Let and be density matrices that commute

PHYSICS · Multi-part · Fermi University Physics · Quantum Information & Technology · Quantum information theory and communication · Quantum information theory and communication

Quantum Information and Technology, Ch Entropy and compression

Problem

Let

\rho

and

\sigma

be density matrices that commute, so in a common eigenbasis

\rho=\mathrm{diag}(p_1,\dots,p_d)

and

\sigma=\mathrm{diag}(q_1,\dots,q_d)

. ((a)) Using

\ln x\le x-1

, prove Klein's inequality

S(\rho\|\sigma)=\sum_i p_i\log_2(p_i/q_i)\ge0

for this commuting case. ((b)) Evaluate

S(\rho\|\sigma)

for

\rho=\mathrm{diag}(0.60,0.40)

and

\sigma=\mathrm{diag}(0.75,0.25)

in this basis, and check the sign against part (a).

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