No-signaling

Alice and Bob share a pair of entangled qubits. The famous example is the singlet state:

In this state, the two qubits are perfectly anti-correlated. If Alice measures hers in the Z basis and gets |0⟩, Bob will measure |1⟩ with certainty. If Alice gets |1⟩, Bob gets |0⟩. The same anti-correlation holds in any basis they both choose.

They're in different cities. They each have one qubit. Light from Alice to Bob takes some non-zero time.

Here's the protocol that would work, if physics allowed it:

1. They agree in advance: Alice will measure in the Z basis to send a 0 bit, or in the X basis to send a 1 bit.
2. Bob always measures in the Z basis.
3. Bob looks at his outcomes and works out which basis Alice used.

If Bob can tell what basis Alice picked, then Alice has sent him a bit of information, instantly, regardless of distance. That would let them communicate faster than light.

Each person, looking only at their own qubit, sees a perfectly random 50/50 sequence of outcomes. That's true whether or not the other person measured. It's true whether or not the other person measured in the same basis. It's true whether or not the other person's qubit even exists.

From Alice's point of view, nothing about her local measurement statistics depends on what Bob does. From Bob's point of view, nothing about his local measurement statistics depends on what Alice does.

The correlations are only visible when they later get together (by classical channel, at light speed) and compare notes.

The clean way to see it uses the marginal density matrix. Don't panic; the idea is simple. Alice's qubit, considered on its own, has some probability distribution over outcomes. That distribution is captured by a small object called ρA, computed from the full two-qubit state.

For the singlet, the marginal on either side is:

I is the identity matrix and ½ I is what physicists call the maximally mixed state. It means: in any measurement basis you can pick, the outcome is 50/50. No bias toward 0 or 1. No bias toward + or −. No bias anywhere.

Here's the key fact: any operation Bob performs on his qubit leaves Alice's marginal unchanged. He can apply any gate, any measurement, any sequence of operations. When you trace it through, ρA stays exactly ½ I.

Alice's observable statistics depend only on ρA. Since ρA never changes, Alice's statistics never change. Therefore no information is transmitted.

No-signaling has a few important consequences.

• Quantum mechanics is compatible with relativity. Even though entanglement creates instantaneous correlations, you cannot use those correlations to send signals. The cosmic speed limit holds.
• Classical communication is still required. Every quantum protocol that does something useful with entanglement (quantum teleportation, superdense coding, key distribution) requires a classical channel between the parties. That classical channel is the rate-limiting step.
• It rules out many would-be protocols. If someone claims a quantum-mechanical scheme that sends information faster than light, no-signaling is what tells you the math has to be wrong somewhere.
• It's built into the formalism. No-signaling isn't a separate principle that quantum mechanics has to satisfy; it follows automatically from how density matrices and partial traces work. Quantum mechanics couldn't signal faster than light even if it wanted to.

Once you've internalized this, you'll see why the protocols in the next chapter are designed the way they are. They use entanglement to do things classical protocols can't, but every protocol still needs ordinary communication for the final step. Entanglement is a resource for correlation, never for signaling.