t b a b a

Suppose the state we want to copy is | ψ〉+ | φ〉, it follows that,
However,
U(| ψ〉+ | φ〉) | e〉 = U | ψe〉 + U | φe〉 =| ψψ〉+ | φφ〉
| ψψ〉+ | φφ〉 �= (| ψ〉+ | φ〉) ⊗ (| ψ〉+ | φ〉)
So, the operation U failed to copy | ψ〉+ | φ〉. This theorem means that, to protect quantum
information from errors, we can not simply make backup copies of the original quantum
states.
The process of extracting information from a set of qubits is called quantum measurement.
We describe quantum measurements of a single qubit in the state α | 0〉 + β | 1〉 as yielding
the results 0 or 1 and leaving the qubit in the corresponding states | 0〉 or | 1〉, with respective
probabilities |α| 2 and |β| 2 .
Quantum measurements are subject to Born’s rule. This fundamental postulate of quan-
tum mechanics is typically formulated as:
Consider a system in state | ψ〉 and an observable A with eigenvectors { | αi〉 } and
eigenvalues { ai }. Then the probability for a measurement of the observable A to yield a
particular value ai is:
p(ai) =| 〈αi | ψ〉 | 2 .
The state then collapses into | αi〉 after the measurement.
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