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Entanglement and Teleportation


Combined Systems
Entanglement Measure (2021)
Hydrogenic Entanglement etc. (2021)
Coherence Time (2019)
Teleportation Experiments
Holographic Space-time, Quantum Entanglement, and Quantum-Gravity (2022)
Wormhole Experiment (2022-12)

Combined Systems

Most mathematical equations are tailored to one particle for simplicity. The next level is the two-particle system, which adds novel
Combined Systems features 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.



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 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.



Correlation of particles in quantum theory is known as "Quantum 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.

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

Figure 03b Monogamy [view large image]

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

"Entanglement Measure" - 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 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".



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 a mixture of 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

Figure 03d Superposition
[view large image]

Figure 03e Decoherence
[view large image]

is called decoherence and can be 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
(See detail of derivation in "Quantum Decoherence, Dirac Notation").

Figure 03f Hilbert Space
[view large image]

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 Schrodinger 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 demonstrated above. This is the reason why we don't perceive such property in everyday life. The popular image
Quantum Nature 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
[view large image]

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).


Entanglement Measure (2021)

Since entanglement involves 2 or more particles, its quantification can only be estimated by using statistical analysis. In classical cases, the method relies on the "Probability Density" in phase space, which counts the averaged number of particles or "how often" these particles is in a small range of phase space at spatial coordinates q = (q1, q2, ... qN) and momenta p = (p1, p2, ... pN) from numerous samplings (See Figure 03h,a). This is referred to as "Statistical (Gibbsian) Ensemble", which essentially is an idealization consisting of a large number of virtual (or mental, or numerical with certain approximation) copies of a system and map out the frequency of occurrence at each point in phase space.

In quantum theory, it is replaced by the "Density Matrix" and ultimately the "Single-Particle Reduced Density Matrix" (Figure 03h,b).

Probability Density

Figure 03h Entanglement Measure

Entanglement of Electrons in Diatomic Molecules

Figure 03j Entanglement of Electrons in Diatomic Molecules

The von Neumann entropy is the quantum version of "Shannon's Measure of Information" :
SMI = - i pilog2(pi), where pi represents the discrete probability distribution of some samples. SMI (aka Shannon Entropy) measures how much information is required to identify one particular sample from that distribution in unit of bit (see example below).

For the case of spinning electrons as in the previous example with p1 = p2 = 1/2, (see illustration : )
SMI = -{(1/2)log2(1/2) + (1/2)log2(1/2)} = log2(2) = 1 bit, which means 1 inquiry of "YES" or "NO" is required to identify a sample. This is the amount of information necessary for the identification. It corresponds to the case of maximal uncertainty in this example, and coincide with the von Neumann entropy for the case of 2 identical electrons in empty space (up to a proportional constant, i.e,
S() = n(2) = n(2) log2(2) = 0.693 log2(2)). The 2 different definitions of entropy do not agree in general.

For p1 = 1, p2 = 0, SMI = -log2(1) = 0. Both samples are in the same partition (one of the cells for marking the distribution). They are certainly in there (no uncertainty) but it is not sure which one, i.e., the information is zero. (see illustration : )

By analogy with "bit", the term "qubit" is the basic unit of quantum information in the von Neumann entropy. An arbitrarily large amount of classical information can be encoded in a qubit. This information can be processed and communicated but at most one qubit can be accessed. The accessible information in a probability distribution (read density matrix) is measured by the von Neumann entropy.

Entanglement Measure Entanglement Measures for the electrons in 5 diatomic molecules has been published in a 2011 paper entitled "Quantum Entanglement and the Dissociation Process of Diatomic Molecules". The post-Hartree–Fock computational method is employed to calculate the wave function and ultimately N as function of the inter-atomic distance R. The Hartree–Fock (HF) method is a numerical approximation for the determination of the wave function and the energy of a quantum many-body system in stationary state. The computation starts with guesssing an one electron wave function for the molecular system with nuclear attractions and a Coulombic repulsion term from a smooth distribution of other electrons. It runs through many iteration cycles to arrive at a minimum value of the energy (as the calculated value turns around from the lowest points) and an acceptable wave function. Post-HF improves the method by replacing the electron cloud with genuine electron-electron interactions.

In the research work above, it states that the single-particle reduced density matrix, necessary for the von Neumann entropy and entanglement measure calculations, was obtained from the correlated molecular wave functions determined according to QCISD and CCSD which are some sorts of post-HF.

Figure 03k Entanglement Measure @ Dissociation & United Atom

Figure 03j shows the Entanglement Measures N between an electron with the rest in 5 different diatomic molecules as function of inter-atomic distance R. Figure 03k is a graph to show the limiting cases of N as R 0 and (at different scale in ratio of 0.1/1).
2021 Update :

Hydrogenic Entanglement The entanglement between the 2 particles in the hydrogenic system is calculated via the Fourier transform of the density matrix. It turns out that the entanglement is proportional to 1/n, where n is the principle quantum number, i.e., the entanglement is getting weaker by increasing separation of the 2 particles. There would be no entanglement as n (see Figure 03p).

Figure 03p Hydrogenic Entanglement

See original article "Hydrogenic Entanglement" and
a readable version "Investigating Hydrogenic Entanglement".
Here's a summary of the mathematics :

In addition to the entanglement criterion in k-sapce reached by the above article, a similar conclusion can be obtained via the entanglement measure defined by :
H Probability

Figure 03q Probability of H Atom

See Figure 03q for a graphical illustration.

The result again shows that entanglement depends on space (separation) and time (coherence duration as shown below). It doesn't happens very often as shown by the 2015 Bell test experiment, in which the team managed to generate 245 entangled pairs of electrons over the course of nine days.

H Probability In retrospective, the transformation from classical to quantum is related to change of the mathematical formulation from non-linear equation to a linear form, which permits the superposition of its solutions resulting in quantum weirdness such as entanglement. The description of system then changes from space and momentum to an un-familiar entity called "wave function", the interpretation of which is "man-made" - another transformation from mathematical reality to subjective reality. The ultimate reality may be something else called

Figure 03r Classical / Quantum Domains [view large image]

See Figure 03r for an illustration of the domains including the non-linear clasical formula for Newtonian mechanics and the linearized quantum version in the form of Schrodinger equation.
In the Newtonian equation of motion, the time t is an independent variable while the spatial coordinate r is a function of t to be determined by this non-linear equation with r appearing differently in different term. In the Schrodinger equation, r becomes an independent variable while the wave function is linearized appearing only once in every term. It is this peculiarity that allows for superposition of its solutions resulting in the quantum weirdness.

The quantum formulation has its origin in the uncertainty principle about the momentum p = mv and r (same for energy E, time t, and other conjugate variables). This assertion was inspired mainly from the observation of particle-wave duality, which shows that "microscopic object" sometimes appears as wave, while in another instance it behaves as particle. In classical theory there is no ambiguity about the position r and momentum p of a particle, i.e., r = 0, and p = 0. In quantum theory, p links to the wave length by the de Broglie relation p = h/, where the Planck constant h = 6.625x10-27 erg-sec, and = h/2; the uncertainty is thus expressed as pr which is equivalent to the commnutation relation rp - pr = [r, p] = i (= 0 for classical, because it is very certain for the value of r and p ). One way to satisfy this relation is to make p into an operator, i.e., p -i(d/dr) operates on a function which is called wave function because it always has an oscillating part (thank to the imaginary variable signified by "i", otherwise the quantum world would either collapse or explode).

The Schrodinger equation follows suit by replacing p in operator form to the kinetic energy T of a particle :
Eigenvalue Def. T + V = mv2/2 + V(r) = p2/2m + V(r) = E and operates on giving
The conservation of energy is just the integration of the Newtonian equation of motion F = ma :

What's more, E is the eigenvalue of the operator on the left-hand-side of the Schrodinger equation. Actually, not all linear differential equations are alike, this property is absent in other cases such as the "classical harmonic oscillator", and the "wave equation". As illustrated in Figure 03s, the eigenvalue is a scalar factor denoting the change of length of the eigenvector by an operator.

Figure 03s Eigenvalue, Definition [view large image]

It is such special property that allows a lot of different states providing rich variety of features such as superposition in the quantum world.

Figure 03s illustrates the definition of eigenvalue with a vector, its matrix representation, and the quantum harmonic oscillator. The left-hand-side of the Schrodinger equation is another form in term of differential equation acting on the wave function (aka eigen-function).
Superposition is the linear combination of the eigen-functions :

Coherence, Spatial Coherence, Temporal
The superposition of 2 quantum states with a constant relative phase is said to be in coherence, i.e., they form a particular pattern that would last until the phase relationship is changed. The pattern can be spatial or temporal. For example, the double-slit interference can be the result of the superpostion of 2 matter waves (Figure 03s2). The interference pattern will gradually disappear as the distance between the slits "d" is getting larger and larger. Another kind involves cyclic variation of its superposition pattern with time as shown in Figure 03s3 for the superposition of 2 quantum states of the "Square Well Potential" with a constant phase (3 - 1). NB Entanglement requires coherence plus two or more independently measurable quantum systems; while coherence does not imply entanglement, cohenence can exist in one single system as well. To reiterate :

Figure 03s2 Coherence, Spatial

Figure 03s3 Coherence, Temporal

The entangled state between such systems requires that the phase relationships are maintained, which means the system must be coherent, but there are more constraints; the most notable being that an entangled state cannot be separable.
An inseparable state in turn is described by mixed state as mentioned earlier. It can be defined generally by a system of N individual molecules (labeled by "i"), which are specified by the same internal basis states n1, n2, n3, ... (or n, m, ...), but the probability of occupying those states (cn)i(cn)i* may vary from molecule to molecule. The states are not orthognal, i.e., < n|m > 0. Such system is shown pictorially in Figure 03s4, the mathematics is summerized below (see "Density Matrix" for the original derivation) :
Density Matrix, General

Figure 03s4 Density Matrix, General [view large image]

BTW, the average over the mixture is similar to transform the density matrix to the "reduced density matrix". See for example, the coherent state of a single particle in "Square Well Potential".

The 2015 article on "Measuring Quantum Coherence with Entanglement" claims that quantum coherence and quantum entanglement are
Coherence / Entanglement the flip side of each other. As shown in Figure 03s5,b, a coherent system can entangle with another system by the so-called "incoherent operation", which distils quantum coherence from a single copy of a coherent state into entanglement with another incoherent system in this case. Thus, coherence and entanglement are "operationally equivalent", that is, equivalent for all practical purposes, though still conceptually distinct. The idea has been somewhat verified by a 2021 experiment (see "Experimental demonstration of one-shot coherence distillation: realizing N-dimensional strictly incoherent operations");

Figure 03s5 Coherence/Entanglement

See a readable version in "Physicists find quantum coherence and quantum entanglement are two sides of the same coin".
It seems that such idea is similar to the "transformation from density matrix to reduced density matrix", but in reverse. In the original scheme, the reduced density matrix is similar to the "coherent input" in Figure 03s5,b; while the entire density matrix (comprised by many systems) is the "incoherent input" in Figure 03s5,b. The reduced density matrix is related to the general density matrix through the averaging of all systems as shown for the case above.


Coherence Time (2019)

The study in the last section for "Entanglement Measure" involves only stationary state. It would not be able to shed light on "Coherence Time" c, which is the duration from beginning to end of entanglement. This is an important factor in quantum computing where
Coherence Time c should be at least 104 times longer than the operation time op (see Figure 03t, also the "Types of Qubit" table + captions for more info, and an up-to-date list of companies involved in quantum computing). In general, it is the interaction with environment that limits the duration of coherence time. Physicists design complex containers with cryogenic temperature, ultra-high vacuum, ... that completely isolate quantum states from the surroundings, while still allowing for state manipulation.

Figure 03t Coherence Time [view large image]

Cu Electronic Configuration The "transition Metal Phthalo-cyanine" (MPc) is organic molecule, its uses were primarily limited to dyes and pigments. It has a central metal atom surrounded by phthalocyanine ring - the 4 ligands (see Figure 03v). Its magnetic and electronic properties are determined by the transition metal's 3d orbitals incorporated in the center. Lately in 2012, it is found that the unpaired electron in copper atom at the center can act as a qubit (see Figure 03u, and "Introducing copper phthalocyanine as a qubit"). A 2016 article on "Tuning of Molecular Qubits" investigates further the influences of various factors on the coherence times of the qubits in some "transition Metal Phthalo-cyanines"

Figure 03u Cu Electronic Configuration [view large image]

(MPc's), the structure of which has been found to be tunable easily. The MPc's tend to aggregate and, thus, have low solubility in common solvents. However, it is found that CuPc can dissolve easily in sulfuric acid (H2SO4).



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

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 3 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.


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
[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).
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

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
Atomic Teleportation 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.


Holographic Space-time, Quantum Entanglement, and Quantum-Gravity

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
Branes Bulk Correspondence to high energy particles 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 (see Figure 22). 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
[view large image]

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, i.e., Anti-de Sitter (space-time)/Conformal (quantum) Field Theory, 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

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 14 Entanglement and Spacetime [view large image]

Figure 15 Entanglement and Wormhole [view large image]

(Figure 15). See original articles in "The Quantum Source of Space-time" and "Entangled Universe".
Figure 16 is a summary of this novel Quantum-Gravity paradigm.
[2022 Update]
    Here' a recap on the chain of rationales for such approach.

  1. ER = EPR -
  2. Thermofield (Thermal Quantum Field. TQF) is the Quantum Field in the exteriors submerging in entangled particles (hence the "Thermo" at certain temperature T, see Figure 20). The energy levels of the Quantum Field En are discrete. The corresponding eigenstates are denoted by |n>L, |n>R. To simplify the notation, the tensor product state |n>L |m>R is labeled by |n, m>. The eternal black hole is described by the entangled state :

    where ß = 1/kT is the inverse temperature of the environment, k = 1.38x10-16 erg/K the Boltzmann constant. The density matrix of each side is a pure thermal density matrix.
  3. Figure 20 Thermofield
    [view large image]

    Note that the entangled state vector = 0 (no more entanglement) when the temperature
    T = 0, i.e., when the exteriors are empty - no particles.

  4. Tensor Network - It is a general mathematical scheme applied to many-body problem. Figure 21 is tailored to show its application to link the entanglement of particles in the body to a special property at the boundary. The followings summarize the steps to arrive at the conclusion.
  5. AdS/CFT - The AdS/CFT correspondence proposed in 1997 has been quoted extensively over the years (see older text and Figure 23). However, there are some lingering doubts about its proliferation to variety of its applications :

[End of 2022 Update]


Wormhole Experiment (2022-12)

The research article "Traversable wormhole dynamics on a quantum processor" was published at the end of 2022 generating sensational headline in the news such as "Scientists create ‘baby’ wormhole as sci-if moves closer to fact". As will be explained in the brief comments below, the experiment uses a quantum computer (see a very simple "2 Qubits, 1-D quantum computer in action") to demonstrate teleportation of a qubit from on side of the computer to the other entangled half. Successful completion of the experiment suggests that the qubit could be considered as traversing a wormhole in higher dimensional space according to the "AdS/CFT correspondence", i.e., it tries to show "EPR = ER"). Followings are some comments on the experiment (see Figure 24).

  1. Hypotheses and Assumptions :

  2. Experimental Setup :

  3. Wormhole Correspondence :

    The space-time interval ds for the regular single (Schwarzschild) black hole is :
    ds2 = -(1 - rs/r) c2dt2 + dr2 / (1 - rs/r) + r2 (sin2 d2 + d2)
    where rs = 2GM/c2 is the event horizon (Schwarzschild radius).

    The space-time interval ds for (1+2)-D eternal black hole is expressed differently in the form :
    ds2 = -{[r2 - (rh)2]/2} c2dt2 + 2dr2 / [r2 - (rh)2] + r2 (d2)
    where rh is the event horizon. There is no singularity at r = 0, and = (1/2) is the cosmological constant with < 0 for AdS.

  4. Results :
  5. Identification of Holography to EPR = ER regradless of so many hypotheses and assumptions (see Figure 30) :

Thus by comparing to the holographic process, the existence of wormhole is just an idea conceived in human mind, albeit very creative.

[End of 2022-12 Update]

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.