Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 48 / 2 / c

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- Solution c

Solution

After the first quantum channel, denote the joint reference-output state by

ω_{R B_{1}}

. Dilate the second CPTP map as an linear isometry

B_{1} \to B_{2} E_{2}

, with the resulting state denoted

τ

. Linear isometries preserve nonzero eigenvalues, both globally and on the transformed output, giving

I_{c} (Φ_{1}, ρ) = S (B_{2} E_{2})_{τ} - S (R B_{2} E_{2})_{τ} .

Tracing out

E_{2}

implements the second channel, so

I_{c} (Φ_{2} \circ Φ_{1}, ρ) = S (B_{2})_{τ} - S (R B_{2})_{τ} .

The Strong subadditivity of Von Neumann entropy for

R, B_{2}, E_{2}

gives

S (R B_{2}) + S (B_{2} E_{2}) \geq S (B_{2}) + S (R B_{2} E_{2})

. Rearranging proves the data-processing inequality for coherent information:

I_{c} (Φ_{1}, ρ) \geq I_{c} (Φ_{2} \circ Φ_{1}, ρ) .

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