How do I represent a Werner state in ket notation?

Question

I want to represent Werner state in the form of ket vector notation. Is there a way in which I can represent it in vector form?

Answer 1

From quantiki we have $$\rho=p_{\mathrm{sym}}\frac{2\hat{P}_{\mathrm{sim}}}{d^2+d}+(1-p_{\mathrm{sym}})\frac{2\hat{P}_{\mathrm{anti-sym}}}{d^2-d}, $$ where $p_{\mathrm{sym}}$ is the probability of being in the symmetric subspace, $d$ is the dimension of the state vectors, the projectors onto the symmetric and anti-symmetric subspaces are given by $$\hat{P}_{\mathrm{sym}}=\frac{1+\hat{P}}{2}$$ and $$\hat{P}_{\mathrm{anti-sym}}=\frac{1-\hat{P}}{2},$$ and, finally, we can connect everything to state vectors through the definition of the exchange operator $$\hat{P}=\sum_{i,j}|i\rangle\langle j|\otimes |j\rangle\langle i|.$$ Here we have used an arbitrary set of basis states for each system.