Past exam of the laser science and quantum information processing course of the University of Oxford / 2006 / 7 / 3 / Solution

The actual resource is a two-component Bell resource in dense coding. Applying the same local encoder $X^v{Z}^u$ togives a superposition of the intended Bell state with amplitude magnitude $2/\sqrt{5}$ and its phase-label partner with amplitude magnitude $1/\sqrt{5}$. A Bell-basis measurement therefore reports the wrong Bell state with probability $1/5$, independently of $uv$. With our labeling, only the phase bit $u$ flips: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

This bit error probability 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 $1/5$ 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 $h_2(p)=-p{\log }_{2}p-(1-p){\log }_2(1-p)$. 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

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. ThusEquivalently, the joint mutual information between the two-bit input and output labels is $2-h_2(1/5)\simeq 1.278072$ 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.