## 2 Two-Way Protocols
### Purifications of Two-Way Protocols  
The approach to the security proof presented in [5] is based on the notion of purification of protocols where the protocols can be described by the measurements made by [PERSON] and [PERSON] on a system provided to them by [PERSON] [13]. Writing the measurements of [PERSON] and [PERSON] as Probability Operator Value Measure  
Figure 2: Schematic diagram for LM05. Arrows represent the distribution of qubits in the channels between [PERSON] and [PERSON]. Note that the arrow in [PERSON]’s station branches off with probability \(c\) or \(1-c\) for EM or CM respectively.  
(POVM) maps \(M_{A}^{\text{EM}}\) and \(M_{B}^{\text{EM}}\) respectively in EM (or \(M_{A}^{\text{CM}}\) and \(M_{B}^{\text{CM}}\) in CM), the state after the measurements is given by  
\[M_{A}^{i}\otimes M_{B}^{i}(\rho_{ABE}) \tag{1}\]  
with \(i=\text{EM}\) or CM. In the purified version of both the SDC and the LM05 as reported in [5], \(\rho_{ABE}\) is a state in the Hilbert space \(\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{E}\) ([PERSON]'s, [PERSON]'s and [PERSON]'s respectively) and \(\mathcal{H}_{A}\) and \(\mathcal{H}_{B}\) are Hilbert spaces for two qubit systems each. [PERSON]'s measurement \(M_{A}^{\text{EM}}\), in EM which acts on a two qubit system must be in such a way that it is equivalent to her encoding operation in the protocols. If we recall how in both protocols, [PERSON] receives a state from [PERSON] and resends after a unitary encoding on the state, \(M_{A}^{\text{EM}}\) can be understood as a (POVM) acting on the received state with half of some entangled state so as the other half of the entangled state (after the measurement \(M_{A}^{\text{EM}}\)) is the same as the output from [PERSON]'s encoding to be sent to [PERSON] [14]. Thus we can imagine \(\rho_{ABE}\) as that pure state which distributes a pair of qubits to [PERSON] and [PERSON] each. [PERSON]'s measurement on the two received qubits, \(\text{tr}_{BE}(\rho_{ABE})\) would ensure that [PERSON]'s measurement, \(M_{B}^{\text{EM}}\) on his received pair, \(\text{tr}_{AE}(\rho_{ABE})\) is equivalent to [PERSON] making a unitary transformation on [PERSON]'s prepared qubit for his subsequent decoding measurement. With regards to [PERSON]'s measurements, \(M_{B}^{\text{EM}}\), it is instructive to note that in SDC, \(M_{B}^{\text{EM}}\) is really the [PERSON] measurement for decoding purposes while in LM05, [PERSON]'s measurement on the first half of \(\text{tr}_{AE}(\rho_{ABE})\) effectively prepares the qubit state sent to [PERSON] (in the forward path) while his other measurement is on the other half of \(\text{tr}_{AE}(\rho_{ABE})\). The measurements \(M_{A}^{\text{CM}}\) and \(M_{B}^{\text{CM}}\) would then correspond to relevant local measurements of each qubit in the qubit pairs in CM (detailed in the ensuing section). Given these, the main ingredient in the security proof for the purified protocols is based on bounding [PERSON]'s information gain given [PERSON]'s using an uncertainty relation which measures the overlap of [PERSON]'s ([PERSON]'s) measurements, _i.e._, \(M_{B}^{\text{EM}}\),\(M_{B}^{\text{CM}}\) (\(M_{A}^{\text{EM}}\),\(M_{A}^{\text{CM}}\)) in a reverse (direct) reconciliation scenario.