Quantum Teleportation and the No-Communication Theorem

The Paradox: Instant Collapse vs. Slow Information

At the heart of the confusion is the concept of wavefunction collapse. When Alice performs her Bell measurement, the combined state of the three qubits collapses instantly. This collapse affects Bob's qubit, changing it into one of four possible states depending on Alice's measurement outcome. This "spooky action at a distance" is real, but does it carry information?

Let's look at a diagram that clarifies the two different "channels" at play in the protocol.

The key insight is that even though Bob's qubit changes instantly, the state it changes to is, from his perspective, completely random. He has no way of knowing what the original state $|\psi ⟩$ was, or even that a teleportation has occurred, until he receives more information from Alice.

The following video from the Qiskit team explains this critical point clearly.

As the video explains, Bob is left with a qubit that is a "scrambled" version of the original state $|\psi ⟩$. Let's formalize this:

• If Alice measures 00, Bob's qubit is in the state $|\psi ⟩$.
• If Alice measures 01, Bob's qubit is in the state $X|\psi ⟩$.
• If Alice measures 10, Bob's qubit is in the state $Z|\psi ⟩$.
• If Alice measures 11, Bob's qubit is in the state $XZ|\psi ⟩$.

Each of these outcomes occurs with a 25% probability. Without knowing which of the four outcomes occurred, Bob has a qubit that is in a random state. If he were to measure it, the result would be random and tell him nothing about $|\psi ⟩$. He absolutely needs the two classical bits from Alice to know which "unscrambling" operation ($I$, $X$, $Z$, or $XZ$) to apply.

Can't We Devise a Cleverer Scheme?

You might argue that a simple measurement isn't the only thing Bob could do. Perhaps there's a more advanced experiment he could perform on his single qubit to deduce what Alice did, without her needing to send a message. This is a common line of thinking, and it's a great question that gets to the heart of the no-signaling principle in quantum mechanics.

This principle states that no local measurement on one part of an entangled system can be used to transmit information to another, distant part of the system. Let's explore why this is true.

The following video does an excellent job of debunking common "loopholes" people propose to get around this limitation.

The video makes the essential point: local measurement statistics are unaffected by distant actions. No matter what measurement Alice performs on her qubit (or even if she does nothing at all), the probabilities for any outcome of any measurement Bob chooses to perform on his qubit remain unchanged. Because he can't see any statistical difference, he can't get any information.

This concept is also clearly explained in the following article.

Tying It All Together: Two Channels are Required

We can now state with confidence why teleportation does not allow for FTL communication. The information required to reconstruct the original quantum state $|\psi ⟩$ on Bob's side is split into two essential pieces:

1. The Quantum Channel: The shared entangled pair allows for non-local correlations to be established. This is what enables the "spooky" part of the protocol, instantly mapping properties of $|\psi ⟩$ onto Bob's qubit, albeit in a scrambled way.
2. The Classical Channel: The two bits from Alice's measurement provide the "key" for Bob to unscramble his qubit and recover the original state. This information must be sent via conventional means (like a radio signal or an optical fiber), and is therefore limited by the speed of light.

Since Bob needs information from both channels to complete the protocol, the total time for the information transfer is bottlenecked by the slower of the two—the classical channel. This is analogous to Alice securely sending a locked box to Bob via an instant courier, but sending the key via regular mail. Bob has the box, but he can't access its contents until the key arrives. The information is not truly "received" until both parts are present.

Conclusion

In this lesson, we have confronted the apparent paradox of quantum teleportation and resolved it. While entanglement involves non-local effects, it cannot be used to transmit information on its own.

Key Takeaways:

• Quantum teleportation is a protocol to transmit a quantum state, not matter or energy.
• The protocol requires both a pre-shared entangled pair (a quantum channel) and classical communication.
• Alice's measurement instantly collapses the wavefunction, but from Bob's local perspective, his qubit is left in a completely random state. He cannot extract any information from it.
• Bob must wait for two classical bits from Alice, sent at or below the speed of light, which tell him how to "unscramble" his qubit to recover the original state.
• This reliance on a classical channel ensures that quantum teleportation respects causality and does not enable faster-than-light communication. This is a manifestation of the no-signaling principle.

Preview of the Next Lesson:
We've just analyzed a protocol that uses entanglement and classical bits to transmit a quantum bit. In the next lesson, we'll explore a fascinating counterpart called superdense coding. This protocol does the reverse: it uses a shared entangled pair to transmit two classical bits of information by sending only one qubit.