Quantum Information Theory - Mathematical Tripos - Exam Paper, Exams of Mathematics

Download Quantum Information Theory - Mathematical Tripos - Exam Paper and more Exams Mathematics in PDF only on Docsity! MATHEMATICAL TRIPOS Part III Thursday, 28 May, 2009 1:30 pm to 4:30 pm PAPER 50 QUANTUM INFORMATION THEORY Attempt no more than FOUR questions. There are FIVE questions in total. The questions carry equal weight. STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS Cover sheet None Treasury Tag Script paper You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator. 2 1 (a) State the necessary and sufficient condition for a quantum error-correcting code (QECC), X , to correct errors in a given set E . (b) Prove that an [[n, k, d]] QECC can correct (d− 1) errors in known locations. (c) Let X be an [[n, k, d]] QECC. The quantum Singleton bound is given by (n− k) > 2(d− 1). Prove this bound, clearly stating any inequalities that you use. [Hint: Let X ⊂ HQ, where HQ is the Hilbert space of a system Q of n qubits. If X is to correct arbitrary errors in a subsystem A of Q, then ρRA = ρR ⊗ ρA, where R is a reference system with Hilbert Space HR ≃ HQ.] (d) Let X denote a non-degenerate [[n, k, d]] QECC. Let t denote the maximum weight of a Pauli operator which can be corrected by X . Express d in terms of t. Find the total number, N(t), of possible errors of weight upto t that can occur on n qubits. The Quantum Hamming bound states that for a non-degenerate code N(t) 6 2n−k. Justify this bound. 2 (a) Prove that any completely positive trace-preserving map Φ, acting on states ρ in a Hilbert space HA, can be written in the Kraus form: Φ(ρ) = ∑ k AkρA † k, where Ak are linear operators satisfying ∑ k A † kAk = I, and I is the identity operator. Hint: Consider a maximally entangled state and use the relative state method, stating clearly what is meant by a relative state and an index state. (b) Using a maximally entangled state and the properties of the swap operator, prove that the transposition operator T is positive but not completely positive. Part III, Paper 50