A biased coin has , .

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

A biased coin has p(1)=0.15p(1)=0.15, p(0)=0.85p(0)=0.85. ((a)) Compute H=h(0.15)H=h(0.15) and the single-symbol variance σ2\sigma^2 of −log⁡2p(xi)-\log_2p(x_i). Using Chebyshev's inequality, evaluate the bound σ2/(nϵ2)\sigma^2/(n\epsilon^2) on P(∣−1nlog⁡2p(xn)−H∣>ϵ)P(|{-\tfrac1n\log_2p(x^n)}-H|>\epsilon) for ϵ=0.10\epsilon=0.10 at n=300n=300 and again at n=900n=900, and comment on how the bound behaves as nn grows. ((b)) For each nn, give the resulting two-sided bound on ∣Tϵn∣|T_\epsilon^n|. ((c)) A shared two-qubit Werner state has singlet weight ww and singlet fraction fs=(3w+1)/4f_{\rm s}=(3w+1)/4; for this state the eigenvalues are fsf_{\rm s} (once) and (1−fs)/3(1-f_{\rm s})/3 (three times), and the reduced state of either qubit is I/2I/2. For fs=0.85f_{\rm s}=0.85, find ww, compute the hashing yield −S(A∣B)-S(A|B) in ebits per pair, and state whether it is positive.

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