Table 1 Protocol 1: The estimation protocol

Protocol parameter

• \(N \in {\Bbb N}\), even: total number of qubits

The protocol

• Alice chooses \(s \in \left\{ {0,1} \right\}^N\) and \(b \in \left\{ {X,Z} \right\}_{N/2}^N\) fully at random and communicates them to Bob, where

\(\{ X,Z \}_{N/2}^N = \{ b \in \{ X,Z \}^N | {\rm X},{\rm Z}, {\rm each}\,{\rm occur}\,N{/}2\,{\rm times}\,{\rm in}\,b \}.\)

• For each qubit slot i = 1, …., N of the channel, Alice prepares a test qubit i in the state S _i with respect to basis \(b_i \in \left\{ {X,Z} \right\}\) and sends it through the channel to Bob.

• For each qubit i = 1, …., N that Bob receives, he measures test qubit i in the basis b _i and records the outcome \(s_i^{\prime} \in \left\{ {0,1} \right\}\).

• Bob determine the error rates

\(e_x = \frac{2}{N}\mathop {\sum}\limits_{i \in I_X} s_i \oplus s{\prime}_i,\quad e_z = \frac{2}{N}\mathop {\sum}\limits_{i \in I_Z} s_i \oplus s{\prime}_i,\)

where

\(\begin{array}{l}I_X = \left\{ {i \in \left\{ {1, \ldots ,N} \right\}\left| {b_i = X} \right.} \right\},\\ I_Z = \left\{ {i \in \left\{ {1, \ldots ,N} \right\}\left| {b_i = Z} \right.} \right\}.\end{array}\)

• Knowing the two error rates e _x and e _z , Bob determines a lower bound on the one-shot quantum capacity according to Theorem 1.