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Alice and Bob Meet Banach: The Interface of Asymptotic Geometric Analysis and Quantum Information Theory [Hardback]

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  • Formāts: Hardback, 414 pages, height x width: 254x178 mm, weight: 893 g
  • Sērija : Mathematical Surveys and Monographs
  • Izdošanas datums: 30-Oct-2017
  • Izdevniecība: American Mathematical Society
  • ISBN-10: 1470434687
  • ISBN-13: 9781470434687
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Alice and Bob Meet Banach: The Interface of Asymptotic Geometric Analysis and Quantum Information TheoryPalielināt
  • Formāts: Hardback, 414 pages, height x width: 254x178 mm, weight: 893 g
  • Sērija : Mathematical Surveys and Monographs
  • Izdošanas datums: 30-Oct-2017
  • Izdevniecība: American Mathematical Society
  • ISBN-10: 1470434687
  • ISBN-13: 9781470434687
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9781470434687.html
Keywords:
The quest to build a quantum computer is arguably one of the major scientific and technological challenges of the twenty-first century, and quantum information theory (QIT) provides the mathematical framework for that quest. Over the last dozen or so years, it has become clear that quantum information theory is closely linked to geometric functional analysis (Banach space theory, operator spaces, high-dimensional probability), a field also known as asymptotic geometric analysis (AGA). In a nutshell, asymptotic geometric analysis investigates quantitative properties of convex sets, or other geometric structures, and their approximate symmetries as the dimension becomes large. This makes it especially relevant to quantum theory, where systems consisting of just a few particles naturally lead to models whose dimension is in the thousands, or even in the billions.

Alice and Bob Meet Banach is aimed at multiple audiences connected through their interest in the interface of QIT and AGA: at quantum information researchers who want to learn AGA or apply its tools; at mathematicians interested in learning QIT, especially the part that is relevant to functional analysis/convex geometry/random matrix theory and related areas; and at beginning researchers in either field. Moreover, this user-friendly book contains numerous tables and explicit estimates, with reasonable constants when possible, which make it a useful reference even for established mathematicians generally familiar with the subject.

Recenzijas

[ This book] will be an invaluable reference to scientists working on the more mathematical aspects of quantum information theory." Mary Beth Ruskai, Zentralblatt MATH

"A wide variety of audiences would be interested in this book: Parts II or III would be suitable for a graduate course on QIT from the perspective of functional analysis, convex geometry, or random matrix theory, or on the applications of AGA. With a mix of classical and recent results, as well as the concise treatment of the subject areas, the book could be used as a reference book for researchers working in this area. Furthermore, the large number of exercises, with an appendix of hints, would make it suitable for an independent study." Sarah Plosker, Mathematical Reviews

List of Tables xiii
List of Figures xv
Preface xix
Part 1 Alice and Bob: Mathematical Aspects of Quantum Information Theory 1(76)
Chapter 0 Notation and basic concepts
3(8)
0.1 Asymptotic and nonasymptotic notation
3(1)
0.2 Euclidean and Hilbert spaces
3(1)
0.3 Bra-ket notation
4(2)
0.4 Tensor products
6(1)
0.5 Complexification
6(1)
0.6 Matrices vs. operators
7(1)
0.7 Block matrices vs. operators on bipartite spaces
8(1)
0.8 Operators vs. tensors
8(1)
0.9 Operators vs. superoperators
8(1)
0.10 States, classical and quantum
8(3)
Chapter 1 Elementary convex analysis
11(20)
1.1 Normed spaces and convex sets
11(7)
1.1.1 Gauges
11(1)
1.1.2 First examples: lp-balls, simplices, polytopes, and convex hulls
12(1)
1.1.3 Extreme points, faces
13(2)
1.1.4 Polarity
15(2)
1.1.5 Polarity and the facial structure
17(1)
1.1.6 Ellipsoids
18(1)
1.2 Cones
18(4)
1.2.1 Cone duality
19(2)
1.2.2 Nondegenerate cones and facial structure
21(1)
1.3 Majorization and Schatten norms
22(7)
1.3.1 Majorization
22(1)
1.3.2 Schatten norms
23(4)
1.3.3 Von Neumann and Renyi entropies
27(2)
Notes and Remarks
29(2)
Chapter 2 The mathematics of quantum information theory
31(36)
2.1 On the geometry of the set of quantum states
31(4)
2.1.1 Pure and mixed states
31(1)
2.1.2 The Bloch ball D(C2)
32(1)
2.1.3 Facial structure
33(1)
2.1.4 Symmetries
34(1)
2.2 States on multipartite Hilbert spaces
35(12)
2.2.1 Partial trace
35(1)
2.2.2 Schmidt decomposition
36(1)
2.2.3 A fundamental dichotomy: Separability vs. entanglement
37(2)
2.2.4 Some examples of bipartite states
39(2)
2.2.5 Entanglement hierarchies
41(1)
2.2.6 Partial transposition
41(2)
2.2.7 PPT states
43(3)
2.2.8 Local unitaries and symmetries of Sep
46(1)
2.3 Superoperators and quantum channels
47(8)
2.3.1 The Choi and Jamiolkowski isomorphisms
47(1)
2.3.2 Positive and completely positive maps
48(2)
2.3.3 Quantum channels and Stinespring representation
50(2)
2.3.4 Some examples of channels
52(3)
2.4 Cones of QIT
55(8)
2.4.1 Cones of operators
55(1)
2.4.2 Cones of superoperators
56(2)
2.4.3 Symmetries of the PSD cone
58(2)
2.4.4 Entanglement witnesses
60(2)
2.4.5 Proofs of Stamer's theorem
62(1)
Notes and Remarks
63(4)
Chapter 3 Quantum mechanics for mathematicians
67(10)
3.1 Simple-minded quantum mechanics
67(1)
3.2 Finite vs. infinite dimension, projective spaces, and matrices
68(1)
3.3 Composite systems and quantum marginals: Mixed states
68(2)
3.4 The partial trace: Purification of mixed states
70(1)
3.5 Unitary evolution and quantum operations: The completely positive maps
71(2)
3.6 Other measurement schemes
73(1)
3.7 Local operations
74(1)
3.8 Spooky action at a distance
75(1)
Notes and Remarks
75(2)
Part 2 Banach and His Spaces: Asymptotic Geometric Analysis Miscellany 77(134)
Chapter 4 More convexity
79(28)
4.1 Basic notions and operations
79(5)
4.1.1 Distances between convex sets
79(1)
4.1.2 Symmetrization
80(1)
4.1.3 Zonotopes and zonoids
81(1)
4.1.4 Projective tensor product
82(2)
4.2 John and Lowner ellipsoids
84(7)
4.2.1 Definition and characterization
84(5)
4.2.2 Convex bodies with enough symmetries
89(2)
4.2.3 Ellipsoids and tensor products
91(1)
4.3 Classical inequalities for convex bodies
91(10)
4.3.1 The Brunn-Minkowski inequality
91(2)
4.3.2 log-concave measures
93(1)
4.3.3 Mean width and the Urysohn inequality
94(4)
4.3.4 The Santalo and the reverse Santalo inequalities
98(1)
4.3.5 Symmetrization inequalities
98(3)
4.3.6 Functional inequalities
101(1)
4.4 Volume of central sections and the isotropic position
101(2)
Notes and Remarks
103(4)
Chapter 5 Metric entropy and concentration of measure in classical spaces
107(42)
5.1 Nets and packings
107(10)
5.1.1 Definitions
107(1)
5.1.2 Nets and packings on the Euclidean sphere
108(5)
5.1.3 Nets and packings in the discrete cube
113(1)
5.1.4 Metric entropy for convex bodies
114(2)
5.1.5 Nets in Grassmann manifolds, orthogonal and unitary groups
116(1)
5.2 Concentration of measure
117(25)
5.2.1 A prime example: concentration on the sphere
119(2)
5.2.2 Gaussian concentration
121(3)
5.2.3 Concentration tricks and treats
124(5)
5.2.4 Geometric and analytic methods. Classical examples
129(7)
5.2.5 Some discrete settings
136(3)
5.2.6 Deviation inequalities for sums of independent random variables
139(3)
Notes and Remarks
142(7)
Chapter 6 Gaussian processes and random matrices
149(32)
6.1 Gaussian processes
149(11)
6.1.1 Key example and basic estimates
150(2)
6.1.2 Comparison inequalities for Gaussian processes
152(2)
6.1.3 Sudakov and dual Sudakov inequalities
154(3)
6.1.4 Dudley's inequality and the generic chaining
157(3)
6.2 Random matrices
160(18)
6.2.1 infinity-Wasserstein distance
161(1)
6.2.2 The Gaussian Unitary Ensemble (GUE)
162(4)
6.2.3 Wishart matrices
166(7)
6.2.4 Real RMT models and Chevet-Gordon inequalities
173(3)
6.2.5 A quick initiation to free probability
176(2)
Notes and Remarks
178(3)
Chapter 7 Some tools from asymptotic geometric analysis
181(30)
7.1 lposition, K-convexity and the MM*-estimate
181(5)
7.1.1 l-norm and l-position
181(1)
7.1.2 K-convexity and the MM*-estimate
182(4)
7.2 Sections of convex bodies
186(21)
7.2.1 Dvoretzky's theorem for Lipschitz functions
186(3)
7.2.2 The Dvoretzky dimension
189(4)
7.2.3 The Figiel-Lindenstrauss-Milman inequality
193(2)
7.2.4 The Dvoretzky dimension of standard spaces
195(5)
7.2.5 Dvoretzky's theorem for general convex bodies
200(1)
7.2.6 Related results
201(4)
7.2.7 Constructivity
205(2)
Notes and Remarks
207(4)
Part 3 The Meeting: AGA and QIT 211(96)
Chapter 8 Entanglement of pure states in high dimensions
213(22)
8.1 Entangled subspaces: Qualitative approach
213(2)
8.2 Entropies of entanglement and additivity questions
215(3)
8.2.1 Quantifying entanglement for pure states
215(1)
8.2.2 Channels as subspaces
216(1)
8.2.3 Minimal output entropy and additivity problems
216(1)
8.2.4 On the 1 -> p norm of quantum channels
217(1)
8.3 Concentration of Ep for p > 1 and applications
218(4)
8.3.1 Counterexamples to the multiplicativity problem
218(2)
8.3.2 Almost randomizing channels
220(2)
8.4 Concentration of von Neumann entropy and applications
222(7)
8.4.1 The basic concentration argument
222(2)
8.4.2 Entangled subspaces of small codimension
224(1)
8.4.3 Extremely entangled subspaces
224(4)
8.4.4 Counterexamples to the additivity problem
228(1)
8.5 Entangled pure states in multipartite systems
229(3)
8.5.1 Geometric measure of entanglement
229(1)
8.5.2 The case of many qubits
230(1)
8.5.3 Multipartite entanglement in real Hilbert spaces
231(1)
Notes and Remarks
232(3)
Chapter 9 Geometry of the set of mixed states
235(28)
9.1 Volume and mean width estimates
236(9)
9.1.1 Symmetrization
236(1)
9.1.2 The set of all quantum states
236(2)
9.1.3 The set of separable states (the bipartite case)
238(2)
9.1.4 The set of block-positive matrices
240(2)
9.1.5 The set of separable states (multipartite case)
242(2)
9.1.6 The set of PPT states
244(1)
9.2 Distance estimates
245(5)
9.2.1 The Gurvits-Barnum theorem
246(1)
9.2.2 Robustness in the bipartite case
247(1)
9.2.3 Distances involving the set of PPT states
248(1)
9.2.4 Distance estimates in the multipartite case
249(1)
9.3 The super-picture: Classes of maps
250(2)
9.4 Approximation by polytopes
252(8)
9.4.1 Approximating the set of all quantum states
252(4)
9.4.2 Approximating the set of separable states
256(2)
9.4.3 Exponentially many entanglement witnesses are necessary
258(2)
Notes and Remarks
260(3)
Chapter 10 Random quantum states
263(12)
10.1 Miscellaneous tools
263(5)
10.1.1 Majorization inequalities
263(1)
10.1.2 Spectra and norms of unitarily invariant random matrices
264(2)
10.1.3 Gaussian approximation to induced states
266(1)
10.1.4 Concentration for gauges of induced states
267(1)
10.2 Separability of random states
268(3)
10.2.1 Almost sure entanglement for low-dimensional environments
268(1)
10.2.2 The threshold theorem
269(2)
10.3 Other thresholds
271(1)
10.3.1 Entanglement of formation
271(1)
10.3.2 Threshold for PPT
272(1)
Notes and Remarks
272(3)
Chapter 11 Bell inequalities and the Grothendieck-Tsirelson inequality
275(24)
11.1 Isometrically Euclidean subspaces via Clifford algebras
275(1)
11.2 Local vs. quantum correlations
276(7)
11.2.1 Correlation matrices
277(3)
11.2.2 Bell correlation inequalities and the Grothendieck constant
280(3)
11.3 Boxes and games
283(11)
11.3.1 Bell inequalities as games
284(1)
11.3.2 Boxes and the nonsignaling principle
285(4)
11.3.3 Bell violations
289(5)
Notes and Remarks
294(5)
Chapter 12 POVMs and the distillability problem
299(8)
12.1 POVMs and zonoids
299(2)
12.1.1 Quantum state discrimination
299(1)
12.1.2 Zonotope associated to a POVM
300(1)
12.1.3 Sparsification of POVMs
300(1)
12.2 The distillability problem
301(4)
12.2.1 State manipulation via LOCC channels
301(1)
12.2.2 Distillable states
302(1)
12.2.3 The case of two qubits
302(2)
12.2.4 Some reformulations of distillability
304(1)
Notes and Remarks
305(2)
Appendix A. Gaussian measures and Gaussian variables 307(4)
A.1 Gaussian random variables
307(1)
A.2 Gaussian vectors
308(1)
Notes and Remarks
309(2)
Appendix B. Classical groups and manifolds 311(10)
B.1 The unit sphere Sn-1- or SCd
311(1)
B.2 The projective space
312(1)
B.3 The orthogonal and unitary groups O(n), U(n)
312(2)
B.4 The Grassmann manifolds Gr(k,Rn), Gr(k, Cn)
314(4)
B.5 The Lorentz group O(1, n - 1)
318(1)
Notes and Remarks
319(2)
Appendix C. Extreme maps between Lorentz cones and the S-lemma 321(4)
Notes and Remarks
324(1)
Appendix D. Polarity and the Santalo point via duality of cones 325(4)
Appendix E. Hints to exercises 329(46)
Appendix F. Notation 375(6)
General notation
375(1)
Convex geometry
375(1)
Linear algebra
376(1)
Probability
377(1)
Geometry and asymptotic geometric analysis
378(1)
Quantum information theory
379(2)
Bibliography 381(28)
Websites
408(1)
Index 409
Guillaume Aubrun, Universite Claude Bernard Lyon 1, Villeurbanne, France.

Stanislaw J. Szarek, Case Western Reserve University, Cleveland, OH.