## Quantum Mechanics

*15 Oct 2014 12:50*

I am *not* going to try to explain quantum mechanics here.

Interpetations --- Copenhagen (pure logical positivism, of course), quantum logic, many worlds. Is there any reason to take the Bohmian (non-local hidden variables) interpretation seriously?

Entanglement (= spooky, non-classical correlations) and coherent states. Decoherence --- the process by which coherent, entangled states lose their coherence due to interaction with other things, particularly very large, complicated, effectively-random things (such as people and measuring devices). There's a growing body of work saying that the rate of decoherence depends on the Lyapunov exponents, which are classical measures of sensitivity to initial conditions and information production. This is very suggestive, and I need to look into it.

Quantum computing; i.e., computing with entangled quantum bits ("qubits").
Problems for which quantum computing is more powerful than classical computing.
(*To check:* nothing classically uncomputable is quantum-computable, is
it?) Avoiding decoherence of the quantum computer. Can we test different
interpretations by experiments on quantum computers? Quantum information theory.

Quantum general relativity, a.k.a. loop quantum gravity, is really interesting. But reading the introductory surveys (especially Thiemann's, below) reminds me of the reason I got out of fundamental theoretical physics in the first place (viz., I'm not smart enough).

The many-worlds interpretation should probably be called something more like "many histories". I like it, because it completely avoids the awful notion of the wave function collapsing, and denies any role whatsoever to measurement, consciousness, etc. And it makes the success of quantum computers remarkably sensible, which I don't think we can say for any of the other interpretations. The only exception I can think of is the Ithaca interpretation, which merges very nicely with many-worlds. This is, roughly, the idea that the wave-function is a real, objective thing, as are correlations between systems, and that "measurement" is just a particular case of decoherence. While on the subject of measurement, I need to learn about algebras of observables.

*Things to think about:*
Quantum computational mechanics.
Can entanglement be treated as a thermodynamic resource? What does the quantum
version of large deviations theory look
like?

*Things far beyond my ken:* Where does that weird complex-valued
not-quite-a-probability measure come from
anyway?

See also Field Theory.

- Recommended:
*Non-technical:*- Nick Herbert, Quantum Reality [Popular treatment of rival interpretations of QM, circa 1985. Herbert's other books are crankish, but this one is OK.]
- David Linsdey, Where Does the Weirdness Go? Why Quantum Mechanics Is Strange, but not as Strange as You Think [Popularization of the measurement problem and decoherence]
- Oliver Morton, "The Computable Cosmos of David Deutsch,"
American Scholar
**69**(Summer 2000): 7ff. [Why the many-worlds interpretation is not, in fact, insane, and how it all relates to quantum computation. Remarkably clear.]

*Technical, overviews:*- David Z. Albert, Quantum Mechanics and Experience [On the interpretation of QM; very good. I like what he calls the "bare theory," as he does, and don't understand his objections to it. (Or rather, after reading Wallace, below, I think I understand, but also think they have no force.) Supposedly this is non-technical --- he claims to teach the formal structure of QM to anyone with high-school math --- but I'd be flabbergasted if such readers really could pick it up from his exposition. That's not his fault or theirs, of course.]
- B. H. Bransden and C. Joachain, Quantum Mechanics [Undergrad-level textbook; comprehensive, and introduces Dirac notation pretty early. Makes a good reference.]
- P. A. M. Dirac, Principles of Quantum Mechanics [Incredibly elegant; hard to learn from.]
- David Griffiths, Introduction to Quantum Mechanics [The best-written introductory book; but then, it's by Griffiths. Good chapter on measurement, Bell's inequalities, and the Aspect experiment.]
- quant-ph, the online archive for papers in QM.
- qubit.org, Web site for quantum computing
- R. F. Streater, "Classical and Quantum Probability," math-ph/0002049 ["There are few mathematical topics that are as badly taught to physicists as probability theory." Let me take this opportunity to recommend the rest of Prof. Streater's excellent website, esepcially Lost Causes in Theoretical Physics and Regained Causes in Theoretical Physics]

*Technical, close-ups:*- K. M. R. Audenaert, J. Calsamiglia, R. Munoz-Tapia,
E. Bagan, Ll. Masanes, A. Acin, and F. Verstraete, "Discriminating States: The
Quantum Chernoff Bound",
Physical
Review Letters
**98**(2007): 160501 - Richard D. Gill, "Teleportation into Quantum Statistics", math.ST/0405572 [Presupposes some familiarity with ordinary statistics, e.g., what the Cramer-Rao inequality is.]
- Domenico Giulini, Erich Joos, Claus Kiefer, Joachim Kupsch, Ion-Olimpiu Stamatescu and H. Dieter Zeh, Decoherence and the Appearance of a Classical World in Quantum Theory
- Meir Hemmo and Orly Shenker, "Quantum Decoherence and the
Approach to Equilibrium", Philosophy of
Science
**70**(2003): 330--358 - Erich Joos, "Elements of Environmental Decoherence," quant-ph/9908008
- Specer R. Klein and Joakim Nystrand, "Does particle decay
cause wave function collapse: an experimental test," Physics Letters
A
**308**(2003): 323--328 - M. Merkli, I. Im. Sigal and G. P. Berman, "Decoherence and
Thermalization", Physical
Review Letters
**98**(2007): 130401, quant-ph/0608181 - N. David Mermin, "The Ithaca Intepretation of Quantum
Mechanics," Pramana
**51**(1998): 549--565, quant-ph/9609013 - Sandu Popescu, Anthony J. Short, and Andreas Winter, "Entanglement and the Foundations of Statistical Mechanics", quant-ph/0511225
- James M. Robins, Tyler J. VanderWeele, Richard D. Gill, "A proof of Bell's inequality in quantum mechanics using causal interactions", arxiv:1207.4913
- Geoffrey Sewell, Quantum Mechanics and Its Emergent Macrophysics
- Max Tegmark, "The Importance of Quantum Decoherence in
Brain Processes," Physical Review E
**61**(2000): 4194--4206, quant-ph/9907009 [Sufficient to rule out all quantum theories of consciousness (except*perhaps*Mitchell Porter's ideas about using topological defects for quantum computing).] - Thomas Thiemann, "Lectures on Loop Quantum Gravity," gr-qc/0210094 [90 page introduction, assuming familiarity with ordinary field theory and general relativity. Nicely presents the motivating idea, which is to take both those theories seriously, and find the minimal mathematical structures which will let us combine them.]
- David Wallace, "Everett and Structure," Studies In
History and Philosophy of Science Part B: Studies In History and Philosophy of
Modern Physics
**34**(2003): 87--105, quant-ph/0107144 [Using ideas from Dennett and decoherence to defend the many-worlds interpretation.] - W. H. Zurek, "Quantum Darwinism and Envariance", quant-ph/0308163

- K. M. R. Audenaert, J. Calsamiglia, R. Munoz-Tapia,
E. Bagan, Ll. Masanes, A. Acin, and F. Verstraete, "Discriminating States: The
Quantum Chernoff Bound",
Physical
Review Letters

- To read, statistical inference and large deviations in quantum settings:
- A. Acin, E. Jané and G. Vidal, "Optimal Estimation of
Quantum Dynamics," quant-ph/0012015 [
*Not*Gore Vidal] - O. E. Barndorff-Nielsen and Richard D. Gill, "Fisher Information in Quantum Statistics", quant-ph/9808009
- O. E. Barndorff-Nielsen, Richard D. Gill and P. E. Jupp, "On Quantum Statistical Inference"
- I, quant-ph/0307189 [Longer, original version]
- II, quant-ph/0307191 [Shorter, published version]

- V.P.Belavkin, "Quantum Diffusion, Measurement and Filtering", quant-ph/0510028 [Published in Probability Theory and Its Applications in 1993-1994]
- Igor Bjelakovic, Jean-Dominique Deuschel, Tyll Krueger, Ruedi
Seiler, Rainer Siegmund-Schultze and Arleta Szkola, "A quantum version of
Sanov's
theorem", quant-ph/0412157
= Communications in
Mathematical Physics
**260**(2005): 659--671 [Quantum large deviations!] - Igor Bjelakovic, Tyll Krueger, Rainer Siegmund-Schultze and Arleta
Szkola
- "The Shannon-McMillan Theorem for Ergodic Quantum Lattice Systems," math.DS/0207121
- "Chained Typical Subspaces - a Quantum Version of Breiman's Theorem," quant-ph/0301177

- Luc Bouten and Ramon van Handel, "Quantum filtering: a reference probability approach", math-ph/0508006 ["These notes are intended as an introduction to noncommutative (quantum) filtering theory."]
- Richard D. Gill
- "Accardi contra Bell (cum mundi): The Impossible Coupling", quant-ph/0110137
- "Asymptotics in Quantum Statistics", math.ST/0405571
- "On an Argument of David Deutsch", quant-ph/0307188
- "Time, Finite Statistics, and Bell's Fifth Position", quant-ph/0301059

- Anna Jencova and Denes Petz, "Sufficiency in quantum statistical
inference", math-ph/0412093
= Communications in
Mathematical Physics
**263**(2006): 259--276 [Sounds cool! Among other, more important, things, a quantum version of sufficiency would open the way to a quantum computational mechanics.] - Michael Keyl, "Quantum state estimation and large deviations", quant-ph/0412053
- Cris Moore and Jim Brink, "Inferring Quantum Dynamics from Classical Time Series" [If they ever get off their butts and write it]
- K. Netocny and F. Redig, "Large deviations for quantum spin
systems", math-ph/0404018
= Journal of Statistical Physics
**117**(2004): 521--547 - Wim van Dam, Peter Grunwald and Richard Gill, "The statistical strength of nonlocality proofs", quant-ph/0307125

- To read, decoherence and the physics of measurement:
- Yury Adamov, I. V. Gornyi and A. D. Mirlin, "Loschmidt Echo and Lyapunov Exponent in a Quantum Disordered System," cond-mat/0212065
- Eleonora Alfinito, Rosario G. Viglione and Giuseppe Vitiello, "The decoherence criterion," quant-ph/0007020
- Armen E. Allahverdyan, Roger Balian, Theo M. Nieuwenhuizen, "Phase transitions and quantum measurements", quant-ph/0508162 ["In a quantum measurement, a coupling $g$ between the system S and the apparatus A triggers the establishment of correlations, which provide statistical information about S. Robust registration requires A to be macroscopic, and a dynamical symmetry breaking of A governed by S allows the absence of any bias. Phase transitions are thus a paradigm for quantum measurement apparatuses, with the order parameter as pointer variable. The coupling $g$ behaves as the source of symmetry breaking. The exact solution of a model where S is a single spin and A a magnetic dot (consisting of $N$ interacting spins and a phonon thermal bath) exhibits the reduction of the state as a relaxation process of the off-diagonal elements of S+A, rapid due to the large size of $N$. The registration of the diagonal elements involves a slower relaxation from the initial paramagnetic state of A to either one of its ferromagnetic states."]
- A. Buchleitner, C. Viviescas and M. Tiersch (eds.), Entanglement and Decoherence: Foundations and Modern Trends
- Gabriel G. Carlo, Giuliano Benenti, and Dima L. Shepelyansky, "Dissipative quantum chaos: transition from wave packet collapse to explosion", quant-ph/0503081
- Martiao Castagnino and Roberto Laura and Olimpia Lombardi, " A General Conceptual Framework for Decoherence in Closed and Open Systems", phil-sci/2950
- Mario Castagnino and Olimpia Lombardi, "Self-induced Decoherence: A
New Approach", Studies in the History and Philosophy of Modern
Physics
**35**(2004): 73--107 = phil-sci/801 - Doron Cohen, "Quantum Chaos, Irreversibility, dissipation and dephasing," quant-ph/0201088
- F. M. Cucchietti, H. M. Pastawski and R. Jalabert, "Dynamical
Origin of Decoherence in Clasically Chaotic Systems," cond-mat/0002207 =
Physica A
**283**(2000): 285--289 - Alexandre Giraud and Julien Serreau, "Decoherence and
Thermalization of a Pure Quantum State in Quantum Field
Theory", Physical
Review Letters
**104**(2010): 230405 ["real-time evolution of a self-interacting O(N) scalar field initially prepared in a pure, coherent quantum state. ... nonequilibrium quantum dynamics from a 1/N expansion of the two-particle-irreducible effective action at next-to-leading order, which includes scattering and memory effects. ... restricting one's attention (or ability to measure) to a subset of the infinite hierarchy of correlation functions, one observes an effective loss of purity or coherence and, on longer time scales, thermalization. .... physics of decoherence is well described by classical statistical field theory."] - Sheldon Goldstein, Joel L. Lebowitz, Roderich Tumulka, and Nino
Zanghi, "Canonical
Typicality", Physical
Review Letters
**96**(2006): 050403 - J. J. Halliwell, "Decoherent Histories and the Emergent
Classicality of Local Densities," Physical Review Letters
**83**(1999): 2481--2485 - Yoseph Imry, "Elementary explanation of the inexistence of decoherence at zero temperature for systems with purely elastic scattering," cond-mat/0202044
- Rodolfo A. Jalabert and Horacio M. Pastawski, "Environment-independent decoherence rate in classically chaotic systems," cond-mat/0002207 [or is it cond-mat/0010094?]
- Joseph F. Johnson, "Statistical Mechanics of Amplifying Apparatus", quant-ph/0502044 ["We implement Feynman's suggestion that the only missing notion needed for the puzzle of Quantum Measurement is the statistical mechanics of amplifying apparatus. We define a thermodynamic limit of quantum amplifiers which is a classically describable system in the sense of Bohr, and define macroscopic pointer variables for the limit system. Then we derive the probabilities of Quantum Measurement from the deterministic Schroedinger equation by the usual techniques of Classical Statistical Mechanics."]
- Johannes Kofler and Caslav Brukner, "Classical World Arising out of
Quantum Physics under the Restriction of Coarse-Grained Measurements",
Physical Review Letters
**99**(2007): 180403 = arxiv:quant-ph/0609079 - Fernando C. Lombardo and Paula I. Villar, "Decoherence in composite quantum open systems: the effectiveness of chaotic degrees of freedom", quant-ph/0507107
- Michael B. Mensky, "Evolution of an Open System as a Continuous
Measurement of This System by Its Environment," Physics Letters
A
**307**(2003): 85--92 - Diana Monteoliva and Juan Pablo Paz, "Decoherence and the Rate of
Entropy Production in Chaotic Quantum Systems" Physical Review E
**85**(2000): 3373--3376 - Arjendu K. Pattanayak, "Lypaunov Exponents, Entropy Production,
and Decoherence," Physical Review Letters
**83**(1999): 4526--4529 - Juan Pablo Paz and Wojciech Hubert Zurek, "Quantum Limit of
Decoherence: Environment Induced Superselection of Energy Eigenstates,"
Physical Review Letters
**82**(1999): 5181--5185 - Tomaz Prosen and Thomas H. Seligman, "Decoherence of spin echoes," nlin/0201038
- Maximilian Schlosshauer, "Decoherence, the measurement problem, and
interpretations of quantum mechanics", Reviews of
Modern Physics
**76**(2005): 1267--1305 = quant-ph/0312059 - Lev Vaidman, "The Meaning of the Interaction-Free Measurements," quant-ph/0103081
- Wheeler and Zurek (eds.), Quantum Mechanics and Measurement

- To read, interpretations of quantum mechanics:
- John S. Bell, Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Mechanics
- Harvey R. Brown and David Wallace, "Solving the measurement problem: de Broglie-Bohm loses out to Everett", phil-sci/1659
- Julian Brown, Minds, Machines, and the Multiverse
- Jeffrey Bub, Interpreting the Quantum World
- Craig Callender (ed.), Physics Meets Philosophy at the Planck Scale: Contemporary Theories in Quantum Gravity
- John G. Cramer, "The Transactional Interpretation of Quantum
Mechanics," Reviews of Modern Physics
**58**(1986): 647--688 [Online. Thanks to Josh Buermann for alerting me to this, which had somehow completely passed me by. This despite Cramer having written a decent hard SF novel, Twistor.] - David Deutsch, The Fabric of Reality
- Robert B. Griffiths, Consistent Quantum Theory
- Michael J. W. Hall, "Incompleteness of trajectory-based
interpretations of quantum mechanics", Journal of Physics
A
**37**(2004): 9549--9556 = quant-ph/0406054 - R. Jackiw and A. Shimony, "The Depth and Breadth of John Bell's Physics," physics/0105046
- Adrian Kent, "One world versus many: the inadequacy of Everettian accounts of evolution, probability, and scientific confirmation", arxiv:0905.0624
- Nicolaas P. Landsman, "Macroscopic observables and the Born rule. I. Long run frequencies", phil-sci/4012
- N. David Mermin
- "What Do These Correlations Know About Reality? Nonlocality and the Absurd," quant-ph/9807055
- "What Is Quantum Mechanics Trying to Tell Us?" quant-ph/9801057

- Roland Omnès, "Consistent interpretations of quantum mechanics", Reviews of Modern Physics
**64**(1992): 339--382 - Dmitry A. Slavnov
- "Quantum mechanics as a complete physical theory,"
Theoretical and Mathematical Physics,
**132**(2002): 1262--1274 = quant-ph/0211053 - "Quantum measurements and Kolmogorovian probability theory," quant-ph/0301027

- "Quantum mechanics as a complete physical theory,"
Theoretical and Mathematical Physics,
- Robert Van Wesep, "Many worlds and the emergence of probability in quantum mechanics", quant-ph/0506024 ["The interpretation of the squared norm as probability and the apparent stochastic nature of observation in quantum mechanics are derived from the strong law of large numbers and the algebraic properties of infinite sequences of simultaneous quantum observables. It is argued that this result validates the many-worlds view of quantum reality."]
- David Wallace
- "Epistemology Quantized: circumstances in which we should come to believe in the Everett interpretation", phil-sci 2368
- "In Defence of Naivete: The Conceptual Status of Lagrangian QFT," quant-ph/0112148
- "The Quantization of Gravity: An Introduction," gr-qc/0004005
- "Quantum Probability and Decision Theory, Revisited," quant-ph/0211104
- "Worlds in the Everett Interpretation," quant-ph/0103092

- To read, quantum information and computation:
- Robert Alicki, "Information-theoretical meaning of quantum dynamical entropy," quant-ph/0201012
- Alp Atici and Rocco A. Servedio, "Improved Bounds on Quantum Learning Algorithms", quant-ph/0411140
- Valerio Cappellini, "Quantum Dynamical Entropies and Complexity in Dynamical Systems", math-ph/0403035
- Jim Crutchfield and Cris Moore, "Quantum Automata and
Quantum Grammars," Theoretical Computer Science
**237**(2000): 275--306 [online] - David Deutsch, Artur Ekert and Rossella Lupacchini, "Machines, Logic and Quantum Physics," math.LO/9911150
- Filippo Giraldi and Paolo Grigolini, "Quantum Entanglement and Entropy," cond-mat/0101479
- Michal Horodecki, Jonathan Oppenheim, Ryszard Horodecki, "Are the laws of entanglement theory thermodynamical?," quant-ph/0207177
- Oliver Johnson and Yuri Suhov, "The von Neumann Entropy and Information Rate for Ideal Quantum Gibbs Ensembles," math-ph/0109023
- Michael Nielsen, "Quantum information science as an approach to complex quantum systems," quant-ph/0208078
- Michael Nielsen and Isaac Chuang, Quantum Computation and Information [Web site]
- Michael A. Nielsen, Christopher M. Dawson, Jennifer L. Dodd, Alexei Gilchrist, Duncan Mortimer, Tobias J. Osborne, Michael J. Bremner, Aram W. Harrow and Andrew Hines, "Quantum dynamics as a physical resource," quant-ph/0208077
- Jonathan Oppenheim, Micha Horodecki, and Ryszard Horodecki, "Are
There Phase Transitions in Information Space?," Physical Review
Letters
**90**(2003): 010404 - Daowen Qiu and Mingsheng Ying, "Characterization of quantum
automata," Theoretical Computer
Science
**312**(2004): 479--489 - Eleanor Riefeel and Wolfgang Polak, "An Introduction to Quantum Computing for Non-Physicists," quant-ph/9809016
- Orly R. Shenker and Meir Hemmo
- "The Von Neumann Entropy: A Reconsideration", phil-sci/2256
- "Von Neumann's Entropy Does Not Correspond to Thermodynamic Entropy", phil-sci/3716

- Alexander Stotland, Andrei A. Pomeransky, Eitan Bachmat and Doron Cohen, "The information entropy of quantum mechanical states", quant-ph/0401021
- Paul M. B. Vitanyi, "Quantum Kolmogorov Complexity Based on Classical Descriptions," quant-ph/0102108

- To read, other:
- Heinz-Peter Breuer and Francesco Petruccione, The Theory of Open Quantum Systems
- Olivier Darrigol, From c-Numbers to q-Numbers: The Classical Analogy in the History of Quantum Theory
- N. D. Hari Dass, S. Kalyana Rama and B. Sathiapalan, "On the Emergence of the Microcanonical Description from a Pure State," cond-mat/0112439
- Hans F. de Groote, "Observables", math-ph/0507019
- L. Kaplan, "Correlation function bootstrapping in quantum chaotic
systems", Physical
Review E
**71**(2005): 056212 - Sang Pyo Kim, "Nonequilibrium Quantum Evolution of Open Systems," cond-mat/9912472
- Elliott H. Lieb, "Quantum Mechanics, the Stability of Matter and Quantum Electrodynamics", math-ph/0401004
- David T. Pegg, "Causality in quantum mechanics",
Physics
Letters A
**349**(2006): 411--414 - Miklos Redei and Stephen J. Summers, "Local Primitive Causality and the Common Cause Principle in Quantum Field Theory," quant-ph/0108023 [I should probably learn what a von Neumann algebra is before reading this...]
- Manfred Requardt and Sisir Roy, "(Quantum) Space-Time as a Statistical Geometry of Fuzzy Lumps and the Connection with Random Metric Spaces," gr-qc/0011076
- Thomas Thiemann
- "Introduction to Modern Canonical Quantum General Relativity," gr-qc/0110034 [300 pages]
- "Lectures on Loop Quantum Gravity," gr-qc/0210094 [90 pages]
- Modern Canonical Quantum General Relativity

- G. Vidal, "Entanglement Renormalization",
Physical Review
LetterS
**99**(2007): 220405 - Ulrich Weiss, Quantum Dissipative Systems
- H. D. Zeh, "There is no `first' quantization," Physics Letters
A
**309**(2003): 329--334 - Steve Zelditch, "Quantum ergodicity and mixing", quant-ph/0503026 ["an expository article for the Encyclopedia of Mathematical Physics"]