How Do You Compute the Density Matrix of a Bipartite State?

[Rayan]

TL;DR Summary

What is the easiest way to compute a density matrix of bipartite states?

If we for example have such a bipartite state:

$$ | \phi > = \frac{1}{2} [ |0>|0> + |1>|0> + |0>|1> + |1>|1> ] $$

What is the easiest way to compute a density matrix of bipartite states? Should I just compute it as it is? i.e:

$$ \rho = | \phi > < \phi | $$

Or should I convert to matrix form first? Any advice appreciated!

I tried to convert it to matrix form and got the following:

$$ | \phi > = \frac{1}{2}
\begin{bmatrix}
1 \\
1 \\
1 \\
1
\end{bmatrix} $$

and

$$ < \phi | = \frac{1}{2}
\begin{pmatrix}
1 & 1\\
1 & 1
\end{pmatrix}
$$

But then I don't think it is possible to compute the following outer product?

$$ \rho = \frac{1}{4}
\begin{bmatrix}
1 \\
1 \\
1 \\
1
\end{bmatrix} \cdot
\begin{pmatrix}
1 & 1\\
1 & 1
\end{pmatrix}
$$

 

[Haborix]

Things went wonky when you calculated the dual vector (bra) of the state. How did the complex conjugate transpose turn a column vector into a matrix?

 

[Demystifier]

Should I just compute it as it is? i.e:
$$ \rho = | \phi > < \phi | $$

Yes.