# Can the entropy of a subsystem exceed the maximum entropy of the system in quantum mechanics?

Quantum mechanics has a peculiar feature, entanglement entropy, allowing the total entropy of a system to be less than the sum of the entropies of the individual subsystems comprising it. Can the entropy of a subsystem exceed the maximum entropy of the system in quantum mechanics?

What I have in mind is eternal inflation. The de Sitter radius is only a few orders of magnitude larger than the Planck length. If the maximum entropy is given by the area of the boundary of the causal patch, the maximum entropy can't be all that large. Suppose a bubble nucleation of the metastable vacuum into another phase with an exponentially tiny cosmological constant happens. After reheating inside the bubble, the entropy of the bubble increases significantly until it exceeds the maximum entropy of the causal patch.

If this is described by entanglement entropy within the bubble itself, when restricted to a subsystem of the bubble, we get a mixed state. In other words, the number of many worlds increases exponentially until it exceeds the exponential of the maximum causal patch entropy. Obviously, the causal patch itself can't possibly have that many many-worlds. So, what is the best way of interpreting these many-worlds for this example?

Thanks a lot!

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Take state $|\psi\rangle=(|00\rangle+|11\rangle)/\sqrt{2}$. It is a pure state, so its (von Neumann) entropy is 0. But both of its one-particle states have entropy equal to 1 bit, as they are completely mixed states of the dimension two.