Entanglement and Teleportation

Contents

Combined Systems

unknown to single particle system even though the combined system is derived by merging each one. For example, let's consider two single systems SA and SB (personified by the two computer geeks Alice and Bob respectively) each with a set of basis vectors |a's and |b's. In particular, SA could be represented by a coin with basis vectors |H (for head) and |T (for tail), while SB is a dice with basis vectors labeled by 1, 2, 3, 4, 5, 6. The combined system (called SAB) has basis vectors shown as the table entries in Figure 01, e.g., |H1, ... |T6. This is the tensor product from merging two vector spaces. Any operator in SA can only act on the first label, same is for SB on the second label. Superposition state in SAB is written as :

Figure 01 Combined System [view large image]

where |aibj is the basis vector in SAB with associated probability c*ij cij .

Correlation

Correlation is about the dependence between two kinds of things labeled as x and y as shown in Figure 02. Some of them pair up randomly showing no discernible pattern (rxy = 0), while the other extreme would display a graph in the form of a straight line. Thus, different system would exhibit different degrees of correlation, which can be computed by formulas such as the Spearman`s Rank Correlation. On the other hand, the Chi Square Test would provide just a "yes" or "no" after running the data with the procedure. The formula in Figure 02 is another way to estimate the degrees of correlation. It can be verified simply with the case of rxy = 1 by taking just 2 points (n = 2) at (0,1), (4,3) and the average (2,2).

Figure 02 Correlation [view large image]

The quantum correlation is defined by the averages of observables which imply no correlation if the average of the products is equal to the product of the averages (Figure 02).

Entanglement

Figure 03 Entanglement [view large image]

Teleportation

		Although it is not an entry in most dictionaries, teleportation is very popular in science fictions. One scheme uses a transporter in which persons or non-living items are placed on the pad and dismantled particle by particle by a beam, with their atoms being patterned in a computer buffer and converted into another beam that is directed toward the destination where the things would be reassembled back into their original form (usually with no mistakes, Figure 04).
Figure 04 Teleportation, Fictional [view large image]	Figure 05 Teleportation, Quantum [view large image]	Quantum teleportation is possible in theory and lately (up to 2015) in practice with photons and partial atom, i.e., transporting only the electron shells without the nucleus.

The following illustrates the principle with three spin spaces entangled together in mathematical formulas and a diagram (Figure 05) :


1. Entanglement Generation - Four maximally entangled states (Bell States) |S_AB, |T_1AB, |T_2AB, |T_3AB are generated between systems A and B as shown in the section about "Entanglement". The subscript AB etc. is now necessary to avoid confusion with the presence of more than two spin spaces.


2. State Preparation - The spin state to be teleported is prepared by Alice with the label "C" : |_C = a |1_C + b |0_C .


3. Joint Bell State Measurement (BSM) - This step merges all the three spin spaces together. For example, Alice can choose the singlet state |S_AB to entangle with |_C . By using the identities :
   
   |00 = (|T₂ - |T₃)/, |01 = (|T₁ - |S)/, |10 = (|T₁ + |S)/, and |11 = (|T₂ + |T₃)/,
   
   |S_AB|_C =
   |S_AC (a |1_B + b |0_B) +
   |T1_AC (-a |1_B + b |0_B) +
   |T2_AC (a |0_B - b |1_B) +
   |T3_AC (a |0_B + b |1_B) .
   
   This formula reveals that the two-spin entanglement has been transferred from system AB to AC with all the four possible Bell states linking to four possible superpositions of the original state vector |_C now labeled under B. Bob knows there are four possibilities but doesn't know exactly which one. Alice then performs a measurement (Joint BSM) on the AC Bell states yielding one of the |S_AC, |T1_AC, |T2_AC, or |T3_AC basis vector.


4. Conditional Transform - Alice and Bob agree on a two-bits code for each of the four AC Bell state, e.g., (00) for |S_AC, (01) for |T1_AC, (10) for |T2_AC, and (11) for |T3_AC. She would send the code corresponding to the measurement to Bob via a classical channel.


5. Teleported State - When Bob has received the code, he would proceed to run the corresponding operation : I, ₃, -i₂, ₁ to the associated |_B state vectors to recover the original in the form of |_B = a |1_B + b |0_B , where the 's are the Pauli matrices

	In principle, Alice can pick any one of the |SAB, |T1AB, |T2AB, or |T3AB basis vectors to entangle with |C , but the resulting relationship would be re-arranged.
Figure 06 Teleportation [view large image]	Actually, there is no transfer of matter involved. The object of system C has not been physically moved to the location of system B; only its state has been conveyed over.

Teleportation Experiments

The entangled photon pairs (0,1) and (2,3) are created in the beta-barium borate (BBO) crystal by a pulsed UV laser. Photon 0 serves as the trigger. Photons 1 and 2 are guide into a optical-fibre beam splitter (BS) connected to the polarizing beam splitters (PBS) for Bell-state measurement (BSM). Photon 3 goes to Bob. Alice's logic electronics identify the Bell state and convey the result through the microwave channel (RF unit) to Bob's electro-optic modulator (EOM). Depending on the message, it either leaves the photon state unaltered or changes it to the opposite state.

Figure 07 Teleportation over River Danube [view large image]

A 2-ms pulse of light is sent through the atomic sample at Bob's location and becomes entangled with the atoms. This is to initialize system 3, which consists of atoms initially optical pumped into the hyperfine energy level F = 4, mF = 4 state with a 4-ms pulse (see Figure 08). The pulse travels 0.5 m to Alice's location and entangle systems 2 and 3. System 2 is entangled on a beamsplitter (BS) with the object of teleportation (system 1) - a few-photon coherent pulse of light - generated by electro-optical modulator (EOM). A Bell measurement is performed, and the results are sent via a classical communication channel to Bob. There they are used to complete the teleportation onto atoms by shifting the atomic collective spin state with a pulse of radio-frequency (RF) magnetic field of 0.2-ms duration. After a delay of 0.1 ms, a verifying pulse is sent to read out the atomic state, in order to prove the successful teleportation.

Figure 08 Teleportation of Light to Atom [view large image]

Note : The interaction between electron and nuclear spins splits the energy level by a small amount (~ 10-6 ev) forming the hyperfine structure (Figure 09).

	In essence, the polarization state of the photons is conveyed from Alice to Bob's location, where it is converted to the spin state of the electron (in the atoms, Figure 09). There is no teleportation of matter. The experiment was performed with 1012 caesium atoms in coherent spin state. It demonstrates the possibility of teleporting the state in moving carrier to stationary object for storage.
Figure 09 Hyperfine Sturcture [view large image]	See original paper "Quantum teleportation between light and matter" for detail.

Teleportation of atomic state in Ca+ ions has also been performed in ion trap. The spin up and down states are replaced by the two atomic states |1 = S1/2 , |0 = D5/2 (see Figure 10). Ion 2 and 3 are entangled in one of the four Bell states. The teleported state is one of |1, |0, (|1 + |0)/, or (|1 + i|0)/. The actual experimental set-up is different from the other experiments, but the outcome is similar, i.e., the teleportation is logical instead of material. The mathematical formulas are implemented by electronic devices. This work is important for future development of quantum computing.

Figure 10 Atomic Teleportation [view large image]

See original paper "Deterministic quantum teleportation with atoms" for detail.