Download Quantum Information Theory M. Phil Exam - June 2004 and more Exams Statistics in PDF only on Docsity! M. PHIL. IN STATISTICAL SCIENCE Friday 4 June, 2004 9 to 12 QUANTUM INFORMATION THEORY Attempt FOUR questions. There are six questions in total. The questions carry equal weight. You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator. 2 1 Define a classical linear (N, k) code C in terms of its generator and parity-check matrices. What is the dual code C⊥? How are the generator and parity-check matrices related? Explain how the concept of a classical dual code is used to define a quantum error- correcting code (CSS code). Discuss the error-correcting properties of a CSS code. Give an example of such a code. 2 Define a quantum operation (superoperator) on the set of density matrices. What are the properties to be satisfied so that a quantum operation would represent an allowed physical process? State the representation for a quantum operation referring to a coupling of a system and an environment. Draw a diagram illustrating this representation. State the operator-sum (Kraus) representation for a quantum operation. Prove the equivalence of the two representations. Prove that any quantum operation written in the Kraus form is linear, completely positive and trace-preserving. 3 a) State the generalised measurement postulate. When is a generalised measure- ment reduced to a projective measurement? Define a POVM and show how it is related to a generalised measurement. What is a pure POVM? State and prove the duality between POVMs and maximally mixed states. b) Give an example of a positive map which does not satisfy the complete positivity property. Justify your answer. 4 Define the von Neumann entropy S(ρ) of a density matrix ρ acting in a finite- dimensional Hilbert space K. State and prove the bounds indicating the range of values of S(ρ). Comment on the cases of equality in these bounds. Prove that S(ρ) is a concave function of ρ, i.e. S(p1ρ1 + . . . + pmρm) > p1S(ρ1) + . . . + pmS(ρm) for any collection of density matrices ρ1, . . . , ρm and probabilities p1, . . . , pm. Prove also that S(p1ρ1 + . . . + pmρm) 6 p1S(ρ1) + . . . + pmS(ρm)− m∑ i=1 pi log2 pi . 5 State and prove Schumacher’s noiseless coding theorem. QUANTUM INFORMATION THEORY
