## 4 Discussion of Qubit Certification  
Classical certification [20; 21] is a well-known constraint that a quantum cryptographic protocol guarantees to all the other parties that one party is restricted so that, to avoid being detected as cheating, his quantum inputs must take the form of pure states of the basis elements of a public fixed basis \(\{|i\rangle\}_{i=1}^{d}\) of the appropriate \(d\)-dimensional input space. Here, as the qubit set \(\{|\psi_{k}\rangle_{\text{X}}\}_{k=1}^{d}\) in the quantum qubit commitment is not always a set of orthonormal bases in \(H_{\text{X}}\) space, we define qubit certification, a new constraint similar to classical certification, that a protocol guarantees to all the other parties that one party is restricted so that his quantum inputs must be a specific qubit of the public fixed set, which can be described as, in a quantum qubit commitment, that any unveiled qubit commitments necessarily have a definite committed qubit choosing from the public set. Like all technologically unconstrained quantum bit commitment protocols, [9; 12] do not force [PERSON] to commit to a specific bit, the protocol here does not force to commit to a specific qubit, as it does not prevent [PERSON] from preparing states in \(H_{\text{X}}\) entangled with other system \(H_{\text{C}}\) with the form \(\sum_{k}\sqrt{p_{k}}|k\rangle_{\text{C}}|\psi_{k}\rangle_{\text{X}}\), where \(\{|k\rangle_{\text{C}}\}_{k=1}^{d}\) is a set of orthonormal bases. In the commitment, [PERSON] can choose to keep the states in quantum entanglement, and input \(|\,\rangle_{\text{X}}\) as the committed qubit and store the entangled state \(|\,\rangle_{\text{C}}\) and then measure it in the computational basis until unveiling. In this case there is a definite committed qubit without being detected cheating, but from [PERSON]'s perspective the unveiled qubit remains unknown with subjective probabilities obeying \(\sum_{k}p_{k}=1\). It does not rule out the case that [PERSON]'s measurement result of qubit C is \(|k\rangle_{\text{C}}(k_{1}\
eq k)\) but she successfully unveils qubit \(|\psi_{k}\rangle_{\text{X}}\), or even the case that she prepares states entangling the subset \(\{|\psi_{k}\rangle_{\text{X}}\}_{k=1}^{d_{1}}\) of \(\{|\psi_{k}\rangle_{\text{X}}\}_{k=1}^{d_{1}}\) (\(d_{1}<d\)) with \(\{|k\rangle_{\text{C}}\}_{k=1}^{d_{1}}\), i.e., \(\sum_{k=1}^{d_{1}}\sqrt{p_{k}}|k\rangle_{\text{C}}|\psi_{k}\rangle_{\text{X}}\) satisfying \(\sum_{k=1}^{d_{1}}p_{k}=1\), but successfully unveils a qubit \(|\psi_{k}\rangle_{\text{X}}\
otin\{|\psi_{k}\rangle_{\text{X}}\}_{k=1}^{d_{1}}\), with a probability \(\delta\), which can be made arbitrarily small by choosing the security parameters appropriately as there is no any measurement result that could act as an evidence to support [PERSON] to obtain a successful probability of unveiling this qubit higher than the successful probability of just guessing by chance. Hence, we say a quantum qubit commitment protocol with a security parameter \(N\) is unconditionally secure when it guarantees \(\sum_{k}p_{k}<1+\delta(N)\) with \(\delta(N)\to 0\) as \(N\rightarrow\infty\). It is the complete mathematical definition of unconditionally secure of a quantum qubit commitment that we presented above.