## #016: Characterize quantum projections and entropy maximization processes with constraints

Category: Quantum mechanics - Tags: Quantum mechanics, Quantum information

Show that quantum projections are processes that maximize the entropy while keeping not just the expectation constant, but all other variables as well.

Mathematical problem. We want to show that quantum projection can be understood as thermodynamic processes of entropy maximization. In a typical entropy maximization problem, the constraint is the expectation of a random variable. The idea here is that the distribution over the random variable is kept constant (this can be understood as a constraint on the expectation of all powers - all moments).

The problem is that this entropy maximization has to be, in a sense, minimal. Only the entropy relative to the conjugate of the quantity being measured has to increase. Finding the right formulation is the core of the issue.

Physical significance. The idea is that a quantum projection is essentially going from one equilibrium to another. As an analogy, suppose you have an open container of water. The number of water molecules in the container will fluctuate, therefore there is no single well defined. As we close the container, we change equilibrium (from grand canonical to canonical). Now the number of water molecules is well defined. The idea is to frame quantum projections in the same way.