Thread by @fgrosshans

Now Chiara Machiavello from@unipv on “Witnessing quantum capcities of noisy communication channels”
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Chiara Machiavello: a channel has several capacities:
The classical capacity C,
Entanglement assisted classcial capacity C_E,
P private capacity: negligile info can be obtained by a third party
and
Q: Quantum capacity
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Chiara Machiavello computes Q:
Channel described by CPTP map ℰ₁, N uses ℰ₁^⊗N

Q=lim Q_N/N, N→∞
Q_N=max_ρ Ic(ρ,ℰ_N) (coherent information)
Ic(ρ,ℰ)=S(ℰ(ρ))-S(ℰ,ρ), where the latter is the entropy exchange
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Chiara MAchiavello: Full process tomography need ∝d⁴ measurements.
An accessible lower bound Q_DET uses d² measurement, to measure teh Von Neumann entropy and bound the entropy exchange:
Q≥Q₁≥I_c(ρ,ℰ₁)≥Q_DET

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Chirar Machiavello prepares a bipartite (S-R) state |ψ_ρ⟩ and send it through the channel ℰ_S⊗ℐ_R and measures a set of d² observables tomographically complete for S to evaluates the entropies.
She then optimizes the various possible choices classically.
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Chiara Machiavello looks in details the single qubit case.
For Pauli channels, the optimized evaluation are input Bell states and Bell measurements. It actually detects the Hashing bound.
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Chiara Machiavello: For the amplitude damping channel, the optimised measurement is better than the Bell measurement. Actually Q-Q_DET≤0.005: it is indistiguishable on the plot from the real Q
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Chiara MAchiavello has similar results (Q-Q_DET<.005) for qubit channels with two Kraus operators
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Chiara Machiavello move on to two-qubit correlated channels. The degree of correlation is μ: 0 for memory less channel, 1 when the same nose operator is applied on the to qubits
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Chiara Machiavello: for depolarizing channel, Q_DET coincides with the best kown lower bound, and Q_DET=Q when μ=1
For amplitude damping channels, the bound is quite far. We don’t understand exactly why.
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Chiara MAchiavello. Q_DET is also useful for other capacities:
C_E≥log₂d + Q_DET and
P≥Q₁≥Q_DET
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Chiara Macchiavello: The procedure was experimentally demonstrated. arxiv.org/abs/1612.07754 doi.org/10.1103/PhysRe…

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