Editing Trap Code for Quantum Authentication (section)

==Protocol Description==
'''Input:''' <math>\rho</math>, pair of secret classical keys <math>k=(k_1, k_2)</math></br></br>
'''Output:''' Quantum state <math>\rho^\prime</math> if the protocol accepts; fixed quantum state <math>\Omega</math> if the protocol aborts
*'''''Encoding:'''''
#<math>\mathcal{S}</math> applies an <math>[[n,1,d]]</math> error correction code
#<math>\mathcal{S}</math> appends an additional trap register of <math>n</math> qubits in state <math>|0\rangle\langle 0|^{\otimes n}</math>
#<math>\mathcal{S}</math> appends a second additional trap register of <math>n</math> qubits in state <math>|+\rangle\langle +|^{\otimes n}</math>
#<math>\mathcal{S}</math> permutes the total <math>3n</math>-qubit register by <math>\pi_{k_1}</math> according to the key <math>k_1</math>
#<math>\mathcal{S}</math> applies a Pauli encryption <math>P_{k_2}</math> according to key <math>k_2</math>
*'''''Mathematical Encoding Description:''''' </br>Mathematically, the encoding process is given by <math display=block>\mathcal{E}_k: \rho \mapsto P_{k_2}\pi_{k_1}\left( \text{Enc}(\rho) \otimes |0\rangle\langle 0|^{\otimes n} \otimes |+\rangle\langle +|^{\otimes n}\right)\pi_{k_1}^\dagger P_{k_2}.</math> In the above, <math>\text{Enc}(\rho)</math> denotes the quantum message <math>\rho</math> after applying the error correction code for encoding (see step 1).
*'''''Decoding:'''''
#<math>\mathcal{A}</math> applies <math>P_{k_2}</math> according to key <math>k_2</math>
#<math>\mathcal{A}</math> applies inverse permutation <math>\pi_{k_1}^\dagger</math> according to the key <math>k_1</math>
#<math>\mathcal{A}</math> measures the last <math>n</math> qubits in the Hadamard basis <math>\{|+\rangle, |-\rangle\}</math>
#<math>\mathcal{A}</math> measures the second last <math>n</math> qubits in the computational basis <math>\{|0\rangle, |1\rangle\}</math></br></br>a. If the two measurements in step 3 and 4 result in <math>|+\rangle\langle +|</math> and <math>|0\rangle\langle 0|</math>, an additional flag qubit in state <math>|\mathrm{ACC}\rangle\langle\mathrm{ACC}|</math> is appended and the quantum message is decoded according to the error correction code </br>b. Otherwise, an additional flag qubit in state <math>|\mathrm{REJ}\rangle\langle\mathrm{REJ}|</math> is appended and the (disturbed) encoded quantum message is replaced by a fixed state <math>\Omega</math>
*'''''Mathematical Decoding Description:''''' </br>Mathematically, the decoding process is given by <math display=block>\mathcal{D}_k: \rho^\prime \mapsto \text{Dec }\mathrm{tr}_{0,+}\left( \mathcal{P}_\text{acc} \pi_{k_1}^\dagger P_{k_2}(\rho^\prime) P_{k_2} \pi_{k_1} \mathcal{P}_\text{acc}^\dagger \right) \otimes |\mathrm{acc}\rangle \langle \mathrm{acc}| + \mathrm{tr}_{0,+} \left(\mathcal{P}_\text{rej} \pi_{k_1}^\dagger P_{k_2}(\rho^\prime) P_{k_2} \pi_{k_1} \mathcal{P}_\text{acc}^\dagger \right) \Omega \otimes |\text{rej}\rangle\langle \text{rej}|.</math> In the above, <math>\text{Dec}</math> refers to decoding of the error correction code (see step 4a) and <math>\mathrm{tr}_{0,+}</math> denotes the trace over the two trap registers. Moreover, <math>\mathcal{P}_\text{acc}</math> and <math>\mathcal{P}_\text{rej}</math> refer to the measurement projectors that determine whether the protocol accepts or aborts the received quantum message. It is <math display=block>\mathcal{P}_\text{acc} = I^{\otimes n}\otimes |0\rangle\langle 0|^{\otimes n}\otimes |+\rangle\langle +|,</math> and <math display=block>\mathcal{P}_\text{rej} = I^{\otimes 3n} - \mathcal{P}_\text{acc}.</math>