Table 2 Protocol 2: The verification protocol

Protocol parameters

• \(N \in {\Bbb N}\): number of data qubits

• \(e_x,e_z \in [0,1]\): tolerated error rate in X, Z

The protocol

• Alice chooses \(s \in \left\{ {0,1} \right\}^{3N}\) and \(b \in \left\{ {X,Z,D} \right\}_N^{3N}\) fully at random and communicates them to Bob, where

\(\left\{ {X,Z,D} \right\}_N^{3N} = \{ {b \in \left\{ {X,Z,D} \right\}^{3N} \left| {X,Z,D\,{\rm{occur}}\,N\,{\rm{times}}\,{\rm{in}}\,b} \right.} \}.\)

• For each qubit slot i = 1, …, 3N of the channel, if \(b_i \in \left\{ {X,Z} \right\}\), 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. If b _i  = D, Alice uses the slot for a data qubit.

• For each qubit i=1, …, 3N that Bob receives, if \(b_i \in \left\{ {X,Z} \right\}\), Bob measures test qubit I in the basis b _i and records the outcome \(s{\prime}_i \in \left\{ {0,1} \right\}\). If b _i  = D, Bob leaves the data qubit untouched.

• They determine the error rates

\(\gamma = \frac{1}{N}\mathop {\sum}\limits_{i \in I_X} s_i \oplus s_i^\prime ,\quad \lambda = \frac{1}{N}\mathop {\sum}\limits_{i \in I_Z} s_i \oplus s_i^\prime ,\)

where

\(\begin{array}{l}I_X = \left\{ {i \in \left\{ {1, \ldots ,3N} \right\}\left| {b_i = X} \right.} \right\},\\ I_Z = \left\{ {i \in \left\{ {1, \ldots ,3N} \right\}\left| {b_i = Z} \right.} \right\}.\end{array}\)

If \(\gamma \le e_x\) and \(\lambda \le e_z\), they continue with the conclusion below. Otherwise, they abort the protocol.

• They conclude that the one-shot quantum capacity of the channel Λ on the N data qubits is bounded as in Theorem 2.