Imagine Bob works at a qubit recycling plant. He receives qubits, rotates them by 45 degrees, and measures them. Whenever he gets 1 as a result, he’s happy.
On Monday Bob gets a bunch of qubits in state |0>. Rotating them by 45 degrees gives ( |0> + |1> ) / √2. When measured, half of those yield 0 and half yield 1. So on Monday Bob is happy 50% of the time.
On Tuesday Bob gets a bunch of qubits in state |1>. Rotating them by 45 degrees gives ( |1> - |0> ) / √2. When measured, half of those yield 0 and half yield 1, just like on Monday. So on Tuesday Bob is happy 50% of the time.
On Wednesday Bob gets a bunch of qubits in state ( |0> + |1> ) / √2. Isn’t that just a mixture of qubits from Monday and Tuesday, so Bob should be happy 50% of the time? Nope! Rotating this state by 45 degrees gives |1>, which always yields 1 when measured. So on Wednesday Bob is happy all the time.
That’s why amplitudes are different from probabilities. If Wednesday’s qubits were a probability-based mixture of Monday’s and Tuesday’s, only half of them would make Bob happy. But with an amplitude-based mixture, all of them make Bob happy.
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crabman asked:
Imagine Bob works at a qubit recycling plant. He receives qubits, rotates them by 45 degrees, and measures them. Whenever he gets 1 as a result, he’s happy.
On Monday Bob gets a bunch of qubits in state |0>. Rotating them by 45 degrees gives ( |0> + |1> ) / √2. When measured, half of those yield 0 and half yield 1. So on Monday Bob is happy 50% of the time.
On Tuesday Bob gets a bunch of qubits in state |1>. Rotating them by 45 degrees gives ( |1> - |0> ) / √2. When measured, half of those yield 0 and half yield 1, just like on Monday. So on Tuesday Bob is happy 50% of the time.
On Wednesday Bob gets a bunch of qubits in state ( |0> + |1> ) / √2. Isn’t that just a mixture of qubits from Monday and Tuesday, so Bob should be happy 50% of the time? Nope! Rotating this state by 45 degrees gives |1>, which always yields 1 when measured. So on Wednesday Bob is happy all the time.
That’s why amplitudes are different from probabilities. If Wednesday’s qubits were a probability-based mixture of Monday’s and Tuesday’s, only half of them would make Bob happy. But with an amplitude-based mixture, all of them make Bob happy.