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GATE 2026 PH – Question 21

Mathematical Physics · Linear vector spaces: basis, orthogonality and completeness · 1 mark · Multiple choice

Given $$|v_1\rangle=\frac1{\sqrt2}\begin{pmatrix}1\\i\end{pmatrix},\quad |v_2\rangle=\frac1{\sqrt2}\begin{pmatrix}1\\-i\end{pmatrix},$$ the tensor product $|v_1\rangle\otimes|v_2\rangle$ is

  1. $$\frac12\begin{pmatrix}1\\-i\\i\\1\end{pmatrix}$$
  2. $$\frac12\begin{pmatrix}1\\i\\-i\\1\end{pmatrix}$$
  3. $$\frac12\begin{pmatrix}1&i\\i&-1\end{pmatrix}$$
  4. $$\frac12\begin{pmatrix}1&i\\-i&1\end{pmatrix}$$

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Correct answer: (A) $$\frac12\begin{pmatrix}1\\-i\\i\\1\end{pmatrix}$$

Explanation

The tensor (Kronecker) product of two column vectors $\begin{pmatrix}a\\b\end{pmatrix}\otimes\begin{pmatrix}c\\d\end{pmatrix}$ is the 4-component vector $(ac,\ ad,\ bc,\ bd)^T$.

With $|v_1\rangle=\dfrac1{\sqrt2}\begin{pmatrix}1\\i\end{pmatrix}$ and $|v_2\rangle=\dfrac1{\sqrt2}\begin{pmatrix}1\\-i\end{pmatrix}$:
- $ac=1\cdot1=1$
- $ad=1\cdot(-i)=-i$
- $bc=i\cdot1=i$
- $bd=i\cdot(-i)=1$

Include the factor $\dfrac1{\sqrt2}\cdot\dfrac1{\sqrt2}=\dfrac12$:
$$|v_1\rangle\otimes|v_2\rangle=\frac12\begin{pmatrix}1\\-i\\i\\1\end{pmatrix}\quad(\text{option A}).$$

(Options C and D are $2\times2$ matrices, which would be an outer product, not a tensor product vector.)