# A Quantum Detectable Byzantine Agreement Protocol Using Only EPR Pairs
## 3 The 3 Player EPRQDBA Protocol
### Agreement Phase  
The EPRQDBA protocol achieves detectable agreement during its third and last phase, aptly named agreement phase. [PERSON], [PERSON] and [PERSON] initiate the agreement phase by measuring their quantum registers. The distribution scheme, as analyzed in Section 3.1, leads to some crucial correlations among the contents of [PERSON], [PERSON] and [PERSON]'s registers. **Definition 3**. : _Let the bit vectors \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) denote the contents of [PERSON], [PERSON] and [PERSON]'s registers after the measurement, and assume that their explicit form is as given below:_  
\[\mathbf{a} =\underbrace{a_{2m-  
\[\mathbf{c}=\underbrace{c_{2m-1}c_{2m-2}}_{\text{pair }m-1}\ \underbrace{c_{2m-3}c_{2m-4}}_{ \text{pair }m-2}\ \cdots\ \underbrace{c_{3}c_{2}}_{\text{pair }1}\ \underbrace{c_{1}c_{0}}_{\text{pair }0}. \tag{9}\]  
_The \(k^{th}\) pair of \(\mathbf{a}\), \(0\leq k\leq m-1\), is the pair of bits \(a_{2k+1}a_{2k}\), and is designated by \(\mathbf{a}_{k}\). Similarly, the \(k^{th}\) pairs of \(\mathbf{b}\) and \(\mathbf{c}\) are designated by \(\mathbf{b}_{k}\) and \(\mathbf{c}_{k}\). Hence, \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) can be written succinctly as:_  
\[\mathbf{a} =\mathbf{a}_{m-1}\ \mathbf{a}_{m-2}\ \dots\ \mathbf{a}_{1}\ \mathbf{a}_{0}\, \tag{10}\] \[\mathbf{b} =\mathbf{b}_{m-1}\ \mathbf{b}_{m-2}\ \dots\ \mathbf{b}_{1}\ \mathbf{b}_{0}\,\] (11) \[\mathbf{c} =\mathbf{c}_{m-1}\ \mathbf{c}_{m-2}\ \dots\ \mathbf{c}_{1}\ \mathbf{c}_{0}. \tag{12}\]  
_We also define a special type of pair, termed the uncertain pair, that is denoted by \(\mathbf{u}=\shuffle\), where \(\shuffle\) is a new symbol, different from \(0\) and \(1\). In contrast, pairs consisting of bits \(0\) and/or \(1\) are called definite pairs._  
When we want refer to either [PERSON]'s or [PERSON]'s bit vector, but without specifying precisely which one, we will designate it by  
\[\mathbf{l} =\mathbf{l}_{m-1}\ \mathbf{l}_{m-2}\ \dots\ \mathbf{l}_{1}\ \mathbf{l}_{0}\] \[=\underbrace{l_{2m-1}l_{2m-2}}_{\text{pair }m-1}\ \underbrace{l_{2m-3}l_{2m-4}}_{\text{pair }m-2}\ \cdots\ \underbrace{l_{3}l_{2}}_{\text{pair }1}\ \underbrace{l_{1}l_{0}}_{\text{pair }0}. \tag{13}\]  
According to the distribution scheme, [PERSON] and [PERSON] share the odd-numbered \(|\Psi^{+}\rangle=\frac{|0\rangle_{A}|1\rangle_{B}+|1\rangle_{A}|0\rangle_{B}} {\sqrt{2}}\) pairs, and [PERSON] and [PERSON] share the even-numbered \(|\Psi^{+}\rangle=\frac{|0\rangle_{A}|1\rangle_{C}+|1\rangle_{A}|0\rangle_{C}} {\sqrt{2}}\) pairs. Therefore, the next Lemma 1 holds. Its proof is trivial and is omitted. **Lemma 1** (Pair Differentiation Property). : _The next property, termed pair differentiation property, characterizes the corresponding bits and pairs of the bit vectors \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\)._  
1. For every odd-numbered bit \(b_{k}\) in \(\mathbf{b}\), where \(k=1,3,\dots,2m-1\), it holds that \[b_{k}=\overline{a_{k}},\ k=1,3,\dots,2m-1\.\] (14)
2. For every even-numbered bit \(c_{k}\) in \(\mathbf{c}\), where \(k=0,2,\dots,2m-2\), it holds that \[c_{k}=\overline{a_{k}},\ k=0,2,\dots,2m-2\.\] (15)
3. Every \(\mathbf{a}_{k}\) pair, where \(0\leq k\leq m-1\), differs from the corresponding pairs \(\mathbf{b}_{k}\) and \(\mathbf{c}_{k}\). \[\mathbf{a}_{k}\
eq\mathbf{b}_{k}\ \text{ and }\ \mathbf{a}_{k}\
eq\mathbf{c}_{k}\,\text{ for every }k,\ 0\leq k\leq m-1\.\] (16) In the above relations (14) and (15), \(\overline{a_{k}}\) is the negation of \(a_{k}\). Visually, the situation regarding the correlations among corresponding bits is shown in Figure 2. Again bits resulting from the same EPR pair are shown in the same color. **Definition 4** (Command Vectors). : _[PERSON] sends to [PERSON] and [PERSON] either the command \(0\) or the command \(1\). Besides her command, as a \"proof\", she also sends an appropriate command vector. The idea is that the command vector for [PERSON] is always different from the command vector for [PERSON], even when the command is the same.