Quantum entanglement in Measurement Quantization

A single composite preparation, realized as correlated measurements. Subsystems A and B are measured using linear-polarization analyzer angles alpha and beta. Outcomes s and t take values plus or minus one. P subscript MQ is the normalized contextual joint outcome probability in the specified ideal model. The displayed sign convention gives positive same-setting correlation. It is not a universal law for every entangled state or a singlet anticorrelation formula.

Composite preparation and realization

MQ treats the selected entangled pair as one composite relational preparation. Its encoded state is not a product of independent subsystem states. The Internal Frame is a nonphysical count-configuration domain. The Frames mapping realizes observable normalized joint probabilities in the System Frame. Spatial separation concerns the realized measurements, not a physical lattice or medium in the configuration domain. Non-product probability structure alone is not an entanglement witness, since classical correlated distributions can also be non-product. The explicit polarization law and Bell scenario supply the relevant content here.

Joint probabilities

P_{MQ} (+, + | α, β) = \frac{1}{2} \cos^{2} (α - β)

P_{MQ} (-, - | α, β) = \frac{1}{2} \cos^{2} (α - β)

P_{MQ} (+, - | α, β) = \frac{1}{2} \sin^{2} (α - β)

P_{MQ} (-, + | α, β) = \frac{1}{2} \sin^{2} (α - β)

MQ0742 records the same-outcome cosine-squared law. The complementary sine-squared laws follow the symmetric dichotomic Appendix L construction, normalization and its balanced local marginals, with DEF0503 and CLM3297 retaining the selected polarization scope. Adding all four joint probabilities gives one; all are nonnegative. Summing over either partner outcome gives one half.

Local marginals

P_{MQ} (s | α, β) = P_{MQ} (t | α, β) = \frac{1}{2}

MQ0738 and MQ0739 supply marginal additivity. DEF0336 and CLM3295-CLM3296 distinguish local marginal independence from non-product joint statistics. The explicit model laws establish these uniform marginals directly, without relying on the ambiguous partner-setting transcription in MQ0740. Marginal independence prevents controllable signaling by changing the distant analyzer setting. The conceptual connecting curves do not represent a communication channel.

Correlation and CHSH

E (α, β) = \sum_{s,t = \pm 1} s t P_{MQ} (s, t | α, β) = \cos [2 (α - β)]

S_{CHSH} = E (α, β) + E (α, β′) + E (α′, β) - E (α′, β′)

| S_{CHSH} | = 2 \sqrt{2}

E is the expected product of the two outcome signs. S subscript CHSH is the displayed four-correlator combination; alpha prime and beta prime are alternative analyzer angles. MQ0673 and MQ0694 give this definition. MQ0674 and MQ0695 state the quantum upper bound, and DEF0503, CLM1162 and CLM3297 retain its saturation for the ideal construction. For this sign convention, angles alpha=0 degrees, alpha prime=45 degrees, beta=22.5 degrees and beta prime=minus 22.5 degrees give the displayed optimal value. Angles may equivalently be expressed in radians. The result is not asserted for arbitrary settings or imperfect experiments.

The Bell-local CHSH bound is two under Bell factorization and measurement-independence assumptions. A non-product joint distribution or no-signaling condition alone does not prove either Bell violation or the Tsirelson bound. Here both results are checked from the explicit probabilities and selected settings. The source picture’s unqualified position-density factorization inequality has therefore been replaced by operational outcome marginals.

The agreement is with standard quantum polarization predictions within MQ’s conditional model, not an independent derivation from count ontology alone or an experiment uniquely verifying MQ. Peer-reviewed Bell experiments support quantum correlations. Source [1]: Giustina and colleagues, Significant-Loophole-Free Test of Bell’s Theorem with Entangled Photons, Physical Review Letters 115, 250401 (2015), https://doi.org/10.1103/PhysRevLett.115.250401 . Numerical and semantic MQ authority is MQKB523.zip. The artwork is an edited conceptual illustration, not measured apparatus geometry or data.

Quantum Entanglement

Entanglement as a Non-Factorizable Realization of Conserved Measurement Availability

Jointly prepared quantum systems can produce correlations that violate a Bell inequality under suitable measurement settings. Experiments with distant electron spins and with entangled photons have observed Bell violations. ^[1] ^[2] How does a joint preparation give rise to separated detector records?

Measurement Quantization (MQ) begins with one axiom. Measure is discrete. The Internal Frame is the discrete relational domain of count configurations and adjacency. The Frames mapping relates those configurations to the System Frame, the non-discrete relational encoding in which continuous geometry and observables are realized.

MQ uses count relationships to geometrically anchor an expanding frame of information. The cosmological and fundamental-measure construction uses four physical inputs to select and normalize the realization for our universe. The fine-structure constant calibrates the dimensionless radial coefficient. The speed of light links realized length and time and has an exact numerical value in SI. The electron mass and Bohr radius enter the cross-dimensional normalization. The cosmic microwave background temperature serves separately as an epoch selector.

In the MQ quantum construction, an entangled preparation is represented by one composite count configuration. Detector separation, analyzer orientations, and recorded outcomes are described in the System Frame.

This distinction changes the physical question. MQ does not ask how one spatially separated particle instantaneously communicates a measurement result to another. It asks how a single non-factorizable preparation is resolved into spatially separated measurement records when the underlying preparation is not itself defined by those System Frame spatial coordinates.

The Composite System Comes First

Consider subsystems A and B. Their joint preparation occupies the relational domain ω′_AB. Each relational cell carries a nonnegative integer availability count at a given update. The update orders count configurations. Realized time enters through the System Frame.

The essential point is that ω'_AB is a joint domain. It is not constructed by first assigning independent physical states to A and B and then adding a correlation between them. The composite count configuration is primary.

Under the Frames mapping, the relational domain is realized as a bipartite System Frame domain Ω_AB. When this realized measure admits a density representation, let ρ_S(r_A,r_B,t) denote the joint encoded availability density, where r_A and r_B are the realized coordinates of the two subsystems and t labels the realized temporal coordinate.

The total mapped availability N₂(t) is the integral of this density over the bipartite configuration space. When this integral is finite and positive, division by it produces the joint probability density ρ_P(r_A,r_B,t).

For the joint availability structure considered here, the normalized joint density differs from the product of its subsystem marginals.

ρ_{P} (r_{A}, r_{B}, t) \neq ρ_{P 1} (r_{A}, t) ρ_{P 2} (r_{B}, t)

The symbols ρ_P1 and ρ_P2 denote the normalized marginal densities of A and B. Each is obtained by integrating over the other subsystem’s coordinates. The inequality describes the joint density as a whole.

A non-product density establishes statistical dependence. Classical preparations can also have dependent outcomes. Bell violation requires the pattern of correlations across measurement settings.

Why Spatial Separation Does Not Split the Preparation

Ordinary intuition encourages us to imagine two particles separating in space while each carries a complete local specification of what it will later do. Bell's theorem shows why that picture cannot reproduce all quantum predictions when the relevant assumptions, including Bell-local factorization and measurement independence, are imposed.

The reason follows directly from the distinction between the Internal Frame and System Frame. Distance, orientation, and ordinary spacetime coordinates belong to the realized System Frame. The primitive Internal Frame configuration is relational and contains no intrinsic spatial metric assigning a conventional distance between the two components of the preparation.

Consequently, increasing the realized distance between detectors does not transform the original joint count configuration into two independent primitive configurations. Spatial separation is a property of the realized description. It is not an operation that factorizes the underlying preparation.

The model uses adjacency-local count redistribution in the Internal Frame. For a closed region, redistribution preserves its total count. Operational no-signaling concerns the System Frame marginals. Bell local causality imposes a separate requirement on the joint probabilities conditioned on a complete hidden state.

Measurement Settings Define Contextual Partitions

Let a and b be the analyzer settings. Write the two output labels as s and t, each taking the values ±1. The label t in an outcome pair is distinct from the realized time argument of an availability function.

The settings define a contextual partition of the joint availability into four mutually exclusive outcome regions corresponding to ++, +-, -+, and --.

In the equation below, the numerator denotes the encoded availability of outcome (s, t) at the selected settings. The denominator denotes the total across all four joint outcomes at the same realized time.

For positive total contextual availability, the realized joint probability, denoted P_MQ, is given by the following ratio.

P_{MQ} (s, t | a, b) = \frac{N_{s t}^{a b} (t)}{N_{a b}^{tot} (t)}

The analyzer settings therefore do not rewrite the prior preparation. They determine how the shared preparation is partitioned into distinguishable System Frame outcomes.

The benchmark holds the preparation fixed across analyzer settings and assumes the setting choices are statistically independent of the source preparation. The timing of the choices alone cannot establish that independence. The realized partition depends on both settings.

Entanglement Does Not Require Predetermined Outcomes

With a specified deterministic redistribution rule, the current Internal Frame configuration fixes the next one. The model describes recorded System Frame outcomes through the completed measurement context.

A completed measurement gives one stable record in the MQ account. The context supplies mutually exclusive record classes. For repeated preparations measured in the same context, normalized availability gives the model’s predicted distribution across those classes.

This is why the broader MQ treatment of determinism and quantum behavior does not reduce entanglement to a conventional local hidden-variable model.

From Joint Availability to Bell Correlations

The Bell benchmark requires a specific dependence of the joint probabilities on the analyzer settings.

The ideal linear-polarization model uses two real amplitude coefficients for the analyzer channels after removal of a common phase.

Let α and β be the two analyzer-axis angles in the System Frame. Their relative angle is Δ = α − β. The two channel coefficients are η₊ and η₋.

The two-channel amplitude has unit norm.

{η_{+}}^{2} (Δ) + {η_{-}}^{2} (Δ) = 1

At zero relative angle the channel transformation is the identity. Successive rotations compose continuously and preserve the norm.

These conditions give rotations in the real two-channel amplitude space. The ideal linear-polarization analyzer representation identifies the channel rotation angle with the physical relative angle. With alignment at zero relative angle, the coefficients are cos(Δ) and sin(Δ), up to the channel sign convention.

Each channel weight is the square of its amplitude coefficient.

For the symmetric maximally correlated polarization preparation, the two equal-outcome channels have the same probability.

P_{MQ} (+, + | α, β) = P_{MQ} (-, - | α, β) = \frac{1}{2} \cos^{2} (α - β)

The two unequal-outcome channels also have equal probabilities.

P_{MQ} (+, - | α, β) = P_{MQ} (-, + | α, β) = \frac{1}{2} \sin^{2} (α - β)

The four probabilities sum to one. The joint distribution is non-product at settings where the equal- and unequal-outcome channel weights differ.

The correlation E is the expectation of the product of the two outcome labels.

E (α, β) = \sum_{s, t = \pm 1} s t P_{MQ} (s, t | α, β)

For the ideal analyzer model, this expectation simplifies to

E (α, β) = \cos [2 (α - β)]

The Bell-CHSH Test

The Clauser–Horne–Shimony–Holt (CHSH) formulation tests a constraint on four correlations. The bound for a Bell-local model assumes factorization at fixed hidden state and measurement independence. ^[3]

Let a and a' denote two analyzer settings available to the first measurement station, and b and b' two settings available to the second. The CHSH combination S_CHSH is constructed from the four corresponding correlations.

For the signed sum that subtracts the correlation at (a′, b′), take a = 0, a′ = π/4, b = π/8 and b′ = −π/8, with angles in radians. The ideal polarization model then gives

| S_{CHSH} | = 2 \sqrt{2}

This exceeds the Bell-local limit of 2 and reaches the quantum Tsirelson bound. ^[4] It is an ideal prediction for this preparation and analyzer model.

A CHSH violation rules out a measurement-independent Bell-local factorization for these joint statistics. Agreement with the ideal quantum prediction tests the benchmark. The Bell experiments establish the observed correlations and leave the interpretation of the preparation open.

No-Signaling Without Bell Factorization

For the ideal polarization model, the local outcome frequencies are independent of the distant setting. A distant observer therefore cannot encode a message in the choice of analyzer angle and recover it from local outcomes alone.

Summing over the distant outcome gives the local probability.

\sum_{t = \pm 1} P_{MQ} (s, t | a, b) = P_{MQ} (s | a)

Summing over the first outcome gives the corresponding probability at the second detector. In this symmetric model, each local outcome has probability one-half for every pair of settings.

In this model, at settings with nonzero correlation, the joint probability differs from the product of the local probabilities.

P_{MQ} (s, t | a, b) \neq P_{MQ} (s | a) P_{MQ} (t | b)

What Happens When One Side Is Measured?

After a record with positive probability has been obtained, MQ restricts the realized System Frame encoding to the corresponding outcome region and renormalizes it for subsequent predictions.

Conditioning on one result can change the predicted distribution of the other result. The unconditioned marginal remains the same, so the distant observer needs the record to sort trials by that result.

In the MQ account, the count configuration, the normalized predictive encoding, and the completed record play distinct roles.

Why Entanglement Is Natural in MQ

The primitive description is relational. A composite preparation can therefore contain physically relevant joint count structure that has no decomposition into independent subsystem measures. The separate particles are recognizable as separate realized subsystems in the System Frame, but that does not imply that the earlier relational preparation was assembled from independent primitive states.

MQ interprets these correlations through the joint preparation and its realized records.

The Frames mapping can be many-to-one, so some details of the primitive configuration need not survive in its realized encoding. The angular benchmark additionally depends on contextual partitioning, positive normalization, and the ideal analyzer response.

A Different Meaning of Quantum Nonlocality

Bell experiments constrain measurement-independent Bell-local models. The ideal probabilities above separately show how Bell violation and operational no-signaling can coexist.

Entanglement as a Test of the MQ Architecture

The ideal polarization benchmark brings together several parts of the MQ quantum construction.

The angular correlation and optimal CHSH value agree with the standard quantum prediction for the stated preparation. MQ interprets them as statistics of spatially separated records from a single composite relational preparation.

The detector records can be spatially separated in the System Frame while the preparation retains its joint structure in the Internal Frame.

The distinction connects quantum entanglement directly to the broader MQ description of frames of reference, determinism and quantum behavior, and the foundational Measurement Quantization framework. The current count-based quantum measurement derivation, including its Bell construction and measurement-conditioning analysis, is also available through the Institute's externally archived research record, Measurement Quantization: A Count-Based Resolution of the Quantum Measurement Problem.

Source Notes

1. B. Hensen et al., Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres. Nature 526, 682–686 (2015). Publisher record

2. M. Giustina et al., Significant-Loophole-Free Test of Bell’s Theorem with Entangled Photons. Physical Review Letters 115, 250401 (2015). Publisher record

3. J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Proposed Experiment to Test Local Hidden-Variable Theories. Physical Review Letters 23, 880–884 (1969). Publisher record

4. B. S. Cirel’son, Quantum generalizations of Bell’s inequality. Letters in Mathematical Physics 4, 93–100 (1980). Publisher record

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