quant-ph/0001076.

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

[430]  Toshiyuki Takasaki,et al.  On the geometry of positive maps in matrix algebras , 1983 .

[431]  U. Fano,et al.  Pairs of two-level systems , 1983 .

[432]  D. Bruß,et al.  Construction of quantum states with bound entanglement , 1999, quant-ph/9911056.

[433]  A. Uhlmann The Metric of Bures and the Geometric Phase. , 1992 .

[434]  H. Briegel,et al.  Persistent entanglement in arrays of interacting particles. , 2000, Physical review letters.

[435]  R. Kippenhahn Über den Wertevorrat einer Matrix , 1951 .

[436]  Pingxing Chen,et al.  A lower bound on entanglement of formation of 2⊗n system , 2002 .

[437]  M. Steiner,et al.  Preserving entanglement under perturbation and sandwiching all separable states , 2000, quant-ph/0009090.

[438]  N. Mermin,et al.  The Ithaca interpretation of quantum mechanics , 1996, quant-ph/9609013.

[439]  Arun K. Pati,et al.  Generalization of the geometric phase to completely positive maps , 2003 .

[440]  Claude E. Shannon,et al.  A mathematical theory of communication , 1948, MOCO.

[441]  DiVincenzo Two-bit gates are universal for quantum computation. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[442]  P. Seymour,et al.  Averaging sets: A generalization of mean values and spherical designs , 1984 .

[443]  J. Eisert,et al.  A comparison of entanglement measures , 1998, quant-ph/9807034.

[444]  I. M. Gelʹfand,et al.  Discriminants, Resultants, and Multidimensional Determinants , 1994 .

[445]  Sumiyoshi Abe,et al.  A note on the q-deformation-theoretic aspect of the generalized entropies in nonextensive physics , 1997 .

[446]  K. Mccrimmon,et al.  Jordan algebras and their applications , 1978 .

[447]  Ashish V. Thapliyal,et al.  Optimal decompositions of barely separable states , 1999, quant-ph/9904005.

[448]  S. Braunstein GEOMETRY OF QUANTUM INFERENCE , 1996 .

[449]  A. Miranowicz,et al.  A comparative study of relative entropy of entanglement, concurrence and negativity , 2004, quant-ph/0409153.

[450]  W. Wootters,et al.  Optimal state-determination by mutually unbiased measurements , 1989 .

[451]  K. Życzkowski,et al.  Indicators of quantum chaos based on eigenvector statistics , 1990 .

[452]  Wojciech Słomczyński,et al.  The Monge distance between quantum states , 1997, quant-ph/9711011.

[453]  S. Sternberg,et al.  Symplectic Techniques in Physics , 1984 .

[454]  Pawel Horodecki,et al.  Beyond the standard entropic inequalities: Stronger scalar separability criteria and their applications , 2007, 0707.4315.

[455]  H. Bechmann-Pasquinucci,et al.  Bell inequality, Bell states and maximally entangled states for n qubits , 1998, quant-ph/9804045.

[456]  Han-Dong Chen,et al.  Geometry of the three-qubit state, entanglement and division algebras , 2003 .

[457]  C. Fuchs,et al.  Unknown Quantum States: The Quantum de Finetti Representation , 2001, quant-ph/0104088.

[458]  M. Fannes,et al.  Continuity of quantum conditional information , 2003, quant-ph/0312081.

[459]  K. Życzkowski,et al.  Random unitary matrices , 1994 .

[460]  A. Jamiołkowski,et al.  Geometric Phases in Classical and Quantum Mechanics , 2004 .

[461]  L. Kantorovich On the Translocation of Masses , 2006 .

[462]  S. Szarek,et al.  LETTER TO THE EDITOR: On the structure of the body of states with positive partial transpose , 2005, quant-ph/0509008.

[463]  Dénes Petz,et al.  Means of Positive Numbers and Matrices , 2005, SIAM J. Matrix Anal. Appl..

[464]  M. Ozawa Entanglement measures and the Hilbert-Schmidt distance , 2000, quant-ph/0002036.

[465]  R. Werner,et al.  On Some Additivity Problems in Quantum Information Theory , 2000, math-ph/0003002.

[466]  A. Shimony,et al.  Bell's theorem. Experimental tests and implications , 1978 .

[467]  A. Ashtekar,et al.  Quantum fields in curved space-times , 1975, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

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

[469]  Kevin Cahill,et al.  DENSITY OPERATORS AND QUASIPROBABILITY DISTRIBUTIONS. , 1969 .

[470]  P. Leboeuf Phase space approach to quantum dynamics , 1991 .

[471]  H. Everett "Relative State" Formulation of Quantum Mechanics , 1957 .

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

[473]  B. Moor,et al.  A comparison of the entanglement measures negativity and concurrence , 2001, quant-ph/0108021.

[474]  M. Partovi Universal measure of entanglement. , 2003, Physical review letters.

[475]  Andrzej Kossakowski,et al.  A Class of Linear Positive Maps in Matrix Algebras , 2003, Open Syst. Inf. Dyn..

[476]  M. Horodecki,et al.  BOUND ENTANGLEMENT CAN BE ACTIVATED , 1998, quant-ph/9806058.

[477]  F. Verstraete,et al.  General monogamy inequality for bipartite qubit entanglement. , 2005, Physical review letters.

[478]  Marek Zukowski,et al.  Experimentally friendly geometrical criteria for entanglement. , 2007, Physical review letters.

[479]  S. G. Hoggar 64 Lines from a Quaternionic Polytope , 1998 .

[480]  P. Shor,et al.  Unextendible Product Bases, Uncompletable Product Bases and Bound Entanglement , 1999, quant-ph/9908070.

[481]  Sánchez-Ruiz,et al.  Simple proof of Page's conjecture on the average entropy of a subsystem. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[482]  David E. Evans Quantum dynamical semigroups, symmetry groups, and locality , 1984 .

[483]  Carlton M. Caves,et al.  Mathematical techniques for quantum communication theory , 1995 .

[484]  G. Lindblad Completely positive maps and entropy inequalities , 1975 .

[485]  V. Bargmann NOTE ON WIGNER'S THEOREM ON SYMMETRY OPERATIONS , 1964 .

[486]  O. Barndorff-Nielsen,et al.  Fisher information in quantum statistics , 1998, quant-ph/9808009.

[487]  Uniwersytet Jagiello Time-reversal symmetry and random polynomials , 1997 .

[488]  E. Størmer Positive linear maps of operator algebras , 2012 .

[489]  Eqab M. Rabei,et al.  BARGMANN INVARIANTS AND GEOMETRIC PHASES : A GENERALIZED CONNECTION , 1999, quant-ph/9907080.

[490]  P. Algoet,et al.  ONE-TO-ONE PARAMETRIZATION OF QUANTUM CHANNELS , 1999 .

[491]  I. N. Sanov On the probability of large deviations of random variables , 1958 .

[492]  P. Hayden,et al.  Renyi-entropic bounds on quantum communication , 2002 .

[493]  P. Horodecki,et al.  Method for direct detection of quantum entanglement. , 2001, Physical review letters.

[494]  C. R. Rao,et al.  Information and the Accuracy Attainable in the Estimation of Statistical Parameters , 1992 .

[495]  T. Constantinescu,et al.  Parametrizing Quantum States and Channels , 2003, Quantum Inf. Process..

[496]  M. K. Bennett Affine and Projective Geometry , 1995 .

[497]  Yicheng Zhang,et al.  On the Anti-Wishart distribution , 2001, cond-mat/0112497.

[498]  A. Jamiołkowski An effective method of investigation of positive maps on the set of positive definite operators , 1974 .

[499]  E. Jonckheere,et al.  Convexity of the joint numerical range: topological and differential geometric viewpoints , 2004 .

[500]  M. Kus,et al.  Concurrence of mixed bipartite quantum states in arbitrary dimensions. , 2004, Physical review letters.

[501]  M. Talagrand New concentration inequalities in product spaces , 1996 .

[502]  E. Lubkin Entropy of an n‐system from its correlation with a k‐reservoir , 1978 .

[503]  A. J. Scott Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions , 2003, quant-ph/0310137.

[504]  M. Horodecki,et al.  Mixed-State Entanglement and Quantum Communication , 2001, quant-ph/0109124.

[505]  Rob Clifton,et al.  Bipartite-mixed-states of infinite-dimensional systems are generically nonseparable , 1999, quant-ph/9908028.

[506]  G. Vidal,et al.  Computable measure of entanglement , 2001, quant-ph/0102117.

[507]  Kil-Chan Ha,et al.  Entangled states with positive partial transposes arising from indecomposable positive linear maps , 2003, quant-ph/0305005.

[508]  B. Reed,et al.  Recent advances in algorithms and combinatorics , 2003 .

[509]  P. M. Mathews,et al.  STOCHASTIC DYNAMICS OF QUANTUM-MECHANICAL SYSTEMS , 1961 .

[510]  E. B. Davies Quantum theory of open systems , 1976 .

[511]  A. Wald,et al.  Probability, statistics and truth , 1939 .

[512]  H. Sommers,et al.  Induced measures in the space of mixed quantum states , 2000, quant-ph/0012101.

[513]  Satoshi Ishizaka,et al.  Maximally Entangled Mixed States in two Qubits , 2002 .

[514]  W. Munro,et al.  Bounds on entanglement in qudit subsystems , 2002, quant-ph/0203037.

[515]  Schumacher,et al.  Quantum data processing and error correction. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[516]  Charles H. Bennett,et al.  Purification of noisy entanglement and faithful teleportation via noisy channels. , 1995, Physical review letters.

[517]  G. Vidal,et al.  Approximate transformations and robust manipulation of bipartite pure-state entanglement , 1999, quant-ph/9910099.

[518]  Charles H. Bennett,et al.  Concentrating partial entanglement by local operations. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[519]  H. Sommers,et al.  Truncations of random unitary matrices , 1999, chao-dyn/9910032.

[520]  Jeroen van de Graaf,et al.  Cryptographic Distinguishability Measures for Quantum-Mechanical States , 1997, IEEE Trans. Inf. Theory.

[521]  H. Woerdeman The separability problem and normal completions , 2004 .

[522]  V. Vedral,et al.  Entanglement in many-body systems , 2007, quant-ph/0703044.

[523]  H. Weyl The Classical Groups , 1939 .

[524]  F D Murnaghan,et al.  On the Field of Values of a Square Matrix. , 1932, Proceedings of the National Academy of Sciences of the United States of America.

[525]  J. Cortese Relative Entropy and Single Qubit Holevo-Schumacher-Westmoreland Channel Capacity , 2002, quant-ph/0207128.

[526]  S. Woronowicz Positive maps of low dimensional matrix algebras , 1976 .

[527]  G. Vidal,et al.  LOCAL DESCRIPTION OF QUANTUM INSEPARABILITY , 1998 .

[528]  D. Bures An extension of Kakutani’s theorem on infinite product measures to the tensor product of semifinite *-algebras , 1969 .

[529]  D. Gross Hudson's theorem for finite-dimensional quantum systems , 2006, quant-ph/0602001.

[530]  Charles H. Bennett,et al.  Exact and asymptotic measures of multipartite pure-state entanglement , 1999, Physical Review A.

[531]  J. Kowski Linear transformations which preserve trace and positive semidefiniteness of operators , 1972 .

[532]  W. Wootters,et al.  Entanglement of a Pair of Quantum Bits , 1997, quant-ph/9703041.

[533]  Nihat Ay,et al.  Dually Flat Manifolds and Global Information Geometry , 2002, Open Syst. Inf. Dyn..

[534]  White,et al.  Density matrix formulation for quantum renormalization groups. , 1992, Physical review letters.

[535]  R. F. Werner,et al.  Meaner king uses biased bases , 2007 .

[536]  Horodecki Information-theoretic aspects of inseparability of mixed states. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[537]  棚橋 浩太郎,et al.  INDECOMPOSABLE POSITIVE MAPS IN MATRIX ALGEBRAS , 1988 .

[538]  N. B. E. Sá Uncertainty for spin systems , 2000, quant-ph/0009033.

[539]  Varga,et al.  Universal classification scheme for the spatial-localization properties of one-particle states in finite, d-dimensional systems. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[540]  P. Bertet,et al.  Measurement of a negative value for the Wigner function of radiation , 2000 .

[541]  M. Lewenstein,et al.  Relations between Entanglement Witnesses and Bell Inequalities , 2005, quant-ph/0504079.

[542]  N. Čencov Statistical Decision Rules and Optimal Inference , 2000 .

[543]  D. Petz Quantum Information Theory and Quantum Statistics , 2007 .

[544]  Dominik Endres,et al.  A new metric for probability distributions , 2003, IEEE Transactions on Information Theory.

[545]  J. Klauder The action option and a Feynman quantization of spinor fields in terms of ordinary c-numbers , 1960 .

[546]  G. Lindblad Entropy, information and quantum measurements , 1973 .

[547]  G. Kimura The Bloch Vector for N-Level Systems , 2003 .

[548]  C. Caves,et al.  Concurrence-based entanglement measures for isotropic states , 2003 .

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

[550]  K. Nomizu,et al.  Foundations of Differential Geometry , 1963 .

[551]  R. Schack,et al.  A Simple Algorithm for Local Conversion of Pure States , 2000, quant-ph/0006049.

[552]  L. Rodman,et al.  The numerical range of 3 × 3 matrices , 1997 .

[553]  M. Horodecki,et al.  Reversible transformations from pure to mixed states and the unique measure of information , 2002, quant-ph/0212019.

[554]  Armin Uhlmann,et al.  Dissipative motion in state spaces , 1981 .

[555]  Beck,et al.  Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum. , 1993, Physical review letters.

[556]  A. Winter,et al.  “Squashed entanglement”: An additive entanglement measure , 2003, quant-ph/0308088.

[557]  E. Gutkin,et al.  The Toeplitz-Hausdorff Theorem Revisited: Relating Linear Algebra and Geometry , 2004 .

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

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

[560]  P. Parrilo,et al.  Distinguishing separable and entangled states. , 2001, Physical review letters.

[561]  Noga Alon,et al.  Unextendible Product Bases , 2001, J. Comb. Theory A.

[562]  F. Mezzadri How to generate random matrices from the classical compact groups , 2006, math-ph/0609050.

[563]  G. Milburn,et al.  Universal state inversion and concurrence in arbitrary dimensions , 2001, quant-ph/0102040.

[564]  Guang-Can Guo,et al.  Compatible conditions, entanglement, and invariants , 2004 .

[565]  A. Kossakowski Remarks on positive maps of finite dimensional simple Jordan algebras , 2000 .

[566]  Richard Von Mises,et al.  Probability, statistics and truth , 1939 .

[567]  Robert Lockhart,et al.  Geometry and product states , 2002, Quantum Inf. Comput..

[568]  M. Ledoux The concentration of measure phenomenon , 2001 .

[569]  H. Weinfurter,et al.  Observation of three-photon Greenberger-Horne-Zeilinger entanglement , 1998, quant-ph/9810035.

[570]  D. J. Newman,et al.  A simplified proof of the partition formula. , 1962 .

[571]  C. T. Lee Wehrl's entropy of spin states and Lieb's conjecture , 1988 .

[572]  C. Helstrom Quantum detection and estimation theory , 1969 .

[573]  A. J. Goldman,et al.  Some geometric results in semidefinite programming , 1995, J. Glob. Optim..

[574]  Terhal,et al.  Entanglement of formation for isotropic states , 2000, Physical review letters.

[575]  A. O. Pittenger,et al.  Geometry of entanglement witnesses and local detection of entanglement , 2002, quant-ph/0207024.

[576]  R. Jozsa,et al.  SEPARABILITY OF VERY NOISY MIXED STATES AND IMPLICATIONS FOR NMR QUANTUM COMPUTING , 1998, quant-ph/9811018.

[577]  A. Horn Doubly Stochastic Matrices and the Diagonal of a Rotation Matrix , 1954 .

[578]  H. Breuer Optimal entanglement criterion for mixed quantum states. , 2006, Physical review letters.

[579]  J. Kidder,et al.  Light and Color in Nature and Art , 1983 .

[580]  Douglas R. Stinson,et al.  Combinatorial designs: constructions and analysis , 2003, SIGA.

[581]  C. Witte,et al.  A NEW ENTANGLEMENT MEASURE INDUCED BY THE HILBERT-SCHMIDT NORM , 1998, quant-ph/9811027.

[582]  F. Bloore Geometrical description of the convex sets of states for systems with spin-1/2 and spin-1 , 1976 .

[583]  W. Zurek Environment-assisted invariance, entanglement, and probabilities in quantum physics. , 2002, Physical review letters.

[584]  G. Vidal,et al.  Robustness of entanglement , 1998, quant-ph/9806094.

[585]  R. Werner,et al.  Why two qubits are special , 1999, quant-ph/9910064.

[586]  V. Vedral,et al.  Entanglement measures and purification procedures , 1997, quant-ph/9707035.

[587]  S. Ishizaka Analytical formula connecting entangled states and the closest disentangled state , 2003, quant-ph/0301107.

[588]  M. Horodecki,et al.  Limits for entanglement measures. , 1999, Physical review letters.

[589]  P. M. Alberti A note on the transition probability over C*-algebras , 1983 .

[590]  C. Tsallis Entropic nonextensivity: a possible measure of complexity , 2000, cond-mat/0010150.

[591]  M. Ruskai,et al.  Monotone Riemannian metrics and relative entropy on noncommutative probability spaces , 1998, math-ph/9808016.

[592]  D. Meyer,et al.  Global entanglement in multiparticle systems , 2001, quant-ph/0108104.

[593]  R. Jozsa,et al.  Lower bound for accessible information in quantum mechanics. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[594]  Oliver Rudolph On the cross norm criterion for separability , 2003 .

[595]  M. Steiner Generalized robustness of entanglement , 2003, quant-ph/0304009.

[596]  A. Knutson The symplectic and algebraic geometry of Horn's problem , 1999, math/9911088.

[597]  H. Paul,et al.  Measuring the quantum state of light , 1997 .

[598]  K. Życzkowski,et al.  CP n , OR, ENTANGLEMENT ILLUSTRATED , 2001, quant-ph/0108064.

[599]  J. Řeháček,et al.  Quantification of entanglement by means of convergent iterations. , 2002, Physical review letters.

[600]  D. Mumford Tata Lectures on Theta I , 1982 .

[601]  Lorenza Viola,et al.  Hadamard products of product operators and the design of gradient-diffusion experiments for simulating decoherence by NMR spectroscopy , 2000, quant-ph/0009010.

[602]  O. Bohigas,et al.  Quantum chaotic dynamics and random polynomials , 1996 .

[603]  D. W. Robinson,et al.  The finite group velocity of quantum spin systems , 1972 .

[604]  O. Gühne,et al.  Covariance matrices and the separability problem. , 2006, Physical review letters.

[605]  H. Sommers,et al.  Statistical properties of random density matrices , 2004, quant-ph/0405031.

[606]  M. Horodecki,et al.  Inseparable Two Spin- 1 2 Density Matrices Can Be Distilled to a Singlet Form , 1997 .

[607]  A. Winter,et al.  A New Inequality for the von Neumann Entropy , 2004, quant-ph/0406162.

[608]  H.,et al.  Convex Trace Functions and the Wigner-Yanase-Dyson Conjecture , 2022 .

[609]  Lee,et al.  Quantum source of entropy for black holes. , 1986, Physical review. D, Particles and fields.

[610]  Maassen,et al.  Generalized entropic uncertainty relations. , 1988, Physical review letters.

[611]  M. Plenio,et al.  Entanglement-Assisted Local Manipulation of Pure Quantum States , 1999, quant-ph/9905071.

[612]  W. Stinespring Positive functions on *-algebras , 1955 .

[613]  A. Rényi On Measures of Entropy and Information , 1961 .

[614]  H. Lawson Lectures On Minimal Submanifolds , 1980 .

[615]  B. Mielnik Nonlinear quantum mechanics: a conflict with the Ptolomean structure? , 2001 .

[616]  Tarrach,et al.  Generalized schmidt decomposition and classification of three-quantum-Bit states , 2000, Physical Review Letters.

[617]  M. Kus,et al.  Universality of eigenvector statistics of kicked tops of different symmetries , 1988 .

[618]  R. Bhatia,et al.  How and Why to Solve the Operator Equation AX−XB = Y , 1997 .

[619]  G. Agarwal Relation between atomic coherent-state representation, state multipoles, and generalized phase-space distributions , 1981 .

[620]  Berthold-Georg Englert,et al.  Remarks on 2–q-bit states , 2001 .

[621]  A. Harrow,et al.  Quantum dynamics as a physical resource , 2002, quant-ph/0208077.

[622]  Robert Gingrich Properties of Entanglement Monotones for Three-Qubit Pure States , 2002 .

[623]  S. Massar,et al.  Optimal Quantum Cloning Machines , 1997, quant-ph/9705046.

[624]  Kodi Husimi,et al.  Some Formal Properties of the Density Matrix , 1940 .

[625]  Jim Freeman Probability Metrics and the Stability of Stochastic Models , 1991 .

[626]  Giorgio Parisi,et al.  Maximally multipartite entangled states , 2007, 0710.2868.

[627]  C. Adami,et al.  Quantum extension of conditional probability , 1999 .

[628]  R. Hildebrand Positive partial transpose from spectra , 2007 .

[629]  Apostolos Vourdas,et al.  Quantum systems with finite Hilbert space , 2004 .

[630]  E. Rieffel,et al.  Quantum operations that cannot be implemented using a small mixed environment , 2001, quant-ph/0111084.

[631]  A. Klyachko Stable bundles, representation theory and Hermitian operators , 1998 .

[632]  R. Picken The Duistermaat-Heckman integration formula on flag manifolds , 1990 .

[633]  R. Schatten,et al.  A theory of cross-spaces , 1950 .

[634]  A. Zeilinger,et al.  Speakable and Unspeakable in Quantum Mechanics , 1989 .

[635]  A. Voros,et al.  Chaos-revealing multiplicative representation of quantum eigenstates , 1990 .

[636]  O. Toeplitz Das algebraische Analogon zu einem Satze von Fejér , 1918 .

[637]  S. Szarek The volume of separable states is super-doubly-exponentially small , 2003, quant-ph/0310061.

[638]  M. Barbieri,et al.  Detection of entanglement with polarized photons: experimental realization of an entanglement witness. , 2003, Physical review letters.

[639]  Julian Schwinger,et al.  Quantum Mechanics: Symbolism of Atomic Measurements , 2001 .

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

[641]  F. Hiai,et al.  The proper formula for relative entropy and its asymptotics in quantum probability , 1991 .

[642]  P. Hayden,et al.  Quantum state transformations and the Schubert calculus , 2004, quant-ph/0410052.

[643]  J. Eisert Entanglement in quantum information theory , 2006, quant-ph/0610253.

[644]  S. Berceanu COHERENT STATES, PHASES AND SYMPLECTIC AREA OF GEODESIC TRIANGLES , 1999 .

[645]  Frank Verstraete,et al.  Matrix product state representations , 2006, Quantum Inf. Comput..

[646]  Andreas Klappenecker,et al.  Beyond stabilizer codes I: Nice error bases , 2002, IEEE Trans. Inf. Theory.

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

[648]  John E. Harriman,et al.  Geometry of density matrices. I. Definitions,Nmatrices and 1 matrices , 1978 .

[649]  Yong-Sheng Zhang,et al.  Entanglement detection beyond the computable cross-norm or realignment criterion , 2007, 0709.3766.

[650]  M. Fannes A continuity property of the entropy density for spin lattice systems , 1973 .

[651]  Eric M. Rains A semidefinite program for distillable entanglement , 2001, IEEE Trans. Inf. Theory.

[652]  W. Hunziker SYMMETRY OPERATIONS IN QUANTUM MECHANICS. , 1972 .

[653]  Markus Neuhauser,et al.  An explicit construction of the metaplectic representation over a finite field. , 2002 .

[654]  B. Moor,et al.  Asymptotic relative entropy of entanglement for orthogonally invariant states , 2002, quant-ph/0204143.

[655]  G. Lindblad Quantum entropy and quantum measurements , 1991 .

[656]  C. Adami,et al.  Reduction criterion for separability , 1999 .

[657]  B. Moor,et al.  Four qubits can be entangled in nine different ways , 2001, quant-ph/0109033.

[658]  W. Munro,et al.  Maximizing the entanglement of two mixed qubits , 2001, quant-ph/0103113.

[659]  M. Horodecki,et al.  Continuity of relative entropy of entanglement , 1999, quant-ph/9910002.

[660]  Julia Kempe,et al.  Multiparticle entanglement and its applications to cryptography , 1999 .

[661]  Derek W. Robinson,et al.  Mean Entropy of States in Quantum‐Statistical Mechanics , 1968 .

[662]  H. Araki A Characterization of the State Space of Quantum Mechanics , 1980 .

[663]  Barbara M. Terhal,et al.  Rank two bipartite bound entangled states do not exist , 2003, Theor. Comput. Sci..

[664]  Elliott H. Lieb The classical limit of quantum spin systems , 1973 .

[665]  H. Umegaki Conditional expectation in an operator algebra. IV. Entropy and information , 1962 .

[666]  D. Bruss,et al.  Separability and distillability in composite quantum systems-a primer , 2000 .

[667]  W. Wootters Entanglement of Formation of an Arbitrary State of Two Qubits , 1997, quant-ph/9709029.

[668]  Vladimir I. Arnold,et al.  Symplectic geometry and topology , 2000 .

[669]  M. Murray,et al.  Differential Geometry and Statistics , 1993 .

[670]  Jaromir Fiurasek Structural physical approximations of unphysical maps and generalized quantum measurements , 2002 .

[671]  L. Molnár Fidelity preserving maps on density operators , 2001, math/0108060.

[672]  R. Paley On Orthogonal Matrices , 1933 .

[673]  Laflamme,et al.  Perfect Quantum Error Correcting Code. , 1996, Physical review letters.

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

[675]  N. B. E. Sá Decomposition of Hilbert space in sets of coherent states , 2000, quant-ph/0009022.

[676]  P. K. Aravind Geometry of the Schmidt decomposition and Hardy’s theorem , 1996 .

[677]  R. Lockhart,et al.  Optimal ensemble length of mixed separable states , 2000 .

[678]  A. Shimony Degree of Entanglement a , 1995 .

[679]  A. Steane,et al.  Introduction to quantum error correction , 1998, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[680]  Frances Kirwan,et al.  Convexity properties of the moment mapping, III , 1984 .

[681]  Arvind,et al.  A generalized Pancharatnam geometric phase formula for three-level quantum systems , 1996, quant-ph/9605042.

[682]  Lloyd,et al.  Almost any quantum logic gate is universal. , 1995, Physical review letters.

[683]  B. C. Carlson,et al.  EIGENVALUES OF DENSITY MATRICES , 1961 .

[684]  N. Christensen,et al.  Potential multiparticle entanglement measure , 2000, quant-ph/0010052.

[685]  Oliver Rudolph A new class of entanglement measures , 2000, math-ph/0005011.

[686]  F. Martini,et al.  Experimental Realization of Teleporting an Unknown Pure Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels , 1997, quant-ph/9710013.

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

[688]  K. B. Oldham,et al.  An Atlas of Functions. , 1988 .

[689]  J. Luque,et al.  Algebraic invariants of five qubits , 2005, quant-ph/0506058.

[690]  G. Vidal,et al.  Operational criterion and constructive checks for the separability of low-rank density matrices , 2000, quant-ph/0002089.

[691]  R. Wald Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics , 1994 .

[692]  Joos Vandewalle,et al.  A Multilinear Singular Value Decomposition , 2000, SIAM J. Matrix Anal. Appl..

[693]  R. Ingarden Information geometry in functional spaces of classical and quantum finite statistical systems , 1981 .

[694]  Michael A. Nielsen,et al.  Majorization and the interconversion of bipartite states , 2001, Quantum Inf. Comput..

[695]  Michal Horodecki,et al.  Entanglement measures , 2001, Quantum Inf. Comput..

[696]  Jeremy Gray,et al.  The Hilbert Challenge , 2001 .

[697]  J. Bell On the Problem of Hidden Variables in Quantum Mechanics , 1966 .

[698]  Benjamin Schumacher,et al.  A new proof of the quantum noiseless coding theorem , 1994 .

[699]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[700]  A. Uhlmann Fidelity and Concurrence of conjugated states , 1999, quant-ph/9909060.

[701]  Sven Gnutzmann,et al.  Coherent states and the classical limit on irreducible representations , 1998 .

[702]  Chandler Davis The Toeplitz-Hausdorff Theorem Explained , 1971, Canadian Mathematical Bulletin.

[703]  D. L. Macadam Visual Sensitivities to Color Differences in Daylight , 1942 .

[704]  H. Korsch,et al.  On the zeros of the Husimi distribution , 1997 .

[705]  F. Topsøe BOUNDS FOR ENTROPY AND DIVERGENCE FOR DISTRIBUTIONS OVER A TWO-ELEMENT SET , 2001 .

[706]  Teiji Takagi,et al.  On an Algebraic Problem Reluted to an Analytic Theorem of Carathéodory and Fejér and on an Allied Theorem of Landau , 1924 .

[707]  V. Giovannetti,et al.  Characterizing the entanglement of bipartite quantum systems , 2002, quant-ph/0210155.

[708]  Jan Myrheim,et al.  Geometrical aspects of entanglement , 2006 .

[709]  Mátyás A. Sustik,et al.  On the existence of equiangular tight frames , 2007 .

[710]  Joe W. Harris,et al.  Principles of Algebraic Geometry , 1978 .

[711]  H. Jehle,et al.  Albert Einstein: Philosopher-Scientist. , 1951 .

[712]  Armin Uhlmann,et al.  On Bures Distance and *-Algebraic Transition Probability between Inner Derived Positive Linear Forms over W*-Algebras , 2000 .

[713]  Wineland,et al.  Optimal frequency measurements with maximally correlated states. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[714]  J. Dalibard,et al.  Many-Body Physics with Ultracold Gases , 2007, 0704.3011.

[715]  L. Ballentine,et al.  Probabilistic and Statistical Aspects of Quantum Theory , 1982 .

[716]  Gavin K. Brennen An observable measure of entanglement for pure states of multi-qubit systems , 2003, Quantum Inf. Comput..

[717]  E. Lieb,et al.  Proof of the strong subadditivity of quantum‐mechanical entropy , 1973 .

[718]  A. Shimony,et al.  Bell’s theorem without inequalities , 1990 .

[719]  P. Slater A priori probabilities of separable quantum states , 1998, quant-ph/9810026.

[720]  W. Wootters,et al.  Optimal detection of quantum information. , 1991, Physical review letters.

[721]  F. Haake Quantum signatures of chaos , 1991 .

[722]  Pawel Wocjan,et al.  New construction of mutually unbiased bases in square dimensions , 2005, Quantum Inf. Comput..

[723]  A. Peres HIGHER ORDER SCHMIDT DECOMPOSITIONS , 1995, quant-ph/9504006.

[724]  E. Schrödinger Der stetige Übergang von der Mikro- zur Makromechanik , 1926, Naturwissenschaften.

[725]  R. Glauber Coherent and incoherent states of the radiation field , 1963 .

[726]  R. Hagedorn Note on symmetry operations in quantum mechanics , 1959 .

[727]  M. Talagrand A new look at independence , 1996 .

[728]  T. Andô Majorizations and inequalities in matrix theory , 1994 .

[729]  Guifre Vidal Entanglement monotones , 1998, quant-ph/9807077.

[730]  Stephanie Frank Singer,et al.  Symmetry in mechanics , 2001 .

[731]  E. Schrödinger Probability relations between separated systems , 1936, Mathematical Proceedings of the Cambridge Philosophical Society.

[732]  R. Laflamme,et al.  Exponential speedup with a single bit of quantum information: measuring the average fidelity decay. , 2003, Physical review letters.

[733]  E. Lieb Proof of an entropy conjecture of Wehrl , 1978 .

[734]  Pablo Arrighi,et al.  A note on the correspondence between qubit quantum operations and special relativity , 2002 .

[735]  Yuriy Makhlin Nonlocal Properties of Two-Qubit Gates and Mixed States, and the Optimization of Quantum Computations , 2002, Quantum Inf. Process..

[736]  R. Dicke Coherence in Spontaneous Radiation Processes , 1954 .

[737]  R. Rossignoli,et al.  Generalized entropic criterion for separability , 2002, 1505.03608.

[738]  V. Vedral The role of relative entropy in quantum information theory , 2001, quant-ph/0102094.

[739]  P. Horodecki,et al.  Quantum redundancies and local realism , 1994 .

[740]  K. Życzkowski Localization of eigenstates and mean Wehrl entropy , 1999, quant-ph/9910088.

[741]  N. Tsing,et al.  Inclusion relations between outhostochastic matrices and products of pinching matrices , 1987 .

[742]  M. Horodecki,et al.  Concurrence in arbitrary dimensions , 2001, quant-ph/0107147.

[743]  Anders Karlsson,et al.  Security of quantum key distribution using d-level systems. , 2001, Physical review letters.

[744]  Jan Bouda,et al.  Purification and correlated measurements of bipartite mixed states , 2001, quant-ph/0112015.

[745]  E. Størmer DECOMPOSABLE POSITIVE MAPS ON C*-ALGEBRAS , 1982 .

[746]  S. Nonnenmacher Crystal properties of eigenstates for quantum cat maps , 1997, chao-dyn/9701019.

[747]  R. Mosseri,et al.  Geometry of entangled states, Bloch spheres and Hopf fibrations , 2001, quant-ph/0108137.

[748]  A. O. Pittenger,et al.  Convexity and the separability problem of quantum mechanical density matrices , 2001, quant-ph/0103038.

[749]  K. Życzkowski,et al.  Wehrl entropy, Lieb conjecture, and entanglement monotones , 2003, quant-ph/0307169.

[750]  Howard Barnum,et al.  Separable balls around the maximally mixed multipartite quantum states , 2003 .

[751]  Erwin Schrödinger,et al.  Space-time structure , 1951 .

[752]  M. Lewenstein,et al.  Separability and entanglement of composite quantum systems , 1997, quant-ph/9707043.

[753]  Richard D. Gill Time, Finite Statistics, and Bell's Fifth Position , 2003 .

[754]  S. Popescu,et al.  Classical analog of entanglement , 2001, quant-ph/0107082.

[755]  D. Chruściński,et al.  Quantum states with strong positive partial transpose , 2007, 0710.1934.

[756]  D. Petz INFORMATION-GEOMETRY OF QUANTUM STATES , 1998 .

[757]  E. Schmidt Zur Theorie der linearen und nichtlinearen Integralgleichungen , 1907 .

[758]  Peter Harremoës,et al.  Inequalities between entropy and index of coincidence derived from information diagrams , 2001, IEEE Trans. Inf. Theory.

[759]  G. Masala,et al.  A geometrical interpretation of the shape invariant for geodesic triangles in complex projective spaces , 1994 .

[760]  Lloyd R. Welch,et al.  Lower bounds on the maximum cross correlation of signals (Corresp.) , 1974, IEEE Trans. Inf. Theory.

[761]  E. Wigner Normal Form of Antiunitary Operators , 1960 .

[762]  Alexander S. Holevo,et al.  The Capacity of the Quantum Channel with General Signal States , 1996, IEEE Trans. Inf. Theory.

[763]  W. Munro,et al.  Upper bound on the region of separable states near the maximally mixed state , 2000, quant-ph/0002002.

[764]  R. Hartley Transmission of information , 1928 .

[765]  AAsa Ericsson Separability and the stella octangula , 2002 .

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

[767]  R. Penrose A spinor approach to general relativity , 1960 .

[768]  Jozef Gruska,et al.  Quantum Computing , 2008, Wiley Encyclopedia of Computer Science and Engineering.

[769]  Hoi Fung Chau,et al.  Unconditionally secure key distribution in higher dimensions by depolarization , 2004, IEEE Transactions on Information Theory.

[770]  J. Eisert,et al.  Multiparty entanglement in graph states , 2003, quant-ph/0307130.

[771]  B. Terhal A family of indecomposable positive linear maps based on entangled quantum states , 1998, quant-ph/9810091.

[772]  A. Sudbery,et al.  Multipartite generalization of the Schmidt decomposition , 2000, quant-ph/0006125.

[773]  D. Bruß,et al.  A two-way algorithm for the entanglement problem , 2004, quant-ph/0407179.

[774]  Y. Aharonov,et al.  The mean king's problem: Prime degrees of freedom , 2001, quant-ph/0101134.

[775]  L. Campbell An extended Čencov characterization of the information metric , 1986 .

[776]  F. Verstraete,et al.  Fidelity of mixed states of two qubits , 2002, quant-ph/0203073.

[777]  S. Lloyd,et al.  Complexity as thermodynamic depth , 1988 .

[778]  David A. Yopp,et al.  On completely copositive and decomposable linear transformations , 2000 .

[779]  Schumacher,et al.  Sending entanglement through noisy quantum channels. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[780]  D. Leibfried,et al.  Experimental demonstration of a controlled-NOT wave-packet gate. , 2002, Physical Review Letters.

[781]  P. Parrilo,et al.  Complete family of separability criteria , 2003, quant-ph/0308032.

[782]  F. Shultz,et al.  Geometry Of State Spaces Of Operator Algebras , 2002 .

[783]  S. Szarek,et al.  An analysis of completely positive trace-preserving maps on M2 , 2002 .

[784]  S. Virmani,et al.  Ordering states with entanglement measures , 1999, quant-ph/9911119.

[785]  P'eter L'evay,et al.  On the geometry of four qubit invariants , 2006, quant-ph/0605151.

[786]  William K. Wootters,et al.  Entanglement of formation and concurrence , 2001, Quantum Inf. Comput..

[787]  F. Shultz,et al.  State Spaces of Operator Algebras , 2001 .

[788]  E. K. Lenzi,et al.  Average entropy of a subsystem from its average Tsallis entropy. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[789]  P. Knight,et al.  Entangled quantum systems and the Schmidt decomposition , 1995 .

[790]  J. Cirac,et al.  Optimal creation of entanglement using a two-qubit gate , 2000, quant-ph/0011050.

[791]  R. Balian,et al.  Dissipation in many-body systems: A geometric approach based on information theory , 1986 .

[792]  A. Osterloh,et al.  ENTANGLEMENT MONOTONES AND MAXIMALLY ENTANGLED STATES IN MULTIPARTITE QUBIT SYSTEMS , 2005, quant-ph/0506073.

[793]  S. Popescu,et al.  Generic quantum nonlocality , 1992 .

[794]  T. Andô,et al.  Means of positive linear operators , 1980 .

[795]  Shun-ichi Amari,et al.  Differential-geometrical methods in statistics , 1985 .

[796]  Aharonov,et al.  Geometry of quantum evolution. , 1990, Physical review letters.

[797]  Karl Löwner Über monotone Matrixfunktionen , 1934 .

[798]  K. Kraus,et al.  States, effects, and operations : fundamental notions of quantum theory : lectures in mathematical physics at the University of Texas at Austin , 1983 .

[799]  W. Majewski,et al.  On a characterization of positive maps , 2001 .

[800]  A. Jenčová Generalized Relative Entropies as Contrast Functionals on Density Matrices , 2004 .