Darren Jung's Short Thoughts

References

http://en.wikipedia.org/wiki/Superdense_coding

http://www.quantiki.org/wiki/index.php/Super-dense_coding

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Superdense coding and quantum  teleportation are maybe the most popular motivational example of quantum information theory.

I’ll first talk about the superdense coding.

For superdense coding, consider Alice and Bob (two most popular people in the world of quantum computing examples)

Suppose Alice is sending one qubit to Bob.

Alice >>>>>>>>>>(qubit)>>>>>>>>>>> Bob

Can Alice convey two classical bits using just one qubit?

The answer is NO, and Holevo’s theorem says this is impossible.

Then, what makes a qubit more interesting than classical bits?

Now we can say that Bob, is allowed to send one qubit to Alice before Alice send a qubit to Bob.

1. Alice <<<<<<<<<<<(qubit)<<<<<<<<<<<< Bob

2. Alice >>>>>>>>>>>(qubit)>>>>>>>>>>>> Bob

Then it turns out that Alice can convey two classical bits to Bob.

How can this help? It doesn’t make much sense logically since allowing Bob to send a qubit to Alice seems to have no impact on the information that Bob is receiving from Alice. However, with entanglement, sending a qubit to Alice can somehow help on sending richer information to Bob.

But we need a setup. Bob should have 2-qubit state beforehand.

Assume that Bob’s two qubits are entangled with state |00> + |11>. Bob sends the first qubit to Alice.

Then Alice can apply unitary rotation on that qubit. Specifically, if first bit is 1, then apply X = [0 1; 1 0] to the qubit. If second bit is 1 then apply Z = [1 0; 0 -1].

Now Alice can send that qubit back to Bob.

Bob can measure the qubit in Bell basis to figure out two bits.

• state is |00> + |11> then first bit = 0, second bit = 0
• state is |00> – |11> then first bit = 0, second bit = 1
• state is |01> + |10> then first bit = 1, second bit = 0
• state is |01> – |10> then first bit = 1, second bit = 1

The intuition here is that applying rotation on Bob’s first qubit have some impact on his second qubit as well since two are entangled. Then we can rotate the two-qubit system on two orthogonal spaces to make distinction between all 4 cases.

Quantum Computing

I’ll start off with the very basic unit of quantum computing… qubit.

reference information : http://en.wikipedia.org/wiki/Qubit

The information here is solely by my understanding and may contain some incorrect ideas.

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What’s the opposite word for the quantum bit?

yes — it’s a classical bit. Classical bits are the most basic unit of information. It can be represented by either 0 or 1.

Quantum bit, or qubit, is pretty much the same — it’s the most basic unit of quantum information.

However, the difference here is that qubit is represented as a linear superposition of two states, |0> and |1>.

Classical bit of 0 is same as the quantum state |0>, and same for 1 and |1>.

Something like <1|, |0>, <1|2> are called a bra-ket notation. They basically represent certain states.

We go back to the qubit… for qubit, we understand it as combination of two states, with probability of being a state |0> or |1>.

Unlike classical bits, the value of qubit is unknown until we measure it. Once we measure the qubit, it can be either |0> or |1>, in either case the state is fully determined, which means that the value of qubit is now known as a 0 or 1 and there is no uncertainty.

One can imagine a black-box, in which there is either white or black ball in it. Nobody knows which ball is in the box, but once someone open it (measurement) we all know which ball is in the blackbox, and you can’t put it back to the blackbox to play the game again because we already know the colour of a ball and it’s determined.

For qubits, of course the probability of being state |0> and the probability of being state |1> should add up to 1.

There’s another strange property of qubits, which is called entanglement. Entangled qubits means two or more qubits are somehow tied up together, and cannot be separated. For example |00>+|11> is a two-qubit system, but this is an entangled state since we cannot factor out two states.

In contrast, the example of non-entangled state would be |00> + |01> + |10> + |11> since it can be factored out as (|0> + |1>) (|0> + |1>).

We can do many things that classical bits cannot do using this strange property of entanglement.

Quantum Computing

Quantum computing is either all or nothing.

That’s why I like it.

It could be extremely useful and yield a revolutionary change for the study of computer science,

Or it could not be useful at all and just be a garbage.

Under this category I’ll post something that I think interesting regarding to the quantum information theory.

Quantum Computing