Show that the line state is separable

[maxi123]

Homework Statement

Consider the "line" state $rho=\frac{1}{d}\sum_{k=0}^{d-1}P_{0,k}$. Show for arbitrary d that the state is separable.

Relevant Equations

$P_{k,j}=|\Omega_{k,j}><\Omega_{k,j}|$ with $|\Omega_{k,j}>=(W_{k,j}\otimes \mathbb{1})\sum_{s=0}^{d-1}|s,s>$. $(W_{k,l}$ are Weyl-Operators)

I introduced the unitary transformation $U=U_a \otimes U_b$ with $(U_a\otimes 1):\;|s,s> \rightarrow\,\frac{1}{\sqrt{d}}\sum_t \omega^{ts}|t,s>$ und $(1\otimes U_b):\;|s,s> \rightarrow\,\frac{1}{\sqrt{d}}\sum_t \omega^{-ts}|s,t>$ $(\omega=e^{2\pi i/d}$) and let it act on the state in the following way $\frac{1}{d}\sum_{k=0}^{d-1}UP_{0,k}U^\dagger$. By doing so i showed that my density matrix is diagonal and therefore the state is separable. I'm not sure if I can do this transformation (under which the Bellstates are invariant) without changing the separablity/entanglement of the state.

 

[Nate810]

Yes, you can do this transformation without changing the separability/entanglement of the state. This is because the unitary transformation preserves the entanglement of the state. In particular, if a state is separable before the transformation, it will remain separable after the transformation.