Quantum Entanglement
Explore spooky action at a distance with interactive qubits
🔗 The Strangest Phenomenon in Physics
Imagine two particles that are mysteriously linked — measure one, and you instantly know the state of the other, no matter how far apart they are. This is quantum entanglement, what Einstein famously called "spooky action at a distance."
💡 Core Concept
When two qubits are entangled, they form a single quantum system. Measuring one qubit instantly determines the state of the other, even if they're separated by galaxies. This isn't about sending information faster than light — it's about shared quantum correlations that defy classical physics.
1. Creating Entangled Pairs
Entangled qubits are prepared in special states called Bell states. These are the maximally entangled two-qubit states.
🔗 Understanding Bell States
What Are Bell States?
Bell states (named after physicist John Stewart Bell) are the four maximally entangled two-qubit states. "Maximally entangled" means the qubits have the strongest possible quantum correlation—you cannot describe them as independent particles. They must be treated as a single quantum system. These four states form a complete basis for two-qubit systems and are the foundation of quantum information protocols.
The Four Bell States: Mathematical Definitions
Equal superposition of both qubits being 0 or both being 1
Like |Φ⁺⟩ but with negative phase on |11⟩
Equal superposition of opposite states (anti-correlated)
Like |Ψ⁺⟩ but with negative phase on |10⟩
Key insight: The 1/√2 factor ensures normalization (total probability = 1). The states are written in the computational basis {|00⟩, |01⟩, |10⟩, |11⟩}, where |ab⟩ means qubit A is in state a and qubit B is in state b.
Why Are They "Maximally Entangled"?
A state is entangled if you cannot write it as a product of individual qubit states. For example, you cannot write |Φ⁺⟩ as (α|0⟩ + β|1⟩)_A ⊗ (γ|0⟩ + δ|1⟩)_B. The qubits are fundamentally correlated. Bell states are maximally entangled because:
How to Create Bell States: The Circuit
Bell states are created using a simple 2-gate quantum circuit. Here's the recipe for |Φ⁺⟩:
How CNOT creates entanglement: The CNOT flips qubit B only if A is |1⟩. So |0⟩⊗|0⟩ stays |00⟩, but |1⟩⊗|0⟩ becomes |11⟩. This creates the correlation!
Creating the Other Three Bell States
The X gate flips |0⟩↔|1⟩, and Z gate adds phase to |1⟩. These simple modifications transform between Bell states.
Entanglement Entropy: Measuring "How Entangled"
We quantify entanglement using von Neumann entropy. For a pure state of two qubits:
where ρ_A is the reduced density matrix of subsystem A
For Bell states, each individual qubit has maximum entropy (S=1), meaning you have zero information about it alone—all the information is in the correlations!
Why Bell States Are Building Blocks
Bell states are used everywhere in quantum information: quantum teleportation (transfer quantum states), superdense coding (send 2 classical bits using 1 qubit), quantum key distribution (unbreakable cryptography), and quantum error correction (protect quantum information). They're the quantum equivalent of classical bit pairs like 00 or 11, but with the magic of quantum correlation that enables protocols impossible in classical computing.
⚛️ Interactive: Choose a Bell State
Selected: |Φ+⟩ state - Both qubits will have the same measurement outcome
🎯 The EPR Paradox and "Spooky Action at a Distance"
Why Does Measuring One Qubit Affect the Other?
This is the heart of quantum entanglement's mystery. When you measure an entangled qubit, you don't just learn about that qubit—you instantaneously determine the state of its entangled partner, no matter how far apart they are. For example, in the |Φ⁺⟩ = (|00⟩ + |11⟩)/√2 state, measuring Alice's qubit and getting |0⟩ means Bob's qubit must be |0⟩ too. The moment Alice measures, the entire two-qubit system collapses from superposition to a definite state.
The EPR Paradox (1935)
In 1935, Einstein, Podolsky, and Rosen (EPR) published a famous paper arguing quantum mechanics was incomplete. Their thought experiment:
Two particles in an entangled state, then separated by a large distance
Alice measures her particle and gets a result (e.g., |0⟩)
Bob's particle instantly has a definite state (e.g., |0⟩), perfectly correlated
EPR's argument: How can Bob's particle "know" what Alice measured instantly? Either (1) information travels faster than light (violates relativity), or (2) the particles had predetermined values all along (hidden variables), and quantum mechanics just doesn't tell us what they are. EPR chose option 2, concluding quantum mechanics was incomplete.
Einstein's "Spooky Action at a Distance"
Einstein called entanglement "spukhafte Fernwirkung" (spooky action at a distance) and deeply distrusted it. He believed in local realism:
Objects are only influenced by their immediate surroundings. No instantaneous influences across space.
Physical properties exist independently of measurement. A particle has definite values even when unobserved.
Quantum mechanics violates one or both! Bell's theorem (1964) and subsequent experiments proved hidden variables cannot explain quantum correlations. Nature is fundamentally non-local.
Measurement Collapse for Entangled Pairs
Before measurement, the two-qubit system exists in a superposition—both outcomes are simultaneously possible. The act of measurement:
Both particles in superposition, perfectly correlated
Both particles collapsed to definite state—Bob's must be |0⟩
The collapse happens to the entire entangled system, not just Alice's qubit. This is why Bob's state is determined—they're not separate particles, but a single quantum system.
Why No Information Travels Faster Than Light
Despite the "instantaneous" correlation, entanglement cannot be used to send information faster than light. Here's why:
When Alice measures, she gets a random result (50% |0⟩, 50% |1⟩). She cannot control what she gets.
From Bob's perspective, his measurements also look random (50/50). He can't tell if Alice has measured yet.
Only when Alice and Bob compare results (using classical communication) do they see the perfect correlation. This comparison cannot happen faster than light.
Analogy: Imagine two magic coins that always land the same way when flipped. Alice flips hers and gets heads. Instantly, Bob's coin "becomes" heads too. But Bob can't tell if Alice has flipped yet—his coin still looks random to him. Only when they call each other (light-speed communication) do they discover the correlation. No message was sent via the coins themselves.
Distance Doesn't Matter: Verified Experiments
Experiments have confirmed entanglement works over vast distances:
Geneva: Entanglement across fiber optic cable
Canary Islands: Free-space entanglement
China's Micius satellite: Space-based entanglement
No distance limit in principle—just practical challenges
Perfect correlations observed in all cases, confirming entanglement is a fundamental property of nature, not limited by distance.
The Nature of Reality
Entanglement reveals that quantum reality is fundamentally non-local. The universe doesn't consist of independent objects with predetermined properties. Instead, quantum systems are interconnected in ways that transcend space. This doesn't mean faster-than-light communication, but it does mean Einstein's local realism is incompatible with nature. The universe is stranger and more interconnected than classical physics ever imagined. As physicist Erwin Schrödinger said, entanglement is "the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought."
🎯 Interactive: Measure the Entangled Pair
Alice's Qubit
Location: EarthBob's Qubit
Location: Mars (0 km away)Measurements performed: 0
2. Understanding Quantum Correlations
The strength of correlation between entangled qubits depends on the measurement angles chosen by Alice and Bob.
📐 Interactive: Adjust Measurement Angles
Correlation Strength
1.000Angle difference: 0° - Perfect correlation!
📊 Interactive: Correlation vs Angle
3. Testing Bell's Theorem
Bell's theorem proves that quantum entanglement cannot be explained by "hidden variables." The CHSH inequality tests this: classical physics says S ≤ 2, but quantum mechanics can reach S = 2√2 ≈ 2.83.
🔬 Bell's Theorem: Proving Quantum Non-Locality
What Are Hidden Variable Theories?
After the EPR paradox, physicists wondered: maybe quantum randomness isn't fundamental. Perhaps particles carry hidden variables—secret instructions determined at creation—that tell them how to behave when measured. Like a pair of magic coins programmed to always show the same face. If this were true, the "spooky action" would be an illusion: the correlation was predetermined, not caused by measurement. Quantum mechanics would just be incomplete, missing these hidden variables.
John Bell's Breakthrough (1964)
Physicist John Stewart Bell asked: can any local hidden variable theory reproduce quantum predictions? His answer: No. He derived mathematical inequalities that any local realistic theory must satisfy, but quantum mechanics violates them. This meant:
Quantum predictions don't match experiments (spoiler: they do, perfectly)
Hidden variables cannot explain quantum correlations—nature is fundamentally non-local
Bell's theorem shifted the debate from philosophy to experiment. We could now test whether nature is local and realistic.
Bell Inequalities: The Mathematics
The simplest Bell inequality compares correlations when Alice and Bob measure at different angles. For any local hidden variable theory:
E(a,b) = correlation when Alice measures at angle a, Bob at angle b
If experiments measure S > 2, hidden variables are ruled out. Quantum mechanics predicts S = 2√2, achievable with optimal measurement angles.
The CHSH Inequality
The Clauser-Horne-Shimony-Holt (CHSH) inequality is a practical version of Bell's theorem:
Any local realistic theory (including hidden variables) cannot exceed this
Maximum achievable in quantum mechanics
These settings give S = 2√2, the maximum quantum violation
What Does Violation Prove?
When experiments measure S > 2, it proves:
Particles don't carry predetermined instructions. The correlation is genuinely created at measurement.
Either locality (no faster-than-light influences) or realism (properties exist before measurement) must be abandoned. Most physicists accept non-locality.
Nature really does work the way quantum mechanics describes—entanglement is real.
Aspect's Experiments (1982)
French physicist Alain Aspect performed the first decisive tests of Bell's theorem:
Created entangled photon pairs, sent them to detectors 12 meters apart, measured polarization at different angles
Changed measurement settings rapidly (every 10 nanoseconds) so detectors couldn't communicate classically during measurement
Aspect won the 2022 Nobel Prize in Physics for these experiments, along with John Clauser and Anton Zeilinger.
Modern Loophole-Free Experiments
Bell tests have faced several "loopholes" (potential ways hidden variables could sneak in):
Detectors miss some particles—maybe hidden variables only affect detected ones
Detectors too close—could communicate at light speed during measurement
Measurement settings chosen by deterministic process
2015: Three independent experiments closed all loopholes at once
Consensus: Bell inequality violations are real. Nature is non-local.
Philosophical Implications
Bell's theorem fundamentally changed our understanding of reality. Einstein's dream of a local, deterministic universe was proven incompatible with experimental results. We must accept that either: (1) Non-locality is real—entangled particles are connected across space in a way that defies classical intuition, or (2) Realism is false—properties don't exist until measured. Most interpretations embrace non-locality while preserving that no information travels faster than light. The universe is not made of independent objects—quantum systems are fundamentally holistic. As John Bell himself said: "The reasonable thing just doesn't work."
🔬 Interactive: CHSH Inequality Test
CHSH Value (S)
Adjust angles to maximize violation
4. Quantum Teleportation Protocol
Using entanglement, we can "teleport" quantum states. This doesn't violate relativity — it requires sending classical information, but the quantum state is transferred without physically moving the qubit.
🚀 Quantum Teleportation: Transferring Quantum States
What is Quantum Teleportation?
Quantum teleportation is a protocol that transfers a quantum state from one location to another without physically moving the particle itself. It's not like Star Trek—no matter is transported, and it's not instantaneous (requires classical communication). Instead, it's a clever use of entanglement and classical bits to reconstruct a quantum state at a distant location. The original state is destroyed in the process (consistent with the no-cloning theorem).
The Three-Qubit System
Teleportation involves three qubits with specific roles:
The state Alice wants to send to Bob. She doesn't know α and β—it's an arbitrary quantum state.
Alice has qubit 2, Bob has qubit 3. They're maximally entangled (prepared earlier).
The complete initial state of all three qubits: |ψ⟩₁ ⊗ |Φ⁺⟩₂₃ = (α|0⟩ + β|1⟩) ⊗ (|00⟩ + |11⟩)/√2
The Protocol: Step by Step
Alice has her unknown qubit |ψ⟩ and half of the entangled pair. Bob has the other half.
Alice performs a joint measurement on her unknown qubit and her half of the entangled pair. This projects them onto one of the four Bell states.
Four possible outcomes (each 25% probability):
Alice sends her 2-bit measurement result to Bob through a classical channel (phone, internet, etc.). This is limited by the speed of light.
Based on the 2 classical bits received, Bob applies a correction operation to his qubit:
After correction: Bob's qubit is now in state |ψ⟩ = α|0⟩ + β|1⟩
Why It Requires Classical Communication
After Alice's measurement, Bob's qubit is in one of four possible states depending on Alice's result. Without knowing which Bell state Alice measured, Bob cannot reconstruct |ψ⟩. The entanglement creates correlation, but the 2 classical bits tell Bob which correction to apply. This classical communication is limited by light speed, so teleportation doesn't violate relativity.
Why It Doesn't Violate Relativity
The protocol cannot complete until Bob receives the 2 classical bits. If Alice and Bob are separated by distance d, there's a minimum delay of d/c (where c = speed of light).
Bob's qubit looks completely random before he applies the correction. He gains no information about |ψ⟩ until Alice's classical message arrives.
The quantum information travels via classical channel, obeying relativity. Entanglement provides the correlation, but classical bits unlock it.
Fidelity: How Well Does It Work?
Fidelity measures how close the teleported state is to the original. In ideal conditions:
Perfect teleportation
Limited by noise, imperfect gates
First demonstration: 1997 (photons). Now routinely achieved with photons, ions, and superconducting qubits.
Connection to the No-Cloning Theorem
The no-cloning theorem states you cannot create an exact copy of an unknown quantum state. Teleportation respects this:
Alice's original qubit is destroyed during the Bell measurement. The state moves from Alice to Bob—it's not copied.
There's never a moment when both Alice and Bob have the quantum state |ψ⟩. Either Alice has it (before measurement) or Bob has it (after reconstruction).
This is why we say the state is "teleported" or "transferred" rather than "copied." It's a move operation, not a copy operation.
Real-World Applications
Transfer quantum states between nodes in a quantum internet
Move quantum information between separate quantum processors
Extend range of quantum communication by "refreshing" entanglement
Move quantum states between physical and logical qubits
Why Teleportation Matters
Quantum teleportation is a cornerstone of quantum information theory. It proves that quantum states can be transmitted using a combination of entanglement and classical communication. This isn't just a curiosity—it's essential for building quantum networks, distributed quantum computers, and long-distance quantum communication. It demonstrates that entanglement is a resource that can be consumed to perform information-theoretic tasks impossible in classical physics. Every quantum network protocol relies on variations of teleportation to move quantum information between nodes.
🚀 Interactive: Quantum Teleportation
Prepare Initial State
Alice has a qubit in an unknown state |ψ⟩ that she wants to send to Bob.
🎓 Key Takeaways
Instant Correlation
Measuring one entangled qubit instantly determines the state of the other, regardless of distance.
Bell States
Four maximally entangled states (|Φ+⟩, |Φ-⟩, |Ψ+⟩, |Ψ-⟩) form the basis of quantum information protocols.
Bell's Theorem
Experimental tests prove quantum correlations exceed classical limits, ruling out hidden variable theories.
Quantum Teleportation
Entanglement enables transferring quantum states using classical communication and shared entanglement.
No Faster-Than-Light
Despite instant correlation, you cannot send information faster than light — you need classical communication.
Real Applications
Powers quantum cryptography (QKD), quantum networks, and distributed quantum computing.