Suppose H|ψ> = 1/√2 (|000> + |111>). What is |ψ> ? Specify the probability amplitude for each component:
|000> =
|001> =
|010> =
|011> =
|100> =
|101> =
|110> =
|111> =
To determine the components of the state |ψ>, we need to apply the operator H to the state |ψ>. The operator H is known as the Hadamard gate.
The Hadamard gate is defined as:
H = 1/√2 (|0> + |1>) (in the standard basis)
Therefore, to apply H to each qubit in |ψ>, we need to apply the Hadamard gate to each corresponding qubit in the state |ψ>.
Let's start with the first qubit:
H|0> = 1/√2 (|0> + |1>) = 1/√2 (|0> + |1>)
H|1> = 1/√2 (|0> - |1>) = 1/√2 (|0> - |1>)
Now, let's apply the Hadamard gate to the second qubit:
H|0> = 1/√2 (|0> + |1>) = 1/√2 (|0> + |1>)
H|1> = 1/√2 (|0> - |1>) = 1/√2 (|0> - |1>)
Finally, let's apply the Hadamard gate to the third qubit:
H|0> = 1/√2 (|0> + |1>) = 1/√2 (|0> + |1>)
H|1> = 1/√2 (|0> - |1>) = 1/√2 (|0> - |1>)
Therefore, the state |ψ> can be written as:
|ψ> = 1/√2 (|0> + |1>) ⊗ 1/√2 (|0> - |1>) ⊗ 1/√2 (|0> + |1>)
= 1/2 (|000> + |010> + |100> + |110> - |001> - |011> - |101> - |111>)
Now, let's determine the probability amplitude for each component:
For |000>, the probability amplitude is 1/2.
For |001>, the probability amplitude is -1/2.
For |010>, the probability amplitude is 1/2.
For |011>, the probability amplitude is -1/2.
For |100>, the probability amplitude is 1/2.
For |101>, the probability amplitude is -1/2.
For |110>, the probability amplitude is 1/2.
For |111>, the probability amplitude is -1/2.
Therefore, the probability amplitude for each component in the state |ψ> is:
|000> = 1/2
|001> = -1/2
|010> = 1/2
|011> = -1/2
|100> = 1/2
|101> = -1/2
|110> = 1/2
|111> = -1/2