Entanglement, Bell Inequalities, and the Limits of Local Realism

1. From Product States to Entanglement

2. Practice: Is This State Entangled?

Use this step to practice distinguishing separable and entangled pure states of two qubits.

Task 1: Visual inspection

Consider these states. For each, decide whether it is separable or entangled. Try to reason before using the algebra test.

1. `|ψ_1⟩ = (|00⟩ + |01⟩) / √2`
2. `|ψ_2⟩ = (|00⟩ + |11⟩) / √2`
3. `|ψ_3⟩ = (|01⟩ + |10⟩) / √2`

Hints:

• For 1, can you factor out a state of A or B?
• For 2 and 3, note that the terms involve correlated bits.

Task 2: Use the algebra test

Write each state in the form

`|ψ⟩ = a|00⟩ + b|01⟩ + c|10⟩ + d|11⟩`

and compute `ad - bc`.

• If `ad - bc = 0`, the state is separable.
• If `ad - bc ≠ 0`, the state is entangled.

Work it out explicitly on paper.

Self-check (do after you try)

1. `|ψ1⟩` is separable. You can write it as `|0⟩A ⊗ (|0⟩ + |1⟩)/√2`.
2. `|ψ_2⟩` is entangled. The correlations between A and B cannot be factored.
3. `|ψ_3⟩` is also entangled (it is one of the Bell states you will see next).

3. Bell States: Canonical Two-Qubit Entanglement

4. Local Realism vs Quantum Nonlocality

5. The CHSH Bell Inequality: Structure and Meaning

6. Quick Check: Understanding CHSH

Test your understanding of the CHSH Bell inequality and what it means.

7. Simulating a Simple CHSH Experiment (Python)

This step walks you through a minimal Python simulation of a CHSH Bell test using a local hidden-variable model and then comparing to the quantum prediction.

You can run this in a Jupyter notebook or any Python 3 environment with NumPy installed.

Part A: Local hidden-variable model

We simulate a simple local model where hidden variables `λ` predetermine outcomes for Alice and Bob.

```python

import numpy as np

Number of trials

N = 200000

Hidden variable lambda: random angle between 0 and 2π

lmbda = 2 np.pi np.random.rand(N)

Define deterministic local response functions

Outcomes are ±1 depending on sign of cos(angle difference)

def outcome(angle, lmbda):

return np.sign(np.cos(angle - lmbda))

Measurement settings (angles) for Alice and Bob

A0, A1 = 0.0, np.pi / 2 # 0 and 90 degrees

B0, B1 = np.pi / 4, 3 * np.pi / 4 # 45 and 135 degrees

Precompute outcomes for all settings

A0_out = outcome(A0, lmbda)

A1_out = outcome(A1, lmbda)

B0_out = outcome(B0, lmbda)

B1_out = outcome(B1, lmbda)

Function to estimate correlation E(Ax, By)

def corr(A, B):

return np.mean(A * B)

EA0B0 = corr(A0out, B0_out)

EA0B1 = corr(A0out, B1_out)

EA1B0 = corr(A1out, B0_out)

EA1B1 = corr(A1out, B1_out)

S = EA0B0 + EA0B1 + EA1B0 - EA1B1

print("Local model S ≈", S)

```

You should find `|S|` not exceeding 2 (often noticeably below, depending on the choice of angles).

Part B: Quantum prediction (analytical)

For the singlet state `|Ψ-⟩` and spin measurements at angles `a` and `b` in a plane, quantum mechanics predicts

`E_Q(a, b) = -cos(a - b)`.

We can compute the CHSH `S` directly from this formula:

```python

def E_Q(a, b):

return -np.cos(a - b)

Use the standard CHSH angles that maximize violation

A0, A1 = 0.0, np.pi / 2

B0, B1 = np.pi / 4, -np.pi / 4

S_Q = (

EQ(A0, B0) + EQ(A0, B1) +

EQ(A1, B0) - EQ(A1, B1)

)

print("Quantum prediction S =", S_Q)

print("2sqrt(2) =", 2 np.sqrt(2))

```

You should see `S_Q = 2√2`, demonstrating the maximal quantum violation of the CHSH inequality.

8. Modern Loophole-Free Bell Tests (Post-2015)

9. Thought Exercise: What Exactly Is Ruled Out?

Use this step to clarify your own interpretation of Bell test results.

Part 1: Reflect

Consider the statement:

"Loophole-free Bell tests show that the world is nonlocal."

Ask yourself:

1. What assumptions go into deriving a Bell inequality?

• Locality
• Realism (hidden variables)
• Freedom of choice (measurement settings are not influenced by hidden variables)

1. Which of these assumptions could, in principle, be relaxed?

Write down your own answers before reading on.

Part 2: Compare

Most physicists agree that local realism, as a combined assumption, is incompatible with experiment. However, there is some interpretational freedom:

• Some interpretations give up realism (e.g. orthodox Copenhagen, QBism).
• Some give up strict locality at the level of underlying variables (e.g. de Broglie–Bohm theory is explicitly nonlocal).
• Some question freedom of choice (superdeterminism), but this is controversial and requires highly constrained, non-testable correlations between measurement choices and hidden variables.

Part 3: Your stance

Write a short paragraph (3–5 sentences) answering:

1. Which assumption do you personally find least plausible, and why?
2. How does this influence your preferred interpretation of quantum mechanics (if any)?

This reflection will help you connect the formalism to conceptual questions.

10. Key Term Review

Flip through these flashcards to reinforce the main concepts from this module.