Conservation of information and determinism?

I'm having a hard time wrapping my head around the conservation of information principle as formulated by Susskind and others. From the most upvoted answers to this post, it seems that the principle of conservation of information is a consequence of the reversibility (unitarity) of physical processes.

Reversibility implies determinism: Reversibility means that we have a one to one correspondence between a past state and a future state, and so given complete knowledge of the current state of the universe, we should be able to predict all future states of the universe (Laplace's famous demon).

But hasn't this type of determinism been completely refuted by Quantum Mechanics, the uncertainty principle and the probabilistic outcome of measurement? Isn't the whole point of Quantum Mechanics that this type of determinism no longer holds?

Moreover, David Wolpert proved that even in a classical, non-chaotic universe, the presence of devices that perform observation and prediction makes Laplace style determinism impossible. Doesn't Wolpert's result contradict the conservation of information as well?

So to summarize my question: How is the conservation of information compatible with the established non-determinism of the universe?

Answer

The short answer to this question is that the Schrödinger equation is deterministic and time reversible up to the point of a measurement. Determinism says that given an initial state of a system and the laws of physics you can calculate what the state of the system will be after any arbitrary amount of time (including a positive or negative amount of time). Classically, the deterministic laws of motion are given by Newton's force laws, the Euler-Lagrange equation, and the Hamiltonian. In quantum mechanics, the law that governs the time evolution of a system is the Schrödinger equation. It shows that quantum states are time reversible up until the point of a measurement, at which point the wave function collapses and it is no longer possible to apply a unitary that will tell you what the state was before, deterministically. However, it should be noted that many-world interpreters who don’t believe that measurements are indeterministic don’t agree with this statement, they think that even measurements are deterministic in the grand scheme of quantum mechanics. To quote Scott Aronson:

Reversibility has been a central concept in physics since Galileo and Newton. Quantum mechanics says that the only exception to reversibility is when you take a measurement, and the Many-Worlders say not even that is an exception.

The reason that people are loose with the phrasing “information is always conserved” is because the “up until a measurement” is taken for granted as background knowledge. In general, the first things you learn about in a quantum mechanics class or textbook is what a superposition is, the Heisenberg uncertainty principle and then the Schrödinger equation.

For an explanation of the Schrödinger equation from Wolfram:

The Schrödinger equation is the fundamental equation of physics for describing quantum mechanical behavior. It is also often called the Schrödinger wave equation, and is a partial differential equation that describes how the wavefunction of a physical system evolves over time.

The Schrödinger equation explains how quantum states develop from one state to another. This evolution is completely deterministic and it is time reversible. Remember that a quantum state is described by a wave function $|\psi\rangle$, which is a collection of probability amplitudes. The Schrödinger equation states that any given wave function $|\psi_{t_0}\rangle$ at moment $t_0$ will evolve to become $|\psi_{t_1}\rangle$ at time $t_1$ unless a measurement is made before $t_1$ . This is a completely deterministic process and it is time reversible. Given $|\psi_{t_1}\rangle$ we can use the equation to calculate what $|\psi_{t_0}\rangle$ is equal to.

If the electron is in a superposition then the wave function will be so:

$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ where $\alpha$ and $\beta$ are equal to $\frac{1}{\sqrt{2}}$.

The state of an electron that is spin up is $|\psi\rangle = 1|1\rangle$. Clearly, a quantum state that is in a superposition of some observables is a valid ontological object. It behaves in a way completely different than an object that is collapsed into only one of the possibilities via a measurement. The problem of measurements, what they are and what constitutes one, is central to the interpretations of quantum mechanics. The most common view is that a measurement is made when the wave function collapses into one of its eigenstates. The Schrödinger equation provides a deterministic description of a state up to the point of a measurement.

Information, as defined by Susskind here, is always conserved up to the point of a measurement. This is because the Schrödinger equation describes the evolution of a quantum state deterministically up until a measurement.

The black hole information paradox can be succinctly stated as this:

Quantum states evolve unitarily governed by the Schrödinger equation. However, when a particle passes through the event horizon of a black hole and is later radiated out via Hawking radiation it is no longer in a pure quantum state (meaning a measurement was made). A measurement could not have been made because the equivalency principle of general relativity assures us that there is nothing special going on at the event horizon. How can all of this be true?

This paradox would not be a paradox if the laws of quantum mechanics didn't give a unitary, deterministic, evolution for quantum states up to a measurement. The reason being, if measurements are the only time unitarity breaks down and the equivalency principle tells us a measurement cannot be happening at the horizon of a black hole, how can unitarity break down and cause the Hawking radiation to be thermal and therefore uncorrelated with the in-falling information? Scott Aaronson gave a talk about quantum information theory and its application to this paradox as well as quantum public key cryptography. In it he explains

The Second Law says that entropy never decreases, and thus the whole universe is undergoing a mixing process (even though the microscopic laws are reversible).

[After having described how black holes seem to destroy infomration in contradiction to the second law] This means that, when bits of information are thrown into a black hole, the bits seem to disappear from the universe, thus violating the Second Law.

So let’s come back to Alice. What does she see? Suppose she knows the complete quantum state $|\psi\rangle$ (we’ll assume for simplicity that it’s pure) of all the infalling matter. Then, after collapse to a black hole and Hawking evaporation, what’s come out is thermal radiation in a mixed state $\rho$. This is a problem. We’d like to think of the laws of physics as just applying one huge unitary transformation to the quantum state of the world. But there’s no unitary U that can be applied to a pure state $|\psi\rangle$ to get a mixed state $\rho$. Hawking proposed that black holes were simply a case where unitarity broke down, and pure states evolved into mixed states. That is, he again thought that black holes were exceptions to the laws that hold everywhere else.

The information paradox was considered to be solved via Susskind's proposal of black hole complementarity and the holographic principle. Later AMPS showed that the solution is not as simple as it was stated and further work needs to be done. Currently the field of physics is engaged in an amazingly beautiful collection of ideas and solutions being proposed to solve the black hole information paradox as well as the AMPS paradox. At the heart of all of these proposals, however, is the belief that information is, conserved up to the point of a measurement.