From Conditional Probability Measurements to Global Matrix Representations on Variant Construction – A Particle Model of Intrinsic Quantum Waves for Double Path Experiments

Quantum statistics play a key role in Quantum Mechanics QM [Feynman et al. (1965,1989); Penrose (2004)]. Two types of Double Slit Experiment are used to explore the core mysteries of quantum interactive behaviors. These are standard Double Slit Experiments with correlated signals and Single Photon Experiments that use ultra low intensity and lengthy exposures to demonstrate quanta self-interference patterns. The key significance is that intrinsic wave properties are observed in both environments [Barrow et al. (2004); Hawkingand &Mlodinow (2010)].


Two types of double slit experiments
Quantum statistics play a key role in Quantum Mechanics QM [Feynman et al. (1965[Feynman et al. ( ,1989; Penrose (2004)]. Two types of Double Slit Experiment are used to explore the core mysteries of quantum interactive behaviors. These are standard Double Slit Experiments with correlated signals and Single Photon Experiments that use ultra low intensity and lengthy exposures to demonstrate quanta self-interference patterns. The key significance is that intrinsic wave properties are observed in both environments [Barrow et al. (2004); Hawkingand & Mlodinow (2010)].

Two types of probabilities
Multivariate probabilities acting on multinomial distributions occupy a central role in classical probability theory and its applications. This mechanism has been explored from the early days in the study of modern probability theories [Ash & Doléans-Dade (2000); Durret (2005)]. Conditional probability is a powerful methodology at the heart of classical Bayesian statistics. In the history of probability and statistical developments, there have been long-running debates and a persistent lack of agreement in differentiating between prior distributions and posterior distributions [Ash & Doléans-Dade (2000); Durret (2005)]. It is worthy of note that the uniform distributions or normal distributions of conditional probability are always linked to a relatively large number of probability distributions in non-normal conditions. This points to practical problems with random distributions. 17 www.intechopen.com

Applying the bohr complementarity principle
The Bohr Complementarity Principle BCP, established back in the 1920s brought us the foundations of QM [Bohr (1949)]. In Bohr's statement:"... we are presented with a choice of either tracing the path of the particle; or observing interference effects ... we have here to do with a typical example of how the complementary phenomena appear under mutually exclusive experimental arrangements." It is significant that BCP provided a powerful intellectual basis for Bohr in key debates in the history of QM and especially in his debates with Einstein [Jammer (1974)].

Testing bell inequality
To help decide between Bohr and Einstein on their approaches to wave and particle issues, Bell proposed a set of Bell-Inequations in the 1960s [Bell (1964]. In 1969, CHSH proposed a spin measurement approach [Clauser et al. (1969)] and experiments by Aspect in 1982 did not support local realism [Aspect et al. (1982)].

Current situation
From the 1920s through to the start of the 21st century, there was no significant experimental evidence to show that there were problems with the BCP. However, Afshar's 2001 experimental results are clearly not consistent with the BCP and further experimental results have provided solid evidence against the BCP. [Afshar (2005;2006); Afshar et al. (2007)]. It is interesting to see that neither local realism nor the BCP are validated by the results of modern advanced single photon experiments [Afshar et al. (2007); Aspect (2007)]. It will be a major challenge in this century to redefine the principles on which the quantum approach may now be safely founded.

Chapter organization
Following on from multivariate probability models, this chapter focusses on a conditional approach to illustrate special properties found in conditional probability measurements via global matrix representations on the variant construction. This chapter is organized into nine sections addressing as follows: 3. analysis of key issues of QM 4. conditional construction proposed 5. exemplar results 6. analysis of visual distributions 7. using the variant solution to resolve longstanding puzzles 8. main results 9. final conclusions 2. Wave and particle debates in QM developments

Heisenberg uncertain principle
The Heisenberg Uncertainty Principle HUP was established in 1927 [Heisenberg (1930)]. The HUP represented a milestone in the early development of quantum theory [Jammer (1974)]. It implies that it is impossible to simultaneously measure the present position of a particle while also determining the future motion of a particle or any system small enough to require a Quantum mechanical treatment. From a mathematical viewpoint, the HUP arises from an equation following the methodology of Fourier analysis for the motion [Q, P]=QP − PQ = ih. The later form of HUP is expressed as △p ·△q ≈ h.
This equation shows that the non-commutativity implies that the HUP provides a physical interpretation for the non-commutativity.

Bohr complementarity principle
The HUP provided Bohr with a new insight into quantum behaviors [Bohr (1958)]. Bohr established the BCP to extend the idea of complementary variables for the HUP to energy and time, and also to particle and wave behaviors. One must choose between a particle model, with localized positions, trajectories and quanta or a wave model, with spreading wave functions, delocalization and interferences [Jammer (1974)].
Under the BCP, complementary descriptions e.g. wave or particle are mutually exclusive within the same mathematical framework because each model excludes the other. However, a conceptual construction allowed the HUP, the BCP and wave functions together with observed results to be integrated to form the Copenhagen Interpretation of QM. In the context of double slit experiments, the BCP dictates that the observation of an interference pattern for waves and the acquisition of directional information for particles are mutually exclusive.

Bohr-Einstein debates on wave and particle issues
Bohr and Einstein remained lifelong friends despite their differences in opinion regarding QM [Bohr (1949;1958)]. In 1926 Born proposed a probability theory for QM without any causal explanation. Einstein's reaction is well known from his letter to Born [Born (1971) Perhaps in a spirit of compromise, Bohr then proposed his BCP that emphases the role of the observer over that which is observed. From 1927-1935, Einstein proposed a series of three intellectual challenges to further explore wave and particle issues [Bohr (1935;1949) [Einstein et al. (1935)] by Einstein, Podolsky and Rosen EPR was published in Physical Review.

EPR claims
The key points of the EPR paper are focused on two aspects: either (1) the description of reality given by the wave function in QM is incomplete or (2); the two quantities P and Q cannot have simultaneous reality.
Both operations: P and Q are applied PQ − QP = ih. Such relationships follow the standard Quantum expression.
Property PQ − QP = 0 implies P and Q operations are related without independent computational properties. Under this condition, it is impossible to execute the two operations simultaneously under extant QM frameworks. From a parallel processing viewpoint, Einstein's view is extremely valuable. As such modern parallel computing theories and practices were only developed in the 1970s [Valiant (1975)] it is remarkable that Einstein pioneered such an approach way back in the 1930s. Modern parallel computing theory and practice support the original EPR paper and the conclusion that a QM description of physical that is expressed only in terms of wave functions is incomplete.

Restriction under HUP
For the HUP, different interpretations originate from the equation [Q, P]=QP − PQ = ih, and the later HUP form △q ·△p ≥ h. From a mathematical viewpoint, this type of inequality implies △q, △p ≥ h too. In other words, a minimal grid of a lattice restricts △q and △p → 0 actions. From the HUP expression, [Q, P] = 0 indicates the construction with a discrete intrinsic limitation. Such structures cannot directly apply to continuous infinitesimal operations.
Many quantum problems do not extend to the region of Plank constant limitations. Investigation back in the 1930s tended to rely more on theoretical considerations rather than actual experimentation [Bohr (1949)]. Consequently many issues had to wait until the 1980s to become better understood.
Both Q and P are infinite dimensional matrices, the restriction of [Q, P]=ih comes with a clear meaning today on its discrete properties. We cannot apply a continuous approach to make [Q, P]=0. From an operational viewpoint, Einstein correctly identified the root of the matter. Since Q and P cannot exchange, it is not possible to run a simultaneous process on the standard wave functions. Simply extending discrete variations using continuous approaches presents further difficulties for the HUP.
In addition, the following questions need to be addressed in relation to the practical identification of complementary objects.
• What determines when a pair of objects is indeed a complementary pair?
• Is a pair of complex conjugate objects: a + bi and a − bi, a pair of complementary objects?
• Is a pair of matrices, a hermit matrix H and its complex conjugate matrix H * , a pair of complementary objects? • Why can density matrix operations on infinite dimension be performed without significant errors while a pair of complementary finite matrices must be restricted by the HUP?
In practice, QM computations are mainly applied to wave functions and [Q, P]=ih formula. Intellectual debate on the theoretical considerations is particularly relevant to the HUP. In comparison, the deeper problems of QM cannot be easily explored in the absence of an experimental approach and a viable alternative theoretical construction [Barrow et al. (2004); Jammer (1974)].

Construction of BCP
Inspired by the HUP, Bohr uses a continuous analogy and a classical logical construction to describe Quantum systems. The BCP extends the HUP to handle different pairs of opposites and to restrict them with exclusive properties [Bohr (1958)].
As there were no well refined critical experiments in these days, all the debates between Bohr and Einstein were based on theoretical considerations alone. Compared with Einstein's open-minded attitudes to QM [Einstein et al. (1935)], Bohr and others insisted on the completeness and consistency of the Copenhagen Interpretation on QM [Bohr (1935)]. Such closed attitudes served to distance Bohr and others of like mind from reasonable suggestions made by Einstein and those expressed in the EPR paper and to lead them to treat such suggestions as if they were attacks on their already strongly held views [Bolles (2004)].
The BCP uses a classical logic framework to support dynamic constructions. Underpinned by the BCP, the HUP and a knowledge of wave functions, the Copenhagen Interpretation played a dominant role in QM from the 1930s on as it had by then been accepted as the orthodox point of view [Jammer (1974)].
Meanwhile, the EPR paper emphases that critical evidence must be obtained by real experiments and measurements. It is in the nature of a priori philosophical considerations that they will run into difficulties when actual experimental results fail to corresponded with their expectations.

EPR construction
The EPR position [Einstein et al. (1935)] can be re-visited in the light of modern advances in knowledge and computing theory. From a computing viewpoint, simultaneous properties may be the key with which these long-standing mysteries of QM can at last be unlocked.
Operators P and Q cannot be exchanged, this indicates operational relevances existing in the lower levels of classical QM construction. In addition, there is a requirement of two systems The core EPR model can be shown in Figure 1, listed notations are explained as follows.
Let S 1 be System I, S 2 be System II, IM(S 1 , S 2 |t), t ∈ [0, T] be Interactive Measurements IM for S 1 and S 2 on t ∈ [0, T], SM(S 1 , S 2 |t), t > T be Separate Measurements SM (non-interactive measurements) for S 1 and S 2 on t > T. Einstein's Experimental devices can be described as an EPR measurement quaternion: If an experiment can be expressed in the requisite form for this model, then it can be legitimately claimed as an EPR experiment.

Afshar experimental device
Afshar's experimental results have shown that it is possible to measure both particle and wave interference properties simultaneously in the same experiment with high accuracy [Afshar (2005;2006); Afshar et al. (2007)]. Since this set of experiments has produced results that challenge the BCP at its very core it is pertinent to analyze and compare the model with the requirements for valid EPR devices.
In Afshar's experiments, {ψ 1 , ψ 2 } are two signals input through double slits; σ 1 is the location on the distance f to collect interference measurements of {ψ 1 , ψ 2 }, and σ 2 is the location on the distance f + d to collect separate measurements of {ψ 1 , ψ 2 }. Under this configuration, a 1-1 corresponding map can be established as follows: ( Using quaternion structures,

Conditional variant simulation and representation system
A comprehensive review of the process of variant construction from conditional probability measurements through to global matrix representations is described briefly in this section.
It is hoped that this may offer a convenient path for those seeking to devise and carry out experiments to further explore natural mysteries through the application of sound principles of logic and measurement.
Using variant principles described in the following subsections, with a N bit 0-1 vector X and a given logic function f , all N bit vectors are exhausted, variant measures generate two groups of histograms. The variant simulation and representation system is shown in Fig 2 (a-b). The detailed principles and methods are described in Sections 4.2-4.7 respectively. For multivariate probability conditions, please refer to the chapter of "From local interactive measurements to global matrix representations on variant construction" elsewhere in this book for sample cases and group distributions in multivariate probability environments.

Conditional simulation and representation model
The full measurement and representational architecture as shown in Figure 2(a) has four components: Conditional Meta Measurements CMM, Conditional Interactive Measurements CIM, Statistical Distributions SD and Global Matrix Representations GMR. The key part of the system, the CIM component, is shown in Fig 2(b).

Conditional Meta Measurements
The Conditional Meta Measurement (CMM) component uses N bit 0-1 vector X and a given function J ∈ B 2 n 2 , CMM transfers N bit 0-1 vector under J(X) to generate four Meta-measures, under a given probability scheme, four conditional probability measurements are generated and output as a quaternion signalρ.

Conditional Interactive Measurements
The Conditional Interactive Measurement (CIM) component is the key location for conditional interactions as shown in Figure 2(b) to transfer a quaternion signalρ under symmetry / anti-symmetry and synchronous / asynchronous conditions, under four combinations of time effects namely (Left, Right, Double Particle, Double Wave). Two types of additive operations are identified. Each {ũ,ṽ} signal is composed of four distinct signals.

Statistical Distributions
The Statistical Distribution (SD) component performs statistical activities on corresponding signals. It is necessary to exhaust all possible vectors of X with a total of 2 N vectors. Under this construction, each sub-signal of {ũ,ṽ} forms a special histogram with a one dimensional spectrum to indicate the distribution under function J. A total of eight histograms are generated in the probability conditions.

Global Matrix Representations
The Global Matrix Representation (GMR) component uses each statistical distribution of the relevant probability histogram as an element of a matrix composed of a total of 2 2 n elements for all possible functions {J}. In this configuration, C code schemes are applied to form a 2 2 n−1 × 2 2 n−1 matrix to show the selected distribution group.
Unlike the other coding schemes (SL, W, F, ...), only C code schemes provide a regular configuration to clearly differentiate the Left path as exhibiting horizontal actions and the Right path as exhibiting vertical actions . Such clearly polarized outcomes may have the potential to help in the understanding of interactive mechanism(s) between double path for particles and double path for waves properties.

Two sets of states
For any n-variables let a position j be the selected bit 0 ≤ j < n, x j be the selected variable. Let output variable y and n-variable For all states of x, a set S(n) composed of the 2 n states can be divided into two sets: S 0 (n) and S 1 (n).

Four meta functions
For a given logic function f , input and output pair relationships define four meta logic

Two polarized functions
Considering two standard logic canonical expressions: the AND-OR form is selected from Any logic function can be expressed as a variant logic form.
In f + |x| f − structure, f + selected 1 items in S 0 (n) as the same as the AND-OR standard expression, and f − selecting relevant parts the same as OR-AND expression 0 items in S 1 (n).

Meta measures and conditional probability measurements
Under variant construction, N bits of 0-1 vector X under a function J produce four Meta measures composed of a measure vector N Using four Meta measures, relevant probability measurements can be formulated.

Variant measure functions
Let Δ be the variant measure function For any given n-variable state there is one position in ΔJ(x) to be 1 and other 3 positions are 0.

Variant measures on vector
For any N bit 0-1 vector X, X = X N−1 ...X j ...X 0 ,0≤ j < N, X j ∈ B 2 , X ∈ B N 2 under n-variable function J, n bit 0-1 output vector Y, Let N bit positions be cyclic linked. Variant measures of J(X) can be decomposed
Input and output pairs are 0-1 variables for only four combinations. For any given function J, the quantitative relationship of {⊥, +, −, ⊤} is directly derived from the input/output sequences. Four meta measures are determined.

Four conditional meta measurements
Using variant quaternion, conditional measurements of probability signals are calculated as four meta conditional measurements by following the given equations. For any N bit 0-1 vector X, function J, under Δ measurement:

Conditional Interactive Measurements
Conditional Interactive Measurements (CIM) are divided into three stages: BP, SW and SM respectively. The BP stage selects {ρ − ,ρ + } as sub-signals. The SW component extends two signals into four signals with different symmetric properties; The SM component merges different signals to form two sets of eight signals.

Statistical distributions
The SD component provides a statistical means to accumulate all possible vectors of N bits for a selected signal and generate a histogram. Eight signals correspond to eight histograms respectively. Among these, four histograms exhibit properties of symmetry and the other four histograms exhibit properties of anti-symmetry.

Statistical histograms
For a function J, all measurement signals are collected and the relevant histogram represents a complete statistical distribution.
Distributions are dependant on the data set as a whole and are not sensitive to varying under special sequences. Under this condition, when the data set has been exhaustively listed, then the same distributions are always linked to the given signal set.
The eight histogram distributions provide invariant spectra to represent properties among different interactive conditions.

Global Matrix Representations
After local interactive measurements and statistical process are undertaken for a given function J, eight histograms are generated. The Global Matrix Representation GMR component performs its operations into two stages. In the first stage, exhausting all possible functions for ∀J ∈ B 2 n 2 to generate eight sets, each set contains 2 2 n elements and each element is a histogram. In the second stage, arranging all 2 2 n elements generated as a matrix by C code scheme. Here, we can see Left and Right path reactions polarized into Horizontal and Vertical relationships respectively.

Matrix and its elements
For a given C scheme, let C(J)= J 1 |J 0 , each element

Simulation results
For ease of illustration, as different signals have intrinsic random properties, only statistical distributions and global matrix representations are selected in this section.

Statistical distributions
The From a given function, a set of histograms can be generated as two groups of eight probability histograms. To show their refined properties, it is necessary to represent them in both odd and even numbers. A total of sixteen histograms are required. For convenience of comparison, sample cases are shown in Figures 3(I-III).

Global matrix representations
All possible 2 2 n functions are applied. It is convenient to arrange all the histograms generated into a matrix and a C code scheme of variant logic is applied to organize a set of 2 2 n histograms into a 2 2 n−1 × 2 2 n−1 matrix.
Applying the C code configuration, any given signal of a function determines a matrix element to represent its histogram. There is one to one correspondence among different configurations.
Using this measurement mechanism, eight types of statistical histograms are systematically illustrated. Each element in the matrix is numbered to indicate its corresponding function and the relevant histogram is shown.

Analysis of results
In the previous section, results of different statistical distributions and their global matrix representations were presented. In this section, plain language is used to explain what various visual effects might be illustrated and to discuss local and global arrangements.

Statistical distributions for a given function
It is necessary to analyze the differences among the various statistical distributions for a given function.

Symmetry groups for a function
For the selected function J = 3, four distributions in symmetry groups are shown in Fig 3  (a-d). However, for P H (ũ 1 |J) under synchronous conditions and with the same Left and Right input signals, the simulation shows a D-W exhibiting interferences among the output distributions that are significantly different from the original components.

Anti-symmetry groups for a function
Four distributions are shown in Fig 3 (e-h) as asymmetry groups. A pair of equations P H (ṽ + |J)=P H (1 −ṽ − |J) shows that one distribution is a mirror image of the other. P H (ṽ + |J) distribution is shown in Fig 3 (e) for Left signals and P H (ṽ − |J) distribution is shown Fig 3 (f) for Right signals. Fig 3 (g) for both paths open under asynchronous conditions to simulate a D-P. Compared with (e-f) distributions, it is feasible to identify the same components from the original inputs.
However P H (ṽ 1 |J) is shown in Fig 3 (h) under synchronous condition with both path signals as inputs to simulate a D-W exhibiting interferences among the output distributions that are significantly different from the original components.
To differentiate between even and odd numbers, N = 12 cases are shown in Fig 3 (II, a-h) and N = 13 cases are shown in Fig 3 (III, a-h) respectively.

Global matrix representations
Sixteen matrices are represented in Fig 4-5 (a-h) with eight signals generating two sets of 16 groups for N = {12, 13} respectively.

Anti-symmetry cases
In a similar manner to the symmetry conditions, four anti-symmetry effects can be identified in  Fig 4-5 (g) show additional effects for each distribution according to the relevant position with components that can be identified as corresponding to identifiable inputs in many cases. Anti-symmetry signals are generated in merging conditions. Fig 4-5 (h) show different properties. In general, only one peak can be observed for each element especially for the J ∈{ 10, 12, 3, 5} condition. Spectra appear to be much simpler than the original distributions in Fig 4-5 (e-f), and significant interference properties are observed.

Four symmetry groups
Pairs of relationships can be checked on symmetry matrices in Figs 4-5 (a-d), four groups are identified.  Fig 4 (d). Under this condition, nine or ten classes of different distributions can be identified for Fig 4 (d) and Fig 5 (d) respectively.

Odd and even numbers
From a group viewpoint, only D-P and D-W need to be reviewed as different groups in symmetry conditions. Anti-symmetry conditions are unremarkable.
It is reasonable to suggest that anti-symmetry operations will be much easier to distinguish under experimental conditions, since sixteen groups in D-P conditions and twelve groups in D-W conditions will have significant differences. However, under the symmetry conditions (only) minor differences can be identified.

Single and double peaks
Single and Double peaks can be observed in

Class numbers in different conditions
To summarize over the different classes, 16 matrices are shown in different numbers of identified classes as follows:

Polarized effects and double path results
In order to contrast the different polarized conditions, it is convenient to compare distributions {P H (ũ + |J), P H (ũ − |J} and {P H (ṽ + |J), P H (ṽ − |J} arranged according to the corresponding polarized vertical and horizontal effects. This visual effect is similar to what might be found when using polarized filters in order to separate complex signals into two channels. Different distributions can be observed under synchronous and asynchronous conditions.
The equation is true for different values of N and n.
Four anti-diagonal positions are linked to symmetry and anti-symmetry pairs, twelve other pairs of functions belong to non-symmetry and non-anti-symmetry conditions. Their meta elements can be identified by the relevant variant expressions.

HUP environment
Under the variant construction, variant measurements can be organized into multiple sets of simultaneous measurements. Each element in a N bit vector provides only a small portion of information, collected measurements are independent of special positions. Under this condition, there is no essential HUP environment for the variant construction. 0-1 groups and their measurements are naturally parallel . They can be processed in simultaneous conditions. Considering these properties, such group measurements do not correspond with the requirements of Heisenberg single particle environments. Viewed as a whole, the system of the variant construction has discrete and separate properties that serve to facilitate complex local interactions for any selected group.
From a measurement viewpoint, the parallel parameters of the variant measurements enable them to exist in different interactive models simultaneously. This set of simultaneous properties exhibits significant differences between the original wave functions and the variant construction.

Weakness of BCP
The main weakness of the BCP lies deep in the very logic on which it is founded. In his approach to QM, Bohr applied then extant classical principles of logic using static YES/NO approaches to dynamic particle and wave measurements. However, the complex nature of QM phenomena means that such a classical logic framework cannot fully support this quaternion organization or fully model the dynamic systems involved. This is the main reason why the BCP requires the application of exclusive properties to pairs of opposites.
The variant construction provides quaternion measurement groups. This property naturally supports QM-like structures. Useful configurations can be chosen for further development.
The main experimental evidence following Bohr in rejecting particle models are sets of wave interference distributions generated in long duration and very low intensity single photon experiments. These experiments show intrinsic wave interference patterns under many environments. Understandably, such data have long been held to be strongly indicitive of wave properties within even single quanta. Consequently, it has been deemed natural and necessary to apply wave descriptions and analysis tools in the search for QM solutions.
However, evidence residing within the main visual distributions of this chapter, serves to show that statistical distributions under a conditional probability environment naturally link to intrinsic wave properties in the majority of situations. Nearly all interesting distributions show obvious wave properties. Notably, such intrinsic wave distributions may be sufficient to allow a satisfactory alternative explanation of experimental results generated in long duration and very low intensity single photon experiments.

The BCP for a special subset of QM
We may deduce that there is (only) a special subset of QM for which the BCP is satisfied. Under the variant construction there are six distinct logical configurations that can be used to support 0-1 vectors. Of these six, Bohr's approach is suitable for only the two schemes of pure static YES or NO. Meanwhile, the other four variant, invariant and mixed configurations lie outside the BCP framework. From this viewpoint, Bohr offers insight into important special circumstances of QM rather than provides an all embracing general solution.
Bohr's QM construction is complete and useful in many theoretical and practical environments for static and static-like systems. However, the variant construction provides a more powerful and general mechanism to handle different dynamic systems with variant and invariant properties.

The EPR contribution on variant construction
From EPR proposed experiments and other theoretical considerations, Einstein demonstrated a depth of understanding of weakness inherent in the foundations of the QM approach. He clearly identified two operators with non-communication properties that failed to support simultaneous operations and recognized that this type of mechanism was still not explained in the Copenhagen interpretation.
Using the variant construction, EPR devices have the following correspondence: From this correspondence, many possible configurations of combinations and their subsets are available for future theoretical and experimental exploration.
Using the variant construction, rich configurations can be expressed. From such mapping, it can be seen to be nothing less than astounding that such meta constructions were identified by Einstein as far back as 1935.

Afshar's experiments on variant construction
Afshar's experiments apply anti-symmetry signals making the following correspondence: All Afshar's experiments are a special case of the EPR model.

Main results
Presented as predictions and conjectures:

Predictions
Commensurate with the chapter of local interactive measurements, similar predictions can be described under conditional probability conditions: Prediction 6: Distributions on conditional environments provide essential evidence to support a series of experimental results on quanta self-interference properties.

Conjectures
Presented in relation to milestones in the historical debate underpinning the foundations of QM: Conjecture 1. Einstein may be declared the winner in the Bohr-Einstein debates on QM.
Conjecture 2. EPR construction is a super-powerful model to support different measurements and simulations of quantum behaviors.

Conjecture 3.
The variant construction provides a logical measurement based foundation to support the simulation and visualization of quantum behaviors.
Conjecture 4. The next generation of fundamental development in QM will grow out of further theoretical and experimental exploration based on variant construction.

Conclusion
Long held views on the wave/particle enigma, especially those investigated through single photon experiments may be founded on a special case rather than a general explanation.
Further insight may be found working from conditional probability measurements to global matrix representation on the variant construction.
Applying conditional probability models on interactive measurements and relevant statistical processes, two groups of parameters {ũ β ,ṽ β } describe left path, right path, D-P and D-W conditions with distinguishing symmetry and anti-symmetry properties. {P H (ũ β |J), P H (ṽ β |J)} provide eight groups of distributions under symmetry and anti-symmetry forms. In addition, {M(ũ β ), M(ṽ β )} provide eight matrices to illustrate global behaviors under conditional environments.
The complexity of n-variable function space has a size of 2 2 n and exhaustive vector space has 2 N . Overall simulation complexity is determined by O(2 2 n × 2 N ) as ultra exponent productions. How to overcome the limitations imposed by such complexity and how best to compare and contrast such simulations with real world experimentation will be key issues in future work.
Six predictions and four conjectures are offered for testing by further theoretical and experimental work. Measurement is a multidisciplinary experimental science. Measurement systems synergistically blend science, engineering and statistical methods to provide fundamental data for research, design and development, control of processes and operations, and facilitate safe and economic performance of systems. In recent years, measuring techniques have expanded rapidly and gained maturity, through extensive research activities and hardware advancements. With individual chapters authored by eminent professionals in their respective topics, Advanced Topics in Measurements attempts to provide a comprehensive presentation and in-depth guidance on some of the key applied and advanced topics in measurements for scientists, engineers and educators.

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