Table Of ContentDynamics, Information and Complexity
in Quantum Systems
Theoretical and Mathematical Physics
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W.Beiglbo¨ck,InstituteofAppliedMathematics,UniversityofHeidelberg,Germany
J.-P.Eckmann,DepartmentofTheoreticalPhysics,UniversityofGeneva,Switzerland
H.Grosse,InstituteofTheoreticalPhysics,UniversityofVienna,Austria
M.Loss,SchoolofMathematics,GeorgiaInstituteofTechnology,Atlanta,GA,USA
S.Smirnov,MathematicsSection,UniversityofGeneva,Switzerland
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J.Yngvason,InstituteofTheoreticalPhysics,UniversityofVienna,Austria
Forfurthervolumes:
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Fabio Benatti
Dynamics, Information
and Complexity
in Quantum Systems
123
Dr.FabioBenatti
Universita` Trieste
Dipto.FisicaTeorica
StradaCostiera,11
34014Trieste
Miramare
Italy
benatti@ts.infn.it
ISBN978-1-4020-9305-0 e-ISBN978-1-4020-9306-7
DOI10.1007/978-1-4020-9306-7
LibraryofCongressControlNumber:2008937916
(cid:2)c SpringerScience+BusinessMediaB.V.2009
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To Heide, for her scholarship, for her friendship
Preface
The aim of this book is to offer a self-consistent overview of a series of is-
sues relating entropy, information and dynamics in classical and quantum
physics. My personal point of view regarding these matters is the result of
what I had the good fortune to learn in the course of the years from various
scientists: Heide Narnhofer in the first place, who introduced me to quan-
tum dynamical entropies and was a precious guide ever since, then Robert
Alicki, Mark Fannes, Giancarlo Ghirardi, Andreas Knauf, John Lewis, Ge-
offrey Sewell, Franco Strocchi, Walter Thirring, Armin Uhlmann. To me, all
of them have been a constant example of rigorous mathematics and physical
intuition jointly at work.
Last but not least, my deep gratitude goes to my family and to the many
friends on whom I could always count for support and encouragement with
a special thought for Traude and Wolfgang Georgiades.
Trieste, 6 August 2008 Fabio Benatti
VII
Contents
1 Introduction.............................................. 1
Part I Classical Dynamical Systems
2 Classical Dynamics and Ergodic Theory .................. 9
2.1 Classical Dynamical Systems............................. 10
2.1.1 Shift Dynamical Systems .......................... 20
2.2 Symbolic Dynamics..................................... 25
2.2.1 Algebraic Formulations............................ 29
2.2.2 Conditional Probabilities and Expectations .......... 34
2.2.3 Dynamical Shifts and Classical Spin Chains.......... 37
2.3 Ergodicity and Mixing .................................. 40
2.3.1 K-Systems ...................................... 49
2.3.2 Ergodicity and Convexity ......................... 54
2.4 Information and Entropy ................................ 56
2.4.1 Transmission Channels............................ 57
2.4.2 Stationary Information Sources .................... 59
2.4.3 Shannon Entropy................................. 61
2.4.4 Conditional Entropy .............................. 63
2.4.5 Mutual Information............................... 67
3 Dynamical Entropy and Information...................... 71
3.1 Dynamical Entropy ..................................... 71
3.1.1 Entropic K-systems .............................. 80
3.2 Codes and Shannon Theorems ........................... 86
3.2.1 Source Compression .............................. 90
3.2.2 Channel Capacity ................................ 98
4 Algorithmic Complexity .................................. 105
4.1 Effective Descriptions ................................... 106
4.1.1 Classical Turing Machines ......................... 108
4.1.2 Kolmogorov Complexity........................... 113
4.2 Algorithmic Complexity and Entropy Rate ................ 122
4.3 Prefix Algorithmic Complexity ........................... 127
IX
X Contents
Part II Quantum Dynamical Systems
5 Quantum Mechanics of Finite Degrees of Freedom........ 139
5.1 Hilbert Space and Operator Algebras ..................... 139
5.2 C∗ Algebras ........................................... 143
5.2.1 Positive Operators................................ 148
5.2.2 Positive and Completely Positive Maps.............. 157
5.3 von Neumann Algebras.................................. 166
5.3.1 States and GNS Representation.................... 170
5.3.2 C∗ and von Neumann Abelian algebras ............. 173
5.4 Quantum Systems with Finite Degrees of Freedom.......... 178
5.5 Quantum States........................................ 190
5.5.1 States in the Algebraic Approach................... 208
5.5.2 Density Matrices and von Neumann Entropy......... 213
5.5.3 Composite Systems............................... 218
5.5.4 Entangled States ................................. 222
5.6 Dynamics and State-Transformations ..................... 227
5.6.1 Quantum Operations ............................. 236
5.6.2 Open Quantum Dynamics ......................... 241
5.6.3 Quantum Dynamical Semigroups ................... 247
5.6.4 Physical Operations and Positive Maps.............. 251
6 Quantum Information Theory ............................ 255
6.1 Quantum Information Theory............................ 255
6.2 Bipartite Entanglement ................................. 261
6.3 Relative Entropy ....................................... 287
6.3.1 Holevo’s Bound and the Entropy of a Subalgebra..... 294
6.3.2 Entropy of a Subalgebra and Entanglement of
Formation....................................... 302
7 Quantum Mechanics of Infinite Degrees of Freedom ...... 317
7.1 Observables, States and Dynamics........................ 323
7.1.1 Bosons and Fermions ............................. 325
7.1.2 GNS Representation and Dynamics................. 335
7.1.3 Quantum Ergodicity and Mixing ................... 341
7.1.4 Algebraic Quantum K-Systems .................... 356
7.1.5 Quantum Spin Chains ............................ 362
7.2 von Neumann Entropy Rate ............................. 376
7.3 Quantum Spin Chains as Quantum Sources................ 381
7.3.1 Quantum Compression Theorems................... 383
7.3.2 Quantum Capacities .............................. 399
Contents XI
Part III Quantum Dynamical Entropies and Complexities
8 Quantum Dynamical Entropies ........................... 411
8.1 CNT Entropy: Decompositions of States .................. 413
8.1.1 CNT Entropy: Quasi-Local Algebras................ 429
8.1.2 CNT Entropy: Stationary Couplings................ 433
8.1.3 CNT entropy: Applications........................ 436
8.1.4 Entropic Quantum K-systems ..................... 443
8.2 AFL Entropy: OPUs.................................... 451
8.2.1 Quantum Symbolic Models and AFL Entropy........ 452
8.2.2 AFL Entropy: Interpretation....................... 455
8.2.3 AFL-Entropy: Applications ........................ 457
8.2.4 AFL Entropy and Quantum Channel Capacities...... 475
9 Quantum Algorithmic Complexities ...................... 483
9.1 Effective Quantum Descriptions .......................... 484
9.1.1 Effective Descriptions by qubit Strings .............. 485
9.1.2 Quantum Turing Machines ........................ 486
9.2 qubit Quantum Complexity .............................. 494
9.2.1 Quantum Brudno’s Theorem....................... 497
9.3 cbit Quantum Complexity ............................... 506
References.................................................... 517
Index......................................................... 527
1 Introduction
Thisbookfocussesuponquantumdynamicsfromvariouspointsofviewwhich
areconnectedbythenotionofdynamicalentropyasameasureofinformation
production during the course of time.
For classical dynamical systems, the notion of dynamical entropy was in-
troduced by Kolmogorov and developed by Sinai (KS entropy) and provided
a link among different fields of mathematics and physics. In fact, in the light
of the first theorem of Shannon, the KS entropy gives the maximal com-
pression rate of the information emitted by ergodic information sources. A
theorem of Pesin relates it to the positive Lyapounov exponents and thus to
the exponential amplification of initial small errors, in a word to classical
chaos. Finally, a theorem of Brudno links the KS entropy to the compress-
ibility of classical trajectories by means of computer programs, namely to
their algorithmic complexity, a notion introduced, independently and almost
simultaneously by Kolmogorov, Solomonoff and Chaitin.
In a previous book by the author, the notion of quantum dynamical en-
tropyelaboratedbyA.Connes,H.NarnhoferandW.Thirring(CNTentropy)
waspresentedwithinthecontextofquantumergodicityandchaos.TheCNT
entropy is a particular proposal of how the KS entropy might be extended
from classical to quantum dynamical systems.
AftertheappearanceoftheCNTentropy,otherproposalsofquantumdy-
namicalentropiesappearedwhichingeneralassigndifferententropyproduc-
tions to the same quantum dynamics. The basic reason is that each proposal
is built according to a different view about what information in quantum
systems should mean. Concretely, it is a general fact that, in order to gain
information about a system and its time-evolution, one has to observeit and
aquantumfactthatobservationsmaybeinvasiveandperturbing.Shouldthis
fact be considered inescapable and thus incorporated in any good quantum
dynamical entropy or, rather, should it be avoided as a source of spurious
effects that have nothing to do with the actual quantum dynamics?
This is an unavoidable question and, based on the possible answers, one
is led to different notions of quantum dynamical entropies. These will be
sensitive to different aspects of the quantum dynamics and thus, not unex-
pectedly, not equivalent: the real issue is which these aspects are and what
kind of informational meaning they do posses.
F.Benatti,Dynamics, Information and Complexity in Quantum Systems, 1
TheoreticalandMathematicalPhysics,DOI10.1007/978-1-4020-9306-7 1,
(cid:2)c SpringerScience+BusinessMediaB.V.2009
