# A Quantum Detectable Byzantine Agreement Protocol Using Only EPR Pairs
## 4 The \(n\) Player EPRQDBA Protocol
### The \(n\) Player Agreement Phase  
The EPRQDBA protocol achieves detectable agreement during its third and last phase, aptly named agreement phase. The agreement phase begins when [PERSON] and her \(n-1\) lieutenant generals \(\mathrm{LT}_{0}\),..., \(\mathrm{LT}_{n-2}\) measure their quantum registers. By recalling the distribution scheme of Section 4.1, we see that certain important correlations among [PERSON]'s register and the registers of her lieutenant generals have been established. **Definition 6**. : _Let the bit vectors \(\mathbf{a}\) and \(\mathbf{I}^{i}\) denote the contents of [PERSON] and \(\mathrm{LT}_{i}\)'s, \(0\leq i\leq n-2\), registers after the measurement, and assume that their explicit form is as given below:_  
\[\mathbf{a} =\underbrace{a_{(n-1)m-1}\ldots a_{(n-1)m-(n-1)}}_{(n-1)\text{-tuple} }\;\;\ldots\;\underbrace{a_{n-2}\ldots a_{0}}_{(n-1)\text{-tuple}}\;, \tag{37}\] \[\mathbf{I}^{i} =\underbrace{l^{i}_{(n-1)m-1}\ldots l^{i}_{(n-1)m-(n-1)}}_{(n-1) \text{-tuple}}\;\;\ldots\;\underbrace{l^{i}_{n-2}\ldots l^{i}_{0}}_{(n-1)\text {-tuple}}\;. \tag{38}\]  
_The \(k^{th}\)\((n-1)\)-tuple of \(\mathbf{a}\), \(0\leq k\leq m-1\), is the \((n-1)\)-tuple of bits \(a_{(n-1)k+(n-2)}\)...\(a_{(n-1)k}\), and is designated by \(\mathbf{a}_{k}\). Similarly, the \(k^{th}\)\((n-1)\)-tuple of \(\mathbf{I}^{i}\) is designated by \(\mathbf{I}^{i}_{k}\). Hence, \(\mathbf{a}\) and \(\mathbf{I}^{i}\) can be written succinctly as:_  
\[\mathbf{a} =\mathbf{a}_{m-1}\;\mathbf{a}_{m-2}\;\;\ldots\;\mathbf{a}_{1}\; \mathbf{a}_{0}\;, \tag{39}\] \[\mathbf{I}^{i} =\mathbf{I}^{i}_{m-1}\;\mathbf{I}^{i}_{m-2}\;\ldots\;\mathbf{I}^{ i}_{1}\;\mathbf{I}^{i}_{0}\;. \tag{40}\]  
_We also use the notion of the uncertain \((n-1)\)-tuple, denoted by \(\mathbf{u}=\sqcup\cdots\sqcup\), where \(\sqcup\) is a new symbol, different from \(0\) and \(1\). In contrast, \((n-1)\)-tuples consisting of bits \(0\) and/or \(1\) are called definite tuples._  
Figure 4: Qubits of the same EPR pair are shown in the same color. Green is used to indicate the EPR pairs shared between [PERSON] and \(\mathrm{LT}_{0}\), which occupy those positions \(k\) that satisfy the relation \(k\equiv 0\bmod(n-1)\) in their quantum registers. Analogously, red is used for the EPR pairs shared between [PERSON] and \(\mathrm{LT}_{1}\) occupying positions \(k\) that satisfy the relation \(k\equiv 1\bmod(n-1)\) in their quantum registers, and blue indicates EPR pairs shared between [PERSON] and \(\mathrm{LT}_{n-2}\) occupying positions \(k\) that satisfy the relation \(k\equiv n-2\bmod(n-1)\). The gray qubits represent those in the \(\ket{\star}\) state that occupy the remaining positions in the quantum registers of the lieutenant generals. According to the distribution scheme, each \((n-1)\)-tuple in [PERSON]'s bit vector \(\mathbf{a}\) shares one \(|\Psi^{+}\rangle=\frac{|0\rangle_{A}(1\rangle_{i}+|1\rangle_{A}|0\rangle_{i})}{ \sqrt{2}}\) pair with every lieutenant general \(\mathrm{LT}_{i}\), \(0\leq i\leq n-2\). This is captured by the next Lemma 3. Its proof is obvious and is omitted. **Lemma 3** (Tuple Differentiation Property). : _The next property, termed tuple differentiation property characterizes the corresponding bits and tuples of the bit vectors \(\mathbf{a}\) and \(\mathbf{I}^{i}\), \(0\leq i\leq n-2\)._  
\[l_{k}^{i}=\overline{a_{k}}\,\ \text{iff}\ k\equiv i\ \mathrm{mod}\ (n-1)\,\ k=0, \ldots,(n-1)m-1\,\ 0\leq i\leq n-2. \tag{41}\]  
_Consequently, each \(\mathbf{a}_{k}\) tuple, where \(0\leq k\leq m-1\), is different from the corresponding tuple of every \(\mathbf{I}^{i}\), \(0\leq i\leq n-2\)._  
\[\mathbf{a}_{k}\
eq\mathbf{I}_{k}^{i}\,\ \text{for every}\ k,\ 0\leq k\leq m-1. \tag{42}\]  
**Definition 7** (\(n\) Player Command Vectors). : _[PERSON] sends to every lieutenant general \(\mathrm{LT}_{i}\), \(0\leq i\leq n-2\), either the command \(0\) or the command \(1\). Besides her order, as a \"proof,\" she also sends an appropriate command vector.