Superdense Coding

We're now getting pretty familiar with the fact that a qubit can hold any state in a superposition of $\ket{0}$ and $\ket{1}$. Whereas a classical bit can only occupy one of these two states, a qubit can occupy an infinite contiuum of states between these two. It therefore feels like the qubit contains more information than the classical bit. Is there someway we could somehow extract use out of this information?

In the previous article we discussed how Alice and Bob could use a shared entagled pair of qubits to communicate using quantum teleportation. We found that they would need to communicate two classical bits in order to decode a sent quantum state.

In this article we will look at the opposite scenario. Where Alice and Bob use their entangled qubit pair to allow them to send 2 classical bits of information while only having to send a single qubit.

We'll begin in the familiar scenario where Alice and Bob have entangled two qubits and taken one half of the entangled pair each.

So our system starts in the usual state:

Now, let's say that Alice has 2 classical bits of information that she wants to be able to send to Bob by only sending her qubit. In other words, Alice must apply some operation to her qubit such that Bob can apply some operation to both his qubit and Alice's qubit in order to decode her 2 classical bits.

Remember in our quantum teleporation protocol, Bob had to apply an $X$ and/or a $Z$ gate to his qubit based on the received classical bits. Let's re-use that intuition here, and we'll make Alice apply an $X$ and/or $Z$ gate depending on the 2 classical bits she wants to send.

We'll denote the two classical bits Alice wants to send as $b_0$ and $b_1$. So Alice will apply the operation $U_{b_0b_1}$ to her qubit where:

In plain terms, this means apply an $X$ gate if the second bit is 1 and apply a $Z$ gate if the first bit is 1. The $X$ gate is always applied first.

At this point Alice has operated on her qubit and is ready to send it to Bob. We can describe the complete state of the system as:

The circuit we have so far looks like this:

Notice how this appears very similar to our quantum teleportation circuit but backwards! Instead of Bob applying $X$ and $Z$ at the end of the circuit, Alice applies them at the beginning.

To understand what Bob must now do to retrieve the bits we can get some inspiration from the teleportation circuit and "break" the entanglement by applying a $\text{CNOT}$ gate followed by an $H$ gate:

And here's what the circuit looks like now:

We could start delving into our outer-product operator notation to explicitly calculate what this equals, but it's more helpful to study this in cases. The full outer-product based derivation is left as an exercise at the end.

First off, the most basic case, when both of Alice's bits are 0. i.e. $b_0 = 0$ and $b_1 = 0$.

In this scenario, we see that $U_{00} = I$ So we can continue on with our equation from there:

So by simply measuring his qubits, Bob will recover Alices 2 bits! This is great, but we need to check it works in the other cases before we can be sure this is a valid method. Let's now look at the case when $b_0 = 1$ and $b_1 = 0$:

In this case, $U_{01} = X$:

So once again Bob will recover the correct qubits! We'll leave the other two cases to the exercises.

So we have shown that Alice can send 2 classical bits to Bob by only sending a single qubit. Note, that this doesn't mean we encoded 2 classical bits of information inside a single qubit. Alice and Bob still had to meet physically before hand to create and share a pair of entangled qubits.

Also note that in communicating the 2 classical bits, Alice had to send her qubit and Bob "broke" the superposition. This means that this is a one time use protocol. If Alice wants to send another 2 bits to Bob, they will have to create a new entangled pair of qubits.

So now we have shown two very important facts. That if Alice and Bob share a pair of entangled qubits, they can:

• Communicate a qubit by physically sending 2 classical bits of information
• Communicate 2 classical bits of information by physically sending a qubit

Why is this significant?

Well two classical bits of information can have 4 different states, but a single qubit has a infinite continuum of different possible states. We might have assumed we needed an arbitrarily large amount of information to communicate a qubit, but now we know we can use entanglement to communicate it with only 2 bits. This is a huge reduction in the amount of information needed.

But it comes at the cost of the qubit being destroyed in the process. If the qubit wasn't destroyed we would violate the no-cloning theorem.

Exercises

Exercise 1

Exercise 2