Proposition: Explicit Smolin identification

Let $R_0=H{S}^{†}$ and $T=Y_R\otimes R_{0,1}\otimes I_2\otimes R_{0,3}$. Then

where $\beta$ runs over the four Bell states.

Proof. The right side is $(I+X^{\otimes 4}+Y^{\otimes 4}+Z^{\otimes 4})/16$. Conjugation by $R_0$ cycles $X\mapsto Y\mapsto Z\mapsto X$. The two rail rotations give signs $-,+,-$. Conjugation by $Y_R$ reverses the first and third signs.