Quantum Information Theory

The following sections are included:

• The density matrix

  • The density matrix for a qubit

  • Composite systems

  • The quantum copying machine

• The Schmidt decomposition

• Purification

• The Kraus representation

• Measurement of the density matrix for a qubit

• Generalized measurements

  • Weak measurements

  • POVM measurements

• The Shannon entropy

• Classical data compression

  • Shannon's noiseless coding theorem

  • Examples of data compression

• The von Neumann entropy

  • Example 1: source of orthogonal pure states

  • Example 2: source of non-orthogonal pure states

• Quantum data compression

  • Schumacher's quantum noiseless coding theorem

  • Compression of an n-qubit message

  • Example 1: two-qubit messages

  • Example 2: three-qubit messages

• Accessible information

  • The Holevo bound

  • Example 1: two non-orthogonal pure states

  • Example 2: three non-orthogonal pure states

• Entanglement concentration and von Neumann entropy

• The Peres separability criterion

• Entropies in physics

  • Thermodynamic entropy

  • Statistical entropy

  • Dynamical Kolmogorov–Sinai entropy

• A guide to the bibliography