Werner state

Example

Let $d \ge 2$ and let $F$ be the swap operator on $\C^d \otimes \C^d$, defined by $F(u \otimes v) = v \otimes u$. Let $P_{s} = \tfrac12(I + F)$ and $P_{a} = \tfrac12(I - F)$ be the projections onto the symmetric and antisymmetric subspaces. Their ranks are $d(d+1)/2$ and $d(d-1)/2$. For $0 \le p \le 1$, the Werner state with parameter $p$ is the density matrix $$\rho_p = p \, \frac{2 P_a}{d(d-1)} + (1 - p) \, \frac{2 P_s}{d(d+1)}.$$ It commutes with $U \otimes U$ for every unitary $U$ on $\C^d$, and every state with this symmetry is of this form.

Werner states are separable exactly when $p \le \tfrac12$, and entangled when $p > \tfrac12$. A quick check of the trace of $F \rho_p$: $\tr(F P_a) = -\tr P_a$ and $\tr(F P_s) = \tr P_s$, so $\tr(F \rho_p) = 1 - 2p$. For a product state, $\tr(F \, \sigma \otimes \tau) = \tr(\sigma \tau) \ge 0$, so every separable state $\rho$ has $\tr(F\rho) \ge 0$. Hence $\rho_p$ is entangled for $p > \tfrac12$. For $p \le \tfrac12$ one shows separability by averaging a suitable product state over the group of all $U \otimes U$.