PaperIII_67 - MATHEMATICAL TRIPOS Part III Friday 5 June...

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MATHEMATICAL TRIPOS Part III Friday, 5 June, 2015 9:00 am to 11:00 am PAPER 67 ADVANCED QUANTUM INFORMATION THEORY Attempt no more than TWO questions. There are FOUR 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) For a system of qudits, let AX and BY be operators that act non-trivially only on qudits in the sets X and Y . AX (t) denotes the time-evolution of AX in the Heisenberg picture. Let Z range over k-element subsets of the qudits, and let H = ∑ Z hZ be a k-local Hamiltonian where hZ acts non-trivially only on the subset Z . Assume that H satisfies the Lieb-Robinson bound: [AX (t), BY ] ≤ 2 AX BY min(|X |, |Y |) e−µd(X,Y ) (e2kst − 1), ∥ ∥ where d(X, Y ) denotes interaction distance, and µ and s are constants. You may use without proof the fact that, for any operator OAB B( dA dB ), TrB [OAB ] B = dB dU ( U )OAB ( U †) where the integral∫is over the Haar measure for the unitary group SU (dB ), nor- malised such that dU = 1. Prove that there exists an operator AX (l) acting non-trivially only on qudits within the subset X (l) = {i : d(i, X ) ≤ vt + l}, such that AX (t) − AX (l) (t) ≤ µvt|X | AX e−µl/2 , where v > 0 is a constant, and find an expression for v in terms of the parameters of the system. (b) ∑onsider a 1-dimensional chain of qudits with nearest-neighbour Hamiltonian H = − Pi,i+1), where Pi,i+1 are projectors that act non-trivially only on qudits i i( and i + 1. Assume that H has a unique ground state |ψ0 ? , spectral gap Δ > 0, and is frustration-free, i.e. i : Pi,i+1 |ψ0 = |ψ0 ? . Let Podd := := := PoddPeven . The i P2i−1,2i , Peven
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i P2i,2i+1 , and K Detectability Lemma states that: 1 K |supp H )1/3 .
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