# 6: Evolution of Open Quantum Systems


We have considered mixed states, where the experimenter has incomplete information about the state preparation procedure, and we have also seen that mixing arises in a system when it is entangled with another system. The combined system can still be pure, but the subsystem has become mixed. This phenomenon arises often when we want to describe systems that have some interaction with their environment. The interaction creates entanglement, and the system taken by itself evolves from a pure state to a mixed state. Such a system is called “open”, since it can leak quantum information to the environment. The theory of open quantum systems revolves around the so-called Lindblad equation.

We will derive the Lindblad equation, which is the direct extension of the Heisenberg equation for the density operator, i.e., the mixed state of a system.
• 6.2: Positive and Completely Positive Maps
• 6.3: Bra Vectors and the Inner Product
For each ket vector |ψ⟩, there is a corresponding bra vector ⟨ψ|. We haven’t yet looked into any specific representations of ket vectors beyond just the ket vector itself, so at the moment that’s all you need to know. However, when we do get into specific representations, the rules for converting ket vectors to bra vectors are generally very easy. You always take the complex conjugate of any numbers in the representation going from the ket vector to the bra vector.
• 6.4: Normalization and Orthogonality
Although we aren’t yet going to learn rules for doing general inner products between state vectors, there are two cases where the inner product of two state vectors produces a simple answer. The first is not intrinsic to the mathematical representation, but rather something we will insist for state vectors that properly represent real physical states.
• 6.5: Interpreting the Inner Product

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