A phase-flip channel with error probability has Kraus operators and .

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

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Problem

A phase-flip channel with error probability p=0.25p = 0.25 has Kraus operators A0=0.75 IA_0 = \sqrt{0.75}\,I and A1=0.25 ZA_1 = \sqrt{0.25}\,Z. ((a)) Using the unitary u=12(111−1)u = \frac{1}{\sqrt2}\begin{pmatrix}1&1\\1&-1\end{pmatrix}, form L1=12(A0+A1)L_1 = \frac{1}{\sqrt2}(A_0+A_1) and L2=12(A0−A1)L_2 = \frac{1}{\sqrt2}(A_0-A_1) as explicit 2×22\times2 matrices. ((b)) Check that L1†L1+L2†L2=IL_1^\dagger L_1 + L_2^\dagger L_2 = I, confirming that {L1,L2}\{L_1, L_2\} is an equally valid Kraus decomposition of the same channel.

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