A ground station in a quantum network reports three figures at the end of one…

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 ground station in a quantum network reports three figures at the end of one operating cycle. Solve each part separately. ((a)) The station's classical control channel sends a block of n=200n=200 independent bits, each equal to 11 with probability p=0.2p=0.2 and 00 otherwise. Let H=h(p)H=h(p) be the binary entropy of the source and let ϵ=0.1\epsilon=0.1 bits. Use Chebyshev's inequality applied to the per-symbol surprise −log⁡2p(xi)-\log_2p(x_i) to bound the probability δ\delta that a block of n=200n=200 symbols is not ϵ\epsilon-typical, and hence bound the number ∣Tϵn∣|T_\epsilon^n| of ϵ\epsilon-typical blocks between (1−δ)2n(H−ϵ)(1-\delta)2^{n(H-\epsilon)} and 2n(H+ϵ)2^{n(H+\epsilon)}. Give both bounds as powers of two, quoting the base-2 exponent to one decimal place. ((b)) In the same cycle the station shares N=500N=500 copies of the two-qubit Werner state ρW=fs∣Ψ−⟩⟨Ψ−∣+1−fs3(∣Φ+⟩⟨Φ+∣+∣Φ−⟩⟨Φ−∣+∣Ψ+⟩⟨Ψ+∣)\rho_W=f_{\rm s}|\Psi^-\rangle\langle\Psi^-|+\frac{1-f_{\rm s}}{3}\big(|\Phi^+\rangle\langle\Phi^+|+|\Phi^-\rangle\langle\Phi^-|+|\Psi^+\rangle\langle\Psi^+|\big) with singlet fraction fs=⟨Ψ−∣ρW∣Ψ−⟩f_{\rm s}=\langle\Psi^-|\rho_W|\Psi^-\rangle, so its eigenvalues are fsf_{\rm s} (once) and (1−fs)/3(1-f_{\rm s})/3 (three times) and the hashing yield is −S(A∣B)=1−S(A,B)-S(A|B)=1-S(A,B) with S(A,B)=−fslog⁡2fs−(1−fs)log⁡2 ⁣(1−fs3)S(A,B)=-f_{\rm s}\log_2f_{\rm s}-(1-f_{\rm s})\log_2\!\big(\tfrac{1-f_{\rm s}}{3}\big). Solve S(A,B)=1S(A,B)=1 numerically for the singlet fraction fsf_{\rm s} at which the hashing yield falls to zero, quoting your answer to three decimal places. Then, for a batch with fs=0.90f_{\rm s}=0.90, find the hashing yield and the number of Bell pairs that hashing can asymptotically distil from the NN copies. ((c)) A different sensor prepares, with equal probability 13\tfrac13 each, one of three trine states equally spaced around a great circle of the Bloch sphere: ∣0⟩|0\rangle, ∣ψ1⟩=cos⁡60∘∣0⟩+sin⁡60∘∣1⟩|\psi_1\rangle=\cos60^\circ|0\rangle+\sin60^\circ|1\rangle (Bloch vector at polar angle 120∘120^\circ from +Z+Z), and ∣ψ2⟩=cos⁡120∘∣0⟩+sin⁡120∘∣1⟩|\psi_2\rangle=\cos120^\circ|0\rangle+\sin120^\circ|1\rangle (Bloch vector at polar angle 240∘240^\circ from +Z+Z), and transmits 20002000 such qubits per cycle. Find the eigenvalues of the ensemble density operator ρ=13∣0⟩⟨0∣+13∣ψ1⟩⟨ψ1∣+13∣ψ2⟩⟨ψ2∣\rho=\tfrac13|0\rangle\langle0|+\tfrac13|\psi_1\rangle\langle\psi_1|+\tfrac13|\psi_2\rangle\langle\psi_2|, its von Neumann entropy S(ρ)S(\rho), and the number of qubits into which Schumacher compression can pack the 20002000 letters. Compare this with the number of bits a classical record of which of the three letters was sent would need.

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