A qubit is prepared in a pure state with Bloch vector of length

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

A qubit is prepared in a pure state with Bloch vector of length 11, then sent through a depolarising channel Dp(ρ)=(1−p)ρ+p I/2\mathcal{D}_p(\rho)=(1-p)\rho+p\,I/2 with p=0.40p=0.40. A depolarising channel shrinks any Bloch vector by the factor (1−p)(1-p). ((a)) Find the length of the Bloch vector after the channel, the eigenvalues of the resulting density matrix, and its von Neumann entropy SS in bits. ((b)) Separately, a source repeatedly prepares the two-qubit pure state ∣ψ⟩=0.65 ∣00⟩+0.35 ∣11⟩|\psi\rangle=\sqrt{0.65}\,|00\rangle+\sqrt{0.35}\,|11\rangle and always keeps one qubit (say Bob's) locally, sending the other away. Using the Schmidt coefficients of ∣ψ⟩|\psi\rangle, find the entropy of entanglement E(ψ)E(\psi), and estimate how many qubits are needed to Schumacher-compress 800800 copies of Bob's half with vanishing error.

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