Bell-label Hamming distance and bit errors ($p_{\mathrm{bit}}=ϵw/2$)

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.