Classical and quantum information

719 Pages·2010·5.976 MB·English

by  Marinescu D., Marinescu G.

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Classical and Quantum Information∗
Dan C. Marinescu and Gabriela M. Marinescu
November 18, 2010
∗Copyright 2006, 2007, 2008, 2009, 2010 by Dan C. Marinescu and Gabriela M. Marinescu
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To Vera Rae
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Contents
1 Preliminaries 14
1.1 Elements of Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2 Hilbert Spaces and Dirac Notations . . . . . . . . . . . . . . . . . . . . . . . . 23
1.3 Hermitian and Unitary Operators; Projectors. . . . . . . . . . . . . . . . . . . 27
1.4 Postulates of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.5 Quantum State Postulate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.6 Dynamics Postulate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.7 Measurement Postulate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.8 Linear Algebra and Systems Dynamics . . . . . . . . . . . . . . . . . . . . . . 50
1.9 Symmetry and Dynamic Evolution . . . . . . . . . . . . . . . . . . . . . . . . 52
1.10 Uncertainty Principle; Minimum Uncertainty States . . . . . . . . . . . . . . . 54
1.11 Pure and Mixed Quantum States . . . . . . . . . . . . . . . . . . . . . . . . . 55
1.12 Entanglement; Bell States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
1.13 Quantum Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
1.14 Physical Realization of Quantum Information Processing Systems . . . . . . . 65
1.15 Universal Computers; The Circuit Model of Computation . . . . . . . . . . . . 68
1.16 Quantum Gates, Circuits, and Quantum Computers . . . . . . . . . . . . . . . 74
1.17 Universality of Quantum Gates; Solovay-Kitaev Theorem . . . . . . . . . . . . 79
1.18 Quantum Computational Models and Quantum Algorithms . . . . . . . . . . . 82
1.19 Deutsch, Deutsch-Jozsa, Bernstein-Vazirani, and Simon Oracles . . . . . . . . 89
1.20 Quantum Phase Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
1.21 Walsh-Hadamard and Quantum Fourier Transforms . . . . . . . . . . . . . . . 102
1.22 Quantum Parallelism and Reversible Computing . . . . . . . . . . . . . . . . . 107
1.23 Grover Search Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
1.24 Amplitude Amplification and Fixed-Point Quantum Search . . . . . . . . . . . 123
1.25 Error Models and Quantum Algorithms . . . . . . . . . . . . . . . . . . . . . . 130
1.26 History Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
1.27 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
1.28 Exercises and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
2 Measurements and Quantum Information 142
2.1 Measurements and Physical Reality . . . . . . . . . . . . . . . . . . . . . . . . 144
2.2 Copenhagen Interpretation of Quantum Mechanics . . . . . . . . . . . . . . . 147
2.3 Mixed States and the Density Operator . . . . . . . . . . . . . . . . . . . . . . 149
2.4 Purification of Mixed States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
2.5 Born Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
2.6 Measurement Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
2.7 Projective Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
2.8 Positive Operator Valued Measures (POVM) . . . . . . . . . . . . . . . . . . . 164
2.9 Neumark Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
2.10 Gleason Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
2.11 Mixed Ensembles and their Time Evolution . . . . . . . . . . . . . . . . . . . 172
2.12 Bipartite Systems; Schmidt Decomposition . . . . . . . . . . . . . . . . . . . . 174
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2.13 Measurements of Bipartite Systems . . . . . . . . . . . . . . . . . . . . . . . . 176
2.14 Operator-Sum (Kraus) Representation . . . . . . . . . . . . . . . . . . . . . . 182
2.15 Entanglement; Monogamy of Entanglement . . . . . . . . . . . . . . . . . . . . 185
2.16 Einstein-Podolski-Rosen (EPR) Thought Experiment . . . . . . . . . . . . . . 189
2.17 Hidden Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
2.18 Bell and CHSH Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
2.19 Violation of Bell Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
2.20 Entanglement and Hidden Variables . . . . . . . . . . . . . . . . . . . . . . . . 206
2.21 Quantum and Classical Correlations . . . . . . . . . . . . . . . . . . . . . . . . 208
2.22 Measurements and Quantum Circuits . . . . . . . . . . . . . . . . . . . . . . . 210
2.23 Measurements and Ancilla Qubits . . . . . . . . . . . . . . . . . . . . . . . . . 214
2.24 Measurements and Distinguishability of Quantum States . . . . . . . . . . . . 217
2.25 Measurements and an Axiomatic Quantum Theory . . . . . . . . . . . . . . . 221
2.26 History Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
2.27 Summary and Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . 225
2.28 Exercises and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
3 Classical and Quantum Information Theory 230
3.1 The Physical Support of Information . . . . . . . . . . . . . . . . . . . . . . . 233
3.2 Entropy; Thermodynamic Entropy . . . . . . . . . . . . . . . . . . . . . . . . 236
3.3 Shannon Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
3.4 Shannon Source Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
3.5 Mutual Information; Relative Entropy . . . . . . . . . . . . . . . . . . . . . . 255
3.6 Fano Inequality; Data Processing Inequality . . . . . . . . . . . . . . . . . . . 259
3.7 Classical Information Transmission through Discrete Channels . . . . . . . . . 261
3.8 Trace Distance and Fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
3.9 von Neumann Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
3.10 Joint, Conditional, and Relative von Neumann Entropy . . . . . . . . . . . . . 274
3.11 Trace Distance and Fidelity of Mixed Quantum States . . . . . . . . . . . . . 275
3.12 Accessible Information in a Quantum Measurement; Holevo Bound . . . . . . 282
3.13 No Broadcasting Theorem for General Mixed States . . . . . . . . . . . . . . . 292
3.14 Schumacher Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
3.15 Quantum Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
3.16 Quantum Erasure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
3.17 Classical Information Capacity of Noiseless Quantum Channels . . . . . . . . . 306
3.18 Entropy Exchange, Entanglement Fidelity, and Coherent Information. . . . . . 312
3.19 Quantum Fano and Data Processing Inequalities . . . . . . . . . . . . . . . . . 318
3.20 Reversible Extraction of Classical Information from Quantum Information . . 322
3.21 Noisy Quantum Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
3.22 Holevo-Schumacher-Westmoreland Noisy Quantum Channel Encoding Theorem329
3.23 Capacity of Noisy Quantum Channels . . . . . . . . . . . . . . . . . . . . . . . 334
3.24 Entanglement-Assisted Capacity of Quantum Channels . . . . . . . . . . . . . 338
3.25 Additivity and Quantum Channel Capacity . . . . . . . . . . . . . . . . . . . 342
3.26 Applications of Information Theory . . . . . . . . . . . . . . . . . . . . . . . . 345
3.27 History Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
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3.28 Summary and Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . 348
3.29 Exercises and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
4 Classical Error Correcting Codes 355
4.1 Informal Introduction to Error Detection and Error Correction . . . . . . . . . 357
4.2 Block Codes. Decoding Policies . . . . . . . . . . . . . . . . . . . . . . . . . . 359
4.3 Error Correcting and Detecting Capabilities of a Block Code . . . . . . . . . . 363
4.4 Algebraic Structures and Coding Theory . . . . . . . . . . . . . . . . . . . . . 366
4.5 Linear Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
4.6 Syndrome and Standard Array Decoding of Linear Codes . . . . . . . . . . . . 383
4.7 Hamming, Singleton, Gilbert-Varshamov, and Plotkin Bounds . . . . . . . . . 387
4.8 Hamming Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
4.9 Proper Ordering, and the Fast Walsh-Hadamard Transform . . . . . . . . . . . 394
4.10 Reed-Muller Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
4.11 Cyclic Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
4.12 Encoding and Decoding Cyclic Codes . . . . . . . . . . . . . . . . . . . . . . . 410
4.13 The Minimum Distance of a Cyclic Code; BCH Bound . . . . . . . . . . . . . 421
4.14 Burst Errors. Interleaving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
4.15 Reed-Solomon Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
4.16 Convolutional Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
4.17 Product Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
4.18 Serially Concatenated Codes and Decoding Complexity . . . . . . . . . . . . . 446
4.19 Parallel Concatenated Codes - Turbo Codes . . . . . . . . . . . . . . . . . . . 449
4.20 History Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
4.21 Summary and Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . 454
4.22 Exercises and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
5 Quantum Error Correcting Codes 461
5.1 Quantum Error Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
5.2 A Necessary Condition for the Existence of a Quantum Code . . . . . . . . . . 468
5.3 Quantum Hamming Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
5.4 Scale-up and Slow-down . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
5.5 A Repetitive Quantum Code for a Single Bit-flip Error . . . . . . . . . . . . . 471
5.6 A Repetitive Quantum Code for a Single Phase-flip Error . . . . . . . . . . . . 478
5.7 The Nine Qubit Error Correcting Code of Shor . . . . . . . . . . . . . . . . . 483
5.8 The Seven Qubit Error Correcting Code of Steane . . . . . . . . . . . . . . . . 485
5.9 An Inequality for Representations in Different Bases . . . . . . . . . . . . . . . 490
5.10 Calderbank-Shor-Steane (CSS) Codes . . . . . . . . . . . . . . . . . . . . . . . 494
5.11 The Pauli Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500
5.12 Stabilizer Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
5.13 Stabilizers for Perfect Quantum Codes . . . . . . . . . . . . . . . . . . . . . . 512
5.14 Quantum Restoration Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
5.15 Quantum Codes over GF(pk) . . . . . . . . . . . . . . . . . . . . . . . . . . . 518
5.16 Quantum Reed-Solomon Codes . . . . . . . . . . . . . . . . . . . . . . . . . . 521
5.17 Concatenated Quantum Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
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5.18 Quantum Convolutional and Quantum Tail-Biting Codes . . . . . . . . . . . . 528
5.19 Correction of Time-Correlated Quantum Errors . . . . . . . . . . . . . . . . . 538
5.20 Quantum Error Correcting Codes as Subsystems . . . . . . . . . . . . . . . . . 541
5.21 Bacon-Shor Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544
5.22 Operator Quantum Error Correction . . . . . . . . . . . . . . . . . . . . . . . 549
5.23 Stabilizers for Operator Quantum Error Correction . . . . . . . . . . . . . . . 553
5.24 Correction of Systematic Errors Based on Fixed-Point Quantum Search . . . . 555
5.25 Reliable Quantum Gates and Quantum Error Correction . . . . . . . . . . . . 557
5.26 History Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560
5.27 Summary and Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . 560
5.28 Exercises and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562
6 Physical Realization of Quantum Information Processing Systems 565
6.1 Requirements for Physical Implementations of Quantum Computers . . . . . . 567
6.2 Cold Ion Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573
6.3 First Experimental Demonstration of a Quantum Logic Gate . . . . . . . . . . 583
6.4 Trapped Ions in Thermal Motion . . . . . . . . . . . . . . . . . . . . . . . . . 588
6.5 Entanglement of Qubits in Ion Traps . . . . . . . . . . . . . . . . . . . . . . . 590
6.6 Nuclear Magnetic Resonance - Ensemble Quantum Computing . . . . . . . . . 596
6.7 Liquid-State NMR Quantum Computer . . . . . . . . . . . . . . . . . . . . . . 598
6.8 NMR Implementation of Single-Qubit Gates . . . . . . . . . . . . . . . . . . . 605
6.9 NMR Implementation of Two-Qubit Gates . . . . . . . . . . . . . . . . . . . . 606
6.10 The First Generation NMR Computer . . . . . . . . . . . . . . . . . . . . . . 612
6.11 Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614
6.12 Fabrication of Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . 621
6.13 Quantum Dot Electron Spins and Cavity QED . . . . . . . . . . . . . . . . . . 624
6.14 Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628
6.15 Fractional Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 631
6.16 Alternative Physical Realizations of Topological Quantum Computers . . . . . 641
6.17 Photonic Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643
6.18 Summary and Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . 649
7 Appendix. Observable Algebras and Channels 652
8 Glossary 688
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“I want to know God’s thoughts... the rest are details. ” Albert Einstein.
Preface
Anewdiscipline, QuantumInformationScience, hasemergedinthelasttwodecadesofthe
twentieth century at the intersection of Physics, Mathematics, and Computer Science. Quan-
tum Information Processing (QIP) is an application of Quantum Information Science which
covers the transformation, storage, and transmission of quantum information; it represents a
revolutionary approach to information processing
Wehavewitnessedthedevelopmentofmicroprocessors,high-speedopticalcommunication,
high-densitystoragetechnologies,followedbythewidespreaduseofsensors, andmorerecently
multi- and many-core processors and spintronics technology. We are now able to collect
humongous amounts of information, process the information at high speeds, transmit the
information through high-bandwidth and low-latency channels, store it on digital media, and
share it using numerous applications built around the World Wide Web. Thus, the full cycle
at the heart of information revolution was closed, Figure 1 [285], and this revolution became
a reality that profoundly affects our daily life.
Now, at the beginning of the twenty first century, information processing is facing new
challenges: heat dissipation, leakage, and other physical phenomena limit our ability to build
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SENSORS
DIGITAL CAMERAS
(2000s)
WORLD WIDE WEB (1990s) MICROPROCESSORS (1980s)
GOOGLE, YouTube (2000s) MULTI-CORE MICROPROCESSORS
(2000s)
COLLECT
MILESTONES IN INFORMATION
PROCESSING
BOOLEAN ALGEBRA (1854)
DISSEMINATE DIGITAL COMPUTERS (1940s) PROCESS
INFORMATION THEORY (1948)
Quantum Computing
Quantum Information Theory
COMMUNICATE
STORE
FIBER OPTICS (1990s)
OPTICAL STORAGE
WIRELESS (2000s)
HIGH DENSITY SOLID-STATE
(1990s)
SPINTRONICS (2000s)
Figure 1: Our ability to collect, process, store, communicate, and disseminate information
has increased considerably during the last two decades of the twentieth century. 1980s was
the decade of microprocessors; advances in solid state technologies allowed the increase of the
number of transistors on a chip by three order of magnitude and a substantial reduction of
the cost of a microprocessor. In 1990s we have seen major breakthroughs in optical storage,
high density solid-state storage technologies, fiber optics communication, and the widespread
acceptance of the Word Wide Web. The first decade of the twenty first century is the decade
of sensors, rapid information dissemination, and multi-core microprocessors.
increasingly faster and, implicitly, increasingly smaller solid-state devices; it is very difficult to
ensure the security of our communication; we are overwhelmed by the volume of information
we are bombarded with, and it is increasingly more difficult to extract useful information
from the vast ocean of information surrounding us.
Information, either classical or quantum, is physical; this is the mantra repeated through-
out the book. Therefore, we must understand the physical processes that affect the state
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of the systems used to carry information. The physical processes for the storage, transfor-
mation, and transport of classical information are governed by the laws of classical Physics
which limit our ability to process information increasingly faster using present day solid-state
technologies. The speed of charge carriers in semiconductors is finite; to increase the speed
of the device we have to pack the logic gates as tightly as possible.
The heat dissipated by a device increases with the clock rate to the power of 2 or 3,
depending upon the solid-state technology. Heat removal is a hard problem for densely packed
devices; the heat produced by a solid-state device is proportional to the number of gates
thus, to the volume of the device. If we pack the gates into a sphere, the heat dissipated
is proportional to the volume of the sphere and can be removed through the surface of the
sphere; while the amount of the heat increases as the cube of the radius, our ability to remove
it only increases as the square of the radius of the sphere. We are thus limited in our ability
to increase the speed and density of classical circuits.
These facts provide a serious motivation to search for alternative physical realization of
computing and communication systems. Scientists are now exploring revolutionary means to
overcoming the limitations of computing and communication systems based upon the laws of
classical Physics. Quantum and biological information processing provide a glimpse of hope
in overcoming some of the limitations we mentioned and could revolutionize computing and
communication in the third millennium. DNA computing together with quantum computing
and quantum communication are the most promising avenues explored nowadays. While a
significant progress has been made in understanding the properties of quantum information,
fundamental questions regarding biological information are still waiting for answers. For
example,how to explain the semantic aspect of biological information; how is information
from a damaged region of the brain recovered?
Quantum information is information stored as a property of a quantum system e.g., the
polarization of a photon, or the spin of an electron. Quantum information can be transmitted,
stored, and processed following the laws of Quantum Mechanics. Several physical embodi-
ments of quantum information are possible; for example, quantum communication involves
a source that supplies quantum systems in a given state, a noisy channel that “transports”
the quantum system, and the recipient that receives and decodes the quantum information.
The source could be a laser producing monochromatic photons, the channel could be an op-
tical fiber and the recipient a photocell; the source could also be an ion trap controlled by
laser pulses, the channel a series of trapped ions, and the receiver a photo detector reading
out the state of the ions via laser-induced fluorescence [276]. The diversity of the processes
and technologies to process quantum information gives us hope that practical applications of
quantum information will emerge sooner rather than later.
The physical processes for photonic, ion-traps, quantum dots, NMR, and other quantum
systems are very different and could distract us from the goal of discovering the common
properties of quantum information independent of its physical support. To study the proper-
ties of quantum information we use an abstract model which captures the critical aspects of
quantum behavior; this model, Quantum Mechanics, describes the properties of physical sys-
tems as entities in a finite-dimensional Hilbert space. Therefore, quantum information theory
requires a basic understanding of Quantum Mechanics and familiarity with the mathematical
apparatus used by Quantum Mechanics and information theory.
Quantum information has special properties: the state of a quantum system cannot be
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