Quantum Key Distribution (QKD) Explained

Quantum Key Distribution (QKD): An Intuitive Explanation

This explanation rebuilds the ideas behind Quantum Key Distribution (QKD) from the ground up, focusing on intuition first and math second.

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Big Picture

Quantum Key Distribution is a way for two parties (Alice and Bob) to:

• Create a shared random secret key
• Detect if anyone (Eve) tried to eavesdrop

The core idea: measuring a quantum system changes it.

Everything else in QKD follows from this rule.

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1. What Are Quantum “States”?

A photon can be polarized in different directions:

• Horizontal (↔)
• Vertical (↕)
• Diagonal (↗)
• Other diagonal (↘)

We label these using symbols:

• |0⟩ = horizontal
• |1⟩ = vertical

These form one basis (a way of asking questions).

Another basis is diagonal:

• |+⟩ = 45°
• |−⟩ = −45°

A basis is just a choice of which questions you are allowed to ask the photon.

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2. Why Bases Matter

If you measure a photon in the correct basis, you get a definite answer. If you measure in the wrong basis, the result is random.

Example:

• The photon is diagonal (|+⟩)
• You measure horizontal vs vertical

Result:

• 50% horizontal
• 50% vertical

This is not due to bad equipment — it is fundamental randomness.

|⟨0 | +⟩|² = 1/2

Translation: if the photon was diagonal and you checked horizontal, you get a coin flip.

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3. What Alice and Bob Do (BB84 Protocol)

For each photon:

1. Alice randomly chooses a bit (0 or 1)
2. Alice randomly chooses a basis
3. She sends the photon to Bob
4. Bob randomly chooses a basis to measure

Later, over a public channel:

• Alice and Bob compare bases (not values)
• They keep only the bits where bases matched

Why this works:

• Same basis → correct bit
• Different basis → random → discarded

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4. Why Eavesdropping Is Detectable

Eve does not know which basis Alice used. She must guess.

• Half the time she guesses wrong
• Her measurement disturbs the photon
• Bob later sees errors

This cannot be avoided due to the no-cloning theorem:

There is no operation U such that U|ψ⟩|0⟩ = |ψ⟩|ψ⟩

Measuring is the same as disturbing.

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5. Quantum Bit Error Rate (QBER)

Alice and Bob publicly compare a small sample of bits.

QBER = (number of wrong bits) / (number of compared bits)

• No Eve → very low QBER
• Eve present → QBER increases

For BB84:

• Secure if QBER < 11%
• Insecure above that threshold

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6. Entropy and Information

Entropy measures uncertainty.

• Shannon entropy (classical)
• von Neumann entropy (quantum)

Security condition:

I(Alice : Bob) > I(Alice : Eve)

Meaning: Bob knows more about Alice’s bits than Eve does.

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7. Final Key Length

After error correction and privacy amplification:

ℓ ≈ n [1 − 2H(QBER)]

• n = raw bits
• ℓ = final secure key length

The remaining bits are provably secret.

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Summary: QKD works because physics itself prevents undetectable eavesdropping.