I'm having trouble with the mathematics of tensor products as applied to Bell states.(adsbygoogle = window.adsbygoogle || []).push({});

Say I have the state

[tex]

\begin{align*}

\left|\psi\right> &= \frac{1}{\sqrt{2}} \left(\left|0\right>_A \otimes \left|0\right>_B + \left|1\right>_A \otimes \left|1\right>_B\right)

\end{align*}

[/tex]

How would the tensor products be expanded here? Are the states [itex]\left|0\right>_A[/itex] etc one dimensional?

I'd usually expand tensor products like

[tex]

\left|\psi\right> \otimes\left|\phi\right> = \left(\matrix{\psi_1\\\psi_2}\right) \otimes \left(\matrix{\phi_1\\\phi_2}\right) = \left(\matrix{\psi_1\phi_1\\\psi_1\phi_2\\\psi_2\phi_1\\\psi_2\phi_2}\right)

[/tex]

In this case, is it just

[tex]\left|0\right>_A \otimes\left|0\right>_B = \left(\matrix{0_A0_B}\right)[/tex] ?

And [itex]\left(0_A0_B\right)[/itex] is equivalent to [itex]\left|00\right>[/itex] and so on by convention of notation(this makes me slightly uncomfortable; can any scalar be denoted a ket?), so that the original state can be written

[tex]

\begin{align*}

\left|\psi\right> &= \frac{1}{\sqrt{2}} \left(\left|00\right> + \left|11\right>\right)

\end{align*}

[/tex]

Was my working here correct? Or is there some funky [itex]\left|0\right> = \left(\matrix{1\\0}\right)[/itex] business going on, like I've seen in places?

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# I Mathematics of tensor products in the Bell states

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