- #1

Agrippa

- 78

- 10

The GHZ state is:

[itex] |\psi> = \frac{|000> + |111>}{\sqrt2} [/itex]

To calculate density matrix we go from:

[itex] GHZ = \frac{1}{2}(|000> + |111>)(<000| + <111|) [/itex]

[itex] GHZ = \frac{1}{2}( |000><000| + |111><111| + |111><000| + |000><111|) [/itex]

To:

[itex] GHZ

= 1/2[

\left( \begin{array}{cc}

1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

\end{array} \right)

+

\left( \begin{array}{cc}

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\

\end{array} \right)

+

\left( \begin{array}{cc}

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

\end{array} \right)

+

\left( \begin{array}{cc}

0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

\end{array} \right)

] [/itex]

And finally to:

[itex]

GHZ = 1/2\left( \begin{array}{cc}

1 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

1 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\

\end{array} \right)

[/itex]

But I see another author (p2) separates the Hilbert space into two subsystems GHZ

[itex]

GHZ_A⊗GHZ_{BC} = 1/4\left( \begin{array}{cc}

1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\

\end{array} \right)

[/itex]

Can anyone explain what this final matrix represents, and how one calculates it?

[itex] |\psi> = \frac{|000> + |111>}{\sqrt2} [/itex]

To calculate density matrix we go from:

[itex] GHZ = \frac{1}{2}(|000> + |111>)(<000| + <111|) [/itex]

[itex] GHZ = \frac{1}{2}( |000><000| + |111><111| + |111><000| + |000><111|) [/itex]

To:

[itex] GHZ

= 1/2[

\left( \begin{array}{cc}

1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

\end{array} \right)

+

\left( \begin{array}{cc}

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\

\end{array} \right)

+

\left( \begin{array}{cc}

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

\end{array} \right)

+

\left( \begin{array}{cc}

0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

\end{array} \right)

] [/itex]

And finally to:

[itex]

GHZ = 1/2\left( \begin{array}{cc}

1 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

1 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\

\end{array} \right)

[/itex]

But I see another author (p2) separates the Hilbert space into two subsystems GHZ

_{A}⊗GHZ_{BC}and gets a "reduced" density matrix:[itex]

GHZ_A⊗GHZ_{BC} = 1/4\left( \begin{array}{cc}

1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\

\end{array} \right)

[/itex]

Can anyone explain what this final matrix represents, and how one calculates it?

Last edited: