Consider the one-parameter family of maps on a qubit, _q( )=q , ^T+(1-q) , I 2 , -1 q 1

PHYSICS · Multi-part · Fermi University Physics · Quantum Information & Technology · Open quantum systems, decoherence and error correction · Open quantum systems, decoherence and error correction

Quantum Information and Technology, Ch Noise as a quantum operation

Problem

Consider the one-parameter family of maps on a qubit, Λq(ρ)=q ρT+(1−q) I2,−1≤q≤1,\Lambda_q(\rho)=q\,\rho^T+(1-q)\,\frac{I}{2},\qquad -1\le q\le1, a mixture of the transpose and the completely depolarising map. ((a)) Using the Bloch-vector picture of the transpose (ry→−ryr_y\to-r_y), show that Λq\Lambda_q sends every density operator to another density operator, for every qq in [−1,1][-1,1]. ((b)) Compute the Choi state J(Λq)=(I⊗Λq)(∣Φ+⟩⟨Φ+∣)J(\Lambda_q)=(\mathcal I\otimes\Lambda_q)\big(|\Phi^+\rangle\langle\Phi^+|\big), and show it equals q2 SWAP+1−q4 I\tfrac q2\,\mathrm{SWAP}+\tfrac{1-q}4\,I. ((c)) Using the eigenvalues of SWAP\mathrm{SWAP}, give the two distinct eigenvalues of J(Λq)J(\Lambda_q) and their multiplicities. ((d)) Find the range of qq for which Λq\Lambda_q is completely positive.

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