Decoding quantum information via the Petz recovery map (Q2820905)

It is known that if we use a quantum memory-less channel, \(N\) with probability of error \(\varepsilon\) \(n\) times, then the amount of qubits that can be sent is \(Q^{n,\varepsilon}(N)=n\cdot I_c(N)+o(n)\). The authors show that \(Q^{n,\varepsilon}(N)\geq n\cdot I_c(N)+\sqrt{n\cdot V_\varepsilon(N)}\cdot \Phi^{-1}(\varepsilon)+O(\log(n))\) for some characteristic \(V_\varepsilon(N)\), where \(\Phi^{-1}\) is an inverse function to the cumulative distribution function of the standard Gaussian distribution. Interestingly, the corresponding lower bound has the same form as for the classical communication channels.NEWLINENEWLINEIn contrast to the previous purely existential proofs of similar asymptotic properties, the new result uses an explicitly defined decoder -- namely, Petz recovery map. For this decoder, it was proven that the average error in sending an ensemble of commuting states is at most twice the corresponding minimal possible error.