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Entanglement and Teleportation


Contents

Combined Systems
Correlation
Entanglement
Teleportation
Teleportation Experiments

Combined Systems

Most mathematical equations are tailored to one particle for simplicity. The next level is the two-particle system, which adds novel features
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 .
The content in the following may involve some unfamiliar mathematics, especially the notations usually reserved for quantum theory; see "Short-cut to the Introduction of Quantum Theory" for quick reference.

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Correlation

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).

The statistical nature of correlation always signifies incompleteness of knowledge about the system. For example, the radom appearance of a correlation diagram may be straightened up by knowing other external influences and making corrections accordingly. This concept is so ingrained in our thinking, it prompted Einstein to suggest that "hidden variable" is involved in entanglement between particles in space-like separation and generally in quantum theory. Modern tests on the Bell's Theorem have vindicated the quantum theory (with no hidden variables) to be correct. Actually, the seemingly fast-than-light action does not imply a message or information can be delivered that way, it is just an action involving the whole system. In the graphic example from Figure 03, both Alice and Bob would not know each other's measurement until they have a chance to bring the data together and compare notice if they have not learned the intricacy of entanglement.

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Entanglement

Correlation in quantum theory is different to the classical counterpart since it does not involve hidden variable. This is known as entanglement. In the decay of the pi meson into an electron-positron pair (Figure 03), since the spin for the pi meson is 0, the spin for the electron-positron pair must be opposite according to the conservation of angular momentum. Therefore, no matter how far apart are the members of this pair, if one spin state is measured for one of the member (collapsed from the superposition state), the spin for the other member will be the opposite at precisely the same moment. This non-local influence (non-locality) occur instantaneously. The following description uses these two spin spaces to illustrate some of the mathematical properties. The spin state is labeled as 1, or 0 corresponding to up or down (u, d) in some literatures.
Entanglement

Figure 03 Entanglement [view large image]


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Teleportation

Teleportation, Fictional Teleportation, Quantum 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) |SAB, |T1AB, |T2AB, |T3AB 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 |1C + b |0C .

  3. Joint Bell State Measurement (BSM) - This step merges all the three spin spaces together. For example, Alice can choose the singlet state |SAB to entangle with |C . By using the identities :

    |00 = (|T2 - |T3)/, |01 = (|T1 - |S)/, |10 = (|T1 + |S)/, and |11 = (|T2 + |T3)/,

    |SAB|C =
    |SAC (a |1B + b |0B) +
    |T1AC (-a |1B + b |0B) +
    |T2AC (a |0B - b |1B) +
    |T3AC (a |0B + b |1B) .

    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 |SAC, |T1AC, |T2AC, or |T3AC 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 |SAC, (01) for |T1AC, (10) for |T2AC, and (11) for |T3AC. 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 |1B + b |0B , where the 's are the Pauli matrices
  6. Entanglement

    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.

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Teleportation Experiments

The actual experimental setup for teleportation is shown in Figure 07 completed successfully over a distance of 600 meters across the River Danube. According to the usual convention, Bob's photon 3 was transported inside an 800 meter long optical fibre in a public sewer located underneath the river, where it is exposed to temperature fluctuations and other environmental factors (the real world).

Teleportation over River Danube
  • 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]


Note that because of the reduced velocity of light within the fibre-based quantum channel, the classical signal arrives about 1.5 microseconds before photon 3. Thus, there is enough time to set the EOM correctly before photon 3 arrives. Polarization rotation (which introduces errors) in the fibres is corrected by polarization controllers (PC) before each run of measurements. Polarization stability proved to be better than 10o on the fibre between Alice and Bob, corresponding to an ideal teleportation fidelity of 0.97.
See origin paper "Communications: Quantum teleportation across the Danube" for detail.

Quantum teleportation has only been done between similar objects - from light to light or matter to matter until 2006, when the first step has been taken to teleport the quantum state between a photon and an atom. This technique is critical in transferring the light qubits into atomic storage. The experiment achieved only for a transmission distance of half a meter. The traveling distance can be extended with improvement on the control of signal degradation. Figure 08 shows the experimental set-up for the experiment. As usual again, Alice is the keeper of system (1) to be teleported, and the entangled system (2); while Bob has the entangled system (3) waiting to receive the teleportation. Here's the protocol:



Teleportation of Light to Atom
  • 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
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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).
Hyperfine Structure 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.

Atomic Teleportation 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.