Two-component Bell resource in dense coding ($|\chi ⟩=a|{\Phi }^{+}⟩+b|{\Phi }^{-}⟩,|a{|}^2+|b{|}^2=1$)

With phase/parity labels $(u,v)$ encoded by $X^v{Z}^u$, a resource $a|{\Phi }^{+}⟩+b|{\Phi }^{-}⟩$ gives intended-label probability $|a{|}^2$ and phase-label-flip probability $|b{|}^2$ under a Bell-basis measurement. This follows because the local Pauli operator maps the two orthogonal components into the two Bell states whose phase label differs. The resource is a coherent pure state, even though its measured label statistics form a classical noisy channel.

If a wrong Bell-label partner occurs with probability $ϵ$ and differs from the intended two-bit label in $w$ positions, its contribution to the error probability of a uniformly selected bit is $ϵw/2$. Relabeling the Bell states can change $w$ while preserving the physical wrong-state probability. Thus symbol error probability and bit error probability are different quantities.