Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 33 / 3 / d

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- Solution d

Solution

A unital quantum channel preserves the identity operator:

Φ (I) = I

. In equal input and output dimensions, it therefore preserves the maximally mixed state. For amplitude damping channel,

A_{p} (I) = K_{0} K_{0}^{†} + K_{1} K_{1}^{†} = (1 + p 0 0 1 - p) = I + pZ .

Hence the channel is not unital for $p > 0$ ; it is unital only in the identity case $p = 0$ . Trace preservation involves the opposite products

K_{i}^{†} K_{i}

and does not imply unitality.

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