The Exercise

A quantum system is in a state $\rho$, and $P$ is an orthogonal projector on its state space, a self adjoint operator with $P^2 = P$. A measurement with the two projectors $P$ and $1 - P$, whose outcome is then forgotten, leaves the system in the state $P\rho P + (1-P)\rho(1-P)$, the pinching of $\rho$ by $P$. The exercise is to show that a fair coin does the same: on heads nothing is done, on tails the system evolves under the unitary $1 - 2P$, and the resulting mixture is the pinching.

A Quantum System

A quantum system $S$ is a physical object whose state space is a finite dimensional complex Hilbert space.^✦ A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$, pure or mixed.^✦

A unitary $U$ of $S$ is an operator on the state space of $S$ with $U^\dagger U = U U^\dagger = 1$; it describes a closed evolution of $S$.^✦ The evolution of a state $\rho$ of $S$ under $U$ is the state $$\rho \mapsto U \rho \, U^\dagger .$$(1)^✦ The identity evolution of $S$ is the unitary $1$, under which (1) leaves every state as it is.^✦

The Reflection

For an orthogonal projector $P$ on the state space of $S$, the reflection $R_P$ of $S$ determined by $P$ is the unitary of $S$ with operator $$R_P = 1 - 2P .$$(2)^✦ It reverses every vector in the range of $P$ and fixes every vector in the range of $1 - P$.

The Claim

What Is Checked

The claim (3) is stated over a state $\rho$ of a quantum system $S$ and an orthogonal projector $P$ on its state space, with the two terms of the mixture given as the evolutions (1) of $\rho$ under the identity evolution and under the reflection $R_P$ of (2).

About these constructions the library proves that $R_P$ is a unitary, since it is self adjoint and $P^2 = P$ gives $R_P^2 = 1$, and that conjugating a state by a unitary gives a state. Each evolution (1) is therefore a state of $S$, and the right side of (3) is an equal mixture of two states.