PaperIII_65_3 - MATHEMATICAL TRIPOS Part III Thursday 7...

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MATHEMATICAL TRIPOS Part III Thursday, 7 June, 2012 1:30 pm to 4:30 pm PAPER 65 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 (i) Define the trace distance D(ρ, σ) between two states ρ, σ D(H) and prove that it can be expressed in the form: D(ρ, σ) = 1 (TrQ + TrR) , (1) 2 where Q and R are suitably defined positive semi-definite operators in B(H). (ii) Using the above identity, prove that D(ρ, σ) = max Tr (P (ρ − σ)) , (2) P where the maximisation is over all projection operators P B(H). (iii) Further, prove that D(ρ, σ) = max Tr (T (ρ − σ)) , (3) T where the maximisation is over all positive semi-definite operators T B(H) with eigenvalues less than or equal to unity. (iv) Let ρ be a quantum state and Λ be a linear completely positive trace- preserving map. Prove that Fe (ρ, Λ) ≤ (F (ρ, Λ(ρ)))2 , (4) where Fe(ρ, Λ) denotes the entanglement fidelity, and F (ρ, Λ(ρ)) denotes the fidelity of the states ρ and Λ(ρ).
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