Conservation of Distinctions in Quantum Mechanics

Question

Recently I have been reading Quantum Mechanics The Theoretical Minimum by Leonard Susskind. In the book he mentions the law of conservation of distinctions, i.e. the conservation of information.
He mentions that if two isolated systems start at  different states, they will continue to stay in different states. So say I have two different systems in states $|A⟩$ and $|B⟩$, and after some time they reach states $|A'⟩$ and $|B'⟩$. Since the states were distinct $⟨B|A⟩=0$, and since distinctions are conserved $⟨B'|A'⟩=0$
Does it mean that the measurement we make are also distinct? (for instance let's say that the state A is spin up and state B is spin down initially, so does it mean that the spins of both the systems will be different when we measure them again?)

Entropy · Accepted Answer

In Quantum Mechanics, two systems being in distinct states doesn't necessarily mean that they show distinct observations upon being acted by various operators. Consider the following scenario where the observable $L$ acts on the states $|A'⟩$ and $|B'⟩$. This may be written as follows.
$$L|A'⟩=alpha_i|lambda_i⟩$$
$$L|B'⟩=beta_j|lambda_j⟩$$
Where $|lambda_i⟩$ are the Eigen vectors of the observable $L$. Now on measurement, this would lead to the state to collapse into one of the Eigen vectors, and as there is no condition that none of the Eigen vectors that represents state $|A'⟩$ also represent $|B'⟩$ (this is the important part, make sure you understand this) it means that there is a possibility that the observation of both states might lead to the same state.

Conservation of Distinctions in Quantum Mechanics

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