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We know that measurements and interactions between particles and their environment conserve the net quantum information (the total between the system and the measurement device/environment) in the sense of von Neumann entropy, none is created or destroyed. Do we know much quantum information is contained in a typical macroscopic object, or how much the universe started off with?

I'm imagining a thought experiment where I start off with N fermions all initially in fully pure states, von Neumann entropy = 0. The particles can become entangled, emit photons etc but the net entropy of the system will not change. The system could separate itself back into an unentangled state, either if you wait long enough or maybe some smart algorithm that detangles the system, without requiring net information exchange with the outer world.

In contrast, I can also imagine starting with a system in an non-zero entropy state. You can create small pockets of low entropy (e.g. inside your favorite Stern-Gerlach experiment or quantum computer), but even maximally unentangling your states, reaching the fully unentangled state will be impossible. The von Neumann entropy is > 0, so states where the net entropy = 0 are fundamentally unattainable.

Therefore the question: which of these cases is our universe?

Do we have a way to estimate the net quantum information per particle?

We usually think of impure states as just entanglements with the environment, but is there something that would prevent the universe from being in an inherently impure state (not entangled with anything else), or what would be the consequence of such an impurity?

Daniel Itkis
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It is true that we can find some attempts to define an "universal wave function", like in Wheeler-DeWitt equation and in Many-Worlds Interpretation of quantum mechanics, but I don't think that any of these proposals could fit in your question exactly.

The problem to find a quantum state representing the whole universe as a quantum system is how much amount of data do we have to do it. If we assume that the universe is a quantum system, we could search for the best quantum state that represents it, and the concrete procedure to do it would be Quantum State Tomography.

Now, by Quantum State tomography, we only could attribute a single quantum state if the set of measurements that we use in the process would be Informationally complete. If this is not the case, it could be that two or infinitely other states fits to represent our data, and we don't have any scientific way to decide between then.

The universe is a macroscopic system, and the number of measurements we need to consider in order to have an informationally complete set of measurements is astonishingly big and impractical for any purposes.

But there is a way to circumvent it if we assume that the hole universe is in thermal equilibrium. If in the cosmological scale we assume an average temperature, we could say that the quantum state of the whole universe is the Gibbs State, given by (unormalized) $\rho = e^{-\beta \mathcal H}$, where $\beta = 1/k_BT$ and $\mathcal H$ is the Hamiltonian for the whole universe. Observe that we change one problem for another: if we want to do any meaningful calculation, we should diagonalize such Hamiltonian, but we don't even know what should be such Hamiltonian.

Even if we know a Hamiltonian describing the whole universe and for some miracle we discover how to diagonalize it, the von Neumann entropy will reduce to Boltzmann entropy and the quantum information in the universe will be no different than the classical information in it.

In fact, we neither now how to describe the whole universe with a single state or Hamiltonian. There is no meaning to talk about tomography and the number of possible states representing our knowledge (or lack of it) is uncountable. If I should end this answer providing some quantum state for the whole universe, I would say it is the maximally mixed state: it gives us the maximal entropy which fits well our knowledge about universal quantum properties.

Ruffolo
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