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Topic 5.12

Quantum Entanglement: Non-Local Correlation & Bell States

Explore Einstein’s "Spooky Action at a Distance": Bell states, non-local quantum correlations, and the 2022 Nobel Prize.

Learning Objectives

✓Define quantum entanglement: non-separable multi-qubit states
✓Construct the 4 maximally entangled Bell States (|Phi+>, |Phi->, |Psi+>, |Psi->)
✓Understand the EPR Paradox, Bell’s Inequality, and Aspect’s 2022 Nobel experiment

📖Let's Begin with a Story

In 1935, Albert Einstein, Boris Podolsky, and Nathan Rosen published a paper arguing that quantum mechanics was incomplete. They pointed out a mind-bending consequence: you could create two particles that are so intimately linked that measuring particle A on Earth would instantly dictate the state of particle B on the other side of the galaxy! Einstein scoffed at this idea, famously deriding it as "spooky action at a distance" (spukhafte Fernwirkung). He believed local hidden variables must secretly exist. In 1964, Northern Irish physicist John Stewart Bell devised a mathematical test (Bell’s Inequality) that could experimentally prove whether Einstein or quantum mechanics was right. In the 1980s and 2015, Alain Aspect, John Clauser, and Anton Zeilinger performed the definitive experiments. Einstein was wrong; quantum mechanics was right! The universe is fundamentally non-local. For this proof, Aspect, Clauser, and Zeilinger were awarded the 2022 Nobel Prize in Physics.

Why Should You Learn Quantum Entanglement: Non-Local Correlation & Bell States?

🎯 Practical & Engineering Stakes

The superpower of quantum speed

Without entanglement, an

n

-qubit computer is merely

n

separate, independent classical bits. Entanglement binds the qubits together into a unified computational fabric that classical computers cannot simulate.

🎮 Fun Learning Zone

Grasp complex quantum concepts visually with zero prior physics or math knowledge!

Zero Prerequisites

Visual Concept Storyboard

The Cosmic Magic Dice & The Pair of Dragon Gloves

The Cosmic Dragon Gloves & Interplanetary Dice|Φ+⟩ = (|00⟩ + |11⟩)/√2

📦 Box A on Mars

🧤 Left Glove (50%)

Opens to reveal Left Glove

📦 Box B on Earth

🧤 Right Glove (100%!)

Instantly snaps with zero time delay

Classical: Serial / ConstrainedQuantum: Coherent Wave Parallelism

Visual takeaway: Understand in 5 seconds

Everyday Analogy: The Cosmic Magic Dice & The Pair of Dragon Gloves

Imagine a master craftsperson sews a pair of dragon-scale gloves: one Left glove and one Right glove. Without looking, they place each glove into an identical sealed titanium box. You take Box A onto a rocket ship to Mars, while Box B stays on Earth. When you land on Mars and open Box A, you find the Left glove. Instantly, with zero seconds of delay, you know Box B on Earth contains the Right glove! But in quantum mechanics, it is infinitely weirder: neither glove was Left or Right while traveling through space; both boxes were in a hazy blur of (Left + Right), and the very act of opening Box A forced Box B across the cosmos to instantly snap into the opposite state!

🧪

Interactive Mental Sandbox

Try It Yourself

Interplanetary Entangled Dice

Alice on Mars rolls her die; Bob on Earth reads his instantaneously.

Trials: 0

🚀 Alice on Mars

❓

100% random outcome (1 to 6)

🌍 Bob on Earth

❓

Instant matching correlation (0 ms delay!)

📖 The Thought Experiment Prompt:“Suppose you and your friend each hold a quantum die. Independently, whenever you roll your die, you get a completely random number from 1 to 6 (100% unpredictable). But whenever you both roll them at the exact same instant, even if you are on opposite sides of the Andromeda Galaxy, your dice always land on the exact same matching number! No radio signal could travel between you that fast. That is Einstein’s "spooky action at a distance"—entanglement connects the particles into a single shared reality.”

✨ The "Aha!" Moment

Entangled qubits cannot be described by individual state vectors; they share a single inseparable joint wavefunction

|\Phi^+\rangle = (|00\rangle + |11\rangle)/\sqrt{2}

. Measuring one instantaneously dictates the outcome of the other!

What Is It?

Simple Explanation

Imagine you have a pair of magical magic shoes. You put one shoe in a box and leave it in New York. You put the other shoe in a box and fly to Tokyo. Neither shoe is left or right yet—they are in a quantum blur! The instant you open the box in New York and see a Left Shoe, you instantly know with 100% certainty that the shoe in Tokyo has become a Right Shoe—faster than light! They are not communicating; they are two parts of one single entangled reality.

Technical Explanation

A multi-qubit state

|\psi\rangle \in \mathcal{H}_A \otimes \mathcal{H}_B

is entangled if and only if it cannot be factored as a product state

|\psi\rangle \ne |\phi_A\rangle \otimes |\phi_B\rangle

. The canonical maximally entangled Bell state is:

|\Phi^+\rangle = \frac{|00\rangle + |11\rangle}{\sqrt{2}}

Measuring qubit A yields 0 or 1 with 50% probability, but instantly projects qubit B into the identical state with correlation coefficient 1.0, violating the Bell-CHSH inequality (

S \le 2

classical vs

S = 2\sqrt{2} \approx 2.828

quantum).

📚 Key Terms

Quantum Entanglement

Simple: When two or more qubits become so connected that you cannot describe one without the other.

Technical: Non-separable pure state in composite Hilbert space possessing non-zero entanglement entropy.

Bell State

Simple: The 4 fundamental pairs of perfectly entangled two-qubit states.

Technical: An orthonormal basis of 4 maximally entangled 2-qubit states generated by Hadamard and CNOT gates.

Bell’s Inequality

Simple: A mathematical formula that proved the universe really is quantum and Einstein’s classical view was wrong.

Technical: Mathematical bounds on classical local hidden variable theories violated by quantum mechanical correlation.

Q-Sphere Lab: Quantum Entanglement: Non-Local Correlation & Bell States

IBM Q-Sphere Model

Interactive 3D state vector exploration, unitary transformation gates, and Born rule collapse for Topic 5.12

Azimuth: 45°Elevation: 25°(Click & drag sphere to orbit)

Q-Sphere Phase Color:

0 π/2 π 3π/2

Quantum Statevector

dim(H) = 2

|ψ⟩ = (1.000 + 0.000i)|0⟩
+ (0.000 + 0.000i)|1⟩

Polar Angle (θ):0.0° (0.000π rad)

Phase Angle (ϕ):0.0° (0.000π rad)

Born Rule Probabilities: P(x) = |⟨x|ψ⟩|²Σ P = 1.00

P(|0⟩) = |α|²100.0%

P(|1⟩) = |β|²0.0%

Quantum Measurement

Apply Single-Qubit Unitary Gates

Presets:

Continuous Polar Angle θ (0° = |0⟩, 180° = |1⟩):0.0°

Continuous Phase Angle ϕ (0° to 360°):0.0°

Recent Operations Log:

#1Initial state: |0⟩

📐 Mathematical Foundation

LaTeX KaTeX

|\Phi^+\rangle = \frac{|00\rangle + |11\rangle}{\sqrt{2}} \ne |\psi_A\rangle \otimes |\psi_B\rangle

What does it mean?

The Bell State Φ+ & Non-Separability: Represents two maximally entangled qubits that cannot be factored into two separate individual particles. Neither qubit has a definite state on its own, but measuring one instantly determines the other with 100% correlation regardless of physical distance.

Understanding the Symbols

Symbol / Term	Meaning	Beginner Interpretation
$\|\Phi^+\rangle$	Maximally entangled Bell state	Equal superposition of both qubits being 0 and both qubits being 1
$\|00\rangle$	Both qubits in state 0	Alice measures 0 and Bob measures 0 (50% probability)
$\|11\rangle$	Both qubits in state 1	Alice measures 1 and Bob measures 1 (50% probability)
$\ne \|\psi_A\rangle \otimes \|\psi_B\rangle$	Non-separability (Entanglement)	Mathematically impossible to describe particle A without mentioning particle B

📊 Numerical Example

Mathematical proof of non-separability & instantaneous correlation:

1. Factoring attempt: (a0

|0\rangle

+ a1

|1\rangle

) ⊗ (b0

|0\rangle

+ b1

|1\rangle

) = a0b0

|00\rangle

+ a0b1

|01\rangle

+ a1b0

|10\rangle

+ a1b1

|11\rangle

Matching to

|\Phi+\rangle

requires: a0b1 = 0 AND a1b0 = 0, but a0b0 = 1/√2 AND a1b1 = 1/√2. If a0b1 = 0, either a0 = 0 or b1 = 0. If a0 = 0, then a0b0 = 0 ≠ 1/√2! Contradiction! If b1 = 0, then a1b1 = 0 ≠ 1/√2! Contradiction! Conclusion: No individual states exist! 2. Instantaneous Measurement Correlation: P(Alice measures 0) = |1/√2|^2 = 50% Upon Alice getting 0, the state collapses instantly to

|00\rangle

. P(Bob measures 0 | Alice measured 0) = 1.00 (100% guaranteed!)

Result: Zero individual states exist; their joint reality creates perfect non-local correlation.

⚙️ How Does It Work?

State Prep

Start with 2 independent qubits in state |00>.

Superposition on Q1

Apply Hadamard gate H to qubit 1: (|0> + |1>)/sqrt(2) (x) |0> = (|00> + |10>)/sqrt(2).

Entangle via CNOT

Apply CNOT with Q1 as control and Q2 as target: flips Q2 only when Q1 is 1, producing (|00> + |11>)/sqrt(2)!

🌍 Where Is It Used?

Quantum Teleportation

Problem: Transmitting unknown quantum states between quantum processors across distances.

Solution: Using a shared Bell pair and 2 classical bits, teleporting exact quantum wavefunctions without physical particle transport.

🎯 Try It Yourself

Is the state |psi> = (1/2)|00> + (1/2)|01> + (1/2)|10> + (1/2)|11> entangled?

Interactive activity component

🐍 Python Hands-On: Lesson 8: Two Qubits and Entanglement (Bell State |Phi+>)

Construct a two-qubit Bell state (|Phi+>) using Hadamard and CNOT gates, simulate on Aer simulator, and verify correlated measurements.

Open Colab

1Install Required Packages

For Google Colab or Jupyter notebooks, run this shell command to install the required dependencies:

Terminal / Notebook Install Commandbash

!pip install qiskit qiskit-aer pylatexenc

Packages:

qiskitqiskit-aerpylatexenc

2Import Libraries

Import Statementspython

1 from qiskit import QuantumCircuit
2 from qiskit_aer import Aer

from qiskit import QuantumCircuit → Construct two-qubit quantum circuits

from qiskit_aer import Aer → Aer simulator backend

3Demonstration Code & Logic

Python Execution Codepython

1 # Create a two-qubit circuit
2 qc = QuantumCircuit(2)
3  
4 # Apply Hadamard gate to create superposition on qubit 0
5 qc.h(0)
6  
7 # Applying a CNOT (Controlled-NOT) Gate
8 qc.cx(0, 1)
9  
10 # Measure all qubits
11 qc.measure_all()
12  
13 # Use Qiskit's Aer simulator
14 simulator = Aer.get_backend('aer_simulator')
15  
16 # Run the circuit on the simulator
17 job = simulator.run(qc, shots=1000)
18 result = job.result()
19  
20 # Get the measurement results
21 counts = result.get_counts(qc)
22 print("Bell State Measurement Counts (Lesson 8):", counts)

Step-by-Step Code Breakdown:

# Create a two-qubit circuit

Allocate 2 quantum wires

qc = QuantumCircuit(2)

Initialize q0 and q1 in ground state |00>

# Apply Hadamard gate to create superposition on qubit 0

Creates (|0> + |1>)|0> / sqrt(2)

qc.h(0)

Put control qubit into equal superposition

# Applying a CNOT (Controlled-NOT) Gate

Entangles target qubit 1 with control qubit 0

qc.cx(0, 1)

Flip q1 conditionally when q0 is 1

# Measure all qubits

Add measurement gates to all wires

qc.measure_all()

Measure [q0, q1] into classical register

# Use Qiskit's Aer simulator

Configure aer_simulator backend

simulator = Aer.get_backend('aer_simulator')

Load simulator instance

# Run the circuit on the simulator

Execute 1,000 shots

job = simulator.run(qc, shots=1000)

Run simulation job

result = job.result()

Retrieve results

# Get the measurement results

Extract output counts dictionary

counts = result.get_counts(qc)

Tally results

print("Bell State Measurement Counts (Lesson 8):", counts)

Display correlated counts: {'00': ~500, '11': ~500}

📋 Complete Script (All-in-One)

Click "Copy for Colab", open Google Colab, paste into a single cell, and run.

Google Colab Notebook Scriptpython

1 # =========================================================
2 # Lesson 8: Two Qubits and Entanglement (Bell State |Phi+>)
3 # Prepared for Google Colab (Run with Shift + Enter)
4 # =========================================================
5  
6 # Step 1: Install required packages
7 !pip install qiskit qiskit-aer pylatexenc
8  
9 # Step 2: Import libraries
10 from qiskit import QuantumCircuit
11 from qiskit_aer import Aer
12  
13 # Step 3: Run Demonstration
14 # Create a two-qubit circuit
15 qc = QuantumCircuit(2)
16  
17 # Apply Hadamard gate to create superposition on qubit 0
18 qc.h(0)
19  
20 # Applying a CNOT (Controlled-NOT) Gate
21 qc.cx(0, 1)
22  
23 # Measure all qubits
24 qc.measure_all()
25  
26 # Use Qiskit's Aer simulator
27 simulator = Aer.get_backend('aer_simulator')
28  
29 # Run the circuit on the simulator
30 job = simulator.run(qc, shots=1000)
31 result = job.result()
32  
33 # Get the measurement results
34 counts = result.get_counts(qc)
35 print("Bell State Measurement Counts (Lesson 8):", counts)

⚡ How to Run in Google Colab (3 Quick Steps)

Click "Copy for Google Colab" above.
Click Open New Google Colab Notebook (opens in a new tab).
Click on the first empty code cell, press Ctrl + V (or Cmd + V) to paste, then press Shift + Enter to execute!

Launch Blank Colab Notebook

🔄 Algorithmic Execution Flow

Two-Qubit Allocation

Allocate 2 qubits in the separable ground state |00>.

→

Superposition Creation

Apply Hadamard on qubit 0 to produce (|00> + |10>)/sqrt(2).

→

Entangling CNOT

Apply CNOT(0, 1) mapping |10> to |11>, synthesizing the maximally entangled Bell state (|00> + |11>)/sqrt(2).

→

Correlated Readout

Simulate 1,000 shots on AerSimulator: only outcomes "00" and "11" appear with 50/50 probability, while "01" and "10" never occur.

📥 Input / Initial State

QuantumCircuit(2) with H(0) and CX(0, 1) and measure_all().

📤 Expected Output

Bell State Measurement Counts (Lesson 8): {'00': 496, '11': 504}

💡 What Does This Demonstration Show?

Demonstrates maximal quantum entanglement: measuring qubit 0 immediately determines the state of qubit 1 with 100% correlation, violating classical local separability.

Course Learning Resource • Slide DeckLesson 810 Slides

Lesson 8: Two Qubits and Entanglement

Hadamard, CNOT, Bell-state intuition, and correlated measurements — Presentation by Dr. Syed Muzamil Basha.

Author: Dr. Syed Muzamil Basha• FDP Approved Curriculum

Open in New Tab

📝 Learning Check

Test your understanding with these questions. Remember, this is for learning, not examination!

Question 1: What won the 2022 Nobel Prize in Physics for Alain Aspect, John Clauser, and Anton Zeilinger?

Question 2: What is the most famous entangled two-qubit state?

Question 3: What gate, combined with a Hadamard, creates a Bell state from |00⟩?

Question 4: Can entanglement be used to send information faster than light?

Question 5: What did the 2022 Nobel Prize in Physics experimentally confirm about entanglement?

Previous: Probability Amplitudes: Complex Numbers in Quantum Mechanics Next: Quantum Interference: Constructive and Destructive

1	from qiskit import QuantumCircuit
2	from qiskit_aer import Aer

1	# =========================================================
2	# Lesson 8: Two Qubits and Entanglement (Bell State \|Phi+>)
3	# Prepared for Google Colab (Run with Shift + Enter)
4	# =========================================================
5
6	# Step 1: Install required packages
7	!pip install qiskit qiskit-aer pylatexenc
8
9	# Step 2: Import libraries
10	from qiskit import QuantumCircuit
11	from qiskit_aer import Aer
12
13	# Step 3: Run Demonstration
14	# Create a two-qubit circuit
15	qc = QuantumCircuit(2)
16
17	# Apply Hadamard gate to create superposition on qubit 0
18	qc.h(0)
19
20	# Applying a CNOT (Controlled-NOT) Gate
21	qc.cx(0, 1)
22
23	# Measure all qubits
24	qc.measure_all()
25
26	# Use Qiskit's Aer simulator
27	simulator = Aer.get_backend('aer_simulator')
28
29	# Run the circuit on the simulator
30	job = simulator.run(qc, shots=1000)
31	result = job.result()
32
33	# Get the measurement results
34	counts = result.get_counts(qc)
35	print("Bell State Measurement Counts (Lesson 8):", counts)