# Quantum Information (WS 21/22)

Lecture: PH1010 QST Theory

Lecturer: Prof. Ignacio Cirac

## Content

The course "PH1010 QST Theory: Quantum Information" offers an introduction to the theoretical foundations of Quantum Science and Technology. The course starts with a brief motivation and an introduction to fundamental concepts and the basic formalism (pure/mixed states, evolution, completely positive maps, measurements Schmidt decomposition, tomography, quantum estimation, hypothesis testing). Then the concept of entanglement is discussed in detail, including the distinction between pure and mixed-state entanglement, entanglement entropy, quantification and conversion. Subsequently, some of the revolutionary promises of exploiting entanglement are presented, including dense coding, quantum teleportation and quantum cryptography. Next the Bell inequalities, characterizing the quantum weirdness of entanglement and non-locality, are introduced and discussed in detail. Subsequent chapters cover central applications of quantum information theory: quantum computation, quantum algorithms such as those of Deutsch, Shor and Grover, quantum simulation, and quantum metrology. Final core topics are decoherence, Lindbladian descriptions thereof, and error correction schemes to counteract the consequences of decoherence and protect fragile quantum information. The module will typically also include one or more optional topics, such as many-body entanglement, topological quantum computation, quantum complexity, or tensor networks, which link quantum information theory to many-body physics.

## General Literature

Standard articles, textbooks, and lecture notes on Quantum Information are:

- Quantum Computation Pfeil (Lecture Notes), John Preskill. A famous set of lecture notes, continually being refined throughout the last 20 years.
- Quantum Computation and Quantum Information Pfeil, 10th Anniversary Edition, Michael A. Nielsen, Isaac L. Chuang, Cambridge University Press, 2010. One of the most cited books in physics of all time, providing a general, accessible and wide-ranging introduction to the topic.

## Material

Date | Content |
---|---|

Nov 3th | Chapter 0: Linear Algebra for Q. Information |

Nov 4th |
Chapter 1: Basic Concepts in Q. Information
Part 1: Single systems: Introduction, States |

Nov 5th |
Chapter 1: Basic Concepts in Q. Information
Part 1 (cont): States, Observables & Measurements |

Nov 10th |
Chapter 1: Basic Concepts in Q. Information
Part 1 (cont): Evolution, General Dynamics, Part 2: Composite Systems: Introduction, States |

Nov 12th |
Chapter 1: Basic Concepts in Q. Information
Part 2 (cont): States, Observables & Measurements. |

Nov 17th |
Chapter 1: Basic Concepts in Q. Information
Part 2 (cont): Evolution, General Dynamics. |

Nov 19th |
Chapter 2: Entanglement and non-locality:
EPR "paradox", Bell's theorem, CHSH inequalities |

Nov 24th |
Chapter 2: Entanglement and non-locality:
Bell's theorem without inequalitites, Noncontextuality, Causality |

Nov 26th |
Chapter 3: Quantum Communication:
Part 1: Cryptography: Classical Cryptograph, Quantum Key Distribution |

Dec 1st |
Chapter 3: Quantum Communication:
Part 1 (cont): Eavesdropping |

Dec 3rd |
Chapter 3: Quantum Communication:
Part 2: Complexity: Classical, Quantum |

Dec 8th |
Chapter 3: Quantum Communication:
Part 3: Repeaters: Imperfections, Basic concepts |

Dec 10th |
Chapter 3: Quantum Communication:
Part 3: Repeaters (cont): Entanglement distillation, Repeaters |

Dec 15th |
Chapter 4: Quantum Computing:
Part 1: Basics: Classical computing, Quantum computing basics |

Dec 17th |
Chapter 4: Quantum Computing:
Part 1: Basics (cont): Quantum circuits, Complexity |

Dec 22nd |
Chapter 4: Quantum Computing:
Part 2: Quantum algorithms: with and without oracles |

Jan 7th | Chapter 4: Part 2: Quantum algorithms: Phase estimation, amplitude amplificatin, quantum simulation |

Jan 12th | Chapter 4: Part 2: Quantum algorithms: Adiabatic, variational, other computational models |

Jan 14th | Chapter 4: Part 3: Quantum error correction: Classical, simple QEC: flips and phases, general QEC |

Jan 18th | Chapter 4: Part 3: Quantum error correction: Stabilizer Codes, Fault-tolerant error correction |

Jan 20th | Chapter 5: Quantum metrology: Atomic clocks, Fisher Information, Quantum Fisher Information, Cramer-Rao bounds |

Jan 26th | POVMs: axiomatic & operational description, Naimark's theorem, applications in (unambiguous) state discrimination and information transmission |

Jan 29th | Quantum channels: tpcp maps, Choi-Jamiolkowski matrix, Kraus representation, environment representation, dephasing/entanglement breaking channels |

Feb 2nd | Entanglement: correlations, separability, witnesses, positive map criteria, DeFinetti characterization, basic entanglement measures & their relations, bound entanglement |

Feb 4th | Period finding: Simon's algorithm, collision problem, Shor's algorithm, hidden subgroup problems |

More Material (e.g. Videos of the lectures) can be found on Moodle.