Consider a system that is in a mixed state. The system has 0.3 of probability in a pure state and 0.7 of probability in another pure state . Then the density matrix is

In the most general cases, neither nor is an eigenstate. So we cannot expect that is diagonal.

For example, if each of the pure state and is a superposition of two eigenstates , then

and

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For simplicity, assume that the eigenstates form a complete orthonormal set.

If we use as basis,

— Me@2018.03.12 11:51 AM

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2018.03.14 Wednesday (c) All rights reserved by ACHK