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This means that there is only one classical state, namely H. It is the classical state that corresponds to 'considering the quantum entity under study' and the state does not specify anything more than that. For the classical case, every state is a classical state. It can be proven that nc : C —+ V(il) is an isomorphism 1 ' 11 . This means that if we filter out the classical part and limit the description of our general physical entity to its classical properties and classical states, the description becomes a standard classical physical description.

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Again considering our two examples, it is easy to see that for the quantum case, hence for £ = V(H), we have no nontrivial classical properties. Indeed, for any closed subspace A € H, different from 0 and H, we have rays of "H that are neither contained in A nor contained in A'. These are exactly the rays that correspond to states that are superposition states of states in A and states in A'. It is the superposition principle in standard quantum mechanics that makes that the only classical properties of a quantum entity are the trivial ones, represented by 0 and H.

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Adiabatic quantum transport. by Avron


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