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Note: This thread will be talking only about the fundamental concepts needed to understand superdense coding, entanglement, and quantum teleportation. I will actually be discussing superdense coding and teleportation themselves in the next thread, but it is definitely a good idea to have a decent foundation of what is actually going on.

The Qubit

In a classical understanding of information theory, the bit is the basic unit. It consists either of a 1 or a 0. These are the simplest forms of information that can exist. In Quantum Mechanics, as we all know, this absolute state of a system is often nonsensical. Instead, it is displayed in a linear algebra Bra-Ket (Dirac) notation as shown below. Since quantum states are vectors, the state of one qubit can be shown as a 2-Dimensional vector in a complex coordinate plane:

Simply stated, the quantum state of a system/quibit is represented as a unit vector in a 2D complex coordinate plane. It may also be notable to state that you can never measure the alpha and beta directly. When you measure the state, by the laws of Quantum Mechanics, it will collapse into a single state. This has nothing to do with the precision of our measurement utilities, either. This is simply a fundamental law of the quantum world. Since it is a unit vector, it is subject to the Normalization Constraint where the amplitude of the probability wave is alpha/beta squared and the summation of the two must be equal to one.

Simple Quantum Logic (Single Qubits)

As with classical information theory, qubits need to go through certain computations to be used in quantum computing or transportation. There are several logical operators, so let's go through a few of the major basic ones:

- Not (X) - also known as the X gate. The X gate is passed through a simple 2-D matrix. A state 0 -> 1 and a state 1 -> 0. Any qubit in a superposition will act linearly and the |0> and |1> states will reverse, essentially flipping the alpha/beta amplitude coefficients. This is the equivalent of rotating the unit vector π radians around the X axis.

- Hadamard (H) - The Hadamard gate is meant to essentially change the number of states that a single qubit can be in. If it is in a ground state, it raises the probability to 50% of each state which would mean alpha and beta are both + or - 1/sqrt(2). It is equivalent of rotating π radians around the (x+z)/sqrt(2) axis.
- Y Gate (Y) - The Y gate is another complex matrix with an imaginary number, i. This is equivalent to rotating the vector π radians around the Y axis.
- Z Gate (Z) - The Z gate is the last single-quibit complex matrix gate I'll discuss. This is equivalent to rotating the vector π radians around the Z axis. The |0> state is conserved and the |1> state is changed.

You certainly don't have to understand everything about the quantum computational gates, but what you should take away is that quantum states, which are 2D vectors, can be changed by passing them through matrices (gates). As a matter of fact, this is rather simple. There are also daggers that represent the complex conjugate transposed matrix which, in basic terms, all the rows become columns and columns become rows, and the value of i in any complex polynomials within the matrix is reversed. If you would like to do the math of these simple matrices above, you can see that some of these gates are reversible. For example, running it through the same gate twice like XX and HH (among others) will produce the same quantum state that was passed through.

Unitary Matrices

I apologize to those of you with multiple years of calculus. You may want to skip this boring stuff. This is essentially as advanced as my vector calculus gets, so if you already understand basic vector calculus and matrix transposition and complex conjugation, you're good to go. That being said, let's define our terms:

A Unitary Matrix is any matrix, U, whose transposed complex conjugate (U^dagger), multiplied to the original matrix, will result in the Identity Matrix. What's interesting is that a Quantum State result MUST be normalized because it is a unit vector, so anything that comes in must have the same Norm as its result. This is perfect. The following is a simple proof:

This identity simply means that any vector, after being applied to any unitary matrix will have the same Norm as the original vector. Again, I apologize if you're far ahead of this. As complicated as it looks, it's not all that difficult and this can all be learned and understood in a matter of a week or so. There is a way of writing a proof for this in Sigma notation, but that's simply unnecessary for this thread. Here is a great paper on matrix multiplication for those interested:

Matrices.pdf

As well as a great video:

Matrix multiplication using index notation (MathsCasts)

2 Qubit Logic

This is actually very important because this is an introduction into what Quantum Entanglement will mean. Here we will see just one single double-qubit logical gate called the Controlled NOT gate. First, you must understand all 4 possible states (with a 2 qubit system... possible states are 2^n) which are 00, 01, 10, and 11. If you understood the previous sections (at least the quantum logic part), this will be no problem. In a cNOT gate, you have the control bit (the first one) and the target bit (the second one). In simplest terms, if the control bit is 1, then the target bit takes the NOT value while the first bit stays the same. If the control bit is 0, then the state of both bits stays the same.

Entangled Particles

The very interesting thing comes when we pass one qubit that is a part of a multi-qubit system through a single-bit gate (such as the Hadamard) and then pass both (or all) of the qubits through the CNOT Gate. We come out with particles that are ENTANGLED. The most common way to entangle particles in QC is sending a 00 state through the Hadamard gate, then through the CNOT gate. Our result is the following:

This entangled state means that the two particles are intertwined in such away that if something can be known or changed about one particle, there is immediately something that can be known or changed about the other. This is instantaneous. It is simply IMPOSSIBLE to describe the states separately. For example, the famous Bell States cannot be factored.

I hope you guys enjoyed this introduction. In the next thread, I'll be going over superdense coding (transferring 2 classical bits with a single entangled qubit)