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I am reading this (https://arxiv.org/abs/1209.0062) article about constructing order parameters from reduced density matrix. The author is discussing long-range order by taking antiferromagnetic spin chain as an example. By considering state $$|\psi\rangle=\frac{1}{\sqrt{2}}(|\uparrow\downarrow\cdots\uparrow\downarrow\rangle +|\downarrow\uparrow\cdots\downarrow\uparrow\rangle )$$ reduced density matrix of a single spin is written as $$\rho_1= \begin{bmatrix} 1/2&0\\0&1/2 \end{bmatrix} $$ and reduced density matrix for two spins at sites $i$ and $j$ is written as $$\rho_{2e}= \begin{bmatrix} 1/2&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&1/2\\ \end{bmatrix} $$ for even $|i-j|$ and for odd $|i-j|$ $$\rho_{2o}= \begin{bmatrix} 0&0&0&0\\ 0&1/2&0&0\\ 0&0&1/2&0\\ 0&0&0&0\\ \end{bmatrix} $$ I want to rederive above three matrices.


My attempt:

Density matrix for a single site $i$ is defined as $$\rho_i=\sum_\mu p_\mu|\phi_{i\mu}\rangle \langle \phi_{i\mu} |$$ where $p_\mu$ is probability of pure state $|\phi_{i\mu}\rangle$. And for two sites $i, j$ $$\rho_{i\bigcup j}=\sum_{\mu,\nu} p_{\mu,\nu}|\phi_{i\mu,j\nu}\rangle\langle\phi_{i\mu,j\nu}|$$

For single spin, I was able to understand that the basis set can be {$|\uparrow\rangle,|\downarrow\rangle$}. And probability for each state is $1/2$. So, the reduced density matrix become $$\rho_1\equiv\rho_i=1/2(|\uparrow\rangle \langle \uparrow|+|\downarrow\rangle \langle \downarrow|)$$ and writing this in matrix form gives the required form i.e. $diag(1/2,1/2)$.

For two spins, well, I am unable to rederive it. I thought that the basis {$|\uparrow\uparrow\rangle,|\uparrow\downarrow\rangle,|\downarrow\uparrow\rangle,|\downarrow\downarrow\rangle$} can reproduce above matrices but I get a matrix $diag(1/4,1/4,1/4,1/4)$. Also, I can not understand how even and odd values of $|i-j|$ can have any impact on the matrices?

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    $\begingroup$ Do you know about the partial trace and how it relates to reduced density matrices? $\endgroup$ – DanielSank Oct 15 '18 at 16:11
  • $\begingroup$ @DanielSank I think I know but I do not understand how is it going to help here. Partial trace is basically tracing out the specific degree of freedom. For example, for two density matrices $\rho_A$ and $\rho_B$ the composite density matrix is $\rho_{AB}=\sum_{ijkl} p_{ijkl}|a_i\rangle\langle a_j| \otimes |b_k\rangle \langle b_l|$, then the trace over B will be $\sum_{ijkl} p_{ijkl} |a_i\rangle\langle a_j|\langle b_k | b_l \rangle$. Correct me if I am wrong. I am new to information theory. I would appreciate if you recommend me some books to understand this kind of mathematics. $\endgroup$ – Luqman Saleem Oct 16 '18 at 11:07

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