You seem to be answering your own question. Obviously there is far too much information to represent in a reasonably sized 2d or 3d plot, so no matter what you do, you will be losing something. If you want a plot that shows correlations between particles, you can do just that: For a spinchain of $N$ sites, create a $N\times N$ image, where the value of the pixel at each point $(i,j)$ is given by the expectation value of the operator $\sigma^A_i \sigma^B_j$, where $\sigma^{A/B}_{i/j} \in \{\sigma_X, \sigma_Y, \sigma_Z\}$. This gives you 9 possible plots.

It seems likely that you may only care about the case where $\sigma^A = \sigma^B$, in which case there are only 3 plots, and you could take the expectation values for $XX$, $YY$ and $ZZ$ as the R, G and B values for a colour plot.

This kind of thing produces things very similar to the type of plot you have in your question.

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