Alice and Bob share many independent copies of the partially entangled pure state

PHYSICS · Multi-part · Fermi University Physics · Quantum Information & Technology · Open quantum systems, decoherence and error correction · Quantum information theory and communication

Quantum Information and Technology, Ch Entropy and compression

Problem

Alice and Bob share many independent copies of the partially entangled pure state ∣ψ⟩AB=0.7 ∣00⟩+0.3 ∣11⟩|\psi\rangle_{AB} = \sqrt{0.7}\,|00\rangle + \sqrt{0.3}\,|11\rangle. Before Bob stores his half, each of his qubits passes through an independent depolarising channel Dp(ρ)=(1−p)ρ+p I/2\mathcal{D}_p(\rho) = (1-p)\rho + p\,I/2 with p=0.2p = 0.2, so the state Bob actually holds for each copy is ρB′=Dp(ρB)\rho_B' = \mathcal{D}_p(\rho_B), where ρB=trA∣ψ⟩⟨ψ∣\rho_B = \mathrm{tr}_A|\psi\rangle\langle\psi|. ((a)) Find the eigenvalues of ρB′\rho_B' and its von Neumann entropy S(ρB′)S(\rho_B') in bits. ((b)) Bob wants to Schumacher-compress a block of 500500 of these noisy halves for storage. Using the asymptotic Schumacher rate, estimate the minimum number of qubits needed for the block, rounded up to a whole number of qubits.

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