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The single-qubit depolarizing channel is D p ( ρ ) = ( 1 − p ) ρ + ( p / 3 ) ( X ρ X + Y ρ Y + Z ρ Z ) D_p(\rho) = (1-p)\rho + (p/3)(X\rho X + Y\rho Y + Z\rho Z) D p ( ρ ) = ( 1 − p ) ρ + ( p /3 ) ( X ρX + Y ρ Y + Z ρZ ) . With what probability does each of the three Pauli errors X , Y , Z X, Y, Z X , Y , Z occur, and what makes this channel 'maximally symmetric'? 🔬 try it before you answer
Only Z Z Z occurs, with probability p p p ; X X X and Y Y Y never occur, modelling pure dephasing Each of X , Y , Z X, Y, Z X , Y , Z occurs with probability p / 3 p/3 p /3 ; no Pauli is preferred, and it shrinks the Bloch vector uniformly toward the centre Each of X , Y , Z X, Y, Z X , Y , Z occurs with probability p p p ; the channel applies all three Paulis on every use Each of X , Y , Z X, Y, Z X , Y , Z occurs with probability ( 1 − p ) / 3 (1-p)/3 ( 1 − p ) /3 ; the identity occurs with probability p p p Check answer