The Mathematical Foundations of Quantum Entanglement and Bell's Theorem: A Deep Dive

Quantum entanglement is one of the most fascinating and counterintuitive phenomena in quantum mechanics. It describes a situation where two or more quantum particles become linked together in such a way that they share the same fate, no matter how far apart they are. This shared fate defies classical notions of locality and realism, and Bell's theorem provides a rigorous mathematical framework to test these fundamental assumptions. Let's unpack the mathematical foundations:

1. The Mathematical Language: Hilbert Spaces and Tensor Products

Quantum mechanics is built upon the mathematical foundation of Hilbert spaces. A Hilbert space is a vector space equipped with an inner product that allows us to define notions of length and angle. Crucially, it's complete, meaning that all Cauchy sequences converge to an element within the space.

Quantum States as Vectors: The state of a quantum particle is represented by a vector (usually normalized) within its Hilbert space. These vectors are often denoted using Dirac notation, or "bra-ket" notation, like |ψ⟩.
Superposition: A quantum particle can exist in a superposition of states. Mathematically, this means that the state vector |ψ⟩ can be written as a linear combination of other basis vectors:

|ψ⟩ = c₁|φ₁⟩ + c₂|φ₂⟩ + ... + cₙ|φₙ⟩

where cᵢ are complex coefficients and |φᵢ⟩ are basis vectors. The probability of measuring the particle in state |φᵢ⟩ is given by |cᵢ|².
Tensor Products: Combining Hilbert Spaces: When dealing with multiple particles, we need to combine their individual Hilbert spaces to describe the composite system. This is done using the tensor product. If particle A lives in Hilbert space Hₐ and particle B lives in Hilbert space Hₑ, then the composite system lives in the Hilbert space Hₐ ⊗ Hₑ.
- If |a⟩ ∈ Hₐ and |b⟩ ∈ Hₑ are states of particles A and B, respectively, then the joint state of the system is represented as |a⟩ ⊗ |b⟩ (often written simply as |ab⟩).
- The dimension of the tensor product space is the product of the dimensions of the individual spaces. If Hₐ has dimension m and Hₑ has dimension n, then Hₐ ⊗ Hₑ has dimension m*n.

2. Entangled States: Beyond Simple Products

Entanglement arises when the state of the composite system cannot be written as a simple tensor product of individual particle states. Instead, it must be expressed as a superposition of tensor products.

Separable States: A state |ψ⟩ ∈ Hₐ ⊗ Hₑ is considered separable (or unentangled) if it can be written as:

|ψ⟩ = |a⟩ ⊗ |b⟩

where |a⟩ ∈ Hₐ and |b⟩ ∈ Hₑ. In this case, each particle has a well-defined, independent state.
Entangled States: A state |ψ⟩ ∈ Hₐ ⊗ Hₑ is considered entangled if it cannot be written in the separable form above. This is the key to entanglement. The particles are correlated in a way that goes beyond classical correlations.
Example: The Singlet State (Bell State): A classic example of an entangled state for two spin-1/2 particles (e.g., electrons or photons) is the singlet state:

|Ψ⟩ = (1/√2)(|↑⟩ₐ |↓⟩ₑ - |↓⟩ₐ |↑⟩ₑ)

Here, |↑⟩ represents spin-up and |↓⟩ represents spin-down along a given axis. Subscripts A and B denote the two particles. Notice that this state cannot be written as |a⟩ ⊗ |b⟩ for any individual states |a⟩ and |b⟩. This means that if you measure particle A to be spin-up, you instantly know that particle B must be spin-down, and vice versa, regardless of the distance separating them. This instantaneous correlation is what Einstein famously called "spooky action at a distance."

3. Observables and Measurements

Observables as Operators: In quantum mechanics, physical quantities (e.g., spin, momentum, energy) are represented by Hermitian operators acting on the Hilbert space. The possible values that can be obtained from a measurement are the eigenvalues of the operator.
Measurement Process: When we measure an observable O on a particle in state |ψ⟩, the state "collapses" into an eigenstate |φᵢ⟩ of O with probability |⟨φᵢ|ψ⟩|², where ⟨φᵢ| is the "bra" vector corresponding to |φᵢ⟩. The result of the measurement is the eigenvalue corresponding to that eigenstate.
Measurements on Entangled States: The crucial point is that measuring an observable on one entangled particle immediately affects the possible measurement outcomes on the other particle, even if they are spatially separated. This correlation is stronger than any classical correlation can achieve.

4. The CHSH Inequality and Bell's Theorem

Bell's theorem is a profound result that demonstrates the incompatibility of quantum mechanics with local realism. It relies on deriving an inequality (the CHSH inequality, for example) that must be satisfied by any theory that adheres to local realism. Quantum mechanics violates this inequality, experimentally confirming that at least one of the assumptions of local realism must be false.

Local Realism: Local realism is a combination of two assumptions:
- Realism: Physical properties of objects exist independently of observation. A particle has a definite value for its spin, even if we haven't measured it yet.
- Locality: An object is only directly influenced by its immediate surroundings. The outcome of a measurement on particle A cannot be influenced by what's happening to particle B if they are sufficiently far apart.
Hidden Variable Theories: Local realistic theories often involve "hidden variables" that are assumed to determine the outcome of a measurement. These variables are not directly observable but are assumed to be present and influence the behavior of the particles.
The CHSH Inequality: The Clauser-Horne-Shimony-Holt (CHSH) inequality is a specific inequality that can be derived under the assumptions of local realism. It involves performing measurements on two entangled particles using different measurement settings.

Let:
- A and B be the results of measurements on particles A and B, respectively. These results can take on values of +1 or -1 (e.g., spin up or spin down).
- a and a' be two possible measurement settings for particle A.
- b and b' be two possible measurement settings for particle B.
- E(a, b) be the expectation value of the product of the measurement outcomes when using settings a and b: E(a, b) = ⟨A(a)B(b)⟩
Under the assumptions of local realism, the CHSH inequality states:

|E(a, b) + E(a, b') + E(a', b) - E(a', b')| ≤ 2
Violation by Quantum Mechanics: Quantum mechanics predicts that for certain entangled states and measurement settings, the CHSH inequality is violated. For example, using the singlet state and appropriate measurement angles, the quantum mechanical prediction can give a value of 2√2, which is greater than 2.
Mathematical Derivation of the CHSH Inequality (Simplified): The core idea behind the derivation is to show that under local realism, the expression inside the absolute value can only be +2 or -2. This is often done by considering all possible values of the hidden variables and the corresponding measurement outcomes, and showing that in each case, the expression simplifies to either +2 or -2. This step requires some careful algebraic manipulation based on the assumption that the outcome of measuring particle A only depends on the measurement setting a and the hidden variables, and similarly for particle B.
Experimental Verification: Numerous experiments have been conducted to test Bell's theorem. These experiments have consistently demonstrated a violation of the CHSH inequality, providing strong evidence against local realism and supporting the predictions of quantum mechanics.

5. Implications and Significance

Bell's theorem and the experimental confirmations of quantum entanglement have profound implications for our understanding of the fundamental nature of reality:

Non-Locality: Quantum mechanics is inherently non-local. The act of measurement on one entangled particle can instantaneously affect the state of the other particle, regardless of the distance separating them. This does not violate special relativity because no information is being transmitted faster than light; the correlation exists, but it can't be used to send a signal.
Rejection of Realism: Bell's theorem challenges the notion that physical properties exist independently of measurement. The act of measurement seems to play a more fundamental role in defining the properties of quantum particles.
Quantum Computing and Information: Entanglement is a crucial resource for quantum computing and quantum information processing. Entangled qubits can be used to perform computations that are impossible for classical computers. Quantum cryptography protocols rely on entanglement to guarantee secure communication.

In Summary:

The mathematical foundations of quantum entanglement and Bell's theorem are built upon the framework of Hilbert spaces, tensor products, and linear operators. Entangled states are those that cannot be expressed as simple products of individual particle states. Bell's theorem provides a rigorous test of local realism, and experimental violations of Bell's inequalities demonstrate the non-local and potentially non-realistic nature of quantum mechanics. Entanglement is not just a philosophical curiosity; it's a fundamental aspect of quantum mechanics with potential applications in various technological fields.

Of course. This is a profound and fascinating topic that lies at the heart of what makes quantum mechanics so strange and powerful. Here is a detailed explanation of the mathematical foundations of quantum entanglement and Bell's theorem, broken down into a logical progression from the basics to the deep implications.

Part 1: The Mathematical Framework of Quantum Mechanics (The Prerequisites)

Before we can discuss entanglement, we need to understand how quantum mechanics describes single, isolated systems.

1.1 State Vectors and Hilbert Spaces

In classical physics, the state of a particle is described by its position and momentum. In quantum mechanics, the state of a system is described by a state vector, denoted by a "ket" $|\psi\rangle$. This vector lives in a complex vector space called a Hilbert space, $\mathcal{H}$.

Example: A Qubit: The simplest quantum system is a qubit, which can represent the spin of an electron (spin-up or spin-down). Its Hilbert space is two-dimensional, denoted as $\mathbb{C}^2$. A basis for this space is:
- $|0\rangle \equiv \begin{pmatrix} 1 \ 0 \end{pmatrix}$ (representing spin-up)
- $|1\rangle \equiv \begin{pmatrix} 0 \ 1 \end{pmatrix}$ (representing spin-down)
Superposition: A qubit can exist in a linear combination of these basis states. A general state $|\psi\rangle$ is: $|\psi\rangle = \alpha |0\rangle + \beta |1\rangle$, where $\alpha, \beta$ are complex numbers. The condition that probabilities sum to 1 imposes the normalization constraint: $|\alpha|^2 + |\beta|^2 = 1$.

1.2 Observables and Operators

Physical quantities that we can measure, like spin, position, or momentum, are called observables. In quantum mechanics, every observable is represented by a Hermitian operator (an operator that is equal to its own conjugate transpose, $A = A^\dagger$).

The possible outcomes of a measurement are the eigenvalues of the operator.
The state of the system after the measurement is the corresponding eigenvector.

For spin, the Pauli matrices are the operators. For spin measurement along the z-axis, the operator is $\sigmaz$: $\sigmaz = \begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix}$ * Eigenvalues: +1 (for spin-up) and -1 (for spin-down). * Eigenvectors: $|0\rangle$ (for eigenvalue +1) and $|1\rangle$ (for eigenvalue -1).

1.3 Measurement and Probability (The Born Rule)

If a system is in a state $|\psi\rangle = \alpha |0\rangle + \beta |1\rangle$, and we measure its spin along the z-axis, we don't get a value of "$\alpha$ up and $\beta$ down". The measurement forces the system to choose one of the eigenstates.

The probability of measuring a specific outcome is given by the square of the magnitude of the projection of the state vector onto the corresponding eigenvector.

Probability of measuring spin-up (+1): $P(+1) = |\langle 0 | \psi \rangle|^2 = |\alpha|^2$
Probability of measuring spin-down (-1): $P(-1) = |\langle 1 | \psi \rangle|^2 = |\beta|^2$

After the measurement, the state of the system "collapses" to the eigenvector corresponding to the outcome.

Part 2: The Mathematics of Quantum Entanglement

Entanglement arises when we consider systems of two or more particles.

2.1 Composite Systems and the Tensor Product

To describe a system of two particles (say, Alice's qubit A and Bob's qubit B), we need to combine their individual Hilbert spaces. The mathematical tool for this is the tensor product, denoted by $\otimes$.

If Alice's qubit lives in $\mathcal{H}A$ and Bob's in $\mathcal{H}B$, the combined system lives in $\mathcal{H}{AB} = \mathcal{H}A \otimes \mathcal{H}_B$.

If $\mathcal{H}A$ has dimension 2 (basis $|0\rangleA, |1\rangleA$) and $\mathcal{H}B$ has dimension 2 (basis $|0\rangleB, |1\rangleB$), the composite space $\mathcal{H}_{AB}$ has dimension $2 \times 2 = 4$.
The basis vectors of the composite space are:
- $|00\rangle \equiv |0\rangleA \otimes |0\rangleB$
- $|01\rangle \equiv |0\rangleA \otimes |1\rangleB$
- $|10\rangle \equiv |1\rangleA \otimes |0\rangleB$
- $|11\rangle \equiv |1\rangleA \otimes |1\rangleB$

2.2 Separable vs. Entangled States

Separable (or Product) State: A state is separable if it can be written as a tensor product of the individual states of its subsystems.
- Example: If Alice's qubit is in state $|\psi\rangleA = \alpha|0\rangleA + \beta|1\rangleA$ and Bob's is in state $|\phi\rangleB = \gamma|0\rangleB + \delta|1\rangleB$, the total state is: $|\Psi{sep}\rangle = |\psi\rangleA \otimes |\phi\rangleB = (\alpha|0\rangleA + \beta|1\rangleA) \otimes (\gamma|0\rangleB + \delta|1\rangle_B)$
- In a separable state, the particles have their own independent, well-defined states. Measuring one has no effect on the other.
Entangled State: An entangled state is any state of a composite system that cannot be written as a product of individual states.
- The Canonical Example: The Bell States. The most famous entangled states are the four Bell states. Let's look at the singlet state: $|\Psi^-\rangle = \frac{1}{\sqrt{2}} (|01\rangle - |10\rangle) = \frac{1}{\sqrt{2}} (|0\rangleA \otimes |1\rangleB - |1\rangleA \otimes |0\rangleB)$
There is no way to factor this expression into the form $(...A) \otimes (...B)$. This mathematical inseparability is the definition of entanglement. It means neither particle has a definite state on its own; the state is defined only for the system as a whole.

2.3 The "Spooky" Correlations

Let's see what happens when we measure an entangled pair in the singlet state $|\Psi^-\rangle$.

The State: $|\Psi^-\rangle = \frac{1}{\sqrt{2}} (|01\rangle - |10\rangle)$. This means the system is in a superposition of "Alice has spin-up, Bob has spin-down" and "Alice has spin-down, Bob has spin-up".
Alice's Measurement: Alice measures the spin of her particle along the z-axis. According to the Born rule, she has a 50% chance of getting spin-up ($|0\rangle$) and a 50% chance of getting spin-down ($|1\rangle$).
- Case 1: Alice measures spin-up (+1). The state of the system collapses to the part of the superposition consistent with her result. The $|10\rangle$ term vanishes. The system instantly becomes: $|\Psi'\rangle = |01\rangle = |0\rangleA \otimes |1\rangleB$ Now, if Bob measures his particle, he is guaranteed to find spin-down ($|1\rangle$).
- Case 2: Alice measures spin-down (-1). The state collapses to the other part: $|\Psi''\rangle = |10\rangle = |1\rangleA \otimes |0\rangleB$ Now, if Bob measures his particle, he is guaranteed to find spin-up ($|0\rangle$).

The outcomes are perfectly anti-correlated. This correlation is instantaneous, regardless of the distance between Alice and Bob. This is what Einstein famously called **"spooky action at a distance."**

Part 3: The EPR Paradox and Bell's Theorem

This "spooky" correlation deeply troubled Einstein, Podolsky, and Rosen (EPR). They argued that quantum mechanics must be incomplete. Their reasoning was based on two classical assumptions:

Locality: No influence can travel faster than the speed of light. Alice's measurement here cannot instantaneously affect Bob's particle over there.
Realism: Physical properties of objects exist independent of measurement. The particles must have had definite spin properties all along, we just didn't know them until we measured.

This led to the idea of **Local Hidden Variables (LHV)**. The LHV hypothesis suggests the correlations are not spooky. They are like having a pair of gloves. If you put one in each of two boxes and send them far apart, opening your box and seeing a left-handed glove *instantly* tells you the other box contains a right-handed glove. There's no spooky action; the information (the "handedness") was there all along. For decades, this was a philosophical debate. Then, in 1964, John Bell devised a mathematical way to test it.

3.1 The Goal of Bell's Theorem

Bell's theorem is not a theorem *of* quantum mechanics. It's a theorem that shows that the predictions of quantum mechanics are fundamentally incompatible with the predictions of *any* theory based on local hidden variables. It does this by deriving an inequality—a mathematical constraint—that any local realist theory must obey. He then showed that quantum mechanics predicts a violation of this inequality under certain experimental conditions.

3.2 The CHSH Inequality (A more practical version of Bell's inequality)

Let's set up a testable experiment, as formulated by Clauser, Horne, Shimony, and Holt (CHSH).

Setup: Alice and Bob each receive one particle from an entangled pair. They can each measure the spin along different axes. Alice can choose between two measurement settings (axes) a and a'. Bob can choose between his two settings b and b'. The outcomes are recorded as +1 or -1.
The Logic of Local Realism:
- Assume a hidden variable, $\lambda$, pre-determines the outcome of any measurement. This $\lambda$ contains all the "glove-in-the-box" information.
- The result Alice gets for setting a is a function $A(\mathbf{a}, \lambda) = \pm 1$.
- The result Bob gets for setting b is a function $B(\mathbf{b}, \lambda) = \pm 1$.
- Crucially, $A$ does not depend on b (locality), and $B$ does not depend on a (locality).
Deriving the Inequality: Consider the quantity $S$ defined by a combination of correlations: $S = E(\mathbf{a}, \mathbf{b}) - E(\mathbf{a}, \mathbf{b'}) + E(\mathbf{a'}, \mathbf{b}) + E(\mathbf{a'}, \mathbf{b'})$ where $E(\mathbf{a}, \mathbf{b})$ is the average value of the product of the outcomes $A(\mathbf{a})B(\mathbf{b})$ over many runs.

In a local hidden variable theory, this average is: $E(\mathbf{a}, \mathbf{b}) = \int \rho(\lambda) A(\mathbf{a}, \lambda) B(\mathbf{b}, \lambda) d\lambda$, where $\rho(\lambda)$ is the probability distribution of the hidden variables.

Let's look at the expression for a single run (a single $\lambda$): $s(\lambda) = A(\mathbf{a}, \lambda)B(\mathbf{b}, \lambda) - A(\mathbf{a}, \lambda)B(\mathbf{b'}, \lambda) + A(\mathbf{a'}, \lambda)B(\mathbf{b}, \lambda) + A(\mathbf{a'}, \lambda)B(\mathbf{b'}, \lambda)$ $s(\lambda) = A(\mathbf{a}, \lambda)[B(\mathbf{b}, \lambda) - B(\mathbf{b'}, \lambda)] + A(\mathbf{a'}, \lambda)[B(\mathbf{b}, \lambda) + B(\mathbf{b'}, \lambda)]$

Since $B$ can only be +1 or -1, one of the two terms in brackets must be 0, and the other must be $\pm 2$.
- If $B(\mathbf{b}, \lambda) = B(\mathbf{b'}, \lambda)$, the first term is 0 and the second is $\pm 2 A(\mathbf{a'}, \lambda) = \pm 2$.
- If $B(\mathbf{b}, \lambda) = -B(\mathbf{b'}, \lambda)$, the second term is 0 and the first is $\pm 2 A(\mathbf{a}, \lambda) = \pm 2$.
In all cases, $|s(\lambda)| \le 2$. Since this is true for every single run, the average value $S$ must also be bounded by 2. This gives the CHSH inequality: $|S| = |E(\mathbf{a}, \mathbf{b}) - E(\mathbf{a}, \mathbf{b'}) + E(\mathbf{a'}, \mathbf{b}) + E(\mathbf{a'}, \mathbf{b'})| \le 2$

This is the crucial result: Any theory based on local realism must obey this constraint.

3.3 The Quantum Mechanical Prediction

Now, let's calculate the value of $S$ using the mathematics of quantum mechanics for the singlet state $|\Psi^-\rangle$. The quantum mechanical prediction for the correlation is: $E(\mathbf{a}, \mathbf{b}) = \langle \Psi^- | (\vec{\sigma}A \cdot \mathbf{a}) \otimes (\vec{\sigma}B \cdot \mathbf{b}) | \Psi^- \rangle = -\mathbf{a} \cdot \mathbf{b} = -\cos(\theta{ab})$ where $\theta{ab}$ is the angle between Alice's and Bob's measurement axes.

Let's pick clever angles to maximize $|S|$: * Alice's axis a is at 0°. * Alice's axis a' is at 90°. * Bob's axis b is at 45°. * Bob's axis b' is at 135°.

Now calculate the correlations: * $E(\mathbf{a}, \mathbf{b}) = -\cos(45^\circ) = -1/\sqrt{2}$ * $E(\mathbf{a}, \mathbf{b'}) = -\cos(135^\circ) = -(-\cos(45^\circ)) = +1/\sqrt{2}$ * $E(\mathbf{a'}, \mathbf{b}) = -\cos(45^\circ) = -1/\sqrt{2}$ (angle between 90° and 45° is 45°) * $E(\mathbf{a'}, \mathbf{b'}) = -\cos(-45^\circ) = -1/\sqrt{2}$ (angle between 90° and 135° is -45°)

Plugging these into the expression for $S$: $S = (-1/\sqrt{2}) - (1/\sqrt{2}) + (-1/\sqrt{2}) + (-1/\sqrt{2}) = -4/\sqrt{2} = -2\sqrt{2}$

So, the quantum mechanical prediction is $|S| = 2\sqrt{2} \approx 2.828$.

Part 4: The Conclusion

Local Realism predicts: $|S| \le 2$ Quantum Mechanics predicts: $|S| = 2\sqrt{2} \approx 2.828$

This is a direct, mathematically testable contradiction. Both cannot be right.

Experiments, starting with Alain Aspect in the 1980s and continuing with increasing precision ever since (closing loopholes), have overwhelmingly confirmed the quantum mechanical prediction. The Bell inequality is violated, just as the math of entanglement predicts.

Implications: The experimental violation of Bell's inequality forces us to abandon at least one of the foundational assumptions of local realism. * We must abandon locality (and accept that there is some form of non-local influence, though it cannot be used to send information faster than light). * Or we must abandon realism (and accept that properties like spin are not defined until measured, and the universe is fundamentally probabilistic).

The mathematics of entanglement (tensor products creating inseparable states) and Bell's Theorem (deriving a testable inequality) provide the irrefutable, quantitative foundation for one of the most profound discoveries in the history of science: our classical intuition about how the universe works is wrong. The world is built on a deeper, interconnected, and "spookier" reality.

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The Mathematical Foundations of Quantum Entanglement and Bell's Theorem

Introduction

Quantum entanglement represents one of the most profound departures from classical physics, and Bell's theorem provides the mathematical framework proving that quantum mechanics cannot be explained by local hidden variable theories. Let me walk you through the mathematical foundations systematically.

1. Quantum States and the Hilbert Space Formalism

Single Particle States

In quantum mechanics, the state of a system is represented by a vector in a complex Hilbert space. For a single qubit (two-level system), we can write:

|ψ⟩ = α|0⟩ + β|1⟩

where α, β ∈ ℂ and |α|² + |β|² = 1 (normalization condition).

Composite Systems and Tensor Products

For two particles A and B, the combined system exists in the tensor product space:

ℋtotal = ℋA ⊗ ℋ_B

A general state of this two-particle system can be written as:

|Ψ⟩ = Σᵢⱼ cᵢⱼ |i⟩A ⊗ |j⟩B

2. Entangled vs. Separable States

Separable States

A state is separable if it can be written as:

|Ψ⟩ = |ψ⟩A ⊗ |φ⟩B

This means the particles can be described independently.

Entangled States

A state is entangled if it cannot be written in separable form. Classic examples include the Bell states:

|Φ⁺⟩ = (|00⟩ + |11⟩)/√2

|Φ⁻⟩ = (|00⟩ - |11⟩)/√2

|Ψ⁺⟩ = (|01⟩ + |10⟩)/√2

|Ψ⁻⟩ = (|01⟩ - |10⟩)/√2

These states are maximally entangled and form an orthonormal basis for the two-qubit system.

3. Measurement and Correlations

Born Rule

The probability of obtaining outcome m when measuring observable M is:

P(m) = |⟨m|Ψ⟩|²

Correlation Functions

For two spatially separated measurements on an entangled pair, we define the correlation function:

E(a, b) = ⟨Ψ|A(a) ⊗ B(b)|Ψ⟩

where A(a) and B(b) are measurement operators with settings a and b respectively.

For spin-1/2 particles measured along directions a and b:

E(a, b) = -a* · b = -cos(θ)*

where θ is the angle between measurement directions.

4. Local Hidden Variable Theories

The EPR Argument

Einstein, Podolsky, and Rosen (1935) argued that quantum mechanics must be incomplete, proposing that "hidden variables" λ determine measurement outcomes.

Mathematical Framework of LHV Theories

In a local hidden variable theory:

There exists a hidden variable λ with probability distribution ρ(λ)
Measurement outcomes are predetermined: A(a, λ) = ±1, B(b, λ) = ±1
Locality: A depends only on a and λ; B depends only on b and λ

The correlation function in LHV theories must be:

E_LHV(a, b) = ∫ ρ(λ) A(a, λ) B(b, λ) dλ

5. Bell's Theorem

Bell's Inequality (CHSH Form)

Bell proved that any local hidden variable theory must satisfy:

|S| ≤ 2

where the CHSH parameter is:

S = E(a, b) - E(a, b') + E(a', b) + E(a', b')

Mathematical Proof Sketch

Given locality and realism: - A(a,λ), A(a',λ), B(b,λ), B(b',λ) ∈ {-1, +1}

Then:

A(a,λ)[B(b,λ) - B(b',λ)] + A(a',λ)[B(b,λ) + B(b',λ)]

Since B(b,λ) ± B(b',λ) equals either ±2 or 0:

|A(a,λ)[B(b,λ) - B(b',λ)] + A(a',λ)[B(b,λ) + B(b',λ)]| ≤ 2

Integrating over λ:

|S| = |∫ ρ(λ)[...] dλ| ≤ 2

Quantum Mechanical Violation

Quantum mechanics predicts for optimal angles (22.5° separations):

S_QM = 2√2 ≈ 2.828

This violates Bell's inequality, proving quantum mechanics cannot be explained by local hidden variables.

6. Mathematical Details: Specific Example

The Singlet State

Consider the spin singlet state:

|Ψ⁻⟩ = (|↑↓⟩ - |↓↑⟩)/√2

For measurements along a and b:

EQM(a, b) = ⟨Ψ⁻|(σa ⊗ σ_b)|Ψ⁻⟩ = -a* · b*

Optimal CHSH Configuration

Choose angles: - a = 0°, a' = 45° - b = 22.5°, b' = -22.5°

Then: - E(a,b) = -cos(22.5°) = -√(2+√2)/2 - E(a,b') = -cos(-22.5°) = -√(2+√2)/2 - E(a',b) = -cos(22.5°) = -√(2+√2)/2 - E(a',b') = -cos(67.5°) = +√(2-√2)/2

S = 2√2, violating the classical bound.

7. Mathematical Significance

No-Go Theorem

Bell's theorem is a no-go theorem: it proves impossible to reproduce quantum predictions with: 1. Locality (no faster-than-light influences) 2. Realism (predetermined measurement outcomes) 3. Freedom of choice (independent measurement settings)

Tsirelson's Bound

Quantum mechanics doesn't violate causality arbitrarily. The maximum quantum violation is bounded:

|S_QM| ≤ 2√2

This is Tsirelson's bound, derived from the algebraic structure of quantum operators.

8. Density Matrix Formalism

Mixed States

For mixed states (statistical ensembles), we use density matrices:

ρ = Σᵢ pᵢ |ψᵢ⟩⟨ψᵢ|

Entanglement Measures

Von Neumann Entropy of the reduced density matrix:

E(ρ) = -Tr(ρA log ρA)

where ρA = TrB(ρ) is the partial trace.

For pure states, this quantifies entanglement. For the Bell states, E = 1 (maximum for qubits).

Conclusion

The mathematical foundations of quantum entanglement and Bell's theorem reveal a profound truth: nature operates according to principles fundamentally different from classical intuition. The tensor product structure of quantum Hilbert spaces allows for correlations that cannot be explained by any local classical theory, as rigorously proven by Bell's inequalities. This mathematics has been confirmed by countless experiments and forms the basis for quantum information technologies like quantum cryptography and quantum computing.