Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 60 / 5 / iv

Table of contents
- Solution iv

Solution

Both output states have the same reference marginal, and

S (R) = S (ρ)

. Expand quantum mutual information to obtain the mutual information and coherent information identity

I (R : B_{1})_{σ} = S (ρ) + I_{c} (Λ_{1}, ρ), I (R : B_{2})_{ω} = S (ρ) + I_{c} (Λ_{2} \circ Λ_{1}, ρ) .

The input-entropy term is identical in the two expressions. Subtract them and use the preceding data-processing inequality for coherent information:

I (R : B_{1})_{σ} - I (R : B_{2})_{ω} = I_{c} (Λ_{1}, ρ) - I_{c} (Λ_{2} \circ Λ_{1}, ρ) \geq 0.

Thus the final quantum channel cannot increase the reference-output quantum mutual information, as required by data processing for quantum mutual information.

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