Quantum Computing
Entanglement and Bell States
Цели урока
- Understand entanglement and why it cannot be described as a product state
- Know all 4 Bell states and their correlation properties
- Understand the EPR paradox and what Bell inequality violation proves
- Explain no-cloning and why it makes QKD physically secure
Предварительные знания
2022. The Nobel Committee hands the Physics Prize to three experimentalists - for proving Einstein wrong. Eighty-seven years to settle his intuition. Entanglement turned out to be real. China now pushes entangled photons across 1000 km of fiber for bank-grade cryptography. The phenomenon Einstein called absurd guards financial transactions today.
- **Quantum cryptography (QKD):** banks in China use entangled photons for secure key distribution over 1000+ km optical fiber
- **Quantum computers:** all useful quantum algorithms - Shor, Grover - require entanglement between qubits
- **Quantum internet:** networks of quantum repeaters are being built for global entanglement distribution
From paradox to Nobel Prize
1935 - Einstein, Podolsky, Rosen publish EPR, claiming QM is incomplete. 1964 - John Bell derives a mathematical test: if hidden variables exist, correlations are bounded. 1982 - Alain Aspect runs the first convincing experiment: $|S| \approx 2.7 > 2$. 2015 - loophole-free experiment closes the question permanently. 2022 - Nobel Prize to Aspect, Clauser, Zeilinger. 87 years from paradox to prize.
Quantum entanglement
Two qubits are entangled when no choice of single-qubit states reproduces them as a tensor product. They behave as one object, even with billions of kilometers between them. Measure one and the other's state snaps into place. Not telepathy - a correlation baked in at the moment the pair was created. The math is brutal and unambiguous: the factorization simply does not exist.
**Entanglement** - a quantum correlation with zero classical counterpart. Two qubits in state $(|00\rangle+|11\rangle)/\sqrt{2}$ obey one rule: measuring the first instantly fixes the second - both 0 or both 1. The coin flip itself is random (50/50); the agreement is iron-clad.
The twist: outcomes are correlated yet random. Nothing carries information across the entanglement channel because the sender has no dial to turn. What gets shared is correlation, not a message. This is the **no-signaling theorem**, and it is exactly why QKD coexists peacefully with relativity.
Two qubits are in state $(|00\rangle+|11\rangle)/\sqrt{2}$. Alice measures hers and gets $|1\rangle$. What will Bob get?
The Four Bell States
Exactly **four** maximally entangled two-qubit states exist - the **Bell states**. They form an orthonormal basis of the 4-dimensional two-qubit space and define the unit of entanglement: the ebit. Any two-qubit state decomposes into Bell states the same way any vector decomposes into coordinate axes.
| Bell State | Formula | Measurement correlation | Creation from |00⟩ |
|---|---|---|---|
| |Phi+⟩ | (|00⟩+|11⟩)/sqrt(2) | Match: 00 or 11 | H → CNOT |
| |Phi-⟩ | (|00⟩-|11⟩)/sqrt(2) | Match: 00 or 11 | X → H → CNOT |
| |Psi+⟩ | (|01⟩+|10⟩)/sqrt(2) | Opposite: 01 or 10 | H → CNOT → X |
| |Psi-⟩ | (|01⟩-|10⟩)/sqrt(2) | Opposite: 01 or 10 | X → H → CNOT → X |
**Superdense coding** weaponizes Bell states: sending 1 qubit from an entangled pair encodes 2 classical bits - the choice of which of the 4 Bell states to apply. One qubit, two classical bits. Just the geometry of entanglement.
In state $|\Psi^+\rangle = (|01\rangle+|10\rangle)/\sqrt{2}$ Alice measures $|0\rangle$. What will Bob get?
The EPR Paradox
1935. Einstein, Podolsky, and Rosen publish the paper that shook physics: quantum mechanics is **incomplete**. The argument is sharp - if measuring one qubit instantly fixes another across the galaxy, the outcome must have been predetermined; otherwise locality breaks. Einstein was convinced he had spotted a fatal hole in the theory.
'Spooky action at a distance' (1935)
Einstein called entanglement 'spukhafte Fernwirkung' - spooky action at a distance. He believed nature could not be so strange: 'God does not play dice'. Thirty years later John Bell showed the opposite is true - and in 2022 the Nobel Prize went to the people who proved it experimentally.
**Bell's inequality** (1964) - a mathematical test: if measurement results are predetermined, correlations satisfy $|S| \leq 2$. Quantum mechanics predicts violation: $|S| \leq 2\sqrt{2}$. Experiments confirmed QM - Nobel Prize 2022 (Aspect, Clauser, Zeilinger).
Paradox dissolved: entanglement **does not** carry information faster than light. The outcome is random - Alice cannot dial in 0 or 1. Correlation only surfaces once results are compared over a classical channel (capped at c). The no-signaling theorem formalizes the verdict: Bob's reduced density matrix is untouched by anything Alice does.
Bell's inequality is experimentally violated. What does this prove?
The No-Cloning Theorem
Classical copying is trivial: Ctrl+C, Ctrl+V. Quantum states **cannot be copied**. Not a hardware limitation - a hard law of nature, falling straight out of the linearity of quantum mechanics. The proof fits in five lines of algebra.
| Operation | Classical bit | Qubit |
|---|---|---|
| Reading | Non-destructive | Destroys superposition (measurement) |
| Copying | Free (copy) | Impossible (no-cloning) |
| Deletion | Simple (delete) | Impossible for pure states (no-deleting) |
| Transfer | Possible with copy | Only teleportation (original destroyed) |
**Consequences of no-cloning:** 1. QKD is secure - an eavesdropper cannot copy a qubit without scrambling it. 2. Quantum teleportation respects relativity - the original gets destroyed, so nothing is duplicated. 3. Quantum error correction does its job through entanglement, never through copying.
No-cloning looks like a constraint. It is actually a **weapon**. Eve intercepts a qubit on a quantum channel - copying and forwarding is off the table. Every attempt to peek perturbs the state, and Alice and Bob notice the intrusion. QKD security rests on physics, not on a hardness assumption that may fall tomorrow.
Entanglement allows information to be transmitted instantly over any distance
Entanglement creates correlation but does not transmit information. The measurement result is random and cannot be controlled by the sender.
The result is random (0 or 1 with equal probability) - the sender cannot choose which outcome to get. Only by comparing results over a classical channel (at speed at most c) does correlation appear. Bob's reduced density matrix does not depend on Alice's actions - this is mathematically provable.
Can a qubit in the known state $|0\rangle$ be cloned?
Key ideas
- **Entanglement** - a non-factorable state of two or more qubits; measuring one instantly determines the other
- **4 Bell states** - the maximally entangled basis for 2 qubits: |Phi+/-⟩ (match), |Psi+/-⟩ (opposite)
- **EPR and Bell inequality:** experiments confirmed QM and refuted local hidden variables (Nobel 2022)
- **No-cloning:** an unknown quantum state cannot be copied - the physical foundation of QKD security
Related topics
Entanglement is a resource for quantum protocols and algorithms:
- Quantum gates — H + CNOT create entanglement from unentangled qubits
- Qubits — Entanglement is a property of systems of 2+ qubits, not a single one
- Shor's algorithm — Entanglement between registers is a key resource in the algorithm
Вопросы для размышления
- If entanglement does not transmit information, how is it useful for computation? What role does it play in Shor's algorithm?
- No-cloning forbids copying. Quantum computers use error correction - does that not contradict no-cloning?
- Why did Einstein find entanglement absurd, while today it appears in commercial products?
Связанные уроки
- qc-02 — H and CNOT gates create entanglement - no Bell states without them
- qc-01 — Entanglement is a multi-qubit property, foundations in qc-01
- qc-04 — Grover's algorithm uses entanglement for quadratic speedup
- qc-05 — Shor's algorithm requires entanglement between registers
- sec-01 — QKD is the direct application of no-cloning and entanglement
- it-01 — Shannon information theory and quantum information are parallel frameworks
- nm-01 — Numerical linear algebra for understanding unitary matrices
- prob-05-independence