The linearity of quantum measurements is a fundamental characteristic of quantum mechanics. It is related to the mathematical formalism of quantum theory, where the evolution of quantum systems is described by linear operators acting on wave functions.
In quantum mechanics, the wave function evolves according to the Schrödinger equation, which is a linear partial differential equation. This linearity implies that if you have a system in a state described by a particular wave function, and you apply a linear operation or measurement to that system, the resulting state or measurement outcome will also be a linear combination of the original states or measurement outcomes.
This linearity property is often expressed through the principle of superposition, which states that quantum systems can exist in a combination or superposition of different states. For example, an electron can be in a superposition of spin-up and spin-down states simultaneously.
When a measurement is performed on a quantum system, the wave function collapses to one of the possible measurement outcomes. The probability of obtaining a particular outcome is given by the squared magnitude of the corresponding coefficient in the superposition. This is known as the Born rule.
While linearity is a fundamental aspect of quantum mechanics, it is worth noting that not all functions in other contexts behave linearly over small distances. Linearity in the context of quantum mechanics refers specifically to the linearity of the mathematical formalism and the evolution of wave functions.