Suppose H⊗3|ϕ⟩=12√(|000⟩+|111⟩). What is |ϕ⟩? Specify the probability amplitude for each component:
i have same doubt.
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0.5, 0, 0, 0.5, 0, 0.5, 0.5, 0
To determine the state |ϕ⟩, we need to find the individual components of the state vector and their corresponding probability amplitudes.
Given that H⊗3|ϕ⟩ = 1/2√(|000⟩ + |111⟩), let's expand this expression using the tensor product notation:
H⊗3|ϕ⟩ = 1/2√(|0⟩⊗|0⟩⊗|0⟩ + |1⟩⊗|1⟩⊗|1⟩)
Now, we can rewrite this expression in terms of the individual components:
\[
H⊗3|ϕ⟩ = 1/2√( |0⟩⊗|0⟩⊗|0⟩ + |1⟩⊗|1⟩⊗|1⟩ )
\]
From here, we can separate the tensor products and rewrite them as tensor products on separate qubits:
\[
= 1/2√( |0⟩⊗|0⟩⊗|0⟩ + |1⟩⊗|1⟩⊗|1⟩ )
= 1/2√( |0⟩⊗|0⟩⊗|0⟩ ) + 1/2√( |1⟩⊗|1⟩⊗|1⟩ )
= 1/(2√2)( |0⟩⊗|0⟩⊗|0⟩ ) + 1/(2√2)( |1⟩⊗|1⟩⊗|1⟩ )
= 1/(2√2)|000⟩ + 1/(2√2)|111⟩
\]
To find the probability amplitude for each component, we need to normalize the state vector. We can do this by dividing each component by the square root of the sum of the absolute squares of all components.
Let's calculate the normalization factor first:
\[
\text{Normalization factor} = \sqrt{ \frac{1}{(2\sqrt{2})^2} + \frac{1}{(2\sqrt{2})^2} }
= \sqrt{\frac{1}{8} + \frac{1}{8}}
= \sqrt{\frac{2}{8}}
= \sqrt{\frac{1}{4}}
= \frac{1}{2}
\]
Now, we can divide each component by the normalization factor:
\[
\text{Normalized state vector } |ϕ⟩ = \frac{1}{2√2} \left( |000⟩ + |111⟩ \right)
\]
Hence, the state |ϕ⟩ is given by:
\[
|ϕ⟩ = \frac{1}{2√2} \left( |000⟩ + |111⟩ \right)
\]
The probability amplitude for each component is 1/(2√2).