# Probabilistic Teleportation via Quantum Channel with Partial Information

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Different Conditions of Quantum Channel with Partial Information

_{1}〉 and |1

_{1}〉 are called qubits to signify the new possibilities introduced by quantum physics into information science [41].

^{2}+ |β|

^{2}= 1. The subscript number indicates the owner of given qubit in the context.

^{2}+ |b|

^{2}= 1, and a ≥ |b| > 0. Particle 2 belongs to the sender Alice, while particle 3 belongs to the receiver Bob.

## 3. The Teleportation for Partial Information Quantum Channel

#### 3.1. Alice only Knows the Amplitude Factor a, and Bob only Knows the Phase Factor ϕ

**Step 1**: A particle m who plays an auxiliary function in teleportation with an initial state |0_{m}〉 is introduced by Alice, and then Alice’s state which is composed of particles 1, 2, m and 3 will take the form of the following Equation (3).$$\begin{array}{l}|{\psi}_{12m3}^{0}\rangle =|{\psi}_{12}\rangle \otimes |{0}_{m}\rangle \otimes |{\psi}_{3}\rangle \\ \phantom{\rule{3em}{0ex}}=\alpha a|{0}_{1}{0}_{2}{0}_{m}{0}_{3}\rangle +\alpha \sqrt{1-{a}^{2}}{e}^{i\varphi}|{0}_{1}{1}_{2}{0}_{m}{1}_{3}\rangle +\beta a|{1}_{1}{0}_{2}{0}_{m}{0}_{3}\rangle +\beta \sqrt{1-{a}^{2}}{e}^{i\varphi}|{1}_{1}{1}_{2}{0}_{m}{1}_{3}\rangle \end{array}$$**Step 2**: Following on the heels of step 1, an operation named U_{S}will be performed on all particles including 1, 2 and m by Alice. If the quantum channel is maximal entangle state, the U_{S}could be bypassed. The U_{S}operation is an unitary transformation, could be expressed as Equation (4).$${U}_{s}=\left(\begin{array}{cccc}A\left(a\right)& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& {\sigma}_{z}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& A\left(a\right)& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& {\sigma}_{z}\end{array}\right)$$_{z}and A(a) could be expressed as$${\sigma}_{z}=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)\phantom{\rule{2em}{0ex}}A\left(a\right)\left(\begin{array}{cc}\sqrt{\frac{1-{a}^{2}}{{a}^{2}}}& \sqrt{\frac{2{a}^{2}-1}{{a}^{2}}}\\ \sqrt{\frac{2{a}^{2}-1}{{a}^{2}}}& -\sqrt{\frac{1-{a}^{2}}{{a}^{2}}}\end{array}\right)$$$$\begin{array}{l}|{\psi}_{12m3}^{1}\rangle =\left({U}_{s}|{\psi}_{12m}^{0}\rangle \right)\otimes |{\psi}_{3}^{0}\rangle \\ \phantom{\rule{3.2em}{0ex}}=\sqrt{\frac{1-{a}^{2}}{2}}|{\varphi}_{12}^{+}\rangle \otimes |{0}_{m}\rangle \otimes \left(\alpha |{0}_{3}\rangle +\beta {e}^{i\varphi}|{1}_{3}\rangle \right)+\sqrt{\frac{1-{a}^{2}}{2}}|{\varphi}_{12}^{-}\rangle \otimes |{0}_{m}\rangle \otimes \left(\alpha |{0}_{3}\rangle -\beta {e}^{i\varphi}|{1}_{3}\rangle \right)\\ \phantom{\rule{3.2em}{0ex}}+\sqrt{\frac{1-{a}^{2}}{2}}|{\mathrm{\Psi}}_{12}^{+}\rangle \otimes |{0}_{m}\rangle \otimes \left(\alpha |{1}_{3}\rangle +\beta {e}^{i\varphi}|{0}_{3}\rangle \right)+\sqrt{\frac{1-{a}^{2}}{2}}|{\mathrm{\Psi}}_{12}^{-}\rangle \otimes |{0}_{m}\rangle \otimes \left(\alpha |{1}_{3}\rangle -\beta {e}^{i\varphi}|{0}_{3}\rangle \right)\\ \phantom{\rule{4em}{0ex}}+\sqrt{2{a}^{2}-1}\left(\alpha |{0}_{1}{0}_{2}\rangle +\beta |{1}_{1}{0}_{2}\rangle \right)\otimes |{1}_{m}\rangle \otimes |{0}_{3}\rangle \end{array}$$$$|{\varphi}_{12}^{\pm}\rangle =\frac{1}{\sqrt{2}}\left(|{0}_{1}{0}_{2}\rangle \pm |{1}_{1}{1}_{2}\rangle \right)$$$$|{\mathrm{\Psi}}_{12}^{\pm}\rangle =\frac{1}{\sqrt{2}}\left(|{0}_{1}{1}_{2}\rangle \pm |{1}_{1}{0}_{2}\rangle \right)$$**Step 3**: Subsequently, the auxiliary particle m is measured and particles 1 and 2 are performed in the form of the Bell-state measurements. Then, Alice transmits measurement results information to Bob in the manner of classical channel.**Step 4**: Bob will perform two continuous unitary operators U_{P}and U_{T}on particle 3 to obtain the original state according to the information including the information received from Alice via classical channel and the local phase factor. Table 1 shows the corresponding relations between the outcomes of measurement and the unitary transformation U_{T}for particle 3. The unitary operation U_{P}, relative to the phase factor ϕ of Equation (2), is described as Equation (9).$${U}_{P}=\left(\begin{array}{cc}1& 0\\ 0& {e}^{-i\varphi}\end{array}\right)$$

_{S}performed by the sender Alice is only relative with the amplitude factor a, and U

_{P}can be finished on the condition that the receiver Bob has the phase factor ϕ. Hence, as long as Alice has the amplitude factor a, and Bob has the phase factor ϕ, the probabilistic teleportation could be successfully obtained. On the other hand, the success probability of the teleportation has significant positive correlation with the fidelity of entangle channel, that is to say, the higher of fidelity, the teleportation will be obtained in higher probability. The success probability could be expressed as 2−2a

^{2}. If quantum channel is a maximally entangled state, $|a|=\frac{1}{\sqrt{2}}$, the teleportation will be successful at all times. This result is in agreement with the success probability in [1].

#### 3.2. Alice only Knows the Phase Factor ϕ, and Bob only Knows the Amplitude Factor a

**Step 1**: Alice performs the unitary operation U_{P}on particle 1 shown as Equation (9) using the phase information owned by herself, as a consequence, the total system can be expressed as$$\begin{array}{l}|{\psi}_{123}^{1}\rangle =|{\psi}_{1}\rangle \otimes ({U}_{P}|{\psi}_{2}\rangle )\otimes |{\psi}_{3}\rangle \\ \phantom{\rule{2.5em}{0ex}}=\frac{1}{\sqrt{2}}|{\varphi}_{12}^{+}\rangle \otimes (\alpha a|{0}_{3}\rangle +\beta \sqrt{1-{a}^{2}}|{1}_{3}\rangle )+\frac{1}{\sqrt{2}}|{\varphi}_{12}^{-}\rangle \otimes (\alpha a|{0}_{3}\rangle -\beta \sqrt{1-{a}^{2}}|{1}_{3}\rangle )\\ \phantom{\rule{3.7em}{0ex}}+\frac{1}{\sqrt{2}}|{\mathrm{\Psi}}_{12}^{+}\rangle \otimes (\alpha a|{1}_{3}\rangle +\beta \sqrt{1-{a}^{2}}|{0}_{3}\rangle )+\frac{1}{\sqrt{2}}|{\mathrm{\Psi}}_{12}^{-}\rangle \otimes (\alpha a|{1}_{3}\rangle -\beta \sqrt{1-{a}^{2}}|{0}_{3}\rangle )\end{array}$$**Step 2**: Alice performs the Bell-state measurements on particles 1 and 2. Subsequently, Alice informs Bob of her measurement results using classical channel.**Step 3**: Similar to the condition presented in previous subsection, an auxiliary particle which could be marked as m is introduced necessarily. Then, an unitary transformation U_{F}which can be written as Equation (11) will be performed on particles 3 and m depending on the remote and local information. The unitary transformations ${U}_{F}^{i}\left(i=0,1,2,3\right)$in Table 2 are the 2 × 2 matrix. Table 2 shows the corresponding relations between the measurement results on particles 1, 2 and the unitary transformation U_{F}on particles 3 and m, and then the origin state is appeared.$$\begin{array}{l}{U}_{F}^{\mathbf{0}}=\left(\begin{array}{cc}A\left(a\right)& \mathbf{0}\\ \mathbf{0}& {\sigma}_{z}\end{array}\right)\phantom{\rule{1em}{0ex}}{U}_{F}^{1}=\left(\begin{array}{cc}A\left(a\right)& \mathbf{0}\\ \mathbf{0}& -{\sigma}_{z}\end{array}\right)\\ {U}_{F}^{2}=\left(\begin{array}{cc}\mathbf{0}& {\sigma}_{z}\\ A\left(a\right)& \mathbf{0}\end{array}\right)\phantom{\rule{1em}{0ex}}{U}_{F}^{3}=\left(\begin{array}{cc}\mathbf{0}& -{\sigma}_{z}\\ A\left(a\right)& \mathbf{0}\end{array}\right)\end{array}$$_{z}and A(a) could be expressed as Equation (5).**Step 4**: Subsequently, To obtain the origin state of qubit, only one measurement result of particle m is in need. There are two cases for the result. In case of that the state of m is |1_{m}〉, quantum teleportation fails. In the other case that the state of m is |0_{m}i, the teleportation will be realized with the same probability of $\frac{1-{a}^{2}}{2}$ for four different kinds showed in Table 2, and then the sum of success probability is 2 − 2a^{2}. To put it in another way, the success probability is decided by the entangle state as discussed in Section 3.1.

_{P}and U

_{F}can be performed by Alice and Bob, respectively. Therefore, the novel scheme of this section is effective.

## 4. Discussion and Conclusions

^{2}, also could be expressed as 2|b|

^{2}in our proposals. It is determined by the fidelity of entangle channel which could be realized in various ways, to name only a few, photon polarization, super-radiance, collective spontaneous emission. Proposals mentioned in this paper mainly focus on the mechanism of Alice and Bob without regard to the fidelity entangle channel. Many outstanding proposals [12,13,35,46] could improve the fidelity of entangle channel, and could be joined with our proposals.

## Acknowledgments

**PACS classifications**: 03.67.-a; 03.67.Hk; 03.65.-w

## Author Contributions

## Conflicts of Interest

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**Figure 1.**A sketch of the whole probabilistic teleportaton processes for the first proposal. Particles 1 and 2 belong to the sender Alice, and particle 3 belongs to the receiver Bob, while the auxiliary particle m with an initial state |0

_{m}〉 is introduced by Alice. The unitary operations U

_{S}is performed by Alice, and U

_{P}is performed by Bob. M

_{i}(i = 1, 2, 3) represent single-qubit measurement with the basis {|0〉, |1〉}.

**Figure 2.**A sketch of the whole probabilistic teleportaton processes for the second proposal. Particles 1 and 2 belong to the sender Alice, and particle 3 belongs to the receiver Bob, while the auxiliary particle m with an initial state |0

_{m}〉 is introduced by Bob. The unitary operations U

_{P}is performed by Alice, and U

_{F}is performed by Bob.

Measurement results | State of Particles 3 | Probabilities | U_{T} | |
---|---|---|---|---|

Particle m | Particles 1, 2 | |||

|0_{m}〉 | $|{\varphi}_{12}^{+}\rangle $ | α∣0_{3}〉 + ∣βe^{iϕ}∣1_{3}〉 | $\frac{1-{\alpha}^{2}}{2}$ | I |

$|{\varphi}_{12}^{-}\rangle $ | α∣0_{3}〉 − βe^{iϕ}∣1_{3}〉 | $\frac{1-{\alpha}^{2}}{2}$ | σ_{z} | |

$|{\mathrm{\Psi}}_{12}^{+}\rangle $ | α∣1_{3}〉 + βe^{iϕ}∣0_{3}〉 | $\frac{1-{\alpha}^{2}}{2}$ | σ_{x} | |

$|{\mathrm{\Psi}}_{12}^{-}\rangle $ | α∣1_{3}〉 − βe^{iϕ}∣0_{3}〉 | $\frac{1-{\alpha}^{2}}{2}$ | iσ_{y} | |

|1_{m}〉 | – | – | 2a^{2}−1 | – |

BMRs on particles 1, 2 | U_{F} | Results after the transformation U_{F} | ||
---|---|---|---|---|

Particle m | Particle 3 | Probabilities | ||

$|{\varphi}_{12}^{+}\rangle $ | ${U}_{F}^{0}$ | ∣0_{m}〉 | α∣0_{3}〉 + β∣1_{3}〉 | $\frac{1-{\alpha}^{2}}{2}$ |

∣1_{m}〉 | ∣0_{3}〉 | $\frac{|\alpha {|}^{2}}{2}(2{\alpha}^{2}-1)$ | ||

$|{\varphi}_{12}^{-}\rangle $ | ${U}_{F}^{1}$ | ∣0_{m}〉 | α∣0_{3}〉 + β∣1_{3}〉 | $\frac{1-{\alpha}^{2}}{2}$ |

∣1_{m}〉 | ∣1_{3}〉 | $\frac{|\alpha {|}^{2}}{2}(2{\alpha}^{2}-1)$ | ||

$|{\mathrm{\Psi}}_{12}^{+}\rangle $ | ${U}_{F}^{2}$ | ∣0_{m}〉 | α∣0_{3}〉 + β∣1_{3}〉 | $\frac{1-{\alpha}^{2}}{2}$ |

∣1_{m}〉 | ∣1_{3}〉 | $\frac{|\alpha {|}^{2}}{2}(2{\alpha}^{2}-1)$ | ||

$|{\mathrm{\Psi}}_{12}^{-}\rangle $ | ${U}_{F}^{3}$ | α∣0_{3}〉 + β∣1_{3}〉 | $\frac{1-{\alpha}^{2}}{2}$ | |

∣1_{3}〉 | $\frac{|\alpha {|}^{2}}{2}(2{\alpha}^{2}-1)$ |

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Liu, D.; Huang, Z.; Guo, X.
Probabilistic Teleportation via Quantum Channel with Partial Information. *Entropy* **2015**, *17*, 3621-3630.
https://doi.org/10.3390/e17063621

**AMA Style**

Liu D, Huang Z, Guo X.
Probabilistic Teleportation via Quantum Channel with Partial Information. *Entropy*. 2015; 17(6):3621-3630.
https://doi.org/10.3390/e17063621

**Chicago/Turabian Style**

Liu, Desheng, Zhiping Huang, and Xiaojun Guo.
2015. "Probabilistic Teleportation via Quantum Channel with Partial Information" *Entropy* 17, no. 6: 3621-3630.
https://doi.org/10.3390/e17063621