General Quantum Theory: II ----Measuring & Identical Theorems, Origins & Classifications of Entanglements and Solution to Crisis of Wave Collapse

C. Huang, Yong-Chang Huang and Yi-You Nie 1. Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley CA 94720, USA 2. Department of Physics and Astronomy, Purdue University, 525 Northwestern Avenue, W. Lafayette, IN 47907-2036, USA 3. Institute of Theoretical Physics, Beijing University of Technology, Beijing, 100124, China 4. Institute of Theoretical Physics, Jiangxi Normal University, Nanchang, 330022, China Abstract This paper derives measurement and identical principles, then makes the two principles into measurement and identical theorems of quantum mechanics, plus the three theorems derived earlier, we deduce the axiom system of current quantum mechanics, the general quantum theory no axiom presumptions not only solves the crisis to understand in current quantum mechanics, but also obtains new discoveries, e.g., discovers the velocities of quantum collapse and entanglement are instantaneously infinitely large. We deduce the general Schrȍdinger equation of any n particles from two aspects, and the wave function not only has particle properties of the complex square root state vector of the classical probability density of any n particles, but also has the plane wave properties of any n particles. Thus, the current crisis of the dispute about the origin of waveparticle duality of any n microscopic particles is solved. We display the classical locality and quantum non-locality for any n particle system, show entanglement origins, and discover not only any n-particle wave function system has the original, superposition and across entanglements, but also the entanglements are of interactions preserving conservation or correlation, three kinds of entanglements directly give lots of entanglement sources. This paper discovers, one of two pillars of modern physics, quantum mechanics of any n particle system is a generalization ( mechanics ) theory of the complex square root ( of real density function ) of classical statistical mechanics, any n particle system’s quantum mechanics of being just a generalization theory of the complex square root of classical statistical mechanics is both a revolutionary discovery and key new physics, which are influencing people’s philosophical thinking for modern physics, solve all the crisises in current quantum theories, quantum information and so on, and make quantum theory have scientific solid foundations checked, no basic axiom presumption and no all quantum strange incomprehensible properties, because classical statistical mechanics and its complex square root have scientific solid foundations checked. Thus, all current studies on various entanglements and their uses to quantum computer, quantum information and so on must be further updated and classified by the new entanglements. This and our early papers derive quantum physics, solve all crisises of basses of quantum mechanics, e.g., wave-particle duality & the first quantization origins, quantum nonlocality, entanglement origins & classifications, wave collapse and so on.

information and statistical physics, and ref. [33] systematically studied fundamental principles of theoretical physics, concepts of quasi-averages, quantum protectorate and emergence.
Arrangements of this paper: Sect. 2 deduces the measuring theorem; Sect. 3 shows identical theorem; Sect.4 displays superposition theorem of states, entanglement origins and three kinds of entanglements; Sect. 5 gives discussion and application; Sect. 6 is summary and conclusions.

2.Measuring theorem
It is generally believed that the classical limit of quantum mechanics is classical mechanics, which is a direct extension of classical mechanics to quantum mechanics. However, classical mechanics does not have the problem of measuring collapse, everything is certain, while quantum mechanics does, so there is no way to understand the problem of measuring collapse in quantum mechanics. Thus, we consider that the classical limit of quantum mechanics first is classical statistical mechanics, and then macroscopic limit of classical statistical mechanics is classical mechanics, instead of going straight to classical mechanics. Otherwise, it will lead to many problems that are difficult to understand, for example, it will also lead to the difficult crisis of the current interpretation of quantum mechanics measurement collapse and so on [12,13].
Collapse effect in classical statistical mechanics [2,3]: A density state means that a particle is in a density state of multiple superposition possibilities, but once an observer has observed it, it can only fall into a density state of specific possibility. Although each measurement must collapse to some possible density state, a large number of observations reveal a probability density distribution, which just shows that the matter is described by probability density.
In the current quantum mechanics [29,30], the fourth axiom presumption is the measurement axiom presumption: Quantum systems are generally in the superposition state of various eigenstates of wave functions (i.e., eigenwave functions). When measured, it will lead to the collapse of the wave function of the quantum system and make the wave function of the quantum system jump to a certain eigenstate with a certain probability.
We now reduce the fourth axiom presumption ( the measurement axiom presumption ) to the measurement theorem, which is to prove the measurement theorem from classical statistical mechanics.
Using eq.(2.2) and independence of complex functions  + and  , it follows that They also indicate that: the eigenvalues (i.e., Lagrange multipliers) of all operators are the eigenvalues corresponding to the extreme values of the variational system in the transition from classical statistical mechanics to quantum mechanics [28].
Ref. [28] uses last term to deduce general Hellmann-Feynman theorem in transition from classical statistical mechanics to quantum mechanics. This shows that ordinary Hermitian operators and eigenvalues also satisfy this important theorem.
In particular, the Hellmann-Feynman theorem is established under the condition that not only the eigenvalue equations of all operators are corresponding to the extreme values of their variational systems, but also under the condition that the variational system is taken as the extreme values.
Therefore, the whole variational system takes the extreme value among all possibly taking values, and the equation derived when taking this extreme value is the eigenvalue equation in the transition from classical statistical mechanics to quantum mechanics.
Similar to deduction of eq.(2.1) for generalizing to any n particle system, we can extensively use the below variation † d can be taken as the measure of the momentum representation. They also indicate that the eigenvalue equation of all operators is the eigenvalue equation corresponding to the extreme value of their variational system. Similar to ref. [28], then the general Hellmann-Feynman theorem for any n variables in the transition from classical statistical mechanics to quantum mechanics can be deduced from the last term in eq.(2.4), which will be studied in another paper due to limit of paper's length.
This relation implies that ordinary Hermitian operators and their eigenvalues also satisfy this important relation. In particular, the Hellmann-Feynman theorem for any n particles is established under the condition that not only the eigenvalue equations of all operators are the eigenvalue equations corresponding to the extreme values of their variational system, but also it's the result of taking the extreme value of the variational system for the system of any n particles.   Schrȍdinger equation (2.6) of any n particles not only reflects the particle property of any n particles, but also displays the wave property of any n particles, in other words, eq.(2.5) only reflects the particle properties of any n particles, eq.(2.6) of any n particles is transformed into Schrȍdinger equation in quantum mechanics which reflects particle-wave duality of any n particles.
The complex square root function the characteristics of wave-particle duality. 7 Thus, the crisis of the dispute over the origin of the wave-particle duality of any n microscopic particles, e.g., a crisis of a fundamental interpretation of quantum mechanics has been debated for nearly a century, is solved.
Consequently, eq.(2.5) and eq.(2.6), respectively, show the classical locality and quantum non-locality for any n particle system, complex square root of the classical probability density of any n particles in the complex domain) and eq.(2.7) directly, respectively, shows the classical locality and quantum non-locality for any n particle system. According to classical statistical mechanics [2,3], when a system is not measured, it evolves according to its own classical statistical mechanics. When measured in the sample space of classical statistical mechanics, any physical quantity measurements instantly make the system collapse at infinitely large velocity at a certain probability to observe the state.
Because we have already taken the micro-particle system state described by a general function of complex square root of real density function (It can be called the complex square root of the probability density function, which is projected onto the plane wave and doing their Fourier integration), so we have entered a special system different from the classical statistical mechanics description of the system, we call it a quantum mechanics system. If the quantum system is in the eigenstate of the probability wave function (i.e., the eigenwave function), the result of measuring the quantum mechanics quantity is the eigenvalue of the quantum system. If the quantum system is not in the eigenstate of wave function, the measurement will lead to the wave function collapse of quantum systems ( because the measurement must act to the system through the measurement instrument ( this effect is ignored in classical statistical physical measurements), which causes the system to change to a certain state), namely the measurement for extracting the information of the quantum system must cause some effects on the quantum isolated system, the measurement makes the wave function of the quantum system jump to some eigenstate with certain probability, and this probability can be calculated strictly according to quantum mechanics.
These measurement processes are nonlocal, decoherent, stochastic and irreversible even from the classical statistical point of view, because classical measures cause the taking eigenvalue corresponding to the variational system's choosing extreme value in process transforming classical statistical mechanics to quantum mechanics and in the quantum mechanics so that people can measure the eigenvalues. On the bases of the nonlocal, decoherent, randomness and irreversibility of these classical statistical mechanics ( which have been understood no problem in classical statistical mechanics [3]) , the nonlocal, decoherent, randomness and irreversibility of plane waves of any n particles are superimposed. Therefore, these properties of nonlocal, decoherent, random and irreversible measurement collapse that is caused by quantum mechanics measurement collapse at infinitely large velocity ( because quantum mechanics must be able to return to the limit of classical statistical mechanics ) are thoroughly proved and understood in the systems.
Therefore, we derive the fourth axiom presumption of quantum mechanics, namely the measurement axiom presumption, from the classical statistical mechanics, that is, it should be reduced to the measurement theorem, i.e., we have proved the measurement theorem from classical statistical mechanics, and the quantum theory is just the current quantum mechanics.

Identical Theorem
In classical statistical mechanics, when the identical particles of a system have indiscernible property, the physical state of the system composed of identical particles will not be changed due to the exchange of identical particles, and the particles with all intrinsic properties such as the same mass, charge, spin and isospin can be called identical particles [2,3].
In classical mechanics, in general, it is always possible to distinguish different particles from different orbits of particle movement. When it is impossible to distinguish the characteristics of different particles from different orbits of particle movement, and when the state of the microscopic particle system is described by the general complex function of the complex square root of real density function, the wave function then, in the above studies, we have proved that the system begins to enter a special description system that is different from the classical statistical mechanics system, namely, the system is called as quantum mechanics system. Therefore, in the present quantum mechanics, each particle corresponds to a probability wave, which is the probability state vector of the general complex function obtained by taking the complex square root of the classical density function and projecting it onto the plane wave for integral. It's well known that waves always overlap as they travel, and that plus the identity make it impossible tell which probabilistic wave belongs to which particle, namely, the identical particle is indistinguishable in quantum mechanics. In fact, for the indistinguishability of identical particles, it's actually classical statistical mechanics where people can (or cannot) distinguish different particles from different orbitals of their motions. Since all waves will overlap in the process of propagation, and the identical property makes it impossible distinguish which probability wave belongs to which particle. For the indistinguishability of identical particles in quantum mechanics, because at the same time the state of the microscopic particle system is described by the general complex function of complex square root of the real density function, and which is projected to the plane probability wave with integral. And the system composed of identical particles has symmetry: the Hamiltonian operator of the identical particle system has the invariance of commutative symmetry (due to the indistinguishability of identical particles), and then the indistinguishability of identical particles in quantum mechanics is derived directly from classical statistical mechanics. So now we should not continuously call it an identical axiom presumption, but an identical theorem. For the multi-particle system, we have obtained eq.(2.6) which is Schrȍdinger equation of the multi-particle system in quantum mechanics, in which we may have The Hamiltonian with identical particles is not changed by identical particle exchange.
Since there is a general solution One can remove this constant, then we deduce that using operator ˆi j P to act on Schrȍdinger equation (2.6) of a multi-particle system is invariant. That is, the general multi-particle system remains unchanged under the exchange of any two particles, but its probability wave function can have a variation of eq. Thus, we derive the fifth axiom presumption of quantum mechanics from classical statistical mechanics, the identical axiom presumption is reduced as identical theorem, i.e., we proved the identical theorem from classical statistical mechanics. 12 automatically obtains a specific state.
For simplicity, we first consider flipping a coin. The probability space formed by flipping a coin is the analogous to the probability space formed by neutral 0  meson decaying into positive and negative electrons. In other words, in flipping a coin, when one side is up, the other side must be down. In the similar way, when one measures a system of positive and negative electrons, because of entanglement, when one measures the spin of one electron is up, the spin of the other must be down. The positive and negative electron systems are separable, and the two sides of a coin are tied together, which is the stronger entanglement and is independent of the separation of coordinates, and people can see this be the extreme case of quantum entanglement. Therefore, it is an independent space coordinate system with overall symmetry. This system can be expressed as a function system of momentum representations independent of spacetime coordinates. If we define the system flipping a coin: heads up is spin 1/2 and heads down is negative spin 1/2, then the two systems have a conservation of spin. Considering when the coin is small enough that the Planck constant effect cannot be ignored, and further considering when the two sides of the coin can be separated in such a small case, in this time, we need to introduce spacetime coordinates to represent the separate states. And when the mass of this coin is further reduced to have the property of a plane wave (from wave-particle duality) with spacetime coordinates and superimposed for integration on the process, as eq.(2.7), the object described by classical statistical mechanics is transformed into the object described by quantum mechanics. The effect of flipping the quantum coin is mathematically identical with the effect measuring the collapse effect of the system of positive and negative electrons. So, we get the conservation of spin angular momentum in a quantum coin toss. That is, no matter how far apart they are, they have to preserve the conservation of the quantum spin angular momentum through this entanglement of the quantum coin toss, e.g., for the conserved quantum angular momentum system of the two particles, the angular momentum state of the quantum system is in a superposition of all possible states. When the angular momentum of the one particle is measured, the quantum state of the other particle will collapse to the other certain state in a way of instantly infinitely large velocity, so as to maintain the conservation of spin angular momentum of the whole system.
Especially, the moment the coin is flipped, a set of initial conditions and boundary conditions are selected from the set sample space of the initial conditions and boundary conditions corresponding to heads up or down, the set sample space for the system collapses instantly at infinitely large velocity. The motion of the later coin is the entanglement motion of satisfying Newtonian mechanics with the certain initial and boundary conditions, for the corresponding entanglement interaction is also at infinitely large velocity.
In the quantum case, there is entanglement before measurement, and the collapse during the measurement is the instantaneous collapse that the entanglement is maintained. In the classical case it could be separated, that is, collapse and then entanglement to maintain some quantity conservation, e.g., the angular momentum conservation or correlation of the system. For quantum mechanics, because quantum mechanics wants to go back to the limits of classical statistical physics, by the transition from classical statistical physics to quantum mechanics, we find that the instantaneous collapse measured in quantum mechanics is actually the collapse with instantaneously infinitely large velocity ( e.g., very exceeding hypervelocity) entanglement to maintain the some quantity conservation or correlation of the system. So, it's actually made up of the two processes.
By comparing the classical with the quantum, we can understand the essence of both classical and quantum collapses and entanglements.
Thus, we conclude that the velocities of such quantum collapse and entanglement are instantaneous and infinite big, rather than just only the super light velocity of collapsing and entangling to maintain the corresponding conservation or correlation of the quantum systems.
Consequently, we discover that entanglements are of interactions ( among all wave function state vectors ) that preserves conservation and correlation. The two examples above illustrate two extreme entanglements in order to preserve the conservation and correlation. The entangled interaction in order to maintain some kind of the conservation or correlation is a new quantum phenomenon, not a known quantum phenomenon. We call this new quantum state interaction as the interaction of entangled quantum states to maintain some kind of the conservation or correlation in the system, for short, entanglement interaction of quantum states.
Therefore, collapse phenomena in quantum mechanics should also include those in classical statistical mechanics, because the classical limit of quantum mechanics is classical statistical mechanics.

Summary and conclusions
Following ref. [28], this paper continues to generalize the density function in classical statistical mechanics to a product of a general complex function for any n particles and its complex Hermitian conjugate function, naturally derives the last two axiom presumptions in the five axiom presumptions of quantum mechanics in literature [29,30]: the measurement principle and the identical principle, and naturally makes the two axiom presumptions into the measurement theorem and the identical theorem of quantum mechanics. The two deduced basic theorems not only solve the crisis that has been very difficult to understand in current quantum mechanics, but also obtain important new physics and new discoveries, e.g., this paper discovers that the velocities of the quantum collapse and quantum entanglement are instantaneous and infinite big, rather than just only the super light velocity of collapsing and entangling to maintain the corresponding conservation or correlation of the quantum systems, which are exactly deduced and are not guessed out liking the past, e.g., see Ref. [34].
Therefore, this paper and ref. [28] together not only naturally deduce the axiom system of quantum mechanics [29,30], but also build up general quantum theory no axiom presumption, therefore, naturally deduce quantum physics no all quantum strange incomprehensible properties. This paper uses Lagrange multipliers i  (i=1,2,3,…) to build up a general variational system, the whole variational system takes the extreme value among all possible values, and the equations derived when taking this extreme value are the eigenvalue equations in the transition from classical statistical mechanics to quantum mechanics.
Furthermore, we deduce the general Schrȍdinger equation of any n particle systems from two aspects, from which we see that some keys of deducing the general Schrȍdinger equation Therefore, for a general quantum system of any n ( >1 ) particles, this paper, for the first time, discovers three kinds of entanglements: original, superposition and across entanglements. The three kinds of entanglements directly give lots of entanglement sources for future large practical applications. Thus, all the studies on various entanglements and applications need to be further studied and classified by the three kinds of entanglements. Otherwise, all the current studies on various entanglements aren't perfect or exact.
Analogous to taking the square root of Klein-Gooden equation, we get Dirac equation Fermi system, in terms of the studies in this paper and ref. [28], we discover that one of two pillars of modern physics, quantum mechanics of any n particle system actually is a generalization mechanic theory of the complex square root of real density function of classical statistical mechanics, for short, quantum mechanics of any n particle system actually is just a generalization theory of the complex square root of classical statistical mechanics, which can be seen from the overall deductions in this paper, any n particle system's quantum mechanics of being just a generalization theory of the complex square root of classical statistical mechanics is both a revolutionary discovery and key new physics, which are further influencing people's philosophical thinking for modern physics, especially ref.[] and this paper solve all the crisises in quantum theories, quantum information and so on, and make quantum theory have scientific solid foundations checked and both no axiom presumption and no all quantum strange incomprehensible properties, because classical statistical mechanics and its complex square root have the scientific solid foundations checked.
Finally, ref. [28] and this paper solve a series of crisises of basses of quantum mechanics, e.g., wave-particle duality origin and the first quantization origin, quantum nonlocality, quantum entanglement origins, wave collapse from quantum measurement and so on, following the new deduced general quantum theory, a lot of works related to quantum communications, quantum computer and so on can be further supplied, classified and updated.
The authors declare that there are no competing interests. Each author equally contributed to this paper because of work together. Data Availability: all data is available in this paper, and I have included the appropriate data availability statement in my manuscript.