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I know that 2 qubits are entangled if it is impossible to represent their joint state as a tensor product. But when we are given a joint state, how can we tell if it is possible to represent it as a tensor product? For example, I am asked to tell if the qubits are entangled for each of the following situations:

$$\begin{align} \left| 01 \right>\\ \frac 12(\left| 00 \right> + i\left| 01 \right> - i\left| 10 \right> + i\left| 11 \right> )\\ \frac 12(\left| 00 \right> - \left| 11 \right>)\\ \frac 12(\left| 00 \right> + \left| 01 \right> +i\left| 10 \right> + \left| 11 \right> ) \end{align}$$

Sanchayan Dutta
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PiMan
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2 Answers2

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If you are given a general 2-qubit state $a \mid 00 \rangle + b \mid 01 \rangle + c \mid 10 \rangle + d \mid 11 \rangle$

If it is unentangled, then the coefficients are that of $(\alpha \mid 0 \rangle + \beta \mid 1 \rangle)(\gamma\mid 0 \rangle + \delta \mid 1 \rangle)$ for some $\alpha .. \delta$.

$$ \alpha \gamma = a\\ \alpha \delta = b\\ \beta \gamma = c\\ \beta \delta = d $$

You want to know if those 4 equations are solvable for a given $a,b,c,d$. This question becomes

$$ ad - bc = 0 $$

so if $ad-bc=0$, then you can solve for $\alpha .. \delta$. You don't need to solve for them, you just need to need to know if it is possible.

The generalization for qudits with potentially different values of $d_1$ and $d_2$ are the quadratic polynomials that cut out the Segre embedding as a zero locus.

AHusain
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It is done for a specific state (a Bell state) here, and the same procedure can be used for any other two-qubit state.

user1271772 No more free time
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