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The partial transpose is not the only positive but not completely positive operation that is possible on 2x2 and 2x3 systems. Trivially, any completely positive operation (such as a local unitary) combined with the partial transpose is a different positive operation. The point is that, as wikipedia puts it every such map $\Lambda$ can be written as $...


4

The way that I think about this is to take a decomposition in Paulis, $$ \rho_z=(I+zZ\otimes Z-zX\otimes X-zY\otimes Y)/4. $$ I can group these as $$ ((1-3z)I+z(I+Z\otimes Z)+z(I-X\otimes X)+z(I-Y\otimes Y))/4. $$ Each of the 4 terms is diagonal in a separable basis, and positive semi-definite (provided $z\leq 1/3$). This directly implies a separable ...


4

The short answer is that $(\rho^{\otimes N})^{T_B}=(\rho^{T_B})^{\otimes N}$. More explicitly, if $\rho=\sum_{ii'jj'}\rho_{ii',jj'}|i\rangle\!\langle i'|\otimes |j\rangle\!\langle j'|$, then we can write $$\rho^{\otimes N}=\sum_{I I' JJ'}\rho_{II',JJ'}\bigotimes_{k=1}^N \Big(|i_k\rangle\!\langle i_k'|\otimes |j_k\rangle\!\langle j_k'|\Big),$$ where $I\equiv(...


3

This is certainly how theorists think of this being done. I don't know if there's an experimental reality to compare this to. Whether they actually decompose it in terms of the eigenvectors, or find some other terms to decompose it as. Just as an example of what I mean, let $$ W=\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & ...


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It looks like the only relation they say is $\text{IdentityMatrix}[3^2]=\sum P_{k,l}$. You get a linear combination of $P_{k,l}$. Those are vertices of a $d^2-1$ simplex so the coefficents $c_{k,l}$ are baryocentric coordinates. You can then match with the previous more general definition of $\rho_d$ term by term on each of the $c_{k,l}$. The inequalities ...


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