= Solution
The actual resource is a <two-component Bell resource in dense coding>. Applying the same local encoder $X^vZ^u$ to
$$
|\chi\rangle=\frac{2|\Phi^+\rangle+|\Phi^-\rangle}{\sqrt5}
$$
gives a superposition of the intended <Bell state> with amplitude magnitude $2/\sqrt5$ and its phase-label partner with amplitude magnitude $1/\sqrt5$. A <Bell-basis measurement> therefore reports \b[the wrong Bell state with probability $1/5$], independently of $uv$. With our labeling, only the phase bit $u$ flips:
$$
(\widehat u,\widehat v)=(u\oplus N,v),\qquad
P(N=1)=\frac15.
$$
The first bit of each pair passes through a <binary symmetric channel> with error $1/5$; the second bit is correct. A uniformly chosen bit from the full stream consequently has
$$
\boxed{P_{\rm bit\ error}=\frac12\cdot\frac15=\frac1{10}.}
$$
This bit error probability \b[depends on the Bell labeling]. The <Bell-label Hamming distance and bit errors> rule multiplies the wrong-symbol probability by the number of bit positions changed. For example, relabel $\Phi^+,\Phi^-,\Psi^+,\Psi^-$ as $00,11,01,10$. Both wrong-state partner pairs then differ in two bits, giving bit error probability \b[$1/5$] instead of $1/10$. The physical wrong-Bell-state probability remains $1/5$.
For the entropy-matched independent source in part 2, the transmitted bit stream is fair, and consecutive two-bit labels are uniform. Write the <binary entropy> as $h_2(p)=-p\log_2p-(1-p)\log_2(1-p)$. If a bit position is chosen uniformly and its phase/parity position is forgotten, the pooled corresponding-bit pair is a <binary symmetric channel> with error $1/10$. Its <mutual information> is
$$
\boxed{I(B;\widehat B)=1-h_2(1/10)\simeq0.531004\ \text{bits}.}
$$
The <bit-position information in a binary channel> distinguishes this pooled quantity from the information in the decoded blocks. Alice and Bob know which bit is first in each Bell label. Conditioned on that position $J$, the first bit gives $1-h_2(1/5)$ bits and the second gives one bit. Thus
$$
\boxed{I(B;\widehat B\mid J)=1-\frac12h_2(1/5)\simeq0.639036\ \text{bits per bit}.}
$$
Equivalently, the joint <mutual information> between the two-bit input and output labels is \b[$2-h_2(1/5)\simeq1.278072$ bits per transmitted Bell label] for the prescribed Bell measurement. Pooling without the position loses usable side information; it is not interchangeable with this block rate. The stated fair-input values use the optimal source distribution above; biased source bits require their actual joint distribution.
\Image[/past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2006/7-dense-coding-bit-information.png]
{title=Mutual information for the two Bell-label positions and for known or forgotten bit positions}
{description=Original calculations for independent fair input bits, Bell-readout error $1/5$, and the labeling in the solution. The first bit may flip while the second is noiseless. Knowing which position a bit occupies retains more information than pooling the two channels and discarding that label.}
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