## 3 Practical Weak QOT and QBC  
Definition 1: Random Oblivious Transfer (R-OT) Channel. [PERSON] sends a random bit \(r\) to [PERSON] via a channel, if  
1. [PERSON] obtains the bit value \(r\) with a probability \(p\) satisfying \(0<b<p<a\), \(a<\frac{1}{2}\), where \(a\) and \(b\) are any two real numbers;
2. [PERSON] does not know whether [PERSON] has got the value of her bit. Then, the channel is named as R-OT channel (an extended Rabin's OT channel). To construct a quantum string R-OT protocol, non-orthogonal states are used. There is no measuring apparatus that can distinguish non-orthogonal states with certainty. Only some probabilistic information can be obtained. Let [PERSON] measure a sequence of photons in two quantum states \(|\Psi_{0}\rangle\), \(|\Psi_{1}\rangle\), where \(\langle\Psi_{0}|\Psi_{1}\rangle=\cos\varphi\). Here we choose \(\varphi=\frac{\pi}{6}\). The quantity of the information [PERSON] obtains depends on the measurement he performs. The optimal measurement can differentiate the two non-orthogonal states with a probability of \(1-\cos\varphi\)[56; 57; 58], which is a kind of POVM measurement. Actually, the complicated measurement is not necessary. Even if we construct the protocol with the sub-optimal measurement, the security of the protocols can still be ensured, which will be analyzed in detail in Section 4. Through all of the measurements, we choose the most practical and easiest one. That is, [PERSON] measures photons in two bases, \(B_{0}=\{|\Psi_{0}\rangle,|\Psi_{0}\rangle^{\perp}\}\) and \(B_{1}=\{|\Psi_{1}\rangle,|\Psi_{1}\rangle^{\perp}\}\) randomly. When the states is \(|\Psi_{0}\rangle\), the measurement results may be \(|\Psi_{0}\rangle\), \(|\Psi_{1}\rangle\) or \(|\Psi_{1}\rangle^{\perp}\). When the states is \(|\Psi_{1}\rangle\), the measurement results may be \(|\Psi_{1}\rangle\), \(|\Psi_{0}\rangle\) or \(|\Psi_{0}\rangle^{\perp}\). It can be seen that if [PERSON]'s measurement results in \(|\Psi_{x}\rangle\), he cannot distinguish which state is sent by [PERSON]. If his measurement results in \(|\Psi_{x}\rangle^{\perp}\), which is orthogonal to \(|\Psi_{x}\rangle\), the initial state cannot be \(|\Psi_{x}\rangle\) and therefore is \(|\Psi_{x\oplus 1}\rangle\). In this sub-optimal measurement, although [PERSON] cannot distinguish the non-orthogonal states with 100%, he unambiguously knows that the receiving state must be \(|\Psi_{x\oplus 1}\rangle\) when his measurement results in \(|\Psi_{x}\rangle^{\perp}\). Ideally, the probability of getting a conclusive result is  
\[p_{ideal}=\frac{1}{2}\times\frac{1}{2}(\langle\Psi_{x\oplus 1}|\Psi_{x} \rangle^{\perp}\langle\Psi_{x}|\Psi_{x\oplus 1}\rangle+\langle\Psi_{x}|\Psi_{x \oplus 1}\rangle^{\perp}\langle\Psi_{x\oplus 1}|\Psi_{x}\rangle)=\frac{1}{8}. \tag{1}\]  
Protocol 1: Practical weak quantum R-OT protocol. 1. [PERSON] and [PERSON] agree on three security parameters, \(N\), \(\alpha\), and \(\varepsilon_{set}\). The parameter \(N\) is the length of the qubit string sent by [PERSON]. The parameter \(\alpha\) is the expected fraction of [PERSON]'s successful detection. The parameter \(\epsilon_{set}\) _is the expected error rate._ _The number of photons in a weak coherent pulse with typical average photon number of_ \(\mu_{S}\) _follows Poisson distribution_ \(p_{n}(\mu_{S})=\frac{e^{-\mu_{S}}\mu_{S}^{n}}{n!}\)_. It can be seen that the probability of no photon in a pulse is_ \(p_{0}(\mu_{S})=e^{-\mu_{S}}\)_. Then, the probability of detecting at least one photons in a pulse with typical average photon number_ \(\mu_{S}\) _through a channel with transfer efficiency_ \(\eta_{C}\) _by a detector with quantum efficiency_ \(\eta_{D}\) _is_ \(1-e^{-\mu}\)_._ _They can set the fraction_ \(\alpha\simeq 1-e^{-\mu}\) _which is the probability that [PERSON] expects [PERSON] to detect successfully and set error rate_ \(\epsilon_{set}\simeq\varepsilon\) _or a little bit higher to allow other noise.