I’m a firm believer in the many-worlds interpretation, since one man’s “wavefunction collapse” is another man’s entanglement (i.e. decoherence), but some nagging details have been bothering me recently. In particular, many-worlds does not seem to answer the fundamental question of why humans only perceive one branch of the wavefunction at a time. If we take it as a fundamental postulate of the theory, then we’re in no better position than the Copenhagen interpretation. Is it just something to be chalked up to human perception, to be thrown in some vague realm of philosophy? Well, after thinking for far too long, I realized that it is in general impossible to measure a superposition state, as it violates linearity.

To see this, consider a two-level system in a state , as well as some observer system in a state , so that the combined state before the interaction is

.

In a standard measurement with full decoherence, the wavefunction after measurement would be

Suppose the observer has some way of sensing both branches, i.e. measuring and . Then they can communicate this information by preparing a state . For example, they may send out a photon of wavelength . Then the full interaction is

.

It is important that we can choose the orthogonal for all different values of and (up to normalization, as in the photon case). This interaction clearly violates linearity, as we have

,

,

so that

which is definitely not what we wanted!

Decoherence is not really required; in fact the combined observer plus two-level system could evolve to any state . The crucial point is that the observer’s wavefunction does not depend on and before the interaction, so we can use linearity. This means that we cannot use the same argument for the observer that prepared the two-level system, as their wavefunction must include the knowledge of the system wavefunction!

I feel that this should be some basic well-known theorem, but it doesn’t seem to match with any of the common no-go theorems such as no-cloning.

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