1. Quantum Teleportation

2. Qubits
A unit of quantum information that is analogous to the classical binary bit.
Mathematically expressed as a linear combination of |0 and | 1 :
ψ = α 0 +𝛽| 1
Using a quantum logic gate:
Unitary transformation
Rotations of the qubit vector in the Bloch sphere
Figure 1

3. Theory
Start with an EPR pair(two particles in a Bell entanglement) where one qubit is at location A and the other is at location B. They can be in one of the four entangled states Figure 2.
Alice has another qubit that she would like to teleport, or express, to Bob:| ψ =α | 0 𝐶 + 𝛽| 1 𝐶 . [3]
The state of all three particles is given in the Figure 3.
Figure 2
Figure 3

4. Theory (Cont.)
Alice then makes a local measurement of the Bell states of the two particles she has. These states can be any stated in Figure 4. The total three particle state is described in Figure 5. [3]
Because of the local measurement the three- particle entanglement state collapses to one of the four states in Figure 6.
Bob is also in one of the states in Figure 6. Each of these states are unitary images of the state to be teleported
Figure 4
Figure 5
Figure 6

5. Theory (Cont.)
7. Alice now knows which of the four states her system is in and she can send the results, through classical bits, to Bob. 8. Bob can then use a unitary operation to transform his particle into the desired state: | ψ =α | 0 𝐵 + 𝛽| 1 𝐵 . He does this using the Pauli matrices. [3] (Instead of 2 Bob will use 13= i2) Because Bob uses the information Alice teleported to him to show the quantum state of the particle she had, the no copying rule was not violated.
Figure 7

6. Experimental Evidence
Initial state of particle Alice wants to share with Bob.
and 3. Alice and Bob’s entangled EPR pair.
Figure 8 [2]

7. Experimental Evidence
Pulse of ultraviolet radiation passed through a nonlinear crystal (EPR pair)
Retroflection creates a second pair of photons. One of these is photon 1 and the other will be used as a trigger.
Alice checks for coincidences at f1 and f2.
When a coincidence between photon 1 and photon 2 is found at f1 and f2, Bob knows his photon 3 is in the initial state of photon 1.
Bob uses the Photon Beam Splitter and detectors d1 and d2 to check this result.
Figure 9 [2]

8. Experimental Evidence
Usually, based off of polarization. In this experiment, they were looking for coincidences at +45 degree polarization. Shown in strong decrease at -45 degrees. [2]
Coincidences between f1 and f2 occur 25% of the time. [2]
Teleportation occurs when detector d2, at +45 degrees polarization, is also in coincidence. Need threefold coincidence to confirm.
Figure 10 [2]

9. Experimental Evidence
Measured threefold coincidences for both +45 degree polarization teleportation and -45 degree polarization teleportation. [2]
Figure 11 [2]

10. Experimental Accomplishments
Entanglement-based quantum communication over 144km [4]
Figure 12 [4]

11. Summary and Applications
Quantum teleportation occurs when one particle, Alice, is able to share information on a particle it possesses to another particle Bob, in order for Bob to replicate the particle that Alice was given.
The particle that Alice was given will no longer be in the same state after she checks the state of the particle, avoiding violating the cloning rule.
Experiments have shown that quantum computing is viable and has been proven.
A recent experiment has achieved quantum teleportation of photon polarization at 144km.
Quantum teleportation would be applicable for encryption and quantum computing. [1]

12. References
[1] Barrett, M. D. et al.Deterministic quantum teleportation of atomic qubits. Nature429,737–739 (2004). [2] Bouwmeester, D. et al.Experimental quantum teleportation. Nature390,575– 579 (1997). [3] Shi, R., Huang, L., Yang, W. & Zhong, H. Controlled quantum perfect teleportation of multiple arbitrary multi-qubit states. Science China Physics, Mechanics and Astronomy54,2208–2216 (2011). [4] Ursin, R. et al.Entanglement-based quantum communication over 144km. Nature3,481–486 (2007).

13. Theory
Bob and Alice are created in an “entangled” state, represented here by Bob and Alice holding hands.

14. Theory
Alice is given another particle, represented here as a letter. Because she hasn’t opened the “letter,” Bob and Alice are still in the entangled state.

15. Theory
Alice decides to open the letter, or check the state of the particle. She is unable to do so and still hold hands with Bob, so they are no longer entangled. Both Alice and Bob are in their own states now.

16. This is what is on the letter
Theory
This is what is on the letter
Alice is then able to tell Bob what is on the letter. She cannot give him a copy or directly give him the letter, but she can tell him what is written on it.

17. Theory
Using the information that Alice gave him, Bob can then show the same information that was in the letter.