Qubit Measurement

Why measure?

Until this point, we have talked a lot about qubits, their states, and how they differ from classical bits. But the entire discussion would be meaningless if we did not have a way to extract information from these qubits. To extract information, we measure qubits.

What does a quantum state mean?

Let’s begin by summarizing what we have learned about quantum states up to the last lesson.

Qubits are represented by two-dimensional vectors in a complex vector space. With components in each direction $|0\rangle$ and $|1\rangle$ are weighted by complex numbers, $\alpha$ and $\beta$, also called the amplitudes of the components. These amplitudes have a normalization constraint on them. That is, the sum of the squares of the states’ amplitudes must have a sum of one. Mathematically, for an arbitrary state $|\phi\rangle$, this is described as follows:

$$|\phi\rangle=\alpha|0\rangle+\beta|1\rangle;\space where\space |\alpha|^{2}+|\beta|^{2} = 1$$

Now let us dive further into the meaning of a quantum state. An arbitrary state, say $|\phi\rangle$, of a qubit, is a probability wave or wavefunction. This probability wave describes the probability of finding the qubit in one state out of all the possible states it can be in. The amplitudes described previously are called the probability amplitudes, and they dictate the state we shall find (or measure) our qubit to be in.

This is where the normalization constraint stems from. Probability ranges from 0 through 1, inclusive. Therefore, the sum of squared probability amplitudes, taken together, must always be equal to 1.

What is a measurement?

We are now ready to put together the pieces of information from the preceding text to formulate one of the most fundamental operations in quantum computing – measurement.

Measurement allows us to extract information from qubits in a quantum computer. Consider a qubit in state $|\phi\rangle =$ $\alpha|0\rangle +$ $\beta|1\rangle$. Measuring this system in the computational basis gives us one bit of classical information: the probability of finding the qubit in $|0\rangle$ with probability $\alpha^{2}$, and the probability of finding it in state $|1\rangle$ with probability $\beta^{2}$ ...