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Correlation

Entanglement

Decoherence

Entanglement Measure (2019)

Coherence Time (2019)

Teleportation

Teleportation Experiments

Entanglement and Holographic Space-time

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 S_{A} and S_{B} (personified by the two computer geeks Alice and Bob respectively) each with a set of basis vectors |a's and |b's. In particular, S_{A} could be represented by a coin with basis vectors |H (for head) and |T (for tail), while S_{B} is a dice with basis vectors labeled by 1, 2, 3, 4, 5, 6. The combined system (called S_{AB}) 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 S_{A} can only act on the first label, same is for S_{B} on the second label. Superposition state in S_{AB} is written as :
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## Figure 01 Combined System |
where |a _{i}b_{j} is the basis vector in S_{AB} with associated probability c*_{ij} c_{ij} . |

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 (r_{xy} = 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 r_{xy} = 1 by taking just 2 points (n = 2) at (0,1), (4,3) and the average (2,2).
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## Figure 02 Correlation |
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.

## Figure 03a Entanglement [view large image] |

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|C

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 |T

See possible coupling mechanism in "Entanglement of Spins".

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 |
## Figure 03e Decoherence |
is called decoherence and can be explained mathematically in the following as transfer of entanglement to all kinds of particles in there. |

(See detail of derivation in "Quantum Decoherence, Dirac Notation"). | |

## Figure 03f Hilbert Space |
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. |

"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 world is | |

## Figure 03g Quantum Nature |
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). |

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

## Figure 03h Entanglement Measure |

- The "Reduced" signifies that only the variable for the first particle have been retained (thus, the "Single-Particle"), while all the others are eliminated by integrating over them. The "density" is referred to as something (probability in this case) per unit dimension (whatever it is).
- The reason for using the reduced form of (x,x'), and why we don't care the other particles, is that
**all the particles are identical**. Therefore, all we need to do is to work out the value for one particle and to multiply the result by N times so that the sum of (x,x') is normalized to 1. - The single variable x
_{1}is split into double entity x and x'. This is equivalent to the introduction of the off diagonal matrix elements in expressing the connection to other particles at other location such as x' (see example below). Such formulation is for measuring entanglement between particles. The x' would be just x for calculating expectation value.

Here's the simplest example with 2 identical electrons in empty space (i.e., independent of spatial coordinates) with x' in_{}and x in_{}state respectively (see Figure 03i). Then is just the outer product of the spin state, i.e., =_{}.

#### Figure 03i Superposition of Spin [view large image]

## Figure 03j Entanglement of Electrons in Diatomic Molecules [view large image] |
The von Neumann entropy is the quantum version of "Shannon's Measure of Information" : SMI = - _{i} p_{i}log_{2}(p_{i}), where p_{i} 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 p

SMI = -{(1/2)log

S() = n(2) = n(2) log

For p

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 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). |

- Here's some comments :
- Data in Figures 03j and 03k are taken from the aforementioned paper on Quantum Entanglement. The study investigates the changes of the electronic entanglement (that is, the entanglement between the electrons in the system) during the dissociation processes of some diatomic molecules.
- The potential curve in Figure 03j is important in determining the vibrational and rotational energy levels of the atomic nuclei (Figure 03l). It is evaluated from stationary configurations at a given inter-atomic distance R.
- The study of entanglement measures in diatomic molecules helps to shed some insight into the cause. Although the quantum dot is more versatile (controllable) and useful in quantum computing, it is similarly an aggregate of electrons (with or without the nuclei) etched into wafers of a semiconductor (Figure 03m).
- It has been shown previously that the entanglement measure is expressed in term of the von Neumann Entropy. Disorder is the usual notion on entropy (von Neumann and otherwise). A more useful interpretation in the current context would be in term of "multiplicity", which is the number of different arrangements that can arrive at a same configuration (state).
- Figure 03k shows that entanglement measure decreases rapidly as the number of electrons increases. Such trend may be related to the sharing of entanglement between electrons. Entanglement is at its maximum with monogamy (such as the case with the H
_{2}molecule), shared entanglement is called polygamy which produces weaker entanglement with more partners (see "Degree of Entanglement"). - Figures 03j and 03k also show the total spin state of the outer most electron in the separated atom at dissociation limit (Figure 03o). It seems to indicate that entanglement favors the "Doublet State", and it also permits the merger of electrons in opposite spin orientation leading to more stable molecule.
- The He
_{2}is in extreme unstable state formed by high energy collision. It shows very weak entanglement measure as the electrons in the 2 He atoms have no chance to connect with each others.

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## Figure 03l Diatomic Energy Levels [view large image] |
## Figure 03m Quantum Dot |
Figure 03m shows that the separation of the quantum dots is about 200 times larger than the dissociation limits shown in Figure 03j for the diatomic molecules under investigation. |

## Figure 03n Entropy Increase with Volume [view large image] |
Thus, the general trend of increasing entanglement measure for large R (as shown in Figure 03j) can be understood as increasing entropy with larger volume (Figure 03n). However, it could not explain the bump near the united atom limit for some of the molecules. |

## Figure 03o Total Spin States [view large image] |
The H_{2} molecule is again a very good example (Figure 03o). |

_{c} should be at least 10^{4} times longer than the operation time _{op} (see Figure 03p, 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.
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## Figure 03p Coherence Time [view large image] |

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 03r). 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 03q, 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 03q 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 (H_{2}SO_{4}). |

- Here's a summary of the 2016 research on the MPc's coherence time :
- The measured "coherence time" is actually the relaxation time in Electron Paramagnetic Resonance (EPR), which is similar to NMR in principle with the nuclear spin replaced by electron spin. It is related to the interactions of the electrons with opposite spin state in an external magnetic field (see B
_{0}in Figure 03r,b). Relaxation time is usually referred to the duration from perturbed state back into equilibrium. The spin-spin relaxation time T_{2}is interpreted as the coherence time for the entanglement of the electron spins. An attempt to summarized this rather complicated process is provided in the following for the inquisitive mind.

- (a) EPR componments (see pictorial display in Figure 03r,a) :
- Electromagnet - It provides the magnetic field B
_{0}with field strength of a few T (teslas = 10^{4}Gauss) to split the electronic spin state into two levels. This is crucial for the spin-spin interaction and the production of EPR spectrum. - Klystron - It produces the microwave pulse of few 100 GHz (~ 10
^{-3}ev) to trigger the spin-spin interaction. The output frequency is controlled by the applied voltage. The pulse width is about 2 ns. - Specimen - A cavity holding the sample for examination is placed in the magnetic field.
- Modulation - A small additional oscillating magnetic field is applied to the microwave pulse at a typical frequency of 100 kHz to obtain the desired spectrum profile (see more detail in "Field Modulation").
- Detector - A scilicon crystal detector is used to convert the microwave signal into DC output.

- (b) In the absence of magnetic field, the energy level of spinning electron is degenerate, i.e., the spin up and spin down states have the same energy. The energy level splits into two in presence of a magnetic B

_{0}according to the formula :

E_{}= (1/2)g_{e}_{}_{B}B_{0}, with corresponding energy gap E = g_{e}_{}_{B}B_{0}(see Figure 03r,b)

where g_{e}~ 2 is the Gyromagnetic Ratio,_{}_{B}= 9.3x10^{-21}erg/G is the Bohr magneton for the magnetic moment of an electron. Figure 03r,b shows that E is proportionally getting larger with increasing value of B_{0}.

(c) There would be more electrons in the state of lower energy (the spin up state) in a sample with B_{0}0. A microwave pulse with energy h_{}= E = g_{e}_{}_{B}B_{0}would excite some electrons to the spin down state at higher energy level in the process called resonance. The spin up and down electrons entangle for a while; then the pairs go their separate ways and return back to the original configuration by re-emitting the radiation. Since B_{0}can be altered to B_{0}+B by the surrounding environment, the frequency of the microwave for resonance will be changed correspondingly. One way to re-establish the resonance (for fixed microwave frequency) is to adjust the magnetic field to B' such that B'+B = B_{0}and thus obtains a spectrum with varying B' against the re-emitting intensity from different site in the molecule. The spectrum in Figure 03r,c shows indirectly the structure of the molecule CH_{2}-- O -- CH_{3}, which is a radical with one un-paired electron.

(d) As shown in Figure 03r,d, the relaxation time T_{1}is the duration from perturbation by the microwave to re-establishment of equilibrium; while T_{2}corresponds to the time interval between coupling and de-coupling of the electron pair and isinterpreted as the coherence time for the entanglement.

(e) The molecular structure and the EPR spectrum is irreverent in measuring the coherence time. An additional process called "Spin Echo" is used to remove the clutter of the surrounding environment. The coherence time is determined from the decaying curve of the resulting echo in the form exp(-2t/T_{2}) as shown in Figure 03r,e and the "Spin Echo Animation" below.#### Figure 03r Coherence Time by EPR [view large image] click me

Spin Echo Animation

- Electromagnet - It provides the magnetic field B
- Returning now to the measurement of coherence time for some of the "transition Metal Phthalo-cyanines" (MPc's). In the 1st run,
solutions of CuPc (0.5 mM, M=mol/L) in H _{2}SO_{4}and D_{2}SO_{4}were employed to probe the interaction between solvent matrix (a compound that promotes the formation of ions) and molecular qubit at 7^{o}K. As shown in Figure 03s, the derivated value of T_{2}~ 41_{}s in D_{2}SO_{4}from experimental data is about 5 times longer than in H_{2}SO_{4}.#### Figure 03s Coherence Time in Solvents [view large image]

- The next study is to find out if a change in ligand properties would modify the electron spin-spin relaxation time. The change can be accomplished by replacing the H atoms in the ligands of CuPc with Cl or F atoms and designated as CuPc
^{Cl}and CuPc^{F}. It turns out that in D_{2}SO_{4}solvent, the relaxation time T_{2}~ 41_{}s for all cases, i.e., T_{2}is insensitive to the composition of the ligands (see Figure 03t). - In order to compare the relaxation times of various qubit cores, different MPc's with central transition metals VO
^{2+}, Mn^{2+}, Co^{2+}and Cu^{2+}were investigated. This series provides charge neutral compounds with increasing Spin Orbit Coupling (SOC).Furthermore the compounds possess only one unpaired electron except Mn ^{2+}(S = 3/2). Finally, different coordination geometries can be compared, as VOPc exhibits square-pyramidal shape whereas the others possess square-planar ones. The spin–spin relaxation time T_{2}of CuPc and VOPc are significantly longer than those of MnPc and CoPc (Figure 03u). This is attributed to the influence of the SOMO (Singly Occupied Molecular Orbital) on the spin dynamics.#### Figure 03t Coherence Time by Ligands [view large image]

#### Figure 03u Coherence Time

for MPc's [view large image]After all is said and done, longer coherence time can be achieved when the molecular orbital bearing the electron spin qubit exhibits minimal contact with the environment.

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 |
## 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) :
- Entanglement Generation - Four maximally entangled states (Bell States) |S
_{AB}, |T_{1}_{AB}, |T_{2}_{AB}, |T_{3}_{AB}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. - State Preparation - The spin state to be teleported is prepared by Alice with the label "C" : |
_{}_{C}= a |1_{C}+ b |0_{C}. - 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 = (|T_{2}- |T_{3})/_{}, |01 = (|T_{1}- |S)/_{}, |10 = (|T_{1}+ |S)/_{}, and |11 = (|T_{2}+ |T_{3})/_{},

It can be shown that |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. - 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. - Teleported State - When Bob has received the code, he would proceed to run the corresponding operation :
**I**,,_{3}**-i**,_{2}to the associated |_{1}_{}_{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

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.

In principle, Alice can pick any one of the |S _{AB}, |T1_{AB}, |T2_{AB}, or |T3_{AB} basis vectors to entangle with |_{}_{C} , but the resulting relationship would be re-arranged.
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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. |

- 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.
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## 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 10

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:

- 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, m
_{F}= 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.
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## Figure 08 Teleportation of Light to Atom |
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 10^{12} caesium atoms in coherent spin state. It demonstrates the possibility of teleporting the state in moving carrier to stationary object for storage.
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## Figure 09 Hyperfine Sturcture |
See original paper "Quantum teleportation between light and matter" for detail. |

Teleportation of atomic state in Ca

states |1 = S_{1/2} , |0 = D_{5/2} (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.
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## Figure 10 Atomic Teleportation [view large image] |
See original paper "Deterministic quantum teleportation with atoms" for detail. |

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 AdS_{5}XS^{5}, (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.
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## 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. |

The Robertson-Walker metric for the AdS universe is in the form : ds ^{2} = c^{2}dt^{2} - R(t)^{2} [dr^{2} + w^{2} (d^{2} + sin^{2} d^{2})] 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 :
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## Figure 12 Hyperbolic 2-D Slice |
R(t) = (c/H) sin(Ht), where H = (||/3)^{1/2}c, 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 :

dL

The circumference of a 2-D slice in a sphere at = /2 has a length of L

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 |

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". |

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.