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I've recently been introduced to the basics of finite-dimensional quantum mechanics from a purely mathematical point of view (with a quantum-information theme to it). When discussing quantum teleportation, the professor made a remark about how the fact that the probability of each Bell state measurement outcome is 1/4 independently of the measured state is fundamental for the protocol to work.

However, I fail to see what the mathematical reason for this is. Is there an easy explanation for this fact?

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When the probability is the same for all teleported states, the Bell state measurement does not give us any information about the teleported state. That is an important feature of the quantum teleportation protocols. If we got some partial information about the state, the teleportation fidelity would be lower than in the case of no information being revealed.

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Certainly it is not allowed for the measurement to gain information about the state. However, this by itself doesn't necessarily require equally likely measurement outcomes. –  Dan Stahlke Nov 5 '12 at 13:25

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