Introduction

For the past however-many years, Artur Ekert has been teaching the masters course “Introduction to Quantum Information” at the University of Oxford. During this time, many versions of accompanying lecture notes have come and gone, with constant improvements and changes being made. The version that you will find on this website has been carefully edited by Tim Hosgood into a cohesive “book”, containing additional exercises and topics. Thanks go to Zhenyu Cai for many helpful comments and corrections, and we also appreciate the work of Yihui Xie in developing the Bookdown package with which this document was built.

For more information, see the accompanying website.

Plan

In this series of lectures you will learn how inherently quantum phenomena, such as quantum interference and quantum entanglement, can make information processing more efficient and more secure, even in the presence of noise.

The interdisciplinary nature of this topic, combined with the diverse backgrounds that different readers have, means that some may find some particular chapters easy, while others find them difficult. The following will be assumed as prerequisites: elementary probability theory, complex numbers, vectors and matrices, tensor products, and Dirac bra-ket notation. A basic knowledge of quantum mechanics (especially in the simple context of finite dimensional state spaces, e.g. state vectors, composite systems, unitary matrices, Born rule for quantum measurements) and some ideas from classical theoretical computer science (complexity theory) would be helpful, but is not at all essential. Some of these things are covered at the end of this chapter.

Topics

  • Fundamentals of quantum theory
    • addition of probability amplitudes
    • quantum interference
    • mathematical description of states and evolution of closed quantum systems (Hilbert space, unitary evolution)
    • measurements (projectors, Born rule)
    • Pauli matrices
  • Distinguishability of quantum states
  • The Bloch sphere
    • parametrisation
    • action of quantum gates on the Bloch vector
  • The definition of quantum entanglement (the tensor product structure)
  • The no-cloning theorem, and quantum teleportation
  • Quantum gates
    • phase gate
    • Hadamard
    • controlled-\texttt{NOT}
    • SWAP
    • the Hadamard-phase-Hadamard network
    • phase “kick-back” induced by controlled-U
    • phase “kick-back” induced by quantum Boolean function evaluation
  • Quantum algorithms
    • Deutsch
    • Bernstein-Vazirani
    • Simon
  • Bell’s theorem
    • Quantum correlations
    • CHSH inequality
  • Density matrices
    • partial trace
    • statistical mixture of pure states
    • Born rule for density matrices
    • quantum entanglement in terms of density matrices
  • Completely positive maps
    • Kraus operators
    • the Choi matrix
    • positive versus completely positive maps
    • partial-transpose
  • The simple model of decoherence
  • Quantum error correction of bit-flip and phase-flip errors