How to calculate density matrix for the GHZ state

[Agrippa]

The GHZ state is:

|\psi> = \frac{|000> + |111>}{\sqrt2}

To calculate density matrix we go from:

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

To:

GHZ<br /> = 1/2[<br /> \left( \begin{array}{cc}<br /> 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> \end{array} \right)<br /> +<br /> \left( \begin{array}{cc}<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1\\<br /> \end{array} \right)<br /> +<br /> \left( \begin{array}{cc}<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> \end{array} \right)<br /> +<br /> \left( \begin{array}{cc}<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> \end{array} \right)<br /> <br /> ]

And finally to:

<br /> GHZ = 1/2\left( \begin{array}{cc}<br /> 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1\\<br /> \end{array} \right)<br />

But I see another author (p2) separates the Hilbert space into two subsystems GHZ_A⊗GHZ_BC and gets a "reduced" density matrix:

<br /> GHZ_A⊗GHZ_{BC} = 1/4\left( \begin{array}{cc}<br /> 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1\\<br /> \end{array} \right)<br />

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

 

[jfizzix]

That density matrix represents the mixed state:
\frac{1}{4}\big(|0,0,0\rangle\langle 0,0,0|+|0,1,1\rangle\langle 0,1,1|+|1,0,0\rangle\langle 1,0,0|+|1,1,1\rangle\langle 1,1,1|\big)
or more simply:
\frac{1}{4}\big(|0\rangle\langle 0|+|1\rangle\langle 1|\big)\otimes\big(|0,0\rangle\langle 0,0|+|1,1\rangle\langle 1,1|\big)
To figure that out, system A separates the density matrix into four equal square blocks (of size 4x4)
Of each block, system B separates the density matrix into four equal square sub-blocks (of size 2x2)
Of each sublock, the state of system C associated to those particular amplitudes of A and B are given.

 

[Agrippa]

That density matrix represents the mixed state:
\frac{1}{4}\big(|0,0,0\rangle\langle 0,0,0|+|0,1,1\rangle\langle 0,1,1|+|1,0,0\rangle\langle 1,0,0|+|1,1,1\rangle\langle 1,1,1|\big)

Thanks. I can (roughly, without yet drawing it up) see how you've constructed this expression from the final matrix I drew up.

or more simply:
\frac{1}{4}\big(|0\rangle\langle 0|+|1\rangle\langle 1|\big)\otimes\big(|0,0\rangle\langle 0,0|+|1,1\rangle\langle 1,1|\big)

Hmmm, I'm trying to work through this equation but am getting the wrong result:
<br /> 1/4\left( \begin{array}{cc}<br /> 1 &amp; 0\\<br /> 0 &amp; 1\\<br /> \end{array} \right)<br /> \otimes<br /> 1/4\left( \begin{array}{cc}<br /> 1 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 1\\<br /> \end{array} \right)<br /> = 1/4\left( \begin{array}{cc}<br /> 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1\\<br /> \end{array} \right)<br />
But perhaps I'm just making a simple mistake somewhere. At least, I can kind of see what you've done: you've tried to find a density matrix that corresponds to system A, and a density matrix corresponding to system BC, such that putting them together ($\otimes$) gives you the final matrix from my first post.

To figure that out, system A separates the density matrix into four equal square blocks (of size 4x4)
Of each block, system B separates the density matrix into four equal square sub-blocks (of size 2x2)
Of each sublock, the state of system C associated to those particular amplitudes of A and B are given.

Unfortunately I don't quite follow you here. In particular, the talk of systems A and B separating the matrix into blocks, and the talk of the state of C being given (but not A or B (or A versus BC)?). Does it relate in some way to the calculation I just made above?
My main confusion is this: take the original density matrix for the pure state. I understand that the diagonal terms represent probabilities for measurement outcomes: 0.5 and 0.5. I also understand (to a lesser degree) that the off-diagonal terms represent the system's ability to exhibit interference. So the density matrix for a pure state like GHZ has a clear interpretation. But how does one interpret the "reduced" density matrix for GHZ_A⊗GHZ_BC?

 

[jfizzix]

Hmmm, I'm trying to work through this equation but am getting the wrong result:
<br /> 1/4\left( \begin{array}{cc}<br /> 1 &amp; 0\\<br /> 0 &amp; 1\\<br /> \end{array} \right)<br /> \otimes<br /> 1/4\left( \begin{array}{cc}<br /> 1 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 1\\<br /> \end{array} \right)<br /> = 1/4\left( \begin{array}{cc}<br /> 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1\\<br /> \end{array} \right)<br />
But perhaps I'm just making a simple mistake somewhere.

It is a simple mistake, easily rectified.
What's here the kronecker product in reverse order. If X is the first matrix you have here, and Y is the second matrix, then to get X\otimes Y, you should make a big matrix out of (in this case 4) copies of Y, and then multiply each copy of Y by the corresponding entry in X.
It looks like that you've done instead is make a big matrix out of... 16 copies of X, and then multiplied them by the corresponding entries of Y.
In short, you've found Y\otimes X instead of X\otimes Y.

 

[Agrippa]

It is a simple mistake, easily rectified.
What's here the kronecker product in reverse order. If X is the first matrix you have here, and Y is the second matrix, then to get X\otimes Y, you should make a big matrix out of (in this case 4) copies of Y, and then multiply each copy of Y by the corresponding entry in X.
It looks like that you've done instead is make a big matrix out of... 16 copies of X, and then multiplied them by the corresponding entries of Y.
In short, you've found Y\otimes X instead of X\otimes Y.

Yes I see the mistake now, thanks. What I still don't understand is the relationship between original density matrix and the reduced density matrix. In particular, we know that the density matrix of the GHZ state is:
\frac{1}{2}( |000&gt;&lt;000| + |111&gt;&lt;111| + |111&gt;&lt;000| + |000&gt;&lt;111|)
We now want to break the system down into two systems (A and BC). But how do we know that this:
\frac{1}{4}\big(|0\rangle\langle 0|+|1\rangle\langle 1|\big)\otimes\big(|0,0\rangle\langle 0,0|+|1,1\rangle\langle 1,1|\big)
is a valid way to decompose the system? Why, for example, do we associate $\frac{1}{4}\big(|0\rangle\langle 0|+|1\rangle\langle 1|\big)$ with subsystem A? How do we know that this expression in any way represents subsystem A?

 

[jfizzix]

Yes I see the mistake now, thanks. What I still don't understand is the relationship between original density matrix and the reduced density matrix. In particular, we know that the density matrix of the GHZ state is:
\frac{1}{2}( |000&gt;&lt;000| + |111&gt;&lt;111| + |111&gt;&lt;000| + |000&gt;&lt;111|)
We now want to break the system down into two systems (A and BC). But how do we know that this:
\frac{1}{4}\big(|0\rangle\langle 0|+|1\rangle\langle 1|\big)\otimes\big(|0,0\rangle\langle 0,0|+|1,1\rangle\langle 1,1|\big)
is a valid way to decompose the system? Why, for example, do we associate $\frac{1}{4}\big(|0\rangle\langle 0|+|1\rangle\langle 1|\big)$ with subsystem A? How do we know that this expression in any way represents subsystem A?

The product
\frac{1}{4}\big(|0\rangle\langle 0|+|1\rangle\langle 1|\big)\otimes\big(|0,0\rangle\langle 0,0|+|1,1\rangle\langle 1,1|\big)

is not a valid decomposition of the GHZ state:
\frac{1}{2}( |000&gt;&lt;000| + |111&gt;&lt;111| + |111&gt;&lt;000| + |000&gt;&lt;111|)

The GHZ state is not separable into such a product or any other product. It's about as entangled as three-particle spin states get.

If one were to measure particle A of the GHZ state (in the 0/1 basis), the state one would get as a result would be:
\frac{1}{2}\big(|0\rangle\langle 0|\otimes|0,0\rangle\langle 0,0|+|1\rangle\langle 1|\otimes |1,1\rangle\langle 1,1|\big)

 

[Agrippa]

The GHZ state is not separable into such a product or any other product. It's about as entangled as three-particle spin states get.

But in decoherence theory, isn't it standard to separate an entangled state into system state and environment state? I thought that something like that must be going on here. For example, I thought that GHZ_A⊗GHZ_BC amounted to something like treating systems B&C as the environment. Is that not right? I'm trying to figure out how to calculate the (reduced) density matrix called 'GHZ_A⊗GHZ_BC' from just the density matrix of the GHZ state.

(In case it helps, I'm ultimately trying to understand a particular definition of (integrated) information of a quantum state (see e.g. page 2), which is calculated via a function of the density matrix of the state and the reduced density matrices of its decompositions.)

If one were to measure particle A of the GHZ state (in the 0/1 basis), the state one would get as a result would be:
\frac{1}{2}\big(|0\rangle\langle 0|\otimes|0,0\rangle\langle 0,0|+|1\rangle\langle 1|\otimes |1,1\rangle\langle 1,1|\big)

That's useful, thanks. I've calculated it as follows:
<br /> 1/2\left( \begin{array}{cc}<br /> 1 &amp; 0\\<br /> 0 &amp; 0\\<br /> \end{array} \right)<br /> \otimes<br /> 1/2\left( \begin{array}{cc}<br /> 1 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0\\<br /> \end{array} \right)<br /> +<br /> 1/2\left( \begin{array}{cc}<br /> 0 &amp; 0\\<br /> 0 &amp; 1\\<br /> \end{array} \right)<br /> \otimes<br /> 1/2\left( \begin{array}{cc}<br /> 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 1\\<br /> \end{array} \right)<br /> = 1/2\left( \begin{array}{cc}<br /> 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1\\<br /> \end{array} \right)<br />
I interpret the diagonal values as representing probabilities (0.5 chance for |000> and 0.5 chance for |111>) and the off-diagonal values as capacity for interference, which is zero given that we now have an eigenstate (either |000> or |111>).
But then I wonder what the two additional 1's on the diagonal of GHZ_A⊗GHZ_BC represent; and why the diagonal contains all ones in the case of GHZ_A⊗GHZ_B⊗GHZ_C = I/8.