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What is the splitting structure of a state in thermal equilibrium in the many worlds interpretation? This is a mixed state, but we can perform a purification of it by doubling the system and forming a pure entangled state between both systems. Can any pure state always admit a natural splitting structure?

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A typical state which looks like thermal equlibrium by itself doesn't split, and cannot split. The splitting is a macroscopic notion, and it makes sense only if the state has some entropy production. The definition of splitting is by the experience of some observer, or robot, in the system, defined by Everett as a machine with memories. In order to store those memories, you need entropy production, so you enter new phase space all the time, and you enter this new phase space in branches, each of which has a consistent history of memories for the robot. This branching structure is not universal to quantum mechanics, it requires an embedding of an observing robot in the description.

In your example, however, once you purify the state of the macroscopic system, it is not thermal at all--- it has zero entropy. It is also a (perhaps macroscopic) system which has been entangled with another identical system perfectly, so as to mock up a thermal equilibrium state if you examine one but no the other. Your state is of the form

$$ |psi\rangle = \sum_i e^{-\beta E_i + i\phi_i} |E_i E_i\rangle $$

For some arbitrary phases $\phi$.

If the two systems are separate from each other and from anything else, the notion of splitting in this pure state is meaningless. It emerges if you entangle some other system with these, where there are states which describe a machine with a memory. Two different memory states of the machine are orthogonal, and each memory state includes a history, and is necessarily producing entropy. This is described in Everett's paper and thesis, and is essential to the description--- it picks out the preferred basis.

The embedding of the memory of the observing machine or observing human into the description is what produces the splitting, the splitting is a statement about which memory paths are present in the wavefunction with significant amplitudes. This doesn't make sense without a structure identifying which regions in wavefunction space correspond to which memories, and it is exactly analogous to the classical question "which atoms make up my memory, and which ones are just floating around inside me". Embedding your memory structure (or that of a robot, or a computer) requires mapping an abstract memory state described in terms of bits of memory and computational operations into the physical state space of a system somehow, and this map is what is producing superposed, or "split" observers.

So your question cannot be answered. If you open one system and look at it, and it is macroscopic, you will ruin the superposition with the other system, and the splitting will be incomprehensible and macroscopic. If you measure the energy of one system very precisely by entangling it with macroscopic pointer of some other system not in thermal equilibrium (or else the entanglement doesn't make sense), and look at the pointer to know the energy, you have reduced the unobserved system to a pure energy state relative to the pointer position, then you reduced the other system to a pure energy state. There is no difference in terms of practical consequences from the Copenhagen interpretation, so that I can't see any positivistic difference between Everett and Copenhagen. Everett is just more explicit about where the collapse is coming from when the observers are described by the theory, which is nice.

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In MWI all states are pure, and there are no thermal states. The latter would just be a coarse-grained description.

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The question is asking about a pure state. –  Ron Maimon Jul 23 '12 at 13:33
    
How can a pure state bei in thermal equilibrium? KMS states are not pure. The doubled system is unphysical (energy unbounded below), hence has no interpretation in MWI. –  Arnold Neumaier Jul 23 '12 at 13:34
    
It's two copies of two entangled macroscopic systems, any one of which is in thermal equilibrium if examined separately, $|\psi\rangle = e^{-\beta(E_i+E_j) + i\phi_i} |E_i,E_i\rangle $ where $|E_1,E_2\rangle$ is the state where system 1 has energy E_1 and system 2 has energy E_2. This is why it's an interesting question. –  Ron Maimon Jul 23 '12 at 13:38
    
@RonMaimon: I very much doubt that two macroscopic systems can ever be entangled in this way. –  Arnold Neumaier Jul 23 '12 at 13:51
    
Where did he say macroscopic? You can do this with two H atoms. There is no limit in principle, so you can ask what happens as the system gets larger. The issue here is the preferred basis for splitting, which is always what gets people riled up about many worlds. Everett adresses this explicitly and cogently with his memory-robot, and nobody else mentions this, because it's the mysticism that people don't like to be involved, that map between memory and physics. That's Everett, it's always there. –  Ron Maimon Jul 23 '12 at 14:09
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