My first quantum protocol
Welcome to Introduction to Quantum Computing. I am your guide, Associate Professor Chris Ferrie, a researcher in the UTS Centre for Quantum Software and Information. This is Lecture 6. It would probably be a good idea to have read the previous Lectures before continuing.
What did you learn last week?
In the last few weeks you completed your basic training. You now know about quantum information, how to write it, how to read it, and how to dial it up to 11 with entanglement. Entangled states were those that could not be written as product states (in any basis!). With multiqubit states and gates, you have all the tools you need to start creating quantum algorithms.What will you learn this week?
This week you will be introduced to the first two canonical quantum protocols: superdense coding and teleportation. These demonstrate that entanglement can be used as a resource for some tasks. You’ll have your first taste as designing protocols as well as analysing them.What will you be able to do at the end of this week 5?
At the end of this module, you should be able to answer these questions:
What is superdense coding?
What is quantum teleportation?
What is entanglement swapping?
How is entanglement a useful resource?
Superdense coding by design
Recall the Holevo theorem: a qubit cannot convey more than one bit of information. But… what if it could? Superdense coding is a communication protocol which allows two bits of information to be sent using a single qubit. How? Entanglement, of course!
Simple communication protocols are usually phrased as a game with players named Alice and Bob. The players can have agreed upon strategies beforehand and are often constrained by what information they can pass to each other. There is always some goal, or win condition, that the players are working toward. In the case of superdense coding, Alice can only send a single qubit to Bob, but must convey two bits of information. They can meet beforehand and agree on some strategy, which they obviously must do since Holevo’s theorem proves that only a single bit of information can be conveyed with the qubit Alice sends.
So, first of all, we know that two qubits are required. But Alice can only send one, and so Bob must possess the other. If the state of the two qubits is a product state, Bob’s qubit contains no information about the bits Alice needs to convey. So, the state must be entangled. We know how to do that with the Hadamard and CNOT gate. The first step in the protocol is for Alice and Bob to create a pre-arranged entangled pair of qubits.
Once they are separated, Alice can only send back her qubit to Bob. Alice needs to perform some unitary on her qubit to encode the bits she wants to send. Whatever she ends up doing to her qubit, call it unitary U, we can see that Bob still needs to perform some action to decode the information contained in the pair. Why? Notice that, after Bob possesses both qubits, the state of the pair is U|0⟩⊗|0⟩ + U|1⟩⊗|1⟩, which is still entangled. To get a definitive answer, Bob has to disentangle the state to reduce it to one of the basis states, which depends on both bits. In other words, he must end up with the state |b₁⟩⊗|b₂⟩. Let’s assume he does this by inverting the original entangling operation.
By working backwards through this computation, we can figure out what Alice needs to do for the whole protocol to work out as desired. That is, we start with |b₁⟩⊗|b₂⟩ and apply the inverse of each operation preceding it. In this case, both the Hadamard and CNOT are self-inverse, so we apply them to get.
Now we need to find some operations of the form (U⊗I), where U depends on b₁b₂, to get back to our initial entangled state |0⟩⊗|0⟩ + |1⟩⊗|1⟩. We’ve done half the work in noticing that a Z gate can apply the phase. But at this point you might be thinking we are stuck since it’s Bob’s qubit that depends on b₂. However, a quick check shows that the symmetry of this entangled state saves us: |0⟩⊗|b₂⟩ + |1⟩⊗|¬b₂⟩ =|b₂⟩⊗|0⟩ + |¬b₂⟩⊗|1⟩. We can then see that the bit flipping X gate will get us back to the correct state.
Putting it all together, the protocol looks like this.
As an exercise, step through each gate of the algorithm to prove the the entire circuits acts as |0⟩⊗|0⟩ ↦|b₁⟩⊗|b₂⟩.
It’s worth pausing here and asking, you know, why? Presumably sending two bits is much easier than sending a qubit to someone. But recalling Holevo’s theorem again, imagine that an eavesdropper intercepted the qubit. Could they decode the two-bit message? No! The most they could learn is one bit. Quantum entanglement enables secure quantum communication.
Quantum teleportation
Superdense coding allowed Alice to communicate two bits with one qubit. Quantum teleportation is in some sense the opposite — it allows Alice to instead communicate one qubit with two bits.This should be surprising — how could Alice communicate a qubit over the telephone with just two bits? Again, the answer is entanglement.
Since it is so similar to superdense coding, I’ll just show you the circuit.
Let’s approach this one from the perspective of verifying the circuit. We need to prove that the state in the first register |𝜓⟩ = 𝛼|0⟩ + 𝛽|1⟩ is the same as the same as the final register at the output. To do so, we step through applying one gate at a time.
Not so bad, right? Now, I know you may be wondering what this has to do with teleportation, as seen on TV. Well, I have nothing to say that this webcomic doesn’t already say about that.
Something cool about the teleportation protocol is that it preserves entanglement. That is, if the qubit Alice wants to send to Bob is entangled with another qubit held by, say, Charlie, the qubit Bob ends up with at the end of the protocol will be entangled with Charlie. In circuit form, it looks like this.
But it gets even better! Imagine now Alice and Bob share an entangled pair, Alice and Charlie share an entangled pair, and Charlie and Diane share an entangled pair. So, we have 6 qubits in total and they are paired off. Alice and Bob perform the teleportation protocol, as do Charlie and Diane. Alice and Charlie measure — and hence collapse — each of the qubits they had initially entangled. Those qubits are individually teleported to Bob and Diane, respectively. But, here’s the kicker — Bob and Diane now share entanglement! This protocol is called entanglement swapping and I’m sure you can imagine how it might be useful in a quantum networking scenario.
Additional Resources
The Wikipedia articles (Superdense coding and Quantum teleportation) are quite good and give a different perspective and additional generalisations and references.
Microsoft’s Quantum Katas contains a set of exercises on Superdense Coding and Quantum Teleportation.
IBM’s Qiskit Textbook contains an introductory discussion of both Quantum Teleportation and Superdense Coding in Chapter 3.
