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Entanglement and Teleportation


Contents

Combined Systems
Correlation
Entanglement
Decoherence
Teleportation
Teleportation Experiments
Entanglement and Holographic Space-time

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
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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 03a, 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 03a), 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 03a Entanglement [view large image]



Monogamy Degree of Entanglement The maximally entangled pair of qubits is monogamy. For example, if A (Alice) and B (Bob) are maximally entangled they cannot be correlated with another partner C (Charlie) as illustrated in Figure 03b. However, if the entanglement is not maximal, polygamy is permitted according to the rules depicted in Figure 03c, in which the "E" stands for "Entanglement Measure" -

Figure 03b Monogamy
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Figure 03c Entanglement, Degree of [view large image]

a measurement quantifying the degree of entanglement contained in the system. The formula can be generalized to many qubits :

E(A|B|C1|C2|C3| ... Cn) = E(A|BC1C2C3...Cn) - E(A|B) - E(A|C1) - E(A|C2) - E(A|C3) - ... E(A|Cn) 0.

It shows that A (Alice) can entangle with a lot of qubits such as those in the environment or measuring device provided it is not maximally entangled with any one of them. Those non-maximal states are mixtures of the Bell (pure) states (such as the |S> and |Ti>s). The monogamy has something to do with the "No-cloning Theorem". It is the special case of E(A|B|C) = 0 as illustrated in Figure 03c. See footnote for a copy of the explanation related to monogamy from "Limitations to sharing entanglement".

See possible coupling mechanism in "Entanglement of Spins".

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Decoherence

Superposition Decoherence A quantum system such as the electron in atom often exists in a superposition state, i.e., it is described by a mixture of many eigen-states (the state with a definite eigen-value such as energy). This is depicted in Figure 03d for an atom in an excited and ground state. A similar idea is expressed in an example of "Infinite Square Well" and graphically in Figure 05c (in the same section). As shown in Figure 03e, the superposition is dissolved via interaction with the environment. This is called decoherence and can be

Figure 03d Superposition
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Figure 03e Decoherence
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explained mathematically in the following as transfer of entanglement to all kinds of particles in there.

Using the same notations as defined previously in "Combined System" (also refer to Figure 03f),
Hibert Space

Figure 03f Hilbert Space
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That is, the probabilities become additive corresponding to the OR operation. Each term in the summation represents the probability of measuring the state |i> by an instrument. Since the quantum state |> can be a combined system such as the entanglement of many qubits, the entanglement also undergoes decoherence. It is sometimes referred to as transfer of entanglement to the environment.

The property of superposition of different states has its origin in the linear form of the governing equation. Figure 03g shows an example of the superposition of 2 spin states (up and down) for a single particle. It is also possible to entangle 2 or more particles as shown in the same picture. However, such quantum nature would be dissolved quickly by interacting with the environment in a process called "Decoherence" as
Quantum Nature demonstrated above. This is the reason why we don't perceive such property in everyday life. The popular image of "Schrodinger's Cat" is ill-conceived by identifying a microscopic property to a macroscopic body. The teleportation so enthralles the public is a fiction trying to extend the concept of entanglement to too many particles. There is no quantum weirdness, which is created only by mis-appropriation of the quantum theory. BTW, according to the reductionist's view, description of the

Figure 03g Quantum Nature
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world is separated into many different levels. An absurd scenario can often be concocted with the so-called "thought experiment" by mixing up the various levels (see Effective Theories).

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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
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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)/,

    It can be shown that |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. Incidentally this step demonstrates the occurrence of monogamy in the transfer of maximal entanglement, i.e., the entanglement can be between AB or AC but not both at the same time.

  5. 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.

  6. 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
  7. 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.
N.B. A very important limitation on entanglement is decoherence. The state of entanglement or superposition will dissolve via interaction with the environment in very short time interval from 10-6 to about a few seconds.

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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.

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Entanglement and Holographic Space-time

The conjecture of AdS/CFT correspondence has its origin in 1997 about a stack of D3-branes and the bulk in Superstring Theory. It was found that the dynamics of elementary particles (the open strings) on the D3-branes (which have so many stacks that it becomes a black hole, now called black brane) can be described by the closed strings (the gravitons) moving slowly (as viewed by a distant observer) in the vicinity of the black brane but still within the bulk (Figure 11). Then the gravity of the black brane imparts a curved shape to the bulk in the form of (4+1)-dimensional Anti-de Sitter spacetime. Subsequently, it has also been shown that a black hole in the bulk corresponds to high energy particles
Branes Bulk Correspondence on the boundary. Since then many examples have been discovered to have such correspondence. The most famous one is the equivalence of "Type II String Theory" on the product space AdS5XS5, (i.e., 5 macroscopic AdS dimensions combines to 5 compactified microscopic dimensions), to the "Supersymmetric Yang-Mills Theory" on the 4-D boundary. A mathematical dictionary has been compiled to link the two perspectives. It is similar to the laser, which transforms a 2-D scrambled pattern into a recognizable 3-D image. This bulk to boundary correspondence as demonstrated by the holography invented in 1947, now becomes the "Holographic Principle" embraced by some physicists, who claim that it will become part of the foundations of new physics.

Figure 11 Branes Bulk Correspondence

Since the AdS space has played such a prominent role in the correspondence and its ramification, some of its properties are described briefly in the following.

Hyperbolic 2-D Slice The Robertson-Walker metric for the AdS universe is in the form :

ds2 = c2dt2 - R(t)2 [dr2 + w2 (d2 + sin2 d2)]

where w = sinh(r) has the unit of length as the curvature k = -1 (in unit of cm-2) is hidden in the formalism. It can be shown readily that the scale factor :

Figure 12 Hyperbolic 2-D Slice
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R(t) = (c/H) sin(Ht), where H = (||/3)1/2c, is the cosmological constant and has a negative value signifying an attractive force (see insert in Figure 12).

In the formulation of the AdS/CFT correspondence, the scale factor R and the cosmological constant were not taken into consideration. The main interest is in the surface element :

dL2 = sinh2(r) (d2 + sin2 d2).

The circumference of a 2-D slice in a sphere at = /2 has a length of L1 = 2r (Figure 12, left). However, for the case of hyperbolic space L2 = 2sinh(r) > L1. The geometry can be visualized by lining up the angels (or devils) along the circumference, as shown by the "Circle Limit" in Figure 12 (right) the L2 circumference can accommodate more of them (with invariant size) than the regular one along L1.

AdS Space The AdS space-time in the correspondence is created by stacking up the hyperbolic slices along the time axis (Figure 13) and has nothing to do with the AdS scale factor nor the cosmological constant. In short, the purported AdS space is an empty hyperbolic space, which becomes Minkowski space at the boundary infinitely faraway (similar to a small piece of flat area faraway from the center of the globe). This property is important for prescribing quantum theories, for all of them are formulated on the background of flat space-time. Anyway, when the correspondence has been promoted to the level of principle, it becomes a tool in vogue with quantum-gravity physicists especially about entanglement.

Figure 13 AdS Space
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Entanglement and Spacetime Entanglement and Wormhole One research recently considers entangled quantum particles in different regions at the boundary. It claims that the AdS sapce within would be split in two as the entanglement is reduced to zero. Thus, there is a link between space and entanglement (Figure 14). Such effect of entanglement dependence can also be applied to the wormhole (in the bulk) linking two black hole in the D3-brane. It is in the same vein on ER = EPR or wormhole = entanglement (Figure 15).

Figure 14 Entanglement and Spacetime [view large image]

Figure 15 Entanglement and Wormhole [view large image]

See original articles in "The Quantum Source of Space-time" and "Entangled Universe".


Footnote :
The no-cloning theorem is a result of quantum mechanics that forbids the creation of identical copies of an arbitrary unknown quantum state. As A and B share a maximally entangled two-qubit state, A and B have the requisite quantum resource to teleport an unknown quantum state from one to the other. As shown in Figure 03b, suppose that A and C also share a maximally entangled two-qubit state. Then A can teleport an unknown quantum state to C. This set-up can be exploited to clone an unknown quantum state as follows:

"A" teleports the state to B and to C; thus, this tripartite network has succeeded in copying the state, i.e., B and C each hold a copy now. However, this operation violates the no cloning theorem, which is in turn a direct consequence of the linearity of quantum mechanics. If A and B share a maximally entangled state, even if one of the two parties shares any entanglement whatsoever with the third party C, the no cloning theorem is violated.