Entanglement and Teleportation

Contents

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 .

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).

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 :

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.

	(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.

	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)

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 (the x-x' spatial coordinates in this case).


• 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 x₁ is split into double entities x and x' to represent 2 opposite spin states. Such formulation is for checkubg entanglement between particles. The x' would be just x for calculating probability (see "Density Matrices Visualized" for an explanation of reduced density matrix with graphics).
  
  
  Here's an example of various forms of density matrix for 2 identical electrons in empty space (i.e., independent of spatial coordinates).
  

  		The 2 examples (cases) are pure states with the density matrix defined by : The outer product (tensor product) = , and tr() = 1. It is a projection operator as shown in Figure 03i1. = 2. tr(2) = 1. See Figure 03i2 for the explicit formulas.
  Figure 03i1 Projection Operator	Figure 03i2 Density Matrix [view large image]	The reduced density matrix considers the superposition of states for only 1 of the particles.

  
  
  
  See Figure 03i2.

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).

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).
Figure 03l Diatomic Energy Levels [view large image]	Figure 03m Quantum Dot [view large image]	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.




	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 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 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 H₂ 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.
Figure 03o Total Spin States [view large image]	The H2 molecule is again a very good example (Figure 03o).

• The He₂ 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.

	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.

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.

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]

"Tao". 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.

N.B.

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

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

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.

		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.

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 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".

Coherence Time (2019)

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]

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).

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 B₀ in Figure 03v,b). Relaxation time is usually referred to the duration from perturbed state back into equilibrium. The spin-spin relaxation time T₂ 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 03v,a) :
  1. Electromagnet - It provides the magnetic field B₀ with field strength of a few T (teslas = 10⁴ Gauss) to split the electronic spin state into two levels. This is crucial for the spin-spin interaction and the production of EPR spectrum.
  2. Klystron - It produces the microwave pulse of few 100 GHz (~ 10^-3ev) to trigger the spin-spin interaction. The output frequency is controlled by the applied voltage. The pulse width is about 2 ns.
  3. Specimen - A cavity holding the sample for examination is placed in the magnetic field.
  4. 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").
  5. 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 B₀ according to the formula :
  E = (1/2)g_eBB₀, with corresponding energy gap E = g_eBB₀ (see Figure 03v,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 03v,b shows that E is proportionally getting larger with increasing value of B₀.
  
  (c) There would be more electrons in the state of lower energy (the spin up state) in a sample with B₀ 0. A microwave pulse with energy h = E = g_eBB₀ 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 B₀ can be altered to B₀+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 = B₀ and thus obtains a spectrum with varying B' against the re-emitting intensity from different site in the molecule. The spectrum in Figure 03v,c shows indirectly the structure of the molecule CH₂ -- O -- CH₃, which is a radical with one un-paired electron.
  
  (d) As shown in Figure 03v,d, the relaxation time T₁ is the duration from perturbation by the microwave to re-establishment of equilibrium; while T₂ corresponds to the time interval between coupling and de-coupling of the electron pair and is

  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/T2) 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

  
  
• 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 H2SO4 and D2SO4 were employed to probe the interaction between solvent matrix (a compound that promotes the formation of ions) and molecular qubit at 7o K. As shown in Figure 03w, the derivated value of T2 ~ 41 s in D2SO4 from experimental data is about 5 times longer than in H2SO4.
  Figure 03w 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 D₂SO₄ solvent, the relaxation time T₂ ~ 41 s for all cases, i.e., T₂ is insensitive to the composition of the ligands (see Figure 03x).


• In order to compare the relaxation times of various qubit cores, different MPc's with central transition metals VO²⁺, Mn²⁺, Co²⁺ and Cu²⁺ were investigated. This series provides charge neutral compounds with increasing Spin Orbit Coupling (SOC).

  		Furthermore the compounds possess only one unpaired electron except Mn2+ (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 T2 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 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.

Teleportation

		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) |S_AB, |T_1AB, |T_2AB, |T_3AB 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 |1_C + b |0_C .


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

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.


4. 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.


5. Teleported State - When Bob has received the code, he would proceed to run the corresponding operation : I, ₃, -i₂, ₁ to the associated |_B state vectors to recover the original in the form of |_B = a |1_B + b |0_B , where the 's are the Pauli matrices

	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.

Teleportation Experiments

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]

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).

	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.

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

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.

	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).

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". Figure 16 is a summary of this novel Quantum-Gravity paradigm.

Here' a recap on the chain of rationales for such approach.


1. ER = EPR -
   
   

   	ER refers to the Einstein–Rosen bridges (named after Albert Einstein and Nathan Rosen). They are connections between 2 black holes and part of the vacuum solutions to the General Relativity field equations. These bridges connect two black holes at different points in space-time (see "Properties of Wormhole").
   Figure 16 Quantum Gravity, New Paradigm [view large image]

   • In 1935, Einstein, Podolsky, and Rosen proposed a paradox (hence EPR-paradox) as a thought experiment ("gedankenexperiment" in German) in which the particles can have both their position and momentum accurately measured in violation of the uncertainty principle pq > ; it also violates the principle of locality (signal cannot travels faster than

     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 "non-locality" 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).

   • ER = EPR is a conjecture proposed by Leonard Susskind and Juan Maldacena in 2013 (see original article "Cool horizons for entangled black holes"). They proposed that a wormhole (the ER bridge) is equivalent to a pair of maximally entangled black holes (the EPR entanglement). It is also suggested that the existence of wormhole depends on the entanglement between the two black holes (see Figure 15).

     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 full-fledged 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 [view large image]

     *** 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.

     Anyway, it is not easy to entangle 2 particles; and furthermore, standard definition of entanglement usually neglects the limitation of coherence time. In general, interaction with environment limits its duration to at most few 10 seconds. Complex containers are designed with cryogenic temperature, ultra-high vacuum, ... that completely isolate entanglement from the surroundings, while still allowing for state manipulation.
     
     

     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).

   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.

   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.

2. 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.
   • Originally, the geometrical symbols represent a vector and the straight line generated a tensor product (entanglement) from the 2 vectors. The symbols in this case has been generalized to particles (fermions) and the line as entanglement (Figure 21).
   • Then the bulk (the entity) can be considered as a piece of material, and in particular a space.
   • As shown in Figure 14, it is found that the space dis-integrated as the entanglement is reduced to zero. It followings that the existence of space depends on entanglement.
   • The degree of entanglement is determined by "Entanglement Measure ( entropy)" as shown by Figure 03j. Particularly, on the the "Entanglement Entropy" (von Neumann Entropy for mixed states) at the boundary (see Figure 21 and Figure 16).

   		Figure 21 also reveals that in term of entanglement entropy, a rather small linear length L on the boundary could correspond to a large bulk space inside. This is a hallmark for the AdS/CFT as mentioned proviously. In addition, it is found that good "quantum computing error correction codes" requires lot of physical qubits (in patterns of entanglement among them) similar to the bulk/boundary relation in AdS/CFT and the holography in Figure 22.
   Figure 21 Tensor Network	Figure 22 Holography

3. 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 :
   
   

   It is derived from the "String Theory", which is no longer viable due to lack of observational evidence. The bulk-boundary correspondence does not apply in our Universe, which is not hyperbolic but expanding. Most applications now just equate a simpler version (of whatever theory) in the bulk to a more complicated one on the boundary. They are links by some sort of translation codes regardless of the original intent.

   Figure 23 AdS/CFT correspondence, 1997

Wormhole Experiment (2022-12)

1. Hypotheses and Assumptions :
   
   
   • ER = EPR - This is a conjecture advocated back in 2013. It is proposed that a wormhole (the ER) is equivalent to a pair of maximally entangled particles (the EPR). Since then, the hypothesis is used to explain other hypotheses such as the "black hole complementarity".
   
   
   • Eternal Black Hole - It turns out that the black hole in this case is not the kind from the collapse of massive star or the one at the center of most galaxies. It is the full-fledged solution to Einstein's gravitational field equations (including the black hole, wormhole, and white hole). There is no matter to collapse to in this case. It exists forever from the beginning of the universe, hence dubbed Eternal Black Hole.
   
   
   • AdS/CFT Correspondence - This is a conjectured relationship between one physical theories (e.g, the Conformal Field Theory) in N dimensional space and another physical theory (such as the SuperString Theory) in the (N+1) dimensional anti-de Sitter spaces (AdS), see Figure 23. Since then, lot of applications just equate a simpler version (of whatever theory) in the bulk to a more complicated one on the boundary and vice versa. They are linked by some sort of translation codes regardless of the original intent and limitation. It is a modern example of promoting a hypothesis into a law.
   
   

   	SYK Model - The Sachdev–Ye–Kitaev (SYK) model of N >> 1 Majorana fermions with random interaction between q of them has features suggesting the existence of a gravitational dual in AdS2 (2-D). The experiment uses learning techniques to construct a sparsified (simplified) SYK model that experimentally realized with N = 7 and q = 4 (see Figure 24,a).
   Figure 24 Wormhole Experiment	Majorana fermion - This is a hypothetical particle not yet discovered. Under certain conditions, a fermion in a superconductor can separate in space into two parts known as Majorana Zero Mode (MZM), which would be stable against decoherence and thus could be very useful in

   quantum computing. Each neutrino in such zero mode pair constitutes only 1/2 of a qubit. The SYK model implicitly assumes the zero mode, but is considered to be a regular fermion (= 1 qubit) in this experiment creating lot of confusion.
   
   
2. Experimental Setup :
   
   
   • The quantum computer is divided into 2 halves with 7 Majorana Zero Mode (MZM) fermions in each one to mimic 2 SYK

     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

     The qubit Q is inserted into SYK-L at time t < 0 and entangled with qubit P to form the Bell state (see Figure 24,b) : |PQ = (|0p|0Q + |1P|1Q)/.

   • The qubit Q is then swapped into the SYK-L system which becomes entangled with SYK-R via the coupling operator e^iV at t = 0 (see Figure 24,b).
   
   
   • The entangled system evolves in time scale about the coherent time of the qubit ( ~ second for superconducting qubit).

     	For > 0, the qubit P entangles itself with the multitude of the many-body 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 [view large image]	which is an Alice/Bob cartoon emulating the said process for < 0 with only 2 entangled qubits in transition).



3. Wormhole Correspondence :
   
   The space-time interval ds for the regular single (Schwarzschild) black hole is :
   ds² = -(1 - r_s/r) c²dt² + dr² / (1 - r_s/r) + r² (sin² d² + d²)
   where r_s = 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 - (r_h)²]/²} c²dt² + ²dr² / [r² - (r_h)²] + r² (d²)
   where r_h is the event horizon. There is no singularity at r = 0, and = (1/²) is the cosmological constant with < 0 for AdS.
   
   
   The wormhole correspondence to the experiment :
   
   
   • The SYK-L and -R models are dual to the "Thermal Quantum Field (TQF)" outside the external spaces of the eternal black hole (see Figures 24,b and 27,a). A qubit is moving toward the event horizon of the black hole on the left side at t < 0.
   
   
   • Entanglement of the SYK-L, -R models and the swapping in of the qubits at t = 0 are dual to the qubit entering the left side of the eternal black hole. The coupling operator e^iV in the experiment is equivalent to the coupling between the 2 TQFs of the wormhole causing a negative energy ( < 0) "shockwave (blue)" which enables the wormhole to become traversable.

     	The coupling is expressed in terms of the tensor product of the energy eigen-states |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 Thermo-Quantum Field Double.
     Figure 27 Wormhole Experiment 2	Dual to the time lapse in the experiment, the qubit advances in time interval t, at which point, contact with the shockwave causing it to emerge from the right side of the eternal black hole (see Figures 27,a and 24,a).

4. Results :
   
   
   • The most stringent test for successful transmission of qubit through the eternal black hole is by measuring the Mutual Information (MI) from the experiment :
     I_PT(t) = S_P(t) + S_T(t) - S_PT(t),
     where S is the "measure of entropy".
     For I_PT(t) = 0, the qubit P merges itself into the multitude of the many-body system in a process called "Scrambling". The qubit P makes it across to the other side only when I_PT(t) 0.
   
   
   • Figure 27,b shows the Mutual Information asymmetry I = I_µ<0(t) - I_µ>0(t) for the learned (green) and SYK model (orange) at the low-temperature (solid, ß = 4) and high-temperature (dashed, ß = 0.1). An analytic computation in the large-N limit of the SYK model (black) is shown for low temperatures, demonstrating the agreement with the experiment.
   
   
   • Other consistent observables include :
     • Prefect Size Winding for traversable wormhole - The action of an operator "O" over strings of n fermions (qubits) in TQF (Thermal Quantum Field) (Left and Right) can be expressed in size and phase of each fermions. For an operator represented by 2X2 Hermitian matrix,
       

       Figure 28 Perfect Side Winding	In this experiment n = 1, while the operation is the "shock wave". Figure 29,a shows cP to be in prefect cosine form, i.e., prefect size winding to signify a traversable wormhole.

     • Time Delay by Scattering - Figure 29,b shows time delay in the eternal traversable wormhole protocol caused by scattering in the bulk. The peak shifts right when another qubit is sent through the wormhole in the opposite direction (dashed) compared to sending a single qubit from left to right (solid). The peak height is also reduced due to inelastic scattering.
     
     

     Causal Time-ordering - If EPR = ER is valid, infalling particles to the wormhold should arrive in a causally consistent order, signals must emerge in the same order they enter (time-ordered teleportation). By contrast, teleportation in the fully scrambled regime produces a time-inverted ordering of signals. As shown in Figure 27,a, this experiment produces time-ordered teleportation in consistence with EPR = ER.

     Figure 29 Other Observables [view large image]

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

   	The real object in holography can be identified to the formula for the eternal black hole. The photographic plate is replaced by the experimental setup in paper and in laboratory. The process of creating virtual image from the hologram is the experimental implementation.
   Figure 30 Holography ~ EPR/ER [view large image]	The virtual image of the object becomes the imaginary eternal black hole as interpreted by the experimental results.