Superposition and entanglement

Superposition

One of the properties that sets a qubit apart from a classical bit is that it can be in superposition. Superposition is one of the fundamental principles of quantum mechanics. In classical physics, a wave describing a musical tone can be seen as several waves with different frequencies that are added together, superposed. Similarly, a quantum state in superposition can be seen as a linear combination of other distinct quantum states. This quantum state in superposition forms a new valid quantum state.

A typical example visualizing superposition is the double-slit experiment. This experiment is explained in the following video:

Qubits can be in a superposition of both the basis states $\left\lvert 0 \right\rangle$ and $\left\lvert 1 \right\rangle$. When a qubit is measured (to be more precise: only observables can be measured), the qubit will collapse to one of its eigenstates and the measured value will reflect that state. For example, when a qubit is in a superposition state of equal weights, a measurement will make it collapse to one of its two basis states $\left\lvert 0 \right\rangle$ and $\left\lvert 1 \right\rangle$ with an equal probability of 50%. $\left\lvert 0 \right\rangle$ is the state that when measured, and therefore collapsed, will always give the result 0. Similarly, $\left\lvert 1 \right\rangle$ will always convert to 1.

Quantum superposition is fundamentally different from superposing classical waves. A quantum computer consisting of $n$ qubits can exist in a superposition of $2^n$ states: from $\left\lvert 000... 0 \right\rangle$ to $\left\lvert 111... 1 \right\rangle$. In contrast, playing $n$ musical sounds with all different frequencies, can only give a superposition of $n$ frequencies. Adding classical waves scales linear, where the superposition of quantum states is exponential.

Entanglement

One of the other counter-intuitive phenomena in quantum physics is entanglement. A pair or group of particles is entangled when the quantum state of each particle cannot be described independently of the quantum state of the other particle(s). The quantum state of the system as a whole can be described; it is in a definite state, although the parts of the system are not.

When two qubits are entangled there exists a special connection between them. The entanglement will become clear from the results of measurements. The outcome of the measurements on the individual qubits could be 0 or 1. However, the outcome of the measurement on one qubit will always be correlated to the measurement on the other qubit. This is always the case, even if the particles are separated from each other by a large distance. Examples of such states are the Bell states.

For example, two particles are created in such a way that the total spin of the system is zero. If the spin of one of the particles is measured on a certain axis and found to be counterclockwise, then it is guaranteed that a measurement of the spin of the other particle (along the same axis) will show the spin to be clockwise. This seems strange, because it appears that one of the entangled particles “feels” that a measurement is performed on the other entangled particle and “knows” what the outcome should be, but this is not the case. This happens, without any information exchange between the entangled particles. They could even be billions of miles away from each other and this entanglement would still be present.

Einstein was confused, not the quantum theory - Stephen Hawking

A common misunderstanding is that entanglement could be used to instantaneously send information from one point to another. This is not possible because although it is possible to know the state of the other particle when measuring one, the measurement results of the individual particles are random. There is no way to predetermine the individual result, therefore it is not possible to send a message in this way.

The power of quantum

The fact that qubits can be entangled, makes a quantum computer more powerful than a classical computer. With the information stored in superposition, some problems can be solved exponentially faster.