Monogamy and no-cloning

Bell’s inequality shows that entanglement can produce genuinely non-classical correlations. This lesson covers two theorems that show entanglement is also genuinely restricted in ways classical correlation isn’t. These restrictions are not inconveniences — they are the reason quantum cryptography works at all.

The no-cloning theorem

Classical bits can be copied freely. Given a bit, you can produce as many copies as you like — it’s how every network protocol works. Upload once, propagate everywhere.

Quantum states cannot be copied. This is the no-cloning theorem (Wootters and Zurek, 1982, and independently Dieks that same year). Formally:

There is no unitary $U$ such that $U(|\psi\rangle \otimes |0\rangle) = |\psi\rangle \otimes |\psi\rangle$ for every quantum state $|\psi\rangle$.

In words: there’s no “cloning gate.” No matter how cleverly you design a circuit, it cannot take an unknown quantum state as input and produce two identical copies.

The proof is surprisingly short. Suppose such a $U$ existed. Then for two states $|\psi\rangle$ and $|\phi\rangle$:

$U(|\psi\rangle \otimes |0\rangle) = |\psi\rangle \otimes |\psi\rangle$

$U(|\phi\rangle \otimes |0\rangle) = |\phi\rangle \otimes |\phi\rangle$

Taking the inner product of both sides:

$\langle\psi|\phi\rangle = \langle\psi|\phi\rangle^2$

This equation has only two solutions: $\langle\psi|\phi\rangle = 0$ or $\langle\psi|\phi\rangle = 1$. So $|\psi\rangle$ and $|\phi\rangle$ must be either orthogonal or identical. You cannot clone a general pair of states — only states from a fixed orthogonal set (which is how classical bits already work, so that’s nothing new).

Why no-cloning matters

The no-cloning theorem is the foundation of quantum cryptography. In particular, quantum key distribution (QKD) works because an eavesdropper trying to intercept and copy the quantum states in transit cannot do so without disturbing them. Any attempt to measure introduces detectable errors. The legitimate parties can spot the eavesdropper just by checking the error rate.

No-cloning also explains why quantum error correction is fundamentally harder than classical error correction. In the classical world, you can simply keep three copies of a bit and take the majority. In the quantum world, you’re not allowed to make three copies — so quantum error correction has to use a cleverer trick called encoding, which we’ll see in Module 8.

Monogamy of entanglement

Classical correlations can be shared among arbitrarily many parties. If Alice and Bob have a shared secret string, there’s nothing stopping them from also sharing it with Charlie — just hand him a copy.

Entanglement is different. If Alice and Bob share a maximally entangled Bell pair, they are monogamously entangled: no third party can share in that entanglement. Charlie can have his own qubit, but it will be essentially uncorrelated with the Bell pair. There’s no way to “extend” the correlation.

Formally: if the concurrence between Alice and Bob is $C_{AB}$, and between Alice and Charlie is $C_{AC}$, then:

$C_{AB}^2 + C_{AC}^2 \leq 1$

This is called the Coffman-Kundu-Wootters inequality, and it says: the “total” entanglement Alice can have with others is bounded. If she’s maximally entangled with Bob ($C_{AB} = 1$), then $C_{AC}$ must be 0 — Alice cannot also be entangled with Charlie.

Why monogamy matters

Monogamy gives us a second security foundation. In QKD, after Alice and Bob generate their shared key via measurements on a Bell pair, they can be sure that no third party (even a powerful eavesdropper with arbitrary technology) shares in their entanglement — because monogamy forbids it. The laws of physics, not just the strength of a classical cryptographic algorithm, protect the key.

This is a stronger security guarantee than any classical cryptosystem can offer. Classical cryptography relies on computational hardness assumptions: “we think factoring is hard, so RSA is secure.” Quantum cryptography relies on physical laws: “no-cloning is true, so eavesdroppers are detectable.” The first is an assumption about the best known algorithm; the second is a theorem of physics.

Taking stock

Module 5 is short because these two theorems are short — but they’re foundational. From here forward, assume the following are locked in:

• Entanglement is genuinely non-classical (Bell’s inequality, Lesson 5.1)
• Quantum states cannot be cloned (no-cloning)
• Entanglement is monogamous

These three facts together define what quantum information cannot do — and, paradoxically, are exactly what make quantum cryptography and many quantum protocols possible.

What’s next

Module 6 takes Bell pairs and shows two protocols that demonstrate their usefulness: superdense coding (sending two classical bits per qubit) and quantum teleportation (transmitting a qubit’s state using only two classical bits + a shared Bell pair).