## 5 The Non-Ideal Case in Practical Application  
The protocol proposed above is idealized with some ideal assumptions. In this section, we discuss the non-ideal case in practical application. Realistically, it is not possible for [PERSON] and [PERSON] to have secure access to precisely the same space-time point to exchange a qubit; [PERSON] cannot signal at precisely light speed; and information cannot be processed instantaneously. In a realistic implementation, all the corrections about the idealized assumptions of these aspects will introduce time delays at various stages of the protocol. The effects of these time delays can be discussed same as the other relativistic protocols [9; 12]. If the corrections are small, the protocol remains secure and the only significant effect is that [PERSON] can only persuade [PERSON] that her commitment is binding from some point in the near causal future of \(O\), rather than from \(O\) itself [9; 12]. Realistically, too, perfect state preparations, transmissions, and measurements are impossible with unavoidable channel losses and errors. A low possibility of losses does not affect the security since the security parameter \(N\) of the protocol implies essentially committing \(N\) copies of the qubit, hence [PERSON] can always be able to receive enough evidence qubits to finally check [PERSON]'s unveiling. To deal with the channel errors, we consider that the two qubits \(|\psi_{0}\rangle_{\chi}=\alpha_{0}|0\rangle_{\chi}+\beta_{0}|1\rangle_{\chi}\) and \(|\psi_{1}\rangle_{\chi}=\alpha_{1}|0\rangle_{\chi}+\beta_{1}|1\rangle_{\chi}\) are two arbitrary elements of the public qubit set \(\{|\psi_{k}\rangle_{k=1}^{\chi}\}\). If [PERSON] attempts to cheat to pretend that the committed qubit \(|\psi_{0}\rangle_{\chi}\) is \(|\psi_{1}\rangle_{\chi}\) in a concrete implementation, she needs to persuade [PERSON] that the strings of \(|\varphi_{0}\rangle_{\text{B}}=\alpha_{0}|0\rangle_{\text{B}}\pm\beta_{0}|1 \rangle_{\text{B}}\) and \(\beta_{0}|0\rangle_{\text{B}}\pm\alpha_{0}|1\rangle_{\text{B}}\) within the collapsed qubits B are the strings of \(|\varphi_{1}\rangle_{\text{B}}=\alpha_{1}|0\rangle_{\text{B}}\pm\beta_{1}|1 \rangle_{\text{B}}\) and \(\beta_{1}|0\rangle_{\text{B}}\pm\alpha_{1}|1\rangle_{\text{B}}\), respectively. Let \(p_{i}^{1}=_{\text{B}}<\!\varphi_{1}|\varphi_{0}\!\succ_{\text{B}}\!\!\!<\! \varphi_{0}|\varphi_{1}\!\!\!>_{\text{B}}\!\!\!<\!\!\varphi_{0}|\varphi_{1}\!\! \!>_{\text{B}}\) be the probability of [PERSON] accepting a revelation of \(|\varphi_{1}\rangle_{\text{B}}\) for the \(i\)th \(|\varphi_{0}\rangle_{\text{B}}\). We have \(p_{i}^{1}=|\alpha_{0}^{*}\alpha_{1}+\beta_{0}^{*}\beta_{1}|^{2}\), and the probability that [PERSON] is detected cheating in unveiling \(|\varphi_{1}\rangle_{\text{B}}\) is \(1-p_{i}^{1}\). If the probability of the channel errors \(\varepsilon\) is bigger than \(1-p_{i}^{1}\), [PERSON] can always claim that the error was due to channel errors, even if she did cheat. Therefore, [PERSON] can be guaranteed that [PERSON]'s commitment is binding only when \(1-p_{i}^{1}>>\varepsilon\). Since it can be found that \(p_{i}^{1}=_{\text{X}}<\!\psi_{1}|\psi_{0}\!\succ_{\text{X}}\!\!\!<\!\!\psi_{0}| \psi_{1}\!\!\!>_{\text{X}}\) in our protocol, it finally becomes \(\!\times\!\psi_{1}|\psi_{0}\!\succ_{\text{X}}\!\!\!<\!\!\psi_{0}|\psi_{1}\!\! \succ_{\text{X}}<<\!\psi_{0}|\psi_{1}\!\!\succ_{\text{X}}<<1-\varepsilon\), which gives, to counter the channel errors, a definite constraint in determining the qubit set. When this constraint condition is satisfied, the channel errors make no essential difference as long as they are small and [PERSON] needs only to check that [PERSON]'s revelation is statistically consistent with the output corresponding to the unveiled qubit and statistically inconsistent with the other. In practical application, we think that \(1-_{\text{X}}<\!\psi_{1}|\psi_{0}\!\succ_{\text{X}}\!\!\!<\!\!\psi_{0}|\psi_{1 }\!\!\succ_{\text{X}}\!\!\!\geq 2\varepsilon\) is good enough to implement our protocol.