# Writing state $|\Psi⟩ =\frac{1}{\sqrt{2}}|00⟩+\frac{i}{\sqrt{2}}|01⟩$ as separate qubits (qiskit textbook)

While going through the IBM qiskit textbook online, I came across the following question in section 2.2:

Write the state: $$|\Psi⟩ =\frac{1}{\sqrt{2}}|00⟩+\frac{i}{\sqrt{2}}|01⟩$$ as two separate qubits.

I understand tensor products with qubits, but I don't know how to even begin this problem. Does anyone have some advice on how to separate a state into its constituent qubits?

As is the case with ordinary multiplication, tensor product distributes over addition, so we can pull $$|0\rangle$$ on the first qubit out in front

\begin{align} |\Psi⟩ &= \frac{1}{\sqrt{2}}|\color{red}{0}0\rangle+\frac{i}{\sqrt{2}}|\color{red}{0}1\rangle \\ &= \frac{1}{\sqrt{2}}\color{red}{|0\rangle}\otimes|0\rangle+\frac{i}{\sqrt{2}}\color{red}{|0\rangle}\otimes|1\rangle \\ &= \color{red}{|0\rangle}\otimes\left(\frac{1}{\sqrt{2}}|0\rangle+\frac{i}{\sqrt{2}}|1\rangle\right) \\ &= \color{red}{|0\rangle}\left(\frac{1}{\sqrt{2}}|0\rangle+\frac{i}{\sqrt{2}}|1\rangle\right) \end{align}

and what remains in parenthesis is the state of the second qubit.

Note that people generally tend to make tensor product signs $$\otimes$$ implicit. I marked them explicitly to highlight the distributive law familiar from ordinary multiplication.

Giving $$|\psi \rangle = \dfrac{1}{\sqrt{2}}|00\rangle + \dfrac{i}{\sqrt{2}}|01\rangle$$ we can see that the first qubit is in the state $$|0\rangle$$ so we can rewrite the state $$|\psi\rangle$$ as a tensor product:

$$|\psi \rangle = |0\rangle \otimes \bigg( \dfrac{|0\rangle + i|1\rangle}{\sqrt{2}}\bigg)$$

So the first qubit is in the state $$|0\rangle$$ and the second qubit is in the state $$\dfrac{|0\rangle + i|1\rangle}{\sqrt{2}}$$.

• That makes a lot of sense, actually. Thanks! Commented Jan 11, 2021 at 16:19

Other important methods to check if a state is a separable or entangled are the Peres-Horodecki criterion and Schmidt decomposition.

• Could you please the derivation for this decomposition.
– RSW
Commented Aug 16, 2022 at 7:52