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70-271 Microsoft Official Academic Course: Supporting Users and Troubleshooting a Microsoft Windows XP Operating System Microsoft Official Academic Course Testbank And Solutions Manual
70-272: Supporting Users and Troubleshooting Desktop Applications on a Microsoft Windows XP Operating System Microsoft Official Academic Course Testbank And Solutions Manual
A Brief History of Modern Psychology Ludy T. Benjamin, Jr. Testbank And Solutions Manual
A History of Modern Britain: 1714 to the Present Ellis Wasson Testbank And Solutions Manual
A Meeting Planner's Guide to Catered Events Patti J. Shock Testbank and Solution Manual
A History of Modern Psychology, 4th Edition C. James Goodwin Testbank And Solutions Manual
Accounting and Auditing Research: Tools and Strategies, 8th Edition Thomas R. Weirich, Thomas C. Pearson, Natalie Tatiana Churyk Testbank And Solutions Manual
Accounting for Managers Interpreting Accounting Information for Decision Making, 4e Paul M. Collier testbank and Solution Manual
Accounting Information Systems, 12th Edition International Student Version Mark G. Simkin, Carolyn A. Strand Norman Testbank And Solutions Manual
Accounting Information Systems: Controls and Processes, 1st Edition Leslie Turner, Andrea Weickgenannt Testbank And Solutions Manual
Accounting Principles, 11th Edition International Student Version Jerry J. Weygandt, Paul D. Kimmel, Donald E. Kieso Testbank And Solutions Manual
Accounting Theory, 7th Edition Jayne Godfrey, Allan Hodgson, Ann Tarca, Jane Hamilton, Scott Holmes Testbank And Solutions Manual
Accounting: Tools for Business Decision Makers, 4th Edition Paul D. Kimmel, Jerry J. Weygandt, Donald E. Kieso Testbank And Solutions Manual
Adobe Dreamweaver CS6 Digital Classroom Jeremy Osborn Testbank and Solution Manual
Adobe Flash Professional CS6 Digital Classroom AGI creative team Testbank and Solution manual
Adobe Photoshop CS6 Digital Classroom Jennifer Smith Testbank and Solution manual
Adobe Illustrator CS6 Digital Classroom Jennifer smith Testbank and Solution Manual
Adobe After Effects CS6 Digital Classroom Jennifer Smith Testbank and solution Manual
Adobe InDesign CS6 Digital Classroom Christopher smith Testbank and solution Manual
Adobe Creative Suite 6 Design and Web Premium Digital Classroom Jennifer Smith Testbank and solution manual
Adobe Creative Cloud Design Tools Digital Classroom Jennifer Smith Testbank and solution manual
Adult Development and Aging: Biopsychosocial Perspectives, 3rd Edition Susan Krauss Whitbourne Testbank And Solutions Manual
Advanced Accounting, 5th Edition International Student Version Debra C. Jeter, Paul K. Chaney Testbank And Solutions Manual
Advanced Topics in Finite Element Analysis of Structures: With Mathematica and MATLAB Computations M. Asghar Bhatti Testbank And Solutions Manual
Advanced Engineering Mathematics, International Student Version, 10th Edition Erwin Kreyszig Testbank And Solutions Manual
Advanced Photoshop CC for Design Professionals Digital Classroom Jennifer smith Testbank and Solution Manual
Advanced Engineering Thermodynamics, 3rd Edition Adrian Bejan Testbank And Solutions Manual
Adult Development and Aging: Biopsychosocial Perspectives, 4th Edition Susan Krauss Whitbourne, Stacey N. Whitbourne
Adult Development and Aging: Biopsychosocial Perspectives, 5th Edition Susan Krauss Whitbourne, Stacey N. Whitbourne
After Effects CC Digital Classroom Jerron smith Testbank and solution manual
Algebra and Trigonometry, 2nd Edition Cynthia Y. Young Testbank And Solutions Manual
Algebra: Form and function William G. McCallum, Eric Connally Testbank And Solutions Manual
Algorithm Design: Foundations, Analysis, and Internet Examples Michael T. Goodrich, Roberto Tamassia Testbank And Solutions Manual
ALS Microsoft Visual Basic .NET Programming Essentials Package Microsoft Official Academic Course Testbank And Solutions Manual
An Introduction to Applied Statistical Thermodynamics Stanley I. Sandler Testbank And Solutions Manual
An Introduction to Chemical Engineering Kinetics and Reactor Design Charles G. Hill Testbank and Solution Manual
An Introduction to Fluid Dynamics Principles of Analysis and Design, Stanley Middle man Testbank and Solution Manual
An Introduction to Mass and Heat Transfer Principles of Analysis and Design, Stanley Middle Man Testbank and Solution Manual
An Introduction to Social Psychology 5e Miles Hewstone, Wolfgang Stroebe, Klaus Jonas Testbank And Solutions Manual
Analysis of Economic Data, 4th Edition Gary Koop Testbank And Solutions Manual
Anatomy and Physiology: From Science to Life, 3rd Edition International Student Version Gail jenkins Testbank And Solutions Manual
Anatomy and Physiology of Domestic Animals, 2nd Edition R. Micheal Akers Testbank and Solution Manual
Anatomy and Physiology for Veterinary Technicians and Nurses A Clinical Approach Robin Sturtz Testbank and solution Manual
Anatomy and Physiology for the Manual Therapies, 1st Edition Andrew Kuntzman, Gerard J. Tortora Testbank And Solutions Manual
Applied Atmospheric Dynamics Amanda H. Lynch, John J. Cassano Testbank And Solutions Manual
Applied Behavior Analysis: Principles and Procedures in Behavior Modification Edward P. Sarafino Testbank And Solutions Manual
Applied Business Statistics: Making Better Business Decisions , International Student Version , 6th Edition Ken Black Testbank And Solutions Manual
Applied Business Statistics: Making Better Business Decisions, 7th Edition International Student Version Ken Black Testbank And Solutions Manual
Applied Calculus, 5th Edition Deborah Hudges- Hallett, Patti Frazer Lock, Andrew M. Gleason, Daneil E. Flath, Sheldon P. Gordon, David O.Lomen Testbank And Solutions Manual
Applied Combinatorics, 6th Edition Alan Tucker Testbank And Solutions Manual
Applied Corporate Finance, 3rd Edition Aswath Damodaran Testbank And Solutions Manual
Applied Data Communications: A Business-Oriented Approach 3rd Edition James E. Goldman, Phillip T. Rawles Testbank And Solutions Manual
Applied Econometric Times Series, 3rd Edition Walter Enders Testbank And Solutions Manual
Applied Management Science: Modeling, Spreadsheet Analysis, and Communication for Decision Making, 2nd Edition John A. Lawrence, Barry A. Pasternac Testbank And Solutions Manual
Applied Psychology Graham C. Davey Testbank And Solutions Manual
Applied Research in Hospitality, 1st Edition Frederic B. Mayo Testbank And Solutions Manual
Applied Statistics and Probability for Engineers, 5th Edition SI Version Douglas C. Montgomery Testbank And Solutions Manual
Applying International Financial Reporting Standards, 3rd Edition Ruth Picker, Ken J. Leo, Janice Loftus, Victoria Wise, Kerry Clark, Keith Alfredson Testbank And Solutions Manual
Baking and Pastry Mastering the Art and Craft, 2nd Edition The culinary Institute of America (CIA) Testbank and solution manual
Auditing and Assurance Services: Understanding the Integrated Audit, 1st edition Karen L. Hooks Testbank And Solutions Manual
Basic Animal Nutrition and Feeding, 5th Edition Wilson G. Pond, David C. Church, Kevin R. Pond, Patricia A. Schoknecht Testbank And Solutions Manual
Basic Chemistry, 9th Edition International Student Version Leo J. Malone, Theodore Dolter Testbank And Solutions Manual
Basic Concepts of Chemistry 8th Edition International student version Leo J. Malone test bank and solution manual
Basic Concepts of Chemistry, International Student Version, 9th Edition Leo J. Malone, Theodore Dolter Testbank And Solutions Manual
Basic Strategy in Context: European text and cases Neil Thomson, Charles Baden-Fuller Testbank And Solutions Manual
Because Teaching Matters, 2nd Edition Marleen C. Pugach Testbank And Solutions Manual
Becoming a Master Manager: A Competing Values Approach, 5th Edition Robert E. Quinn, Sue R. Faerman, Michael P. Thompson, Michael McGrath, Lynda S. St. Clair Testbank And Solutions Manual
Becoming a Skilled Negotiator: Concepts and Practices Kathleen Reardon Testbank And Solutions Manual
Beginning ASP.NET 2.0 Chris Hart, John Kauffman, David Sussman, Chris Ullman Testbank And Solutions Manual
Beginning CSS: Cascading Style Sheets for Web Design, 2nd Edition Ian Pouncey, Richard York Testbank And Solutions Manual
Beginning JavaScript, 4th Edition Paul Wilton, Jeremy McPeak Testbank And Solutions Manual
Beneath the Mask: An Introduction to Theories of Personality, 8th Edition Robert N. Sollod, Christopher F. Monte, John P. Wilson Testbank And Solutions Manual
Beginning XML, 3rd Edition, David hunter Testbank And Solutions Manual
Beginning XML, 4th Edition, David hunter Testbank And Solutions Manual
Big C++, 2nd Edition Cay S. Horstmann, Timothy A. Budd Testbank And Solutions Manual
Big Java 4e for Java 7 and 8 Cay. S horstmann Testbank And Solutions Manual
Big Java, 5th Edition International Student Version testbank and solution manual
Big Java, WeL Courseware Student Version Cay S. Horstmann Testbank And Solutions Manual
Big Java: Late Objects Cay S. Horstmann Testbank And Solutions Manual
Biochemistry, 4th Edition Donald Voet, Judith G. Voet Testbank And Solutions Manual
Bioinformatics and Functional Genomics, 2nd Edition Jonathan Pevsner testbank and solution manual
Bioinformatics For Dummies, 2nd Edition Jean Micheal Clairve testbank and solution manual
Brand Relevance: Making Competitor Irrelevent Testbank and Solution Manual
BioInquiry: Making Connections in Biology, 3rd Edition Nancy L. Pruitt, Larry S. Underwood Testbank And Solutions Manual
Bread: A Baker's Book of Techniques and Recipes, 2nd Edition Jeffrey Hamelman Testbank And Solutions Manual
Bread Baking: An Artisan's Perspective Daniel T. Dumizio testbank and solution manual
Business Data Communications Manish Agrawal Testbank And Solutions Manual
Business Ethics: How to Design and Manage Ethical Organizations Denis Collins Testbank And Solutions Manual
Business Ethics: Mistakes and Successes Robert F. Hartley Testbank And Solutions Manual
Business Intelligence Rajiv Sabherwal, Irma Becerra-Fernandez Testbank And Solutions Manual
Business Planning and Control: Integrating Accounting, Strategy, and People Bruce Bowhill Testbank And Solutions Manual
C++ For Everyone, 2nd Edition Cay S. Horstmann Testbank And Solutions Manual
Calculus Late Transcendentals Single Variable 9th Edition Howard Anton, Irl C. Bivens, Stephen Davis Testbank And Solutions Manual
Calculus Late Transcendentals, 10th Edition International Student Version Howard Anton, Irl C. Bivens, Stephen Davis Testbank And Solutions Manual
Calculus: Multivariable, International Student Version, 5th Edition William G. McCallum Solutions Manual
Case Studies in Abnormal Psychology, 9th Edition Thomas F. Oltmanns, Michele T. Martin, John M. Neale, Gerald C. Davison Testbank And Solutions Manual
Cases in Hospitality Management: A Critical Incident Approach, 2nd Edition Timothy R. Hinkin Testbank And Solutions Manual
Casino Operations Management, 2nd Edition Jim kilby Testbank and solution manual
Catering: A Guide to Managing a Successful Business Operation Bruce Mattel
Catering Management, 4th Edition Nancy Loman Scanlon Testbank And Solutions Manual
Cell Biology, 6th Edition, International Student Version Gerald Karp Testbank And Solutions Manual
Cell Biology, 7th Edition International Student Version Gerald Karp Testbank And Solutions Manual
Changing the U.S. Health Care System: Key Issues in Health Services Policy and Management, 4th Edition Gerald F. Kominski Testbank and Solution Manual
Chemical Reaction Engineering, 3rd Edition Octave Levenspiel Testbank And Solutions Manual
Chemical Reactions and Chemical Reactors, 1st Edition George W. Roberts Testbank and Solution Manual
Chemical Reactor Analysis and Design, 3rd Edition Gilbert F. Froment, Kenneth B. Bischoff, Juray De Wilde Testbank And Solutions Manual
Chemical, Biochemical, and Engineering Thermodynamics, 4th Edition Stanley I. Sandler Testbank and Solution Manual
Chemistry Case Studies for Allied Health, 1st Edition Colleen Kelley, Wendy Weeks Testbank And Solutions Manual
Chemistry, 2nd Edition Allan Blackman, Steve Bottle, Siegbert Schmid, Mauro Mocerino, Uta Wille Testbank And Solutions Manual
Chemistry, , 6th Edition, International Student Version James E. Brady, Neil D. Jespersen, Alison Hyslop Testbank And Solutions Manual
Chemistry, Textbook and Student Solutions Manual , 4th Edition John A. Olmsted, Gregory M. Williams Testbank And Solutions Manual
Chemistry: A Guided Inquiry, 5th Edition Richard S. Moog, John J. Farrell Testbank And Solutions Manual
Chemistry: An Everyday Approach to Chemical Investigation Sally Solomon, Susan Rutkowsky, Charles Boritz Testbank And Solutions Manual
Chemistry Case Studies for Allied Health, 1st Edition Colleen Kelley Testbank and Solution Manual
Chemistry: Structure and Dynamics, 5th edition International student version James N. Spencer, George M. Bodner, Lyman H. Rickard Testbank And Solutions Manual
Chemistry: The Study of Matter and Its Changes, 5th Edition International Student Version James E. Brady, Frederick A. Senese Testbank And Solutions Manual
Chemistry, 2nd edition Allan Blackman Testbank and Solution Manual
Chemistry, International Student Version, 6th Edition James E. Brandy Testbank and Solution Manual
Child Psychology: Development in a Changing Society, 5th Edition Robin Harwood, Scott A. Miller, Ross Vasta Testbank And Solutions Manual
Childrens Literature: A Developmental Perspective, 1st Edition Barbara E. Travers, John F. Travers Testbank And Solutions Manual
Chocolates and Confections: Formula, Theory, and Technique for the Artisan Confectioner Peter P. Greweling, The Culinary Institute of America (CIA) Testbank And Solutions Manual
Chocolates and Confections: Formula, Theory, and Technique for the Artisan Confectioner Peter P. Greweling, 2nd edition The Culinary Institute of America (CIA)
Christian History Alister E. McGrath Testbank And Solutions Manual
Christian Theology: An Introduction, 5th Edition Alister E. McGrath Testbank And Solutions Manual
Cinemetrics: Architectural Drawing Today Brian McGrath, Jean Gardner Testbank And Solutions Manual
Classical Mechanics: From Newton to Einstein: A Modern Introduction, 2nd Edition Martin W. M call solution manual
Classroom Management Strategies: Gaining and Maintaining Students' Cooperation, 7th Edition James S. Cangelosi Testbank And Solutions Manual
Classroom Management: Creating a Successful K-12 Learning Community, 5th Edition Paul Burden Testbank And Solutions Manual
Cognition, 6th Edition Margaret W. Matlin Testbank And Solutions Manual
Cognition, First Edition Douglas Whitman Testbank And Solutions Manual
Cooking to the Image A Plating Handbook Elaine Sikorski Testbank and Solution Manual
College Algebra, 2nd Edition Cynthia Y. Young Testbank And Solutions Manual
Commercial Banking: The Management of Risk, 3rd Edition Benton E. Gup, James W. Kolari Testbank And Solutions Manual
Communicating in the 21st Century, 3rd Edition Baden Eunson Testbank And Solutions Manual
Community Nursing in Australia 2nd Edition Debbie Kralik (Editor), Antonia van Loon (Editor) Testbank And Solutions Manual
Comparative Religious Ethics: A Narrative Approach to Global Ethics, 2nd Edition Darrell J. Fasching, Dell deChant, David M. Lantigua Testbank And Solutions Manual
Competitive Success, How Branding Adds Value John A. Davis Testbank And Solutions Manual
CompTIA A+ Complete Deluxe Study Guide: Exams 220-701 (Essentials) and 220-702 (Practical Application) Quentin Docter, Emmett Dulaney, Toby Skandier Testbank And Solutions Manual
CompTIA A+ Complete Study Guide: (Exams 220-601/602/603/604) Quentin Docter, Emmett Dulaney, Toby Skandier Testbank And Solutions Manual
CompTIA A+ Complete Study Guide: (Exams 220-601/602/603/604), Deluxe Edition Quentin Docter, Emmett Dulaney, Toby Skandier Testbank And Solutions Manual
CompTIA A+ Complete Study Guide Authorized Courseware: Exams 220-801 and 220-802, 2nd Edition Quentin docter Testbank and solution manual
Computer Architecture and Organization: An Integrated Approach Miles J. Murdocca, Vincent P. Heuring Testbank And Solutions Manual
Computer Education for Teachers: Integrating Technology into Classroom Teaching, 6th Edition V. Sharp Testbank And Solutions Manual
Computer Networking: Internet Protocols in Action Jeanna Matthews Testbank And Solutions Manual
Computer Forensics JumpStart, 2nd Edition Micheal G. Solomen testbank and solution manual
Computer Networks: Principles, Technologies and Protocols for Network Design Natalia Olifer, Victor Olifer Testbank And Solutions Manual
Consumer Behaviour , 2nd Edition Martin M. Evans, Gordon Foxall, Ahmad Jamal Testbank And Solutions Manual
Contemporary Business, 13th Edition International Student Version Louis E. Boone, David L. Kurtz Testbank And Solutions Manual
Contemporary Business, 15th Edition Louis E. Boone, David L. Kurtz Testbank And Solutions Manual
Contemporary Industrial Organization: A Quantitative Approach Lynne Pepall, Dan Richards, George Norman Testbank And Solutions Manual
Contemporary Issues in Accounting Michaela Rankin, Patricia Stanton, Susan McGowan, Kimberly Ferlauto, Matthew Tilling Testbank And Solutions Manual
Contemporary Strategy Analysis, 8th Edition Robert M. Grant Testbank And Solutions Manual
Convection Heat Transfer, 4th Edition Adrian Bejan Testbank And Solutions Manual
Conversational Spanish Grammar for the Hospitality Classroom Matt A. Casado Testbank And Solutions Manual
Cooking to the Image: A Plating Handbook Elaine Sikorski Testbank And Solutions Manual
Core Concepts of Accounting, 2nd Edition Cecily A. Raiborn Testbank And Solutions Manual
Core Concepts of Government and Not-For-Profit Accounting, 2nd edition Michael H. Granof, Penelope S. Wardlow Testbank And Solutions Manual
Core Concepts of Information Technology Auditing James E. Hunton, Stephanie M. Bryant, Nancy A. Bagranoff Testbank And Solutions Manual
Core Concepts of Organizational Behavior John R. Schermerhorn, James G. Hunt, Richard N. Osborn Testbank And Solutions Manual
Core Concepts of Real Estate Principles and Practices James E. Larsen Testbank And Solutions Manual
Core Concepts of Financial Analysis: A User Approach Gary Giroux Testbank And Solutions Manual
Corporate Finance European Edition Peter moles, Robert Parrino testbank and solution manual
Corporate Finance Laurence Booth, Sean Cleary, Pamela Drake Testbank And Solutions Manual
Corporate Finance: Theory and Practice, 2nd Edition Aswath Damodaran Testbank And Solutions Manual
Corporate Financial Reporting and Analysis, 3rd Edition David Young, Jacob Cohen Testbank And Solutions Manual
Cost Management: Measuring, Monitoring, and Motivating Performance Leslie G. Eldenburg, Susan Wolcott Testbank And Solutions Manual
Cost Management: Measuring, Monitoring, and Motivating Performance, 2nd Edition Leslie G. Eldenburg, Susan K. Wolcott Testbank And Solutions Manual
Creating Your Culinary Career The American Institute of Culinary Arts (CIA) Testbank and Solution Manual
Criminal Justice Essentials, 9th Edition Sue Titus Reid Testbank And Solutions Manual
Culinary Math, 3rd, Revised and Expanded Edition Linda Blocker Testbank and Solution Manual
Culinary Calculations: Simplified Math for Culinary Professionals, 2nd Edition Terri Jones Testbank And Solutions Manual
Data Structures and Algorithms in C++, 2nd Edition Michael T. Goodrich, Roberto Tamassia, David M. Mount Testbank And Solutions Manual
Data Structures and Algorithms in Java 5th Edition Michael T. Goodrich, Roberto Tamassia Testbank And Solutions Manual
Data Structures: Abstraction and Design Using Java, Second Edition Elliot B. Koffman, Paul A. T. Wolfgang Testbank And Solutions Manual
Database Design and Implementation Edward Sciore Testbank And Solutions Manual
Decision Analysis for Management Judgment, 4th Edition Paul Goodwin, George Wright testbank and solution manual
Design and Analysis of Experiments, 8th Edition International Student Version Douglas C. Montgomery Testbank And Solutions Manual
Design and Equipment for Restaurants and Foodservice: A Management View, 3rd Edition Costas Katsigris, Chris Thomas Testbank And Solutions Manual
Design and Equipment for Restaurants and Foodservice: A Management View, 4th Edition Chris Thomas, Edwin J. Norman, Costas Katsigris Testbank And Solutions Manual
Design with Constructal Theory Adrian Bejan Testbank And Solutions Manual
Design and Layout of Foodservice Facilities, 3rd Edition John C. Birchfield, John Birchfield, Jr Testbank And Solutions Manual
Designing the Customer-Centric Organization: A Guide to Strategy, Structure, and Process Jay R. Galbrieth Testbank And Solutions Manual
Designing and Conducting Research in Health and Human Performance Tracy D. Matthews Testbank And Solutions Manual
Detecting Earnings Management Gary Giroux Testbank And Solutions Manual
Development Economics: Theory, Empirical Research, and Policy Analysis Julie Schaffner Testbank And Solutions Manual
Dicho en vivo: Beginning Spanish with Personal Native-Speaker Coaching Kim Potowski, Silvia Sobral Testbank And Solutions Manual
Dicho y hecho: Beginning Spanish, 9th Edition Kim Potowski, Silvia Sobral, Laila M. Dawson Testbank And Solutions Manual
Differential Equations with Boundary Value Problems: Modern Methods and Applications, ISV, 2nd editionJames R. Brannan, William E. Boyce Testbank And Solutions Manual
Dimensions of the Hospitality Industry, 3rd Edition Paul R. Dittmer Testbank And Solutions Manual
Discovering Physical Geography 3rd Edition Alan F. Arbogast Testbank And Solutions Manual
Discovering Physical Geography, 1st Edition Alan F. Arbogast Testbank And Solutions Manual
Discovering Physical Geography, 2nd Edition Alan F. Arbogast Testbank And Solutions Manual
Diversity and Cultural Competence in Health Care: A Systems Approach Janice L. Dreachslin Testbank and Solution Manual
Diversity in U.S. Mass Media Catherine A. Luther, Carolyn Ringer Lepre, Naeemah Clark Testbank And Solutions Manual
Dragons and Tigers: A Geography of South, East, and Southeast Asia, 3rd Edition Barbara A. Weightman Testbank And Solutions Manual
Dreamweaver CS5 Digital Classroom, (Book and Video Training covers CS5 & CS5.5) Jeremy Osborn
Dreamweaver CS4 Digital Classroom, (Book and Video Training) Jeremy Osborn
Dreamweaver CC Digital Classroom Micheal Arguin
Drug Use, Misuse, and Abuse Cecile A. Marczinski Testbank And Solutions Manual
E-Business Technologies Supporting the Net-Enhanced Organization Craig Vann Slyke Testbank and Solution Manual
E-Business: Organizational and Technical Foundations Michael P. Papazoglou, Pieter Ribbers Testbank And Solutions Manual
Econometric Analysis of Panel Data, 5th Edition Badi H. Baltagi Testbank And Solutions Manual
Economics for Investment Decision Makers Micro, Macro, and International Economics Christopher D. Piros Testbank and Solution Manual
Economic Geography: A Contemporary Introduction, 2nd Edition Neil Coe, Philip Kelly, Henry W. C. Yeung Testbank And Solutions Manual
Economics and the Environment, 6th Edition Eban S. Goodstein Testbank And Solutions Manual
Economics and the Environment, 7th Edition Eban S. Goodstein, Stephen Polasky Testbank And Solutions Manual
Economics of Farm Management in a Global Setting Kent Olson Testbank And Solutions Manual
Economics of strategy, 5th edition International Student version David Besanko, David Dranove, Mark Shanley, Scott Schaefer Testbank And Solutions Manual
Economics of Strategy, 6th Edition International Student Version David Besanko, David Dranove, Mark Shanley, Scott Schaefer Testbank And Solutions Manual
Economics: Theory and Practice, 10th Edition Patrick J. Welch, Gerry F. Welch Testbank And Solutions Manual
Educational Assessment: A Practical Introduction Thomas P. Hogan Testbank And Solutions Manual
Educational Psychology Roxana Moreno Testbank And Solutions Manual
Educational Psychology: Reflection for Action, 3rd Edition Angela M. O'Donnell, Johnmarshall Reeve, Jeffrey K. Smith Testbank And Solutions Manual
Educational Research James B. Schreiber, Kimberly Asner-Self Testbank And Solutions Manual
Educational Research: A Contextual Approach Ken Springer Testbank And Solutions Manual
Educational Testing and Measurement: Classroom Application and Practice, 10th Edition Tom Kubiszyn, Gary D. Borich Testbank And Solutions Manual
Electronics with Discrete Components, 1st Edition Enrique J. Galvez Testbank And Solutions Manual
Elementary Linear Algebra with Supplemental Applications, 10th Edition International Student Version Howard Anton, Chris Rorres Testbank And Solutions Manual
Emerging Theories in Health Promotion Practice and Research, 2nd Edition Ralph J. DiClemente, Richard A. Crosby, Michelle Kregler testbank and solution manual
En bonne forme, 8th Edition Simone Renaud, Dominique van Hooff Testbank And Solutions Manual
Energy and the Environment, 2nd Edition Robert A. Ristinen, Jack P. Kraushaar Testbank And Solutions Manual
Engineering and Chemical Thermodynamics, Milo D. Koretsky Testbank And Solutions Manual
Engineering and Chemical Thermodynamics, 2nd Edition Milo D. Koretsky Testbank And Solutions Manual
Engineering Mechanics: Dynamics, 7th Edition International student version J. L. Meriam, L. G. Kraige Testbank And Solutions Manual
Engineering Optimization Theory and Practice, 4th Edition Testbank and Solution Manual
Engineering Fluid Mechanics, 10th Edition SI Version Donald F. Elger Testbank and Solution Manual
Engineering Statistics, SI Version, 6th Edition Douglas C. Montgomery, George C. Runger, Norma F. Hubele Testbank And Solutions Manual
Ensemble: Grammaire en action, 7th Edition Raymond F. Comeau, Normand J. Lamoureux Testbank And Solutions Manual
Entrepreneurship and Small Business, 4th Asia Pacific Edition Michael Schaper, Thierry Volery, Paull Weber, Brian Gibson Testbank And Solutions Manual
Entrepreneurship, 2nd Edition William D. Bygrave, Andrew Zacharakis Testbank And Solutions Manual
Environment, 7th Edition Peter H. Raven, Linda R. Berg, David M. Hassenzahl Testbank And Solutions Manual
Environment, 8th Edition International Student Version Peter H. Raven, Linda R. Berg, David M. Hassenzahl Testbank And Solutions Manual
Environmental Geology Laboratory Manual, 2nd Edition Tom Freeman Testbank And Solutions Manual
Environmental Science: Active Learning Laboratories and Applied Problem Sets, 2nd Edition Travis P. Wagner, Robert Sanford Testbank And Solutions Manual
Environmental Science: Earth as a Living Planet, 7th Edition Daniel B. Botkin, Edward A. Keller Testbank And Solutions Manual
Environmental Science: Earth as a Living Planet, 8th Edition Daniel B. Botkin, Edward A. Keller Testbank And Solutions Manual
Equity Asset Valuation, 2nd Edition Jerald E. Pinto Testbank and Solution Manual
Essential Biochemistry, 3rd Edition Charlotte W. Pratt, Kathleen Cornely Testbank And Solutions Manual
Essential Computational Fluid Dynamics Elag Zikanov Testbank and Solution Manual
Essential Environmental Science, 1st Edition Edward A. Keller, Daniel B. Botkin Testbank And Solutions Manual
Essential Mathematics for Economics and Business, 4th Edition Teresa Bradley Testbank And Solutions Manual
Essential Microbiology, 2nd Edition Stuart Hogg Testbank And Solutions Manual
Essential Statistics for Economics, Business and Management Teresa Bradley Testbank And Solutions Manual
Essentials of Anatomy and Physiology, ISV 8th Edition Gerard J. Tortora, Bryan H. Derrickson Testbank And Solutions Manual
Essentials of Anatomy and Physiology, 9th Edition International Student Version Gerard J. Tortora
Essentials of Business Processes and Information Systems Simha R. Magal, Jeffrey Word Testbank And Solutions Manual
Essentials of Contemporary Business Louis E. Boone, David L. Kurtz Testbank And Solutions Manual
Essentials of Corporate Finance Robert Parrino, David S. Kidwell, Thomas Bates Testbank And Solutions Manual
Essentials of Information Systems for Managers: Text Only Gabe Piccoli Testbank And Solutions Manual
Essentials of Professional Cooking Wayne Gisslen Testbank And Solutions Manual
Essentials of Quality with Cases and Experiential Exercises, 1st Edition V. E. Sower Testbank And Solutions Manual
Ethics and Corporate Social Responsibility in the Meetings and Events Industry Elizabeth V. Henderson, Mariela McIlwraith Testbank And Solutions Manual
Experiments and Exercises in Basic Chemistry, 7th Edition Steven Murov, Brian Stedjee Testbank And Solutions Manual
Explorations in College Algebra, 5th Edition Linda Almgren Kime, Judy Clark, Beverly K. Michael Testbank And Solutions Manual
Exploring Management, 3rd Edition John R. Schermerhorn Testbank And Solutions Manual
Exploring Wine: The Culinary Institute of America's Guide to Wines of the World, Completely Revised 3rd Edition Steven Kolpan, Brian H. Smith Testbank and sm
Evolutionary Optimization Algorithms Dan simon
Festival and Special Event Management 5th Edition Johnny Allen, William O'Toole, Robert Harris, Ian McDonnell Testbank And Solutions Manual
Financial Accounting: A Focus on Fundamentals, 6e, International Student Version Jerry J. Weygandt Testbank And Solutions Manual
Financial Accounting: A Valuation Emphasis John S. Hughes, Frances L. Ayres, Robert E. Hoskin Testbank And Solutions Manual
Financial Accounting : A Global Approach Rod Monger Testbank And Solutions Manual
Financial Accounting 7th Edition, ISV Paul D. Kimmel, Jerry J. Weygandt, Donald E. Kieso Testbank And Solutions Manual
Financial Accounting Micheal J. Jones testbank and solution manual
Financial Accounting in an Economic Context, International Student Version, 8th Edition Jamie Pratt Testbank And Solutions Manual
Financial Accounting, IFRS Edition: 2nd Edition Jerry J. Weygandt, Paul D. Kimmel, Donald E. Kieso Testbank And Solutions Manual
Financial Accounting Theory and Analysis: Text and Cases, 11th Edition Richard G. Schroeder, Myrtle W. Clark, Jack M. Cathey Testbank And Solutions Manual
Financial Accounting with Annual Report, 5th Edition Jerry J. Wangedt testbank and solution manual
Financial Accounting Tools for Business Decision Making, International Student Version, 5th Edition Paul D. Kimmel Testbank and solution manual
Financial Economics Frank J. Fabozzi, CFA, Edwin H. Neave, Guofu Zhou Testbank And Solutions Manual
Financial and Economic Analysis for Engineering and Technology Management, 2nd Edition Henry E. riggs testbank and solution manual
Financial Information Analysis, 2nd Edition Philip O'Regan Testbank And Solutions Manual
Financial Institutions, Markets, and Money, 10th Edition David S. Kidwell, David W. Blackwell, David A. Whidbee, Richard L. Peterson Testbank And Solutions Manual
Finite Mathematics An Applied Approach, Textbook and Student Solutions Manual, 10th Edition Micheal Sullivan testbank and solution manual
Finite Mathematics: An Applied Approach, International Student Version, 11th Edition Michael Sullivan Testbank And Solutions Manual
Financial Reporting and Analysis John Dunn Testbank And Solutions Manual
Financial Reporting and Corporate Governance Thomas A. Lee Testbank And Solutions Manual
Financial Reporting for Managers: A Value-Creation Perspective, 1st Edition Jamie Pratt, D. Eric Hirst Testbank And Solutions Manual
Financial Reporting under IFRS: A Topic Based Approach Wolfgang Dick, Franck Missonier-Piera Testbank And Solutions Manual
Financial Statement Analysis A Practitioner's Guide, 4th Edition Martin S. Fridson Testbank and Solution Manual
Fixed Income Securities: Tools for Today's Markets, 3rd Edition, University Edition Bruce Tuckman, Angel Serrat Testbank And Solutions Manual
Fixed Income Securities: Valuation, Risk, and Risk Management Pietro Veronesi Testbank And Solutions Manual
Flash Professional CS5 Digital Classroom, (Book and Video Training) Fred Gerantabee
Fluid Mechanics, 7th Edition SI Version Bruce R. Munson, Theodore H. Okiishi, Wade W. Huebsch, Alric P. Rothmayer Testbank And Solutions Manual
Fluid Mechanics, International Student Version, 8th Edition Robert W. Fox, Alan T. McDonald, Philip J. Pritchard Testbank And Solutions Manual
Food and Beverage Cost Control, 5th Edition Lea R. Dopson Testbank And Solutions Manual
Foodservice Management Fundamentals Dennis R. Reynolds Testbank And Solutions Manual
Forensic Accounting and Fraud Examination, 1st Edition Mary-Jo Kranacher, Richard Riley, Joseph T. Wells Testbank And Solutions Manual
Foundations and Change in Early Childhood Education, 1st Edition Martha T. Dever, Renee C. Falconer Testbank And Solutions Manual
Foundations of College Chemistry, 14th Edition Morris Hein, Susan Arena Testbank And Solutions Manual
Foundations of Multinational Financial Management, 6th Edition Alan C. Shapiro, Atulya Sarin Testbank And Solutions Manual
Foundations of Net-Enhanced Organizations Detmar Straub Testbank And Solutions Manual
Foundations of Strategy Robert M. Grant, Judith J. Jordan Testbank And Solutions Manual
Fundamental Finite Element Analysis and Applications with Mathematica and Matlab Computations M. Asghar Bhatti Testbank and solution manual
Fundamental Molecular Biology, 2nd Edition Lizabeth A. Allison Testbank And Solutions Manual
Fundamentals of Biochemistry: Life at the Molecular Level, Biochemical Interactions CD Version 2.0 , 2nd Edition Donald Voet
Fundamentals of Business Data Communications, 11th Edition International Student Version Alan Dennis, Alexandra Durcikova Testbank And Solutions Manual
Fundamentals of Business Data Communications, International Student Version, 10th Edition Jerry FitzGerald, Alan Dennis Testbank And Solutions Manual
Fundamentals of Corporate Finance, 2nd Edition Robert Parrino, David S. Kidwell, Thomas Bates Testbank And Solutions Manual
Fundamentals of Database Management Systems, 2nd Edition Mark L. Gillenson Testbank And Solutions Manual
Fundamentals of Fluid Mechanics, SI Version, 6th Edition Bruce R. Munsen Testbank and Solution Manual
Fundamentals of Ground Water Franklin W. Schwartz, Hubao Zhang Testbank And Solutions Manual
Fundamentals of Materials Science and Engineering: An Integrated Approach, International Student Version, 3rd Edition
Fundamentals of Materials Science and Engineering, 4th Edition SI Version William D. Callister, David G. Rethwisch Testbank And Solutions Manual
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Solution for ”Quantum Computation and Quantum Information: 10th Anniversary Edition” by Nielsen and Chuang goropikari August 11, 2018
Copylight Notice: c bna This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Contents 0.1
Eratta list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Introduction to quantum mechanics
2 3
8 Quantum noise and quantum operations
37
9 Distance measures for quantum information
39
1
2
CONTENTS
0.1
Eratta list
∑ ∑ • p.101. eq (2.150) ρ = m p(m)ρm should be ρ′ = m p(m)ρm . ∑ ∑ • p.408. eq (9.49) i pi D(ρi , σi ) + D(pi , qi ) should be i pi D(ρi , σi ) + 2D(pi , qi ).
eqn (9.48) =
∑
pi Tr(P (ρi − σi )) +
i
≤
∑
∑
pi Tr(P (ρi − σi )) +
∑
i
=
∑
=
(pi − qi ) (∵ Tr(P σi ) ≤ 1)
i
∑
pi Tr(P (ρi − σi )) + 2
i
∑
(pi − qi ) Tr(P σi )
i
i (pi
− qi )
2
pi Tr(P (ρi − σi )) + 2D(pi , qi )
i
• p.409. Exercise 9.12. If ρ = σ, then D(ρ, σ) = 0. Furthermore trace distance is nonnegative. Therefore 0 ≤ D(E(ρ), E(σ)) ≤ 0 ⇒ D(E(ρ), E(σ)) = 0. So I think the map E is not strictly contractive. If p ̸= 1 and ρ ̸= σ, then D(E(ρ), E(σ)) < D(ρ, σ) is satisfied. • p.411. Exercise 9.16. eqn(9.73) Tr(A† B) = ⟨m|A ⊗ B|m⟩ should be Tr(AT B) = ⟨m|A ⊗ B|m⟩. ] ] [ [ 1 0 i 0 , In this case, . B= Simple counter example is the case that A = 0 0 0 0 ] ] [ ][ [ −i 0 −i 0 1 0 , = AB= 0 0 0 0 0 0 †
Tr(A† B) = −i,  i 0 A⊗B = 0 0
0 0 0 0
0 0 0 0
 0 0  0 0
⟨m|A ⊗ B|m⟩ = (⟨00| + ⟨11⟩)(A ⊗ B)(|00⟩ + |11⟩) = i. Thus Tr(A† B) ̸= ⟨m|A ⊗ B|m⟩. By using following relation, we can prove. (I ⊗ A) |m⟩ = (AT ⊗ I) |m⟩ Tr(A) = ⟨m|I ⊗ A|m⟩ Tr(AT B) = Tr(BAT ) = ⟨m|I ⊗ BAT |m⟩ = ⟨m|(I ⊗ B)(I ⊗ AT )|m⟩ = ⟨m|(I ⊗ B)(A ⊗ I)|m⟩ = ⟨m|A ⊗ B|m⟩ .
Chapter 2
Introduction to quantum mechanics 2.1
[
] [ ] [ ] [ ] 1 1 2 0 + − = −1 2 1 0
2.2
A |0⟩ = A11 |0⟩ + A21 |1⟩ = |1⟩ ⇒ A11 = 0, A21 = 1 A |1⟩ = A12 |0⟩ + A22 |1⟩ = |0⟩ ⇒ A12 = 1, A22 = 0 ] [ 0 1 ∴A= 1 0
input: {|0⟩ , |1⟩}, output: {|1⟩ , |0⟩} A |0⟩ = A11 |1⟩ + A21 |0⟩ = |1⟩ ⇒ A11 = 1, A21 = 0 A |1⟩ = A12 |1⟩ + A22 |0⟩ = |0⟩ ⇒ A12 = 0, A22 = 1 [ ] 1 0 A= 0 1 2.3 From eq (2.12) A |vi ⟩ =
∑
Aji |wj ⟩
j
B |wj ⟩ =
∑ k
3
Bkj |xk ⟩
4
CHAPTER 2. INTRODUCTION TO QUANTUM MECHANICS
Thus   ∑ BA |vi ⟩ = B  Aji |wj ⟩ =
∑
j
Aji B |wj ⟩
j
=
∑ j,k
=
∑
Aji Bkj |xk ⟩  
∑
 Bkj Aji  |xk ⟩
k
j
k
∑
∑ = (BA)ki |xk ⟩ ∴(BA)ki =
Bkj Aji
j
2.4
I |vj ⟩ =
∑
Iij |vi ⟩ = |vj ⟩ , ∀j.
i
⇒ Iij = δij
2.5 Defined inner product on C n is ((y1 , · · · , yn ), (z1 , · · · , zn )) =
∑
yi∗ zi .
i
Verify (1) of eq (2.13). ( (y1 , · · · , yn ),
∑
) λi (zi1 , · · · , zin )
=
i
∑
 yi∗ 
i
=
∑
=
j
=
∑
 λj zji 
j
yi∗ λj zji
i,j
∑
∑
λj
( ∑
) yi∗ zji
i
λj ((y1 , · · · , yn ), (zj1 , · · · , zjn ))
j
=
∑ i
λi ((y1 , · · · , yn ), (zi1 , · · · , zin )) .
5 Verify (2) of eq (2.13), ( ((y1 , · · · , yn ), (z1 , · · · , zn ))∗ = ( = ( =
∑
)∗ yi∗ zi
i
∑
) yi zi∗
i
∑
(2.1)
(2.2) )
zi∗ yi
(2.3)
i
= ((z1 , · · · , zn ), (y1 , · · · , yn ))
(2.4)
. Verify (3) of eq (2.13), ((y1 , · · · , yn ), (y1 , · · · , yn )) =
∑
yi∗ yi
i
=
∑
|yi |2
i
∑ Since |yi |2 ≥ 0 for all i. Thus i |yi |2 = ((y1 , · · · , yn ), (y1 , · · · , yn )) ≥ 0. From now on, I will show the following statement, ((y1 , · · · , yn ), (y1 , · · · , yn )) = 0 iff (y1 , · · · , yn ) = 0. (⇐) This is obvious. ∑ (⇒) Suppose ((y1 , · · · , yn ), (y1 , · · · , yn )) = 0. Then i |yi |2 = 0. ∑ Since |yi |2 ≥ 0 for all i, if i |yi |2 = 0, then |yi |2 = 0 for all i. Therefore |yi |2 = 0 ⇔ yi = 0 for all i. Thus, (y1 , · · · , yn ) = 0.
2.6 ( ∑
) λi |wi ⟩ , |v⟩
( |v⟩ ,
=
i
[ =
∑
)∗ λi |wi ⟩
i
∑
λi (|v⟩ , |wi ⟩)
i
=
∑
λ∗i (|v⟩ , |wi ⟩)∗
i
=
∑ i
2.7
]∗
λ∗i (|wi ⟩ , |v⟩)
(∵ linearlity in the 2nd arg.)
6
CHAPTER 2. INTRODUCTION TO QUANTUM MECHANICS
] 1 =1−1=0 −1 [ ] |w⟩ |w⟩ 1 1 √ √ = = ∥|w⟩∥ 2 1 ⟨w|w⟩ [ ] |v⟩ |v⟩ 1 1 =√ =√ −1 ∥|v⟩∥ 2 ⟨v|v⟩ [ ] ⟨w|v⟩ = 1 1
[
2.8 If k = 1, |w2 ⟩ − ⟨v1 |w2 ⟩ |v1 ⟩ ∥|w2 ⟩ − ⟨v1 |w2 ⟩ |v1 ⟩∥ ( ) |w2 ⟩ − ⟨v1 |w2 ⟩ |v1 ⟩ ⟨v1 |v2 ⟩ = ⟨v1 | ∥|w2 ⟩ − ⟨v1 |w2 ⟩ |v1 ⟩∥ ⟨v1 |w2 ⟩ − ⟨v1 |w2 ⟩ ⟨v1 |v1 ⟩ = ∥|w2 ⟩ − ⟨v1 |w2 ⟩ |v1 ⟩∥ = 0. |v2 ⟩ =
Suppose {v1 , · · · vn } (n ≤ d − 1) is a orthonormal basis. Then ∑ ) |wn+1 ⟩ − ni=1 ⟨vi |wn+1 ⟩ |vi ⟩ ∑ (j ≤ n) ⟨vj |vn+1 ⟩ = ⟨vj | ∥|wn+1 ⟩ − ni=1 ⟨vi |wn+1 ⟩ |vi ⟩∥ ∑n ⟨vj |wn+1 ⟩ − i=1 ⟨vi |wn+1 ⟩ ⟨vj |vi ⟩ ∑ = ∥|wn+1 ⟩ − ni=1 ⟨vi |wn+1 ⟩ |vi ⟩∥ ∑ ⟨vj |wn+1 ⟩ − ni=1 ⟨vi |wn+1 ⟩ δij ∑ = ∥|wn+1 ⟩ − ni=1 ⟨vi |wn+1 ⟩ |vi ⟩∥ ⟨vj |wn+1 ⟩ − ⟨vj |wn+1 ⟩ ∑ = ∥|wn+1 ⟩ − ni=1 ⟨vi |wn+1 ⟩ |vi ⟩∥ (
= 0. Thus Gram-Schmidt procedure produces an orthonormal basis. 2.9
σ0 = I = |0⟩ ⟨0| + |1⟩ ⟨1| σ1 = X = |0⟩ ⟨1| + |1⟩ ⟨0| σ2 = Y = −i |0⟩ ⟨1| + i |1⟩ ⟨0| σ3 = Z = |0⟩ ⟨0| − |1⟩ ⟨1|
2.10
7
|vj ⟩ ⟨vk | = IV |vj ⟩ ⟨vk | IV ) ( ) ( ∑ ∑ |vq ⟩ ⟨vq | = |vp ⟩ ⟨vp | |vj ⟩ ⟨vk | p
=
∑
q
|vp ⟩ ⟨vp |vj ⟩ ⟨vk |vq ⟩ ⟨vq |
p,q
=
∑
δpj δkq |vp ⟩ ⟨vq |
p,q
Thus (|vj ⟩ ⟨vk |)pq = δpj δkq 2.11 [
] ([ ]) 0 1 −λ 1 X= , det(X − λI) = det = 0 ⇒ λ = ±1 1 0 1 −λ If λ = −1,
[
Thus
1 1 1 1
][ ] [ ] c1 0 = c2 0
[ ] [ ] 1 −1 c1 |λ = −1⟩ = =√ c2 2 1
If λ = 1
[ ] 1 1 |λ = 1⟩ = √ 2 1 [
−1 0 X= 0 1
] w.r.t. {|λ = −1⟩ , |λ = 1⟩}
2.12 ([ ] ) 1 0 − λI = (1 − λ)2 = 0 ⇒ λ = 1 det 1 1 Therefore the eigenvector associated with eigenvalue λ = 1 is [ ] 0 |λ = 1⟩ = 1 [ ] 0 0 Because |λ = 1⟩ ⟨λ = 1| = , 0 1 [ ] [ ] 1 0 0 0 ̸= c |λ = 1⟩ ⟨λ = 1| = 1 1 0 c
8
CHAPTER 2. INTRODUCTION TO QUANTUM MECHANICS
2.13 Suppose |ψ⟩ , |ϕ⟩ are arbitrary vectors in V . ( )∗ (|ψ⟩ , (|w⟩ ⟨v|) |ϕ⟩)∗ = (|w⟩ ⟨v|)† |ψ⟩ , |ϕ⟩ ( ) = |ϕ⟩ , (|w⟩ ⟨v|)† |ψ⟩ = ⟨ϕ| (|w⟩ ⟨v|)† |ψ⟩ . On the other hand, (|ψ⟩ , (|w⟩ ⟨v|) |ϕ⟩)∗ = (⟨ψ|w⟩ ⟨v|ϕ⟩)∗ = ⟨ϕ|v⟩ ⟨w|ψ⟩ . Thus ⟨ϕ| (|w⟩ ⟨v|)† |ψ⟩ = ⟨ϕ|v⟩ ⟨w|ψ⟩ for arbitrary vectors |ψ⟩ , |ϕ⟩ ∴ (|w⟩ ⟨v|)† = |v⟩ ⟨w|
2.14
((ai Ai )† |ϕ⟩ , |ψ⟩) = (|ϕ⟩ , ai Ai |ψ⟩) = ai (|ϕ⟩ , Ai |ψ⟩) = ai (A†i |ϕ⟩ , |ψ⟩) = (a∗i A†i |ϕ⟩ , |ψ⟩) ∴ (ai Ai )† = a∗i A†i
2.15
((A† )† |ψ⟩ , |ϕ⟩) = (|ψ⟩ , A† |ϕ⟩) = (A† |ϕ⟩ , |ψ⟩)∗ = (|ϕ⟩ , A |ψ⟩)∗ = (A |ψ⟩ , |ϕ⟩) ∴ (A† )† = A
2.16
9
P =
∑
|i⟩ ⟨i| .
i
P2 =
( ∑
) |i⟩ ⟨i| 
i
=
∑
∑
 |j⟩ ⟨j|
j
|i⟩ ⟨i|j⟩ ⟨j|
i,j
=
∑
|i⟩ ⟨j| δij
i,j
=
∑
|i⟩ ⟨i|
i
=P
2.17 Proof. (⇒) Suppose A is Hermitian. Then A = A† . Let |λ⟩ be eigenvectors of A with eigenvalues λ, that is, A |⟩ = λ |λ⟩ . Therefore ⟨λ|A|λ⟩ = λ ⟨λ|λ⟩ = λ. On the other hand, λ∗ = ⟨λ|A|λ⟩∗ = ⟨λ|A† |λ⟩ = ⟨λ|A|λ⟩ = λ ⟨λ|λ⟩ = λ. Hence eigenvalues of Hermitian matrix are real. (⇐) Suppose eigenvalues of A are real. From spectral theorem, normal matrix A can be written by ∑ A= (2.5) λi |λi ⟩⟨λi | i
where λi are real eigenvalues with eigenvectors |λi ⟩. By taking adjoint, we get ∑ A† = λ∗i |λi ⟩⟨λi | i
=
∑ i
=A Thus A is Hermitian. 2.18
λi |λi ⟩⟨λi |
(∵ λi are real)
10
CHAPTER 2. INTRODUCTION TO QUANTUM MECHANICS Suppose |v⟩ is a eigenvector with corresponding eigenvalue λ. U |v⟩ = λ |v⟩ . 1 = ⟨v|v⟩ = ⟨v| I |v⟩ = ⟨v| U † U |v⟩ = λλ∗ ⟨v|v⟩ = ∥λ∥2 ∴ λ = eiθ
2.19 [ ][ ] [ ] 0 1 0 1 1 0 X = = =I 1 0 1 0 0 1 2
2.20 U≡
∑
|wi ⟩ ⟨vi |
i ′
Aij = ⟨vi |A|vj ⟩ = ⟨vi |U U † AU U † |vj ⟩ ∑ = ⟨vi |wp ⟩ ⟨vp |vq ⟩ ⟨wq |A|wr ⟩ ⟨vr |vs ⟩ ⟨ws |vj ⟩ p,q,r,s
=
∑
′′
⟨vi |wp ⟩ δpq Aqr δrs ⟨ws |vj ⟩
p,q,r,s
=
∑
′′
⟨vi |wp ⟩ ⟨wr |vj ⟩ Apr
p,r
2.21 Suppose M be Hermitian. Then M = M † . M = IM I = (P + Q)M (P + Q) = P M P + QM P + P M Q + QM Q
Now P M P = λP , QM P = 0, P M Q = P M † Q = (QM P )∗ = 0. Thus M = P M P + QM Q. Next prove QM Q is normal. QM Q(QM Q)† = QM QQM † Q = QM † QQM Q (M = M † ) = (QM † Q)QM Q Therefore QM Q is normal. By induction, QM Q is diagonal ... (following is same as Box 2.2)
11
2.22 Suppose A is a Hermitian operator and |vi ⟩ are eigenvectors of A with eigenvalues λi . Then ⟨vi |A|vj ⟩ = λj ⟨vi |vj ⟩ . On the other hand, ⟨vi |A|vj ⟩ = ⟨vi |A† |vj ⟩ = ⟨vj |A|vi ⟩∗ = λ∗i ⟨vj |vi ⟩∗ = λ∗i ⟨vi |vj ⟩ = λi ⟨vi |vj ⟩ Thus (λi − λj ) ⟨vi |vj ⟩ = 0. If λi ̸= λj , then ⟨vi |vj ⟩ = 0. 2.23 Suppose P is projector and |λ⟩ are eigenvectors of P with eigenvalues λ. Then P 2 = P . P |λ⟩ = λ |λ⟩ and P |λ⟩ = P 2 |λ⟩ = λP |λ⟩ = λ2 |λ⟩ . Therefore λ = λ2 λ(λ − 1) = 0 λ = 0 or 1.
2.24 Def of positive ⟨v|A|v⟩ ≥ 0 for all |v⟩. Suppose A is a positive operator. A can be decomposed as follows. A=
A + A† A − A† +i 2 2i
= B + iC
where B =
A + A† A − A† , C= . 2 2i
Now operators B and C are Hermitian.
⟨v|A|v⟩ = ⟨v|B + iC|v⟩ = ⟨v|B|v⟩ + i ⟨v|C|v⟩ = α + iβ where α = ⟨v|B|v⟩ , β = ⟨v|C|v⟩ . Since B and C are Hermitian, α, β ∈ R. From def of positive operator, β should be vanished because ⟨v|A|v⟩ is real. Hence β = ⟨v|C|v⟩ = 0 for all |v⟩, i.e. C = 0. Therefore A = A† .
12
CHAPTER 2. INTRODUCTION TO QUANTUM MECHANICS
0013
0010
Reference: MIT 8.05 Lecture note by Prof. Barton Zwiebach. https://ocw.mit.edu/courses/physics/8-05-quantum-physics-ii-fall-2013/ lecture-notes/MIT8_05F13_Chap_03.pdf Proposition. 2.0.1. Let T be a linear operator in a complex vector space V . If (u, T v) = 0 for all u, v ∈ V , then T = 0. Proof. Suppose u = T v. Then (T v, T v) = 0 for all v implies that T v = 0 for all v. Therefore T = 0. Theorem. 2.0.1. If (v, Av) = 0 for all v ∈ V , then A = 0. Proof. First, we show that (u, T v) = 0 if (v, Av) = 0. Then apply proposition 2.0.1 Suppose u, v ∈ V . Then (u, T v) is decomposed as [ 1 1 (u, T v) = (u + v, T (u + v)) − (u − v, T (u − v)) + (u + iv, T (u + iv)) 4 i ] 1 − (u − iv, T (u − iv)) . i If (v, T v) = 0 for all v ∈ V , the right hand side of above eqn vanishes. Thus (u, T v) = 0 for all u, v ∈ V . Then T = 0.
0012
2.25
⟨ψ|A† A|ψ⟩ = ∥A |ψ⟩∥2 ≥ 0 for all |ψ⟩ .
Thus A† A is positive.
2.26
1 1 |ψ⟩⊗2 = √ (|0⟩ + |1⟩) ⊗ √ (|0⟩ + |1⟩ 2 2 1 = (|00⟩ + |01⟩ + |10⟩ + |11⟩) 2  1 1 1  =  2 1 1
0011
13
1 1 1 |ψ⟩⊗3 = √ (|0⟩ + |1⟩) ⊗ √ (|0⟩ + |1⟩ ⊗ √ (|0⟩ + |1⟩ 2 2 2 1 = √ (|000⟩ + |001⟩ + |010⟩ + |011⟩ + |100⟩ + |101⟩ + |110⟩ + |111⟩) 2 2   1 1   1    1  1 √ =  2 2 1  1   1 1 2.27 [
] [ ] 0 1 1 0 ⊗ 1 0 0 −1   0 0 1 0 0 0 0 −1  = 1 0 0 0  0 −1 0 0
X ⊗Z =
[ 1 I ⊗X = 0  0 1 = 0 0
] ] [ 0 1 0 ⊗ 1 0 1  1 0 0 0 0 0  0 0 1 0 1 0
[ 0 X ⊗I = 1  0 0 = 1 0
] ] [ 1 0 1 ⊗ 0 1 0  0 1 0 0 0 1  0 0 0 1 0 0
In general, tensor product is not commutable. 2.28 
∗ A11 B · · · A1n B  ..  .. (A ⊗ B)∗ =  ... . .  Am1 B · · · Amn B  ∗ ∗  A11 B · · · A∗1n B ∗  ..  .. =  ... . .  A∗m1 B ∗ · · · A∗mn B ∗ = A∗ ⊗ B ∗ .
14
CHAPTER 2. INTRODUCTION TO QUANTUM MECHANICS

T A11 B · · · A1n B  ..  .. (A ⊗ B)T =  ... . .  Am1 B · · · Amn B   A11 B T · · · Am1 B T   .. .. =  ...  . . T T A1n B · · · Amn B   T A11 B · · · A1m B T   .. .. =  ...  . . · · · Anm B T
An1 B T = AT ⊗ B T .
(A ⊗ B)† = ((A ⊗ B)∗ )T = (A∗ ⊗ B ∗ )T = (A∗ )T ⊗ (B ∗ )T = A† ⊗ B † . 2.29 Suppose U1 and U2 are unitary operators. Then (U1 ⊗ U2 )(U1 ⊗ U2 )† = U1 U1† ⊗ U2 U2† = I ⊗ I. Similarly, (U1 ⊗ U2 )† (U1 ⊗ U2 ) = I ⊗ I. 2.30 Suppose A and B are Hermitian operators. Then (A ⊗ B)† = A† ⊗ B † = A ⊗ B.
(2.6)
Thus A ⊗ B is Hermitian. 2.31 Suppose A and B are positive operators. Then ⟨ψ| ⊗ ⟨ϕ| (A ⊗ B) |ψ⟩ ⊗ |ϕ⟩ = ⟨ψ|A|ψ⟩ ⟨ϕ|B|ϕ⟩ . Since A and B are positive operators, ⟨ψ|A|ψ⟩ ≥ 0 and ⟨ϕ|B|ϕ⟩ ≥ 0 for all |ψ⟩ , |ϕ⟩. Then ⟨ψ|A|ψ⟩ ⟨ϕ|B|ϕ⟩ ≥ 0. Thus A ⊗ B is positive if A and B are positive. 2.32
15 Suppose P1 and P2 are projectors. Then (P1 ⊗ P2 )2 = P12 ⊗ P22 = P1 ⊗ P2 . Thus P1 ⊗ P2 . is also projector. 2.33 [ ] 1 1 1 H=√ 2 1 −1
H ⊗2
(2.7)
  1 1 1 1 [ ] [ ] 1 1 1 1 1 −1 1 −1 1 1 1  =√ ⊗√ =  2 1 1 −1 −1 2 1 −1 2 1 −1 1 −1 −1 1
2.34 ] 4 3 . Suppose A = 3 4 [
det(A − λI) = (4 − λ)2 − 32 = λ2 − 8λ + 7 = (λ − 1)(λ − 7) ] 1 , |λ = 7⟩ = −1
[ Eigenvalues of A are λ = 1, 7. Corresponding eigenvectors are |λ = 1⟩ = [ ] 1 √1 . 2 1 Thus A = |λ = 1⟩⟨λ = 1| + 7 |λ = 7⟩⟨λ = 7| . √
√ A = |λ = 1⟩⟨λ = 1| + 7 |λ = 7⟩⟨λ = 7| [ ] √ [ ] 1 1 −1 7 1 1 = + 2 −1 1 2 1 1 √ √ ] [ 1 1 + 7 −1 + 7 √ √ = 2 −1 + 7 1 + 7
log(A) = log(1) |λ = 1⟩⟨λ = 1| + log(7) |λ = 7⟩⟨λ = 7| [ ] log(7) 1 1 = 1 1 2
√1 2
16
CHAPTER 2. INTRODUCTION TO QUANTUM MECHANICS
2.35
⃗v · ⃗σ =
3 ∑
vi σi
i=1
[
] [ ] [ ] 0 1 0 −i 1 0 = v1 + v2 + v3 1 0 i 0 0 −1 [ ] v3 v1 − iv2 = v1 + iv2 −v3
det(⃗v · ⃗σ − λI) = (v3 − λ)(−v3 − λ) − (v1 − iv2 )(v1 + iv2 ) = λ2 − (v12 + v22 + v32 ) = λ2 − 1 (∵ |⃗v | = 1) Eigenvalues are λ = ±1. Let |λ±1 ⟩ be eigenvectors with eigenvalues ±1. Since ⃗v · ⃗σ is Hermitian, ⃗v · ⃗σ is diagonalizable. Then ⃗v · ⃗σ = |λ1 ⟩⟨λ1 | − |λ−1 ⟩⟨λ−1 | Thus exp (iθ⃗v · ⃗σ ) = eiθ |λ1 ⟩⟨λ1 | + e−iθ |λ−1 ⟩⟨λ−1 | = (cos θ + i sin θ) |λ1 ⟩⟨λ1 | + (cos θ − i sin θ) |λ−1 ⟩⟨λ−1 | = cos θ(|λ1 ⟩⟨λ1 | + |λ−1 ⟩⟨λ−1 |) + i sin θ(|λ1 ⟩⟨λ1 | − |λ−1 ⟩⟨λ−1 |) = cos(θ)I + i sin(θ)⃗v · ⃗σ . ∵ Since ⃗v · ⃗σ is Hermitian, |λ1 ⟩ and |λ−1 ⟩ are orthogonal. Thus |λ1 ⟩⟨λ1 | + |λ−1 ⟩⟨λ−1 | = I.
2.36 ([ 0 Tr(σ1 ) = Tr 1 ([ 0 Tr(σ2 ) = Tr i ([ 1 Tr(σ3 ) = Tr 0
2.37
1 0
])
=0 ]) −i =0 0 ]) 0 =1−1=0 −1
17
Tr(AB) =
∑
⟨i|AB|i⟩
i
=
∑
⟨i|AIB|i⟩
i
=
∑
⟨i|A|j⟩ ⟨j|B|i⟩
i,j
=
∑
⟨j|B|i⟩ ⟨i|A|j⟩
i,j
=
∑
⟨j|BA|j⟩
j
= Tr(BA) 2.38 Tr(A + B) =
∑
⟨i|A + B|i⟩
i
=
∑ (⟨i|A|i⟩ + ⟨i|B|i⟩) i
=
∑
⟨i|A|i⟩ +
i
∑
⟨i|B|i⟩
i
= Tr(A) + Tr(B).
Tr(zA) =
∑
⟨i|zA|i⟩
i
=
∑
z ⟨i|A|i⟩
i
=z
∑
⟨i|A|i⟩
i
= z Tr(A). 2.39 (1) (A, B) ≡ Tr(A† B). (i) ( A,
∑
) λi Bi
[
(
= Tr A†
i
∑
)] λ i Bi
i
= Tr(A† λ1 B1 ) + · · · + Tr(A† λn Bn ) †
†
= λ1 Tr(A B1 ) + · · · + λn Tr(A Bn ) ∑ = λi Tr(A† Bi ) i
(∵ Execise 2.38)
18
CHAPTER 2. INTRODUCTION TO QUANTUM MECHANICS (ii) ( )∗ (A, B)∗ = Tr(A† B)  ∗ ∑ = ⟨i|A† |j⟩ ⟨j|B|i⟩ i,j
=
∑
∗
⟨i|A† |j⟩ ⟨j|B|i⟩∗
i,j
=
∑
∗
⟨j|B|i⟩∗ ⟨i|A† |j⟩
i,j
=
∑
⟨i|B † |j⟩ ⟨j|A|i⟩
i,j
=
∑
⟨i|B † A|i⟩
i
= Tr(B † A) = (B, A). (iii) (A, A) = Tr(A† A) ∑ = ⟨i|A† A|i⟩ i
Since A† A is positive, ⟨i|A† A|i⟩ ≥ 0 for all |i⟩. Let ai be i-th column of A. If ⟨i|A† A|i⟩ = 0, then ⟨i|A† A|i⟩ = a†i ai = ∥ai ∥2 = 0 iff ai = 0. Therefore (A, A) = 0 iff A = 0. (2) (3) 2.40
[X, Y ] = XY [ 0 = 1 [ i = 0 [ 2i = 0 = 2iZ
−YX ][ ] [ ][ ] 1 0 −i 0 −i 0 1 − 0 i 0 i 0 1 0 ] [ ] 0 −i 0 − −i 0 i ] 0 −2i
19
] [ ][ ] [ ][ 1 0 0 −i 0 −i 1 0 − [Y, Z] = i 0 0 −1 0 −1 i 0 [ ] 0 2i = 2i 0 = 2iX
[ ][ ] [ ][ ] 1 0 0 1 0 1 1 0 [Z, X] = − 0 −1 1 0 1 0 0 −1 [ ] 0 −i = 2i i 0 = 2iY
2.41 {σ1 , σ2 } = σ1 σ2 + σ2 σ1 ] ][ ] [ ][ [ 0 −i 0 1 0 1 0 −i + = 1 0 i 0 1 0 i 0 ] ] [ [ −i 0 i 0 + = 0 i 0 −i =0
] ][ ] [ ][ [ 0 −i 1 0 0 −i 1 0 + {σ2 , σ3 } = 0 −1 i 0 0 −1 i 0 =0
] ][ ] [ ][ 0 1 1 0 0 1 1 0 {σ3 , σ1 } = + 1 0 0 −1 0 −1 1 0 [
=0
σ02 = I 2 = I [ ]2 0 1 2 σ1 = =I 1 0 [ ]2 0 −i 2 σ2 = =I i 0 [ ]2 1 0 2 σ3 = =I 0 −1
20
CHAPTER 2. INTRODUCTION TO QUANTUM MECHANICS
2.42 [A, B] + {A, B} AB − BA + AB + BA = = AB 2 2 2.43 From eq (2.75) and eq (2.76), {σj , σk } = 2δjk I. From eq (2.77), [σj , σk ] + {σj , σk } 2 ∑3 2i l=1 ϵjkl σl + 2δjk I = 2 3 ∑ = δjk I + i ϵjkl σl
σj σk =
l=1
2.44 By assumption, [A, B] = 0 and {A, B} = 0, then AB = 0. Since A is invertible, multiply by from left, then
A−1
A−1 AB = 0 IB = 0 B = 0. 2.45 [A, B]† = (AB − BA)† = B † A† − A† B † ] [ = B † , A† 2.46 [A, B] = AB − BA = −(BA − AB) = − [B, A] 2.47 (i [A, B])† = −i [A, B]† [ ] = −i B † , A† = −i [B, A] = i [A, B]
21
2.48 (Positive ) ∑ Since P is positive, it is diagonalizable. Then P = i λi |i⟩⟨i|, (λi ≥ 0). √ ∑ ∑√ √ √ λi |i⟩⟨i| = P. J = P †P = P P = P 2 = λ2i |i⟩⟨i| = i
i
Therefore polar decomposition of P is P = U P for all P . Thus U = I, then P = P . (Unitary) Suppose unitary U is decomposed by U = W J where W is unitary and J is positive, √ † J = U U. √ √ J = U †U = I = I Since unitary operators are invertible, W = U J −1 = U I −1 = U I = U . Thus polar decomposition of U is U = U . (Hermitian) Suppose H = U J. J=
√
H †H =
√
HH =
√ H 2.
√ Thus H = U H 2 . 0013
0010
√
In general, H ̸= H 2 . ∑ From spectral decomposition, H = i λi |i⟩⟨i|, λi ∈ R. √∑ ∑ ∑√ √ H2 = λ2i |i⟩⟨i| = |λi | |i⟩⟨i| ̸= H λ2i |i⟩⟨i| = i
i
i
0012
0011
2.49 ∑ Normal matrix is diagonalizable, A = i λi |i⟩⟨i|. √ ∑ J = A† A = |λi | |i⟩⟨i| . U=
∑
i
|ei ⟩⟨i|
i
A = UJ =
∑
|λi | |ei ⟩⟨i| .
i
2.50 [
] [ ] 1 0 2 1 † Define A = . A A= . 1 1 1 1 Characteristic equation of A† A is det(A† A − λI) = λ2 − 3λ + 1 = 0.] Eigenvalues of A† A are [ √ 2√ . λ± = 3±2 5 and associated eigenvectors are |λ± ⟩ = √ 1 √ 10∓2 5 −1 ± 5
22
CHAPTER 2. INTRODUCTION TO QUANTUM MECHANICS
A† A = λ+ |λ+ ⟩⟨λ+ | + λ− |λ− ⟩⟨λ− | .
J=
√
√ √ λ+ |λ+ ⟩⟨λ+ | + λ− |λ− ⟩⟨λ− | √ √ √ √ [ √ √ [ √ √ ] ] 3+ 5 5− 5 3− 5 5+ 5 4 2 5 − 2 4 −2 5 − 2 √ √ + √ √ = · · 2 5−2 6−2 5 −2 5 − 2 6 + 2 5 2 40 2 40
A† A =
1 1 |λ+ ⟩⟨λ+ | + √ |λ− ⟩⟨λ− | . J −1 = √ λ+ λ− U = AJ −1 I’m tired. 2.51
†
H H=
(
])† ] ] ] ] [ [ [ [ [ 1 1 1 1 1 1 1 1 1 1 2 0 1 1 1 √ √ √ =√ = = I. 2 0 2 2 1 −1 2 1 −1 2 1 −1 2 1 −1
2.52
†
H =
(
])† ] [ [ 1 1 1 1 1 1 √ =√ = H. 2 1 −1 2 1 −1
Thus H 2 = I.
2.53 (
)( ) 1 1 1 det (H − λI) = √ − λ −√ − λ − 2 2 2 1 1 = λ2 − − 2 2 2 =λ −1 ] 1√ . −1 ± 2
[ Eigenvalues are λ± = ±1 and associated eigenvectors are |λ± ⟩ = √ 2.54
1 √ 4∓2 2
23 Since [A, B] = 0, A and B are simultaneously diagonalize, A = ( exp(A) exp(B) =
∑
)( exp(ai ) |i⟩⟨i|
i
=
∑
∑
∑
i ai |i⟩⟨i|,
B=
∑
i bi |i⟩⟨i|.
) exp(bi ) |i⟩⟨i|
i
exp(ai + bj ) |i⟩ ⟨i|j⟩ ⟨j|
i,j
=
∑
exp(ai + bj ) |i⟩⟨j| δi,j
i,j
=
∑
exp(ai + bi ) |i⟩⟨i|
i
= exp(A + B)
2.55
H=
∑
E |E⟩⟨E|
E
) ( ) ( iH(t2 − t1 ) iH(t2 − t1 ) exp U (t2 − t1 )U (t2 − t1 ) = exp − ℏ ℏ ( ) )( ( ) ) ∑( iE(t2 − t1 ) iE ′ (t2 − t1 ) = exp − |E⟩⟨E| exp − |E ′ ⟩⟨E ′ | ℏ ℏ E,E ′ ( ) ) ∑( i(E − E ′ )(t2 − t1 ) = exp − |E⟩⟨E ′ | δE,E ′ ℏ E,E ′ ∑ = exp(0) |E⟩⟨E| †
E
=
∑
|E⟩⟨E|
E
=I Similarly, U † (t2 − t1 )U (t2 − t1 ) = I. 2.56 U=
∑
i λi |λi ⟩⟨λi |
log(U ) =
(|λi | = 1). ∑
log(λj ) |λj ⟩⟨λj | =
j
K = −i log(U ) =
∑
∑
iθj |λj ⟩⟨λj | where θj = arg(λj )
j
θj |λj ⟩⟨λj | .
j
 K † = (−i log U )† = 
∑ j
† θj |λj ⟩⟨λj | =
∑ j
θj∗ |λj ⟩⟨λj | =
∑ j
θj |λj ⟩⟨λj | = K
24
CHAPTER 2. INTRODUCTION TO QUANTUM MECHANICS
2.57 Ll |ψ⟩ |ϕ⟩ ≡ √ ⟨ψ|L†l Ll |ψ⟩ † ⟨ϕ|Mm Mm |ϕ⟩
=
† ⟨ψ|L†l Mm Mm Ll |ψ⟩
⟨ψ|L†l Ll |ψ⟩
√
⟨ψ|L†l Ll |ψ⟩ Mm Ll |ψ⟩ Mm Ll |ψ⟩ Nlm |ψ⟩ √ =√ ·√ =√ =√ † † † † ⟨ψ|L†l Ll |ψ⟩ ⟨ψ|L†l Mm Mm Ll |ψ⟩ ⟨ψ|L†l Mm Mm Ll |ψ⟩ ⟨ψ|Nlm Nlm |ψ⟩ ⟨ϕ|Mm Mm |ϕ⟩ Mm |ϕ⟩
2.58 ⟨M ⟩ = ⟨ψ|M |ψ⟩ = ⟨ψ|m|ψ⟩ = m ⟨ψ|ψ⟩ = m ⟨M 2 ⟩ = ⟨ψ|M 2 |ψ⟩ = ⟨ψ|m2 |ψ⟩ = m2 ⟨ψ|ψ⟩ = m2 deviation = ⟨M 2 ⟩ − ⟨M ⟩2 = m2 − m2 = 0. 2.59 ⟨X⟩ = ⟨0|X|0⟩ = ⟨0|1⟩ = 0 ⟨X 2 ⟩ = ⟨0|X 2 |0⟩ = ⟨0|X|1⟩ = ⟨0|0⟩ = 1 √ standard deviation = ⟨X 2 ⟩ − ⟨X⟩2 = 1 2.60
⃗v · ⃗σ =
3 ∑
vi σi
i=1
[
] [ ] [ ] 0 1 0 −i 1 0 = v1 + v2 + v3 1 0 i 0 0 −1 [ ] v3 v1 − iv2 = v1 + iv2 −v3 det(⃗v · ⃗σ − λI) = (v3 − λ)(−v3 − λ) − (v1 − iv2 )(v1 + iv2 ) = λ2 − (v12 + v22 + v32 ) = λ2 − 1 (∵ |⃗v | = 1) Eigenvalues are λ = ±1. (i) if λ = 1 ⃗v · ⃗σ − λI = ⃗v · ⃗σ − I [ ] v3 − 1 v1 − iv2 = v1 + iv2 −v3 − 1
25 Normalized eigenvector is |λ1 ⟩ =
√
[ 1+v3 2
1 + v3 |λ1 ⟩⟨λ1 | = 2 = = = =
1
]
1−v3 v1 −iv2
.
[
]
1 1−v3 v1 −iv2
[
1
1−v3 v1 +iv2
]
] [ v1 −iv2 1 1 + v3 1+v3 1−v3 v1 +iv2 2 1+v3 1+v3 [ ] 1 1 + v3 v1 − iv2 2 v1 + iv2 1 − v3 ( [ ]) 1 v3 v1 − iv2 I+ v1 + iv2 −v3 2 1 (I + ⃗v · ⃗σ ) 2
(ii) If λ = −1. ⃗v · ⃗σ − λI = ⃗v · ⃗σ + I [ ] v3 + 1 v1 − iv2 = v1 + iv2 −v3 + 1 [ ] √ 1 1−v3 Normalized eigenvalue is |λ−1 ⟩ = . 3 2 − v1+v 1 −iv2 [ ] ] [ 1 − v3 1 3 1 − v1+v |λ−1 ⟩⟨λ−1 | = 1+v3 +iv2 1 − 2 v −iv ] [ 1 2 1 −iv2 1 − v1−v 1 − v3 3 = 1+v3 1 +iv2 − v1−v 2 1−v3 3 ] [ 1 1 − v3 −(v1 − iv2 ) = 1 + v3 2 −(v1 + iv2 ) ( [ ]) 1 v3 v1 − iv2 = I− (v1 + iv2 −v3 2 1 = (I − ⃗v · ⃗σ ). 2 While I review my proof, I notice that my proof has a defect. The case (v1 , v2 , v3 ) = (0, 0, 1), 3 second component of eigenstate, v1−v , diverges. So I implicitly assume v1 − iv2 ̸= 0. Hence 1 −iv2 my proof is incomplete. Since the exercise doesn’t require explicit form of projector, we should prove the problem more abstractly. In order to prove, we use the following properties of ⃗v · ⃗σ • ⃗v · ⃗σ is Hermitian • (⃗v · ⃗σ )2 = I where ⃗v is a real unit vector. We can easily check above conditions. (⃗v · ⃗σ )† = (v1 σ1 + v2 σ2 + v3 σ3 )† = v1 σ1† + v2 σ2† + v3 σ3† = v1 σ1 + v2 σ2 + v3 σ3 = ⃗v · ⃗σ
(∵ Pauli matrices are Hermitian.)
26
CHAPTER 2. INTRODUCTION TO QUANTUM MECHANICS
(⃗v · ⃗σ )2 =
3 ∑
(vj σj )(vk σk )
j,k=1
=
3 ∑
vj vk σ j σ k
j,k=1
=
3 ∑
( vj vk
δjk I + i
j,k=1
=
3 ∑
3 ∑ j=1
=I
) ϵjkl σl
(∵ eqn(2.78) page78)
l=1
vj vk δjk I + i
j,k=1
=
3 ∑
3 ∑
ϵjkl vj vk σl
j,k,l=1
vj2 I  ∵
∑
 vj2 = 1
j
Proof. Suppose |λ⟩ is an eigenstate of ⃗v · ⃗σ with eigenvalue λ. Then ⃗v · ⃗σ |λ⟩ = λ |λ⟩ (⃗v · ⃗σ )2 |λ⟩ = λ2 |λ⟩ On the other hand (⃗v · ⃗σ )2 = I, (⃗v · ⃗σ )2 |λ⟩ = I |λ⟩ = |λ⟩ ∴ λ2 |λ⟩ = |λ⟩ . Thus λ2 = 1 ⇒ λ = ±1. Therefore ⃗v · ⃗σ has eigenvalues ±1. Let |λ1 ⟩ and |λ−1 ⟩ are eigenvectors with eigenvalues 1 and −1, respectively. I will prove that P± = |λ±1 ⟩⟨λ±1 |. In order to prove above equation, all we have to do is prove following condition. (see Theorem 2.0.1) 0013
0010
⟨ψ|(P± − |λ±1 ⟩⟨λ±1 |)|ψ⟩ = 0 for all |ψ⟩ ∈ C2 .
(2.8)
0012
0011
Since ⃗v ·⃗σ is Hermitian, |λ1 ⟩ and |λ−1 ⟩ are orthonormal vector (∵ Exercise 2.22). Let |ψ⟩ ∈ be an arbitrary state. |ψ⟩ can be written as |ψ⟩ = α |λ1 ⟩ + β |λ±1 ⟩ (|α|2 + |β|2 = 1, α, β ∈ C).
C2
27
⟨ψ|(P± − |λ± ⟩⟨λ± |)|ψ⟩ = ⟨ψ|P± |ψ⟩ − ⟨ψ|λ± ⟩ ⟨λ± |ψ⟩ . 1 ⟨ψ|P± |ψ⟩ = ⟨ψ| (I ± ⃗v · ⃗σ )|ψ⟩ 2 1 1 = ± ⟨ψ|⃗v · ⃗σ )|ψ⟩ 2 2 1 1 = ± (|α|2 − |β|2 ) 2 2 1 1 = ± (2|α|2 − 1) (∵ |α|2 + |β|2 = 1) 2 2 ⟨ψ|λ1 ⟩ ⟨λ1 |ψ⟩ = |α|2 ⟨ψ|λ−1 ⟩ ⟨λ−1 |ψ⟩ = |β|2 = 1 − |α|2 Therefore ⟨ψ|(P± − |λ±1 ⟩⟨λ±1 |)|ψ⟩ = 0 for all |ψ⟩ ∈ C2 . Thus P± = |λ±1 ⟩⟨λ±1 |. 2.61 ⟨λ1 |0⟩ ⟨0|λ1 ⟩ = ⟨0|λ1 ⟩ ⟨λ1 |0⟩ 1 = ⟨0| (I + ⃗v · ⃗σ )|0⟩ 2 1 = (1 + v3 ) 2 Post-measurement state is [ ] |λ1 ⟩ ⟨λ1 |0⟩ 1 1 1 + v3 √ =√ · 2 v1 + iv2 1 ⟨0|λ1 ⟩ ⟨λ1 |0⟩ (1 + v ) 3 2 √ [ ] 1 1 (1 + v3 ) v1 +iv2 = 2 1+v3 √ [ ] 1 + v3 1 = 1−v3 2 v1 −iv2 = |λ1 ⟩ .
2.62 † Suppose Mm is a measurement operator. From the assumption, Em = Mm Mm = Mm . Then
⟨ψ|Em |ψ⟩ = ⟨ψ|Mm |ψ⟩ ≥ 0. for all |ψ⟩. Since Mm is positive operator, Mm is Hermitian. Therefore, † 2 Em = Mm Mm = Mm Mm = M m = Mm .
Thus the measurement is a projective measurement.
28
CHAPTER 2. INTRODUCTION TO QUANTUM MECHANICS
2.63 √ † Em Um Um Em √ √ = Em I Em
† Mm Mm =
√
= Em . † Since Em is POVM, for arbitrary unitary U , Mm Mm is POVM.
2.64 Read following paper: Lu-Ming Duan, Guang-Can Guo. Probabilistic cloning and identification of linearly independent quantum states. Phys. Rev. Lett.,80:4999-5002, 1998. arXiv eprint quant-ph/9804064. https://arxiv.org/abs/quant-ph/9804064 2.65
|+⟩ ≡
|0⟩ + |1⟩ √ , 2
|−⟩ ≡
|0⟩ − |1⟩ √ 2
2.66 ( X1 Z2 ( ⟨X1 Z2 ⟩ =
⟨00| + ⟨11| √ 2
|00⟩ + |11⟩ √ 2
)
( X1 Z2
) =
|00⟩ + |11⟩ √ 2
|10⟩ − |01⟩ √ 2
) =
⟨00| + ⟨11| |10⟩ − |01⟩ √ √ · =0 2 2
2.67 Unsolved W ⊂ V → V = W ⊕ W ⊥. U : W → V , U′ : V → V . U ′ |w⟩ = U |w⟩ U ′ ∈ L(V ) U ∈ L(W ) U ′ = U ⊕ I ??? 2.68 √ |ψ⟩ = |00⟩+|11⟩ . 2 Suppose |a⟩ = a0 |0⟩ + a1 |1⟩ and |b⟩ = b0 |0⟩ + b1 |1⟩.
|a⟩ |b⟩ = a0 b0 |00⟩ + a0 b1 |01⟩ + a1 b0 |10⟩ + a1 b1 |11⟩ . If |ψ⟩ = |a⟩ |b⟩, then a0 b0 = 1, a0 b1 = 0, a1 b0 = 0, a1 b1 = 1 since {|ij⟩} is an orthonormal basis.
29 If a0 b1 = 0, then a0 = 0 or b1 = 0. When a0 = 0 , this is contradiction to a0 b0 = 1. When b1 = 0 , this is contradiction to a1 b1 = 1. Thus |ψ⟩ ̸= |a⟩ |b⟩. 2.69 Define Bell states as follows. |ψ1 ⟩ ≡
|00⟩ + |11⟩ √ = 2
|ψ2 ⟩ ≡
|00⟩ − |11⟩ √ = 2
|ψ3 ⟩ ≡
|01⟩ + |10⟩ √ = 2
|ψ4 ⟩ ≡
|01⟩ − |10⟩ √ = 2
  1 1  0  √   2 0 1   1 1  0  √   2 0 −1   0  1 1  √  2 1 0   0 1  1  √   2 −1 0
First, we prove {|ψi ⟩} is a linearly independent basis. a1 |ψ1 ⟩ + a2 |ψ2 ⟩ + a3 |ψ3 ⟩ + a4 |ψ4 ⟩ = 0   a1 + a2 1 a3 + a4  =0 ∴√  2 a3 − a4  a1 − a2  a1 + a2 = 0    a +a =0 3 4 ∴  a3 − a4 = 0    a1 − a2 = 0 ∴ a1 = a2 = a3 = a4 = 0 Thus {|ψi ⟩} is a linearly independent basis. Moreover ∥|ψi ⟩∥ = 1 and ⟨ψi |ψj ⟩ = δij for i, j = 1, 2, 3, 4. Therefore {|ψi ⟩} forms an orthonormal basis. 2.70 For any Bell states we get ⟨ψi |E ⊗ I|ψi ⟩ = 12 (⟨0|E|0⟩ + ⟨1|E|1⟩). Suppose Eve measures the qubit Alice sent by measurement operators Mm . The probability † † that Eve gets result m is pi (m) = ⟨ψi |Mm Mm ⊗ I|ψi ⟩. Since Mm Mm is positive, pi (m) are same values for all |ψi ⟩. Thus Eve can’t distinguish Bell states.
30
CHAPTER 2. INTRODUCTION TO QUANTUM MECHANICS
2.71 From spectral decomposition, ∑ ∑ ρ= pi |ψi ⟩⟨ψi | , pi ≥ 0, pi = 1. i
2
ρ =
∑
i
pi pj |i⟩ ⟨i|j⟩ ⟨j|
i,j
=
∑
pi pj |i⟩⟨j| δij
i,j
=
∑
p2i |i⟩⟨i|
i
( 2
Tr(ρ ) = Tr
∑ i
) p2i |i⟩⟨i|
=
∑ i
∑
p2i Tr(|i⟩⟨i|) =
∑ i
p2i ⟨i|i⟩ =
∑
p2i ≤
i
∑
pi = 1
(∵ p2i ≤ pi )
i
Suppose Tr(ρ2 ) = 1. Then i p2i = 1. If 0 ≤ pi < 1, then p2i < pi . Thus only one pi = 1 and otherwise are 0. Therefore ρ = |ψi ⟩⟨ψi | is pure state. Conversely if ρ is pure, then ρ = |ψ⟩⟨ψ|. Tr(ρ2 ) = Tr(|ψ⟩ ⟨ψ|ψ⟩ ⟨ψ|) = Tr(|ψ⟩⟨ψ|) = ⟨ψ|ψ⟩ = 1. 2.72 ] a b , a, d ∈ R and (1) Since density matrix is Hermitian, matrix representation is ρ = ∗ b d b ∈ C w.r.t. standard basis. Because ρ is density matrix, Tr(ρ) = a + d = 1. Define a = (1 + r3 )/2, d = (1 − r3 )/2 and b = (r1 − ir2 )/2, (ri ∈ R). In this case, ] [ ] [ 1 1 + r3 r1 − ir2 1 a b = = (I + ⃗r · ⃗σ ). ρ= ∗ b d 2 r1 + ir2 1 − r3 2 [
Thus for arbitrary density matrix ρ can be written as ρ = 21 (I + ⃗r · ⃗σ ). Next, we derive the condition that ρ is positive. If ρ is positive, all eigenvalues of ρ should be non-negative. det(ρ − λI) = (a − λ)(b − λ) − |b|2 = λ2 − (a + d)λ + ad − |b2 | = 0 √ (a + d) ± (a + d)2 − 4(ad − |b|2 ) λ= 2 √ ( 2 ) 1−r3 r12 +r22 1± 1−4 − 4 4 = 2 √ 1 ± 1 − (1 − r12 − r22 − r32 ) = 2 √ 1 ± |⃗r|2 = 2 1 ± |⃗r| = 2
31 r| Since ρ is positive, 1−|⃗ r| ≤ 1. 2 ≥ 0 → |⃗ Therefore an arbitrary density matrix for a mixed state qubit is written as ρ = 12 (I + ⃗r · ⃗σ ).
(2) ρ = I/2 → ⃗r = 0. Thus ρ = I/2 corresponds to the origin of Bloch sphere. (3) 1 1 ρ2 = (I + ⃗r · ⃗σ ) (I + ⃗r · ⃗σ ) 2 2 ) ( 3 ∑ ∑ 1 ϵjkl σl  rj rk δjk I + i = I + 2⃗r · ⃗σ + 4 l=1
j,k
) 1( = I + 2⃗r · ⃗σ + |⃗r|2 I 4 1 2 Tr(ρ ) = (2 + 2|⃗r|2 ) 4 If ρ is pure, then Tr(ρ2 ) = 1. 1 1 = Tr(ρ2 ) = (2 + 2|⃗r|2 ) 4 ∴ |⃗r| = 1. Conversely, if |⃗r| = 1, then Tr(ρ2 ) = 14 (2 + 2|⃗r|2 ) = 1. Therefore ρ is pure. 2.73 0013
0010
Theorem 2.6
ρ=
∑
pi |ψi ⟩⟨ψi | =
i
∑
|ψ˜i ⟩⟨ψ˜i | =
i
∑
|φ˜j ⟩⟨φ˜j | =
∑
j
qj |φj ⟩⟨φj | ⇔ |ψ˜i ⟩ =
j
∑
uij |φ˜j ⟩
j
where u is unitary. ∑ Transformation in theorem 2.6, |ψ˜i ⟩ = j uij |φ˜j ⟩, corresponds to [
] [ ] |ψ˜1 ⟩ · · · |ψ˜k ⟩ = |φ˜1 ⟩ · · · |φ˜k ⟩ U T
where k = rank(ρ). 0012
0011 ∑ From spectral theorem, density matrix ρ is decomposed as ρ = dk=1 λk |k⟩⟨k| where d = dim H. Without loss of generality, we ∑ can assume pk > ∑ 0 for k = 1 · · · , l where l √ = rank(ρ) and ˜ k|, ˜ where |k⟩ ˜ = λk |k⟩. pk = 0 for k = l + 1, · · · , d. Thus ρ = lk=1 pk |k⟩⟨k| = lk=1 |k⟩⟨ Suppose |ψi ⟩ is a state in support ρ. Then
|ψi ⟩ =
l ∑ k=1
Define pi = ∑
1
|cik |2 k λk
and uik
cik |k⟩ ,
√ pi cik = √ . λk
∑ k
|cik |2 = 1.
32
CHAPTER 2. INTRODUCTION TO QUANTUM MECHANICS Now ∑
|uik |2 =
∑ pi |cik |2
k
k
λk
= pi
∑ |cik |2 λk
k
= 1.
Next prepare an unitary operator 1 such that ith row of U is [ui1 · · · uik · · · uil ]. Then we can define another ensemble such that [ ] [ ] |ψ˜1 ⟩ · · · |ψ˜i ⟩ · · · |ψ˜l ⟩ = |k˜1 ⟩ · · · |k˜l ⟩ U T √ where |ψ˜i ⟩ = pi |ψi ⟩. From theorem 2.6, ∑ ∑ ˜ k| ˜ = ρ= |k⟩⟨ |ψ˜k ⟩⟨ψ˜k | . k
k
Therefore we can obtain ∑ a 1minimal ensemble for ρ that contains |ψi ⟩. −1 Moreover since ρ = k λk |k⟩⟨k|, ⟨ψi |ρ−1 |ψi ⟩ =
∑ 1 ∑ |cik |2 1 ⟨ψi |k⟩⟨k|ψi ⟩ = = . λk λk pi k
Hence,
1 ⟨ψi |ρ−1 |ψi ⟩
k
= pi .
2.74 ρAB = |a⟩⟨a|A ⊗ |b⟩⟨b|B ρA = TrB ρAB = |a⟩⟨a| Tr(|b⟩⟨b|) = |a⟩⟨a| Tr(ρ2A ) = 1 Thus ρA is pure. 2.75 Define |Φ± ⟩ =
√1 (|00⟩ 2
± |11⟩) and |Ψ± ⟩ =
√1 (|01⟩ 2
± |10⟩).
1 |Φ± ⟩⟨Φ± |AB = (|00⟩⟨00| ± |00⟩⟨11| ± |11⟩⟨00| + |11⟩⟨11|) 2 1 I TrB (|Φ± ⟩⟨Φ± |AB ) = (|0⟩⟨0| + |1⟩⟨1|) = 2 2 1 |Ψ± ⟩⟨Ψ± | = (|01⟩⟨01| ± |01⟩⟨10| ± |10⟩⟨01| + |10⟩⟨10|) 2 1 I TrB (|Ψ± ⟩⟨Ψ± |) = (|0⟩⟨0| + |1⟩⟨1|) = 2 2 2.76 By Gram-Schmidt procedure construct an orthonormal basis {uj } (row vector) with ui = [ui1 · · · uik · · · uil ].   u1  ..   .     Then define unitary U =   ui .  .   ..  1
ul
33 Unsolved. I think the polar decomposition can only apply to square matrix A, not arbitrary linear operators. Suppose A is m × n matrix. Then size of A† A is n × n. Thus the size of U should be m × n. Maybe U is isometry, but I think it is not unitary. I misunderstand linear operator. Quoted from ”Advanced Liner Algebra” by Steven Roman, ISBN 0387247661. A linear transformation τ : V → V is called a linear operator on V .2 Thus coordinate matrices of linear operator are square matrices. And Nielsen and Chaung say at Theorem 2.3, ”Let A be a linear operator on a vector space V .” Therefore A is a linear transformation such that A : V → V . 2.77
|ψ⟩ = |0⟩ |Φ+ ⟩ [ ] 1 = |0⟩ √ (|00⟩ + |11⟩) 2 ] [ 1 = (α |ϕ0 ⟩ + β |ϕ1 ⟩) √ (|ϕ0 ϕ0 ⟩ + |ϕ1 ϕ1 ⟩) 2
where |ϕi ⟩ are arbitrary orthonormal∑states and α, β ∈ C. We cannot vanish cross term. Therefore |ψ⟩ cannot be written as |ψ⟩ = i λi |i⟩A |i⟩B |i⟩C . 2.78
Proof. Former part. If |ψ⟩ is product, then there exist a state |ϕA ⟩ for system A, and a state |ϕB ⟩ for system B such that |ψ⟩ = |ϕA ⟩ |ϕB ⟩. Obviously, this Schmidt number is 1. Conversely, if Schmidt number is 1, the state is written as |ψ⟩ = |ϕA ⟩ |ϕB ⟩. Hence this is a product state. Proof. Later part. (⇒) Proved by exercise 2.74. ∑ ∑ (⇐) Let a pure state be |ψ⟩ = i λi |iA ⟩ |iB ⟩. Then ρA = TrB (|ψ⟩⟨ψ|) = i λ2i |i⟩⟨i|. If ρA is a pure state, then λj = 1 and otherwise 0 for some j. It follows that |ψj ⟩ = |jA ⟩ |jB ⟩. Thus |ψ⟩ is a product state.
2.79
2
According to Roman, some authors use the term linear operator for any linear transformation from V to W .
34
CHAPTER 2. INTRODUCTION TO QUANTUM MECHANICS
0013
0010
Procedure of∑ Schmidt decomposition. √ Goal: |ψ⟩ = i λi |iA ⟩ |iB ⟩ • Diagonalize reduced density matrix ρA = • Derive |iB ⟩, |iB ⟩ =
0012
∑
i λi |iA ⟩⟨iA |.
(I ⊗ ⟨iA |) |ψ⟩ √ λi
• Construct |ψ⟩.
0011
(i) 1 √ (|00⟩ + |11⟩) This is already decomposed. 2 (ii) |00⟩ + |01⟩ + |10⟩ + |11⟩ = 2
(
|0⟩ + |1⟩ √ 2
)
( ⊗
|0⟩ + |1⟩ √ 2
) = |ψ⟩ |ψ⟩ where |ψ⟩ =
(iii) 1 |ψ⟩AB = √ (|00⟩ + |01⟩ + |10⟩) 3 ρAB = |ψ⟩⟨ψ|AB 1 ρA = TrB (ρAB ) = (2 |0⟩⟨0| + |0⟩⟨1| + |1⟩⟨0| + |1⟩⟨1|) ( )( 3 ) 2 1 1 det(ρA − λI) = −λ −λ − =0 3 3 9 1 λ2 − λ + = 0 9√ √ 1 ± 5/3 3± 5 λ= = 2 6 [ √ ] √ 1+ 5 3+ 5 1 2 Eigenvector with eigenvalue λ0 ≡ . is |λ0 ⟩ ≡ √ √ 6 1 5+ 5 2 [ √ ] √ 1− 5 3− 5 1 2 Eigenvector with eigenvalue λ1 ≡ is |λ1 ⟩ ≡ √ √ . 6 1 5− 5 2
ρA = λ0 |λ0 ⟩⟨λ0 | + λ1 |λ1 ⟩⟨λ1 | . (I ⊗ ⟨λ0 |) |ψ⟩ √ λ0 (I ⊗ ⟨λ1 |) |ψ⟩ √ |a1 ⟩ ≡ λ1 |a0 ⟩ ≡
Then |ψ⟩ =
1 √ ∑ λi |ai ⟩ |λi ⟩ . i=0
|0⟩ + |1⟩ √ 2
35 (It’s too tiresome to calculate |ai ⟩) 2.80 ∑ ∑ Let |ψ⟩ = i λi |ψi ⟩A |ψi ⟩B and |φ⟩ = i λi |φi ⟩A |φi ⟩B . ∑ ∑ Define U = i |ψj ⟩⟨φj |A and V = j |ψj ⟩⟨φj |. Then (U ⊗ V ) |φ⟩ =
∑
λi U |φi ⟩A V |φi ⟩B
i
=
∑
λi |ψi ⟩A |ψi ⟩B
i
= |ψ⟩ .
2.81 Suppose ρA = TrR |AR2 ⟩⟨AR2 | =
∑
i λi |i⟩⟨i|.
Define |AR1 ⟩ = (IA ⊗ UR ) |AR2 ⟩.
) ( TrR (|AR1 ⟩⟨AR1 |) = TrR (IA ⊗ UR ) |AR2 ⟩⟨AR2 | (IA ⊗ UR† ) ) ( = TrR |AR2 ⟩⟨AR2 | (IA ⊗ UR† )(IA ⊗ UR ) = TrR (|AR2 ⟩⟨AR2 |) = ρA . Thus |AR1 ⟩ is also a purification of ρA . 2.82 (1) ∑ √ Let |ψ⟩ = i pi |ψi ⟩ |i⟩. TrR (|ψ⟩⟨ψ|) =
∑
pi |ψi ⟩⟨ψi |
i
Thus |ψ⟩ is a purification of ρ. (2) Probability Tr [(I ⊗ |i⟩⟨i|) |ψ⟩⟨ψ|] = ⟨ψ|(I ⊗ |i⟩⟨i|)|ψ⟩ = pi ⟨ψi |ψi ⟩ = pi . Post-measurement state (I ⊗ |i⟩⟨i| |ψ⟩) = √ pi
(3)
√
pi |ψi ⟩ = |ψi ⟩ . √ pi
36
CHAPTER 2. INTRODUCTION TO QUANTUM MECHANICS
∑ √ Suppose |AR⟩ is a purification of ρ such that |AR⟩ = i pi |ψi ⟩ |ri ⟩. By exercise 2.81, the others purification is written as (I ⊗ U ) |AR⟩. ∑√ (I ⊗ U ) |AR⟩ = (I ⊗ U ) pi |ψi ⟩ |ri ⟩ =
∑√
i
pi |ψi ⟩ U |ri ⟩
i
=
∑√
pi |ψi ⟩ |i⟩
i
∑ where U = i |i⟩⟨ri |. By (2), if we measure the system R w.r.t |i⟩, post-measurement state for system A is |ψi ⟩ with probability pi , which prove the assertion. Problem 2.1 From Exercise 2.35, ⃗n · ⃗σ is decomposed as ⃗n · ⃗σ = |λ1 ⟩⟨λ1 | − |λ−1 ⟩⟨λ−1 | where |λ±1 ⟩ are eigenvector of ⃗n · ⃗σ with eigenvalues ±1. Thus f (θ⃗n · ⃗σ ) = f (θ) |λ1 ⟩⟨λ1 | + f (−θ) |λ−1 ⟩⟨λ−1 | ( ) ( ) f (θ) + f (−θ) f (θ) − f (−θ) f (θ) + f (−θ) f (θ) − f (−θ) = + |λ1 ⟩⟨λ1 | + − |λ−1 ⟩⟨λ−1 | 2 2 2 2 f (θ) + f (−θ) f (θ) − f (−θ) = (|λ1 ⟩⟨λ1 | + |λ−1 ⟩⟨λ−1 |) + (|λ1 ⟩⟨λ1 | − |λ−1 ⟩⟨λ−1 |) 2 2 f (θ) − f (−θ) f (θ) + f (−θ) I+ ⃗n · ⃗σ = 2 2 Problem 2.2 Unsolved Problem 2.3 Unsolved
Chapter 8
Quantum noise and quantum operations 8.1 Density operator of initial state is written by |ψ⟩⟨ψ| and final state is written by U |ψ⟩⟨ψ| U † . Thus time development of ρ = |ψ⟩⟨ψ| can be written by E(ρ) = U ρU † . 8.2 From eqn (2.147) (on page 100), ρm =
† Mm ρMm † Tr(Mm Mm ρ)
=
† Mm ρMm † Tr(Mm ρMm )
=
Em (ρ) . Tr Em (ρ)
† † And from eqn (2.143) (on page 99), p(m) = Tr(Mm Mm ρ) = Tr(Mm ρMm ) = Tr Em (ρ).
8.3
8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 37
38
8.14 8.15 8.16 8.17 8.18 8.19 8.20 8.21 8.22 8.23 8.24 8.25 8.26 8.27 8.28 8.29 8.30 8.31 8.32 8.33 8.34 8.35
CHAPTER 8. QUANTUM NOISE AND QUANTUM OPERATIONS
Chapter 9
Distance measures for quantum information 9.1 1 (|1 − 1/2| + |0 − 1/2|) 2( ) 1 1 1 = + 2 2 2 1 = 2
D((1, 0), (1/2, 1/2)) =
1 (|1/2 − 3/4| + |1/3 − 1/8| + |1/6 − 1/8|) 2 1 = (1/4 + 5/24 + 1/24) 2 1 = 4
D ((1/2, 1/3, 1/6), (3/4, 1/8, 1/8)) =
9.2 1 (|p − q| + |(1 − p) − (1 − q)|) 2 1 = (|p − q| + | − p + q|) 2 = |p − q|
D ((p, 1 − p), (q, 1 − q)) =
9.3 F ((1, 0), (1/2, 1/2)) =
√ √ 1 1 · 1/2 + 0 · 1/2 = √ 2
√ √ √ 1/2 · 3/4 + 1/3 · 1/8 + 1/6 · 1/8 √ √ 4 6+ 3 = 12
F ((1/2, 1/3, 1/6), (3/4, 1/8, 1/8)) =
39
40
CHAPTER 9. DISTANCE MEASURES FOR QUANTUM INFORMATION
9.4 Define rx = px − qx . Let U be the whole index set. ∑ ∑ max |p(S) − q(S)| = max px − qx S S x∈S x∈S ∑ = max (px − qx ) S x∈S ∑ = max rx S Since
∑
x∈S
x∈S rx
is written as ∑
rx =
x∈S
∑
∑
∑
rx +
x∈S rx ≥0
rx ,
(9.1)
x∈S rx <0
is maximized when S = {x ∈ U |rx ≥ 0} or S = {x ∈ U |rx < 0}. r x x∈S Define S+ = {x ∈ U |rx ≥ 0} and S− = {x ∈ U |rx < 0}. Now the sum of all rx is 0, ∑ ∑ ∑ rx = rx + rx = 0 x∈U
∴
∑
x∈S+
rx = −
x∈S+
Thus
x∈S−
∑
rx .
x∈S−
∑ ∑ ∑ max rx = rx = − rx . S x∈S
x∈S+
x∈S−
On the other hand, 1∑ |px − qx | 2 x∈U 1∑ |rx | = 2 x∈U 1 ∑ 1 ∑ = |rx | + |rx | 2 2
D(px , qx ) =
x∈S+
x∈S−
1 ∑ 1 ∑ rx − rx = 2 2 x∈S+
x∈S−
1 ∑ 1 ∑ = rx + rx 2 2 x∈S+ x∈S+ ∑ = rx x∈S+
∑ = max rx . S x∈S
(∵ eqn(9.2))
(9.2)
41 ∑ ∑ Therefore D(px , qx ) = maxS x∈S px − x∈S qx = maxS |p(S) − q(S)|. 9.5 ∑ ∑ From eqn (9.1) and (9.2), maximizing x∈S rx is equivalent to maximizing x∈S rx . Hence ( ) ∑ ∑ D(px , qx ) = max(p(S) − q(S)) = max px − qx . S
S
x∈S
x∈S
9.6 Define ρ =
3 4
|0⟩⟨0| + 14 |1⟩⟨1|, σ =
2 3
|1⟩⟨1| + 31 |1⟩⟨1|.
1 Tr |ρ − σ| 2 = D((3/4, 1/4), (2/3, 1/3)) ) ( 1 3 2 1 1 = − + − 2 4 3 4 3 ( ) 1 1 1 = + 2 12 12 1 = 12
D(ρ, σ) =
Define ρ =
3 4
|0⟩⟨0| + 14 |1⟩⟨1|, σ =
2 3
|+⟩⟨+| + 13 |−⟩⟨−|.
1 |+⟩⟨+| = (|0⟩⟨0| + |0⟩⟨1| + |1⟩⟨0| + |1⟩⟨1|) 2 1 |−⟩⟨−| = (|0⟩⟨0| − |0⟩⟨1| − |1⟩⟨0| + |1⟩⟨1|) 2 (
3 1 − 4 2
)
1 ρ−σ = |0⟩⟨0| − (|0⟩⟨1| + |1⟩⟨0|) + 6 1 1 1 = |0⟩⟨0| − (|0⟩⟨1| + |1⟩⟨0|) − |1⟩⟨1| 4 6 4
(
1 1 − 4 2
) |1⟩⟨1|
1 1 1 1 1 1 1 1 |0⟩⟨0| − |0⟩⟨1| + 2 |0⟩⟨0| + |0⟩⟨1| − |1⟩⟨0| + 2 |1⟩⟨1| + |1⟩⟨0| + 2 |1⟩⟨1| 2 4 6 6·4 4·6 6 4·6 4 ( )4 · 6 1 1 = + (|0⟩⟨0| + |1⟩⟨1|) 42 62
(ρ − σ)† (ρ − σ) =
1 Tr |ρ − σ| 2 √ 1 1 = + 2 2 4 6
D(ρ, σ) =
9.7
42
CHAPTER 9. DISTANCE MEASURES FOR QUANTUM INFORMATION Since ρ − σ is Hermitian, we can apply spectral decomposition. Then ρ − σ is written as ρ−σ =
k ∑
n ∑
λi |i⟩⟨i| +
i=1
λi |i⟩⟨i|
i=k+1
where λi are positive ∑ eigenvalues for i = 1, ∑· · · , k and negative eigenvalues for i = k + 1, · · · , n. Define Q = ki=1 λi |i⟩⟨i| and S = − ni=k+1 λi |i⟩⟨i|. Then P and S are positive operator. Therefore ρ − σ = P − S. 0013
0010
Proof of |ρ − σ| = Q + S. |ρ − σ| = |Q − S| √ = (Q − S)† (Q − S) √ = (Q − S)2 √ = Q2 − QS − SQ + S 2 √ = Q2 + S 2 √∑ = λ2i |i⟩⟨i| =
∑
i
|λi | |i⟩⟨i|
i
=Q+S 0012
0011
9.8 ∑ Suppose σ = σi . Then σ = i pi σi . ( ) ( ) ∑ ∑ ∑ D pi ρi , σ = D pi ρi , pi σi i
≤
∑
i
(9.3)
i
pi D(ρi , σi )
(∵ eqn(9.50))
(9.4)
i
=
∑
pi D(ρi , σ). (∵ assumption).
(9.5)
i
9.9 9.10 9.11 9.12 Suppose ρ = 21 (I + ⃗r · ⃗σ ) and σ = 12 (I + ⃗s · ⃗σ ) where ⃗v and ⃗s are real vectors s.t. |⃗v |, |⃗s| ≤ 1. I E(ρ) = p + (1 − p)ρ, 2
I E(σ) = p + (1 − p)σ. 2
43
1 Tr |E(ρ) − E(σ)| 2 1 = Tr |(1 − p)(ρ − σ)| 2 1 = (1 − p) Tr |ρ − σ| 2 = (1 − p)D(ρ, σ) |⃗r − ⃗s| = (1 − p) 2
D(E(ρ), E(σ)) =
Is this strictly contractive? 9.13 Bit flip channel E0 =
√
pI, E1 =
√
1 − pσx .
E(ρ) = E0 ρE0† + E1 ρE1† = pρ + (1 − p)σx ρσx . Since σx σx σx = σx , σx σy σx = −σy and σx σz σx = −σz , then σx (⃗r · ⃗σ ) = r1 σx − r2 σy − r3 σ3 . Thus 1 Tr |E(ρ) − E(σ)| 2 1 = Tr |p(ρ − σ) + (1 − p)(σx ρσx − σx σσx )| 2 1 1 ≤ p Tr |ρ − σ| + (1 − p) Tr |σx (ρ − σ)σx | 2 2 = pD(ρ, σ) + (1 − p)D(σx ρσx , σx σσx )
D(E(ρ), E(σ)) =
= D(ρ, σ)
(∵ eqn(9.21)).
Suppose ρ0 = 12 (I + ⃗r · ⃗σ ) is a fixed point. Then ρ0 = E(ρ0 ) = pρ0 + (1 − p)σx ρ0 σx ∴ (1 − p)ρ0 − (1 − p)σx ρ0 σx = 0 ∴ (1 − p)(ρ − σx ρ0 σx ) = 0 ∴ ρ0 = σx ρ0 σx 1 1 ∴ (I + r1 σx + r2 σy + r3 σz ) (I + r1 σx − r2 σy − r3 σz ) 2 2 Since {I, σx , σy , σz } are linearly independent, r2 = −r2 and r3 = −r3 . Thus r2 = r3 = 0. Therefore the set of fixed points for the bit flip channel is {ρ | ρ = 21 (I + rσx ), |r| ≤ 1, r ∈ R} 9.14
44
CHAPTER 9. DISTANCE MEASURES FOR QUANTUM INFORMATION
√ F (U ρU , U σU ) = Tr (U ρU † )1/2 σ(U ρU † ) √ = Tr U ρ1/2 σρ1/2 U † √ = Tr(U ρ1/2 σρ1/2 U † ) √ = Tr( ρ1/2 σρ1/2 U † U ) √ = Tr ρ1/2 σρ1/2 †
†
= F (ρ, σ) 0013
√
0010
√
I think the fact U AU † = U AU † is not restricted for positive operator. Suppose A is a normal matrix. From spectral theorem, it is decomposed as ∑ A= ai |i⟩⟨i| . i
Let f be a function. Then ∑ f (U AU † ) = f ( ai U |i⟩⟨i| U † ) =
∑ i
= U(
i
f (ai )U |i⟩⟨i| U †
∑
f (ai )U |i⟩⟨i| U † )U †
i
= U f (A)U † 0012
0011
9.15 √ √ |ψ⟩ = (UR ⊗ ρUQ ) |m⟩ is any fixed purification of ρ, and |ϕ⟩ = (VR ⊗ σVQ ) |m⟩ is purification √ √ √ √ √ √ of σ. Suppose ρ σ = | ρ σ|V is the polar decomposition of ρ σ. Then ( ) †√ √ | ⟨ψ|ϕ⟩ | = ⟨m| UR† VR ⊗ UQ ρ σVQ |m⟩ ( ) †√ √ = Tr (UR† VR )T UQ ρ σVQ ( ) †√ √ = Tr VRT UR∗ UQ ρ σVQ ( ) †√ √ ρ σ = Tr VQ VRT UR∗ UQ ( ) † √ √ = Tr VQ VRT UR∗ UQ | ρ σ|V ( ) † √ √ = Tr V VQ VRT UR∗ UQ | ρ σ| √ √ ≤ Tr | ρ σ| = F (ρ, σ) ∗ U † )† we see that equality is attained. Choosing VQ = V † , VRT = (UQ R
9.16
45 I think eq (9.73) has a typo. Tr(A† B) = ⟨m|A ⊗ B|m⟩ should be Tr(AT B) = ⟨m|A ⊗ B|m⟩. See errata list. In order to show that this exercise, I will prove following two properties, Tr(A) = ⟨m|(I ⊗ A)|m⟩ , (I ⊗ A) |m⟩ = (AT ⊗ I) |m⟩ where A is a linear operator and |m⟩ is unnormalized maximally entangled state, |m⟩ = ⟨m|I ⊗ A|m⟩ =
∑
⟨ii|(I ⊗ A)|jj⟩
ij
=
∑
⟨i|I|j⟩ ⟨i|A|j⟩
ij
=
∑
δij ⟨i|A|j⟩
ij
=
∑
⟨i|A|i⟩
i
Suppose A =
= Tr(A)
∑ ij
aij |i⟩⟨j|.

(I ⊗ A) |m⟩ = I ⊗ =
∑
∑
 aij |i⟩⟨j|
∑
ij
|kk⟩
k
aij |k⟩ ⊗ |i⟩ ⟨j|k⟩
ijk
=
∑
aij |k⟩ ⊗ |i⟩ δjk
ijk
=
∑
aij |j⟩ ⊗ |i⟩
ij
=
∑
aji |i⟩ ⊗ |j⟩
ij
  ∑ ∑ (AT ⊗ I) |m⟩ =  aji |i⟩⟨j| ⊗ I  |kk⟩ ij
=
∑
k
aji |i⟩ ⟨j|k⟩ ⊗ |k⟩
ij
=
∑
aji |i⟩ δjk ⊗ |k⟩
ij
=
∑
aji |ij⟩
ij
= (I ⊗ A) |m⟩ Thus Tr(AT B) = Tr(BAT ) = ⟨m|I ⊗ BAT |m⟩ = ⟨m|(I ⊗ B)(I ⊗ AT )|m⟩ = ⟨m|(I ⊗ B)(A ⊗ I)|m⟩ = ⟨m|A ⊗ B|m⟩ .
∑
i |ii⟩.
46
CHAPTER 9. DISTANCE MEASURES FOR QUANTUM INFORMATION
9.17 If ρ = σ, then F (ρ, σ) = 1. Thus A(ρ, σ) = arccos F (ρ, σ) = arccos 1 = 0. If A(ρ, σ) = 0, then arccos F (ρ, σ) = 0 ⇒ cos(arccos F (ρ, σ)) = cos(0) ⇒ F (ρ, σ) = 1 (∵ text p.411, the fifth line from bottom). 9.18 For 0 ≤ x ≤ y ≤ 1, arccos(x) ≥ arccos(y). From F (E(ρ), E(σ)) ≥ F (ρ, σ) and 0 ≤ F (E(ρ), E(σ)), F (ρ, σ) ≤ 1, arccos F (E(ρ), E(σ)) ≥ arccos F (ρ, σ) ∴ A(E(ρ), E(σ)) ≥ A(ρ, σ) 9.19 From eq (9.92) F
( ∑
pi ρi ,
i
∑
) ≥
pi σi
i
∑√ pi pi F (ρi , σi ) i
=
∑
pi F (ρi , σi ).
i
9.20 Suppose σi = σ. Then F
( ∑
) pi ρi , σ
=F
i
( ∑
pi ρi ,
∑
i
=F ≥
( ∑
∑
) pi σ
i
pi ρi ,
∑
i
) pi σi
i
pi F (ρi , σi ) (∵ Exercise9.19)
i
=
∑
pi F (ρi , σ)
i
9.21 1 − F (|ψ⟩ , σ)2 = 1 − ⟨ψ|σ|ψ⟩ (∵ eq(9.60))
D(|ψ⟩ , σ) = max Tr(P (ρ − σ)) (where P is projector.) P
≥ Tr (|ψ⟩⟨ψ| (ρ − σ)) = ⟨ψ|(|ψ⟩⟨ψ| − σ)|ψ⟩ = 1 − ⟨ψ|σ|ψ⟩ = 1 − F (|ψ⟩ , σ)2 . 9.22
47 (ref: QCQI Exercise Solutions (Chapter 9) - めもめも http://enakai00.hatenablog.com/entry/2018/04/12/134722) For all ρ, following inequality is satisfied, d(V U ρU † V † , F ◦ E(ρ)) ≤ d(V U ρU † V † , F(U ρU † )) + d(F(U ρU † ), F ◦ E(ρ)) ≤ d(V U ρU † V † ) + d(U ρU † , E(ρ)) ≤ E(V, F) + E(U, E). First inequality is triangular inequality, second is contractivity of the metric1 and third is from definition of E. Above inequality is hold for all ρ. Thus E(V U, F ◦ E) ≤ E(V, F) + E(U, E). 9.23 (⇐) If E(ρj ) = ρj for all j such that pj > 0, then ∑ ∑ ∑ ∑ F¯ = pj F (ρj , E(ρj ))2 = pj F (ρj , ρj )2 = pj 12 = pj = 1. j
j
j
j
(⇒) Suppose E(ρj ) ̸= ρj . Then F (ρj , E(ρj )) < 1 (∵ text p.411, the fifth line from bottom ). Thus ∑ ∑ F¯ = pj F (ρj , E(ρj ))2 < pj = 1. j
j
Therefore if F¯ = 1, then E(ρj ) = ρj . Problem 1 Problem 2
Problem 3 Theorem 5.3 of ”Theory of Quantum Error Correction for General Noise”, Emanuel Knill, Raymond Laflamme, and Lorenza Viola, Phys. Rev. Lett. 84, 2525 ‒ Published 13 March 2000. arXiv:quant-ph/9604034 https://arxiv.org/abs/quant-ph/9604034
1
Trace distance and angle are satisfied with contractive (eq (9.35), eq (9.91)), but I don’t assure that arbitrary metric satisfied with contractive.