Is Quantum Mechanics Compatible with Conservation of Information?

Question

What is exactly the law of conservation of information? In quantum mechanics we have truly random outcomes in experiments, but doesn't this randomness mean that new information is produced and the law of conservation of information is violated?

Answer 1

Any conservation law -- energy, momentum, you name it, holds only in an isolated system. If a system interacts with its environment, then neither energy nor information associated with the system will be conserved. Of course, you can consider the system and its environment together as an isolated system, to which the conservation laws apply.

Since quantum measurement of a system involves interacting with it, it therefore should not be surprising that conservation of information associated with the system is violated. Of course, you could try to consider the system and the experimenter doing the measurement as a single isolated quantum system. An outside observer -- a "Wigner's friend" -- to the system-experimenter composite will describe this composite system as an isolated system undergoing unitary evolution. The relationship between that, and the subjective experience of the experimenter, is the "measurement problem" which various interpretations of quantum mechanics try to address in various ways.

Answer 2

As Dominic stated, local conservation laws hold in quantum mechanics, i.e. the conservation equation for probability density.

But in a simpler sense, "information is conserved" in that the total probably to measure an observable is always 1. More technically, if $\Psi(x)$ is a function that represents a quantum state in the position basis, then we have that

$$ 1 = \int_{\infty}^{\infty} |\Psi(x)|^{2} dx$$

which is the well known normalization condition. In our context here, it means that we will always find our particle at some position if we measure it. So in this sense, all the information of the wave function is conserved as well.