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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 :  
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).  
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). 
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 : 
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 eigenstates (the state with a definite eigenvalue 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. 
of "Schrodinger's Cat" is illconceived 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 misappropriation of the quantum theory. BTW, according to the reductionist's view, description of the  
Figure 03g Quantum Nature 
world is separated into many different levels. An absurd scenario can often be concocted with the socalled "thought experiment" by mixing up the various levels (see Effective Theories). 
Figure 03h Entanglement Measure _{} 
 
Figure 03i1 Projection Operator _{} 
Figure 03i2 Density Matrix [view large image] 
Figure 03j Entanglement of Electrons in Diatomic Molecules _{} 
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). 
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 postHartree–Fock computational method is employed to calculate the wave function and ultimately _{N} as function of the interatomic distance R. The Hartree–Fock (HF) method is a numerical approximation for the determination of the wave function and the energy of a quantum manybody 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. PostHF improves the method by replacing the electron cloud with genuine electronelectron interactions. In the research work above, it states that the singleparticle 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 postHF.  
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 interatomic 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). 
 
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). 
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 : 
Figure 03q _{} Probability of H Atom 
See Figure 03q for a graphical illustration. 
In retrospective, the transformation from classical to quantum is related to change of the mathematical formulation from nonlinear 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 unfamiliar entity called "wave function", the interpretation of which is "manmade"  another transformation from mathematical reality to subjective reality. The ultimate reality may be something else called  
Figure 03r Classical / Quantum Domains [view large image] 
"Tao". See Figure 03r for an illustration of the domains including the nonlinear clasical formula for Newtonian mechanics and the linearized quantum version in the form of Schrodinger equation. 
T + V = mv^{2}/2 + V(r) = p^{2}/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 lefthandside 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. 
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 doubleslit 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. 
Figure 03s4 Density Matrix, General [view large image] 
the flip side of each other. As shown in Figure 03s5,b, a coherent system can entangle with another system by the socalled "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 oneshot coherence distillation: realizing Ndimensional 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". 
_{c} should be at least 10^{4} times longer than the operation time _{op} (see Figure 03t, also the "Types of Qubit" table + captions for more info, and an uptodate 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, ultrahigh vacuum, ... that completely isolate quantum states from the surroundings, while still allowing for state manipulation.  
Figure 03t Coherence Time [view large image] 
The "transition Metal Phthalocyanine" (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 Phthalocyanines"  
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 (H_{2}SO_{4}). 
interpreted 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 03v,e and the "Spin Echo Animation" below.  
Figure 03v Coherence Time by EPR [view large image] click me 
Spin Echo Animation 
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 03w, 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 03w Coherence Time in Solvents [view large image] 
Furthermore the compounds possess only one unpaired electron except Mn^{2+} (S = 3/2). Finally, different coordination geometries can be compared, as VOPc exhibits squarepyramidal shape whereas the others possess squareplanar ones. The spin–spin relaxation time T_{2} of CuPc and VOPc are significantly longer than those of MnPc and CoPc (Figure 03y). This is attributed to the influence of the SOMO (Singly Occupied Molecular Orbital) on the spin dynamics.  
Figure 03x Coherence Time by Ligands [view large image] 
Figure 03y Coherence Time 
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 nonliving 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. 
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 rearranged.  
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. 
 
Figure 07 Teleportation over River Danube [view large image] 
 
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.  
Figure 09 Hyperfine Sturcture _{} 
See original paper "Quantum teleportation between light and matter" for detail. 
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 + i0)/_{}. The actual experimental setup 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. 
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 YangMills Theory" on the 4D boundary. A mathematical dictionary has been compiled to link the two perspectives. It is similar to the laser, which transforms a 2D scrambled pattern into a recognizable 3D 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. 
The RobertsonWalker 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 :  
Figure 12 Hyperbolic 2D 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). 
The AdS spacetime 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 spacetime. Anyway, when the correspondence has been promoted to the level of principle, it becomes a tool in vogue with quantumgravity 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 D3brane. 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 Spacetime" and "Entangled Universe". Figure 16 is a summary of this novel QuantumGravity paradigm. 
 
Figure 16 Quantum Gravity, 
the speed of light) in special relativity (see Figure 17). They argued that "elements of reality" such as momentum and position are deterministic properties of particles although it is hidden in quantum theory which endows them with probability only. Hence quantum theory is an incomplete theory, there is no need to invoke "nonlocality" and also can do away with the "uncertainty principle" if the hidden cause is included in the theory.  
Figure 17 EPR Paradox [view large image] 
The thought experiment became real with the discovery of entanglement in 1967 (see previous section for details). 
It turns out that the black holes these scientists talk about are not the kind from the collapse of massive star or the one at the center of most galaxies. It is the fullfledged static, spherically symmetric, vacuum solution to Einstein's gravitational field equations. There is no matter to collapse to in this solution. It exists forever from the beginning of the universe, hence dubbed Eternal Black Hole. It is more suitable to have it portrayed with the KS coordinates as shown in Figure 18,a (Figure 18,b is a more conventional view). The whole thing includes black hole 1, and 2 (a white hole), a wormhole and 2 exterior regions. The left exterior is normal but forever inaccessible to us. Now, it is suggested that it can be accessible in the form of entanglement interchangeable with the wormhole.  
Figure 18 Eternal Black Hole 
*** At first glance, the ER = EPR hypothesis would mean quantum systems that become entangled, and therefore enter a superposition, suddenly gain a wormhole  a conjuring trick the superposition principle doesn't obviously allow. 
Regradless of the problem, the novel concept generalizes the black hole 2 (the white hole) by scrambled cloud of Hawking radiation (Figure 19) and the wormhole (ER) becomes a whole bunch of entanglement (EPR). Anyway, it is admitted that the cartoon is speculative, and is based on the assumption that the wormhole has a geometric description. They say that understanding the ER connecting the black hole to the radiation is the key to determining whether the horizons of evaporating black holes are smooth.  
Figure 19 Black Hole and Hawking Radiation _{} 
An even further conjecture would replace the 2 black holes by 2 particles (see Figure 16). 
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.  
Figure 20 Thermofield 
Note that the entangled state vector = 0 (no more entanglement) when the temperature T = 0, i.e., when the exteriors are empty  no particles. 
 
Figure 21 _{} 
Figure 22 _{} 
 
Figure 23 AdS/CFT correspondence, 1997 


Figure 24 Wormhole Experiment _{} 
models, i.e., the L and the R systems (see Figure 24,a). Each fermion in the MZM pair is identified to a qubit in this experiment out of necessity because the existence of MZM is still in the verification stage (not yet available for running the experiment). BTW, the experiment was implemented by the Google Sycamore processor based on superconducting qubits (as shown in Figure 25, not MZM). See "Type of Qubit".  
Figure 25 [view large image] Superconducting Qubit 
_{PQ} = (0_{p}0_{Q} + 1_{P}1_{Q})/_{}. 
For > 0, the qubit P entangles itself with the multitude of the manybody system in a process called "Scrambling". For < 0, the unique identity of qubit P survives and could entangle with qubit T (see Figure 24,b and Figure 26  
Figure 26 Quantum Teleportation 
which is an Alice/Bob cartoon emulating the said process for < 0 with only 2 entangled qubits in transition). 
The coupling is expressed in terms of the tensor product of the energy eigenstates n>_{L,R} : where ß = 1/kT is the inverse temperature of the environment, k = 1.38x10^{16} erg/K the Boltzmann constant (see Figure 27,a) and TFD stands for ThermoQuantum Field Double.  
Figure 27 Wormhole Experiment 2 _{} 
Figure 28 _{} Perfect Side Winding 
In this experiment n = 1, while the operation is the "shock wave". Figure 29,a shows c_{P} to be in prefect cosine form, i.e., prefect size winding to signify a traversable wormhole. 
 
Figure 29 Other Observables 
 
Figure 30 Holography ~ EPR/ER [view large image] 
