# Is Quantum Mechanics Compatible with Conservation of Information?

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?

• Unitary QM conserves probabilities/information. See my answer here: physics.stackexchange.com/a/434912/133418 – Avantgarde Oct 18 '18 at 5:43
• @Avantgarde Measurement as projection operator is not unitary (in fact non invertible). So, does a measurement destroy information? – K_inverse Oct 18 '18 at 8:15
• Information, in contrast to energy or momentum, is usually not present on the basic level of physics, and I see no reason why it should be conserved, or should it? (If I tell you a fact and you learn it, haven't we created new information?) – Luke Oct 18 '18 at 13:21
• I don't think there is a law of conservation of information per se. It is only a formal-looking statement of the determinist stance. – Stéphane Rollandin Oct 19 '18 at 8:45

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.

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.

• I'm not sure I agree with this characterization. In any consistent theory, the total probability has to be 1 -- this is an axiom of probability theory. But classical stochastic processes, when described in terms of a probability distribution, although they obviously satisfy this, clearly do not conserve information since they are not reversible. I think the precise statement of conservation of information is simply equivalent to reversibility. – Dominic Else Oct 20 '18 at 15:34
• Interesting point, but in my answer I am disregarding things like black holes and stochastic processes, since the OP's original question is with regards to basic quantum mechanics, where the unitary time evolution operator determines the evolution of the system while conserving information (since it is invariant under time reversal) and we force the state of the system to be normalized meaning every state is plausibly occupied by the system. – N. Steinle Oct 20 '18 at 16:39
• More to your point, though, you seem to be employing Susskind's defn. of "information conservation" which, arising from the black hole paradox, is a different kind of "information" than what I'm considering here, which is I think why you disagree. – N. Steinle Oct 20 '18 at 16:40
• However, as far as the black hole firewall paradox goes, we're not yet sure what the best definitions are, and so I'm not tied down to what you think the best definition is. Indeed, the axiom of unit measure of probability theory is deeply related to unitarity in quantum physics. But, in principle, one can create a probability theory without the axiom of unit measure – N. Steinle Oct 20 '18 at 16:49
• My definition of "conservation of information" is just that you can always uniquely determine the initial state, given the configuration after time t. I doubt this is original to Susskind and the black hole paradox, though it is used there. I don't know how else you could define it? – Dominic Else Oct 20 '18 at 17:35