# A New Security Proof for Twin-Field Quantum Key Distribution (QKD)
## 2 Preliminaries
The classical error probability of the sampling strategy is defined simply as follows:  
\[\epsilon_{\delta}^{cl}=\max_{q\in\mathcal{A}_{d}^{N}}Pr_{t}(q\
ot\in \mathcal{G}_{t}). \tag{13}\]  
From this definition, it holds that for any word \(q\in\mathcal{A}_{d}^{N}\), if the sampling strategy is performed on it, the probability that it fails (namely that the guess is not \(\delta\) close to the target) is at most \(\epsilon_{\delta}^{cl}\). The main result from [29] was to extend this to the quantum domain. A classical sampling strategy can be promoted to a quantum one in a natural way: given a quantum state \(\ket{\psi}_{AE}\) where the \(A\) register consists of \(N\) qudits, each qudit of dimension \(d\), one chooses a subset \(t\) and measures the qudits, indexed by \(t\), in some \(d\) dimensional orthonormal basis \(\{\ket{0}^{B},\cdots,\ket{d-1}^{B}\}\). This measurement results in a classical outcome \(q_{t}\). Then, according to [PERSON] and [PERSON]'s main result, the unmeasured portion behaves like a superposition of words that are \(\delta\) close to the guess (with respect to the given target). To formalize this, a basis \(B\) is fixed and the following space is considered:  
\[\text{span}(\mathcal{G}_{t})\otimes\mathcal{H}_{E}=\text{span}\Big{\{}\ket{q} ^{B}\ :\ q\in\mathcal{G}_{t}\Big{\}}\otimes\mathcal{H}_{E}. \tag{14}\]  
This subspace is called the ideal subspace; a state within it is called an ideal state. Note that if one is given an ideal state \(\ket{v^{t}}\) (which only makes sense at the moment for a specific subset \(t\) according to this definition and thus the superscript index), and if one performs a measurement in the \(B\) basis on subset \(t\) resulting in outcome \(q_{t}\in\mathcal{A}_{d}^{\ket{t}}\), the post measured state must collapse to one that is of the following form:  
\[\ket{v_{q}^{t}}=\sum_{ \begin{subarray}{c}i\in\mathcal{A}_{d}^{N-\ket{t}}\\ g(q_{t})\sim_{\delta}r(i)\end{subarray} }\alpha_{i}\ket{i}^{B}\otimes\ket{E_{i} }. \tag{15}\]  
Namely, it must collapse to a superposition of words that are \(\delta\) close to the observed value (with respect to the given guess and target functions). The states may not be necessarily ideal; however, the following theorem says that for any quantum state, one can define a collection of ideal states that are \(\epsilon\) close in trace distance to the given state. **Theorem 1** (from [29], but re-worded slightly for our application and approach). : _Let \(\delta>0\), \(B\) be a \(d\)-dimensional orthonormal basis, and \(\ket{\psi}_{AE}\) be a pure quantum state where the \(A\) register consists of \(N\) qudits, each qudit of dimension \(d\). It is assumed that the dimension of each system \(N\) is known. Given a classical sampling strategy with error probability \(\epsilon_{\delta}^{cl}\), there is a collection of ideal states \(\{\ket{v^{t}}\}_{t}\), indexed by all subsets \(t\) such that \(P_{T}(t)>0\):_  
\[\ket{v^{t}}\in\text{span}(\mathcal{G}_{t})\otimes\mathcal{H}_{E} \tag{16}\]  
_Furthermore,_  
\[\frac{1}{2}\bigg{|}\bigg{|}\sum_{t}P_{T}(t)\ket{t}\bra{t}\otimes\ket{\psi} \bra{\psi}-\sum_{t}P_{T}(t)\ket{t}\bra{t}\otimes\ket{v^{t}}\bra{v^{t}}\bigg{|} \bigg{|}\leq\sqrt{\epsilon_{\delta}^{cl}}. \tag{17}\]  
Note that, the original proof of Theorem 1 assumes [PERSON]'s ancilla is finite dimensional. This is without loss of generality in our proof since we are considering ideal sources. Before leaving this section, we discuss an important sampling strategy which we will use later. This strategy was analyzed in [37] for Bell states. Given a word \(q\in\mathcal{A}_{4}^{n+m}\), a subset of size \(m\) is uniformly chosen at random from all \(m\) size subsets of \(\{1,\cdots,n+m\}\)The guess and target functions are simply \(g(x)=r(x)=\frac{1}{\left|x\right|}\#_{1,3}(x)\).