Two qubits
Whatever we have learnt till now is only about one qubit. What happens if we have more than one qubit? It is an important question because any quantum processor that is worth its salt should have quite a few qubits. For example, Google’s Sycamore has \(53\) qubits, and IBM’s Hummingbird has \(65\) qubits. IBM is also planning to build a \(1000-\)plus qubit processor by 2023.
Consider a system of two qubits. The qubits could be physically realised as the left and right polarisations of a photon, or the up and down spins of a spin-\(1/2\) particle, or by means of any other two state quantum mechanical system.
For simplicity, let’s take a step back, and start with two classical bits, which can be in states \(0\) and \(1\). Since they are classical bits, there is no superposition state. The possible states of this two-bit system are: \(00, 01, 10, 11\). Therefore, the computational basis states of the corresponding two-qubit system are: \(|0 \rangle |0 \rangle\), \(|0 \rangle |1 \rangle\), \(|1 \rangle |0 \rangle\), and \(|1 \rangle |1 \rangle\).
The two-qubit system can also exist in a state that is a linear combination of the computational basis states. Such a state can be represented by:
\[\begin{equation} |\psi \rangle = \alpha_{00}|0 \rangle |0 \rangle + \alpha_{01} |0 \rangle |1 \rangle + \alpha_{10} |1 \rangle |0 \rangle + \alpha_{11} |1 \rangle |1 \rangle \end{equation}, \label{eq:two-qubits}\]where \(\alpha_{00}, \alpha_{01}, \alpha_{10}\), and \(\alpha_{11}\) are complex numbers, and are called amplitudes. The normalisation condition now becomes,
\[\begin{equation} |\alpha_{00}|^{2} + |\alpha_{01}|^{2} + |\alpha_{10}|^{2} + |\alpha_{11}|^{2} = 1 \end{equation}.\]From Eq. \eqref{eq:two-qubits}, it is clear that four complex numbers are required to describe the state of a two-qubit system.
Suppose we want to examine the state of our system of two qubits. This obviously involves measuring the state of each qubit, one after the other. What is the probability of finding the first qubit in state \(|0 \rangle\)?
Since we are interested in those states in which the first qubit is in state \(|0 \rangle\), we can neglect the last two terms in Eq.\eqref{eq:two-qubits}. Now, we can write,
\[\begin{equation} \begin{aligned} \left( \begin{array}{c} \text{Probability of} \\ \text{finding the first} \\ \text{qubit in state } |0 \rangle \end{array} \right) &= % \left( \begin{array}{c} \text{Probability of} \\ \text{getting } \\ |0 \rangle |0 \rangle \end{array} \right) + % \left( \begin{array}{c} \text{Probability of} \\ \text{getting } \\ |0 \rangle |1 \rangle \end{array} \right) \\[10pt] % &= |\alpha_{00}|^{2} + |\alpha_{01}|^{2} \end{aligned} \end{equation}.\]After the measurement, the system collapses to the state:
\[\begin{equation} |\psi' \rangle = \frac{\alpha_{00} |0 \rangle |0 \rangle + \alpha_{01} |0 \rangle |1 \rangle} {\sqrt{|\alpha_{00}|^{2} + |\alpha_{01}|^{2}}} \end{equation}.\]That is, the post-measurement state is re-normalised by the factor \(\sqrt{|\alpha_{00}|^{2} + |\alpha_{01}|^{2}}\).
Bell State
Bell state (or EPR Pair) is an interesting two-particle system in quantum mechanics. According to Nielsen and Chuang, “Bell state is responsible for many surprises in quantum computation and quantum information”. It is represented by,
\[\begin{equation} |B \rangle = \frac{1}{\sqrt{2}}|0 \rangle |0 \rangle + \frac{1}{\sqrt{2}}|1 \rangle |1 \rangle \end{equation}.\]If we make a measurement on a two-qubit system in Bell state, then there is a 50% chance of finding both the qubits in state \(|0 \rangle\), and 50% chance of finding both of them in state \(|1 \rangle\).
Here comes the interesting part. If we make a measurement, and find the first qubit to be in state \(|0 \rangle\), then the post measurement state would be \(|0 \rangle |0 \rangle\). Because of this, the measurement of the second qubit always gives the same result as the measurement of first qubit. That is, the measurement outcomes are correlated. Similar thing happens when we find the first qubit to be in state \(|1 \rangle\).
Reference:
- Nielsen and Chuang. Quantum Computation and Quantum Information