Entanglement

One of the most important two qubit states is the Bell state or the EPR pair. Such a state is defined as:

..........(1)

The above state is not a tensor product state . The EPR (Einstein-Podolsky-Rosen). Their experiment demonstrated that a quantum mechanical wavefunction does not provide a complete description of physical reality. Later in 1964, J.S Bell, proved that the EPR argument was incompatible with quantum mechanics.
The above state is very interesting for quantum computation and is mainly responsible for quantum teleportation and superdense coding. In the above state, by performing a measurement on one of the qubits, the state of the other qubit can be determined. If a measurement on qubit 1 is performed and it is found to be in state ∣0〉, then qubit 2 will also be in state ∣0〉. The post measurement state will be ∣00〉. As a result, measurement on the second qubit yields the same result as the first. Thus, by determining the state of the first qubit, the state of second qubit can be determined without performing any measurements on the second. That is, the measurement outcomes are correlated. These correlations are very strong, and such strong correlation could never exist in classical systems. These correlations are also called entangled states. Such correlations between the qubits exists in quantum computers and are responsible for making quantum computers more powerful than classical computers.
Consider an “n” qubit multiple state, ∣x₁, x₂, x₃, x₄…〉. Such a multiple qubit state can be expressed with a 2n computational basis and has 2n amplitudes. This means that 2n complex numbers can be stored by using only “n” qubits. For example, consider n=1000, where the number of complex numbers are 2¹⁰⁰⁰, which is a remarkably a large number. In a classical computer, n bits can store only n computational basis states at a time. Storing this many complex numbers on a classical computer is simply impossible. This phenomenon is called quantum parallelism, and this may result in a quantum computer delivering an exponential increase in processing power as compared to a classical computer.