L5-06 · Quantum channels and complete positivity

Learn¶

A quantum channel maps input density operators to output density operators. A deterministic physical channel is completely positive and trace preserving. Complete positivity means the map remains positive when an untouched reference system is included. Positivity on isolated inputs alone is insufficient.

Kraus operators provide one representation of a channel. Apply each operator on both sides of the input state, then add the results. The trace-preservation condition is a sum of conjugate products equal to the identity. Different Kraus lists can represent the same channel.

A channel can erase information even though total system and environment evolution is reversible. Tracing out the environment hides its record. A channel therefore differs from a unitary gate on the system alone. Preparation, damping, and measurement without a retained outcome are useful examples.

Practice¶

Check your understanding¶

Which condition guarantees trace preservation in a Kraus representation?

• A. The Kraus operators sum to I

• B. The sum of K dagger K equals I

• C. Every Kraus operator is unitary

Sources¶

This lesson draws on Watrous, 2018Preskill, n.d..

Next: L5-07 · Measurements and instruments