In classical computing, a bit can represent one of two states: 0 or 1. However, in quantum computing, a quantum bit, or qubit, can exist in a superposition of multiple states. The number of different states that can be encoded into one qubit depends on its dimensionality.
In a two-level quantum system, which is the most common representation of a qubit, the qubit can exist in a superposition of two states. This means that it can simultaneously represent both 0 and 1 with varying probabilities. Mathematically, a qubit can be expressed as a linear combination of the two basis states, often denoted as |0⟩ and |1⟩:
|ψ⟩ = α|0⟩ + β|1⟩,
where α and β are complex numbers called probability amplitudes. The probabilities of measuring the qubit in the states |0⟩ and |1⟩ are given by the squared magnitudes of the amplitudes, |α|^2 and |β|^2, respectively. The sum of the squared magnitudes of the probability amplitudes must be equal to 1.
So, in a two-level quantum system, one qubit can represent an infinite number of possible states within the complex plane of its probability amplitudes. However, upon measurement, the qubit will collapse to one of the two basis states with a probability determined by the amplitudes.
It's worth noting that in higher-dimensional quantum systems, such as systems with more energy levels or systems composed of multiple qubits, the number of possible states that can be encoded becomes exponentially larger. For example, with two qubits, the system can represent four possible states simultaneously, and with three qubits, it can represent eight states, and so on, with the number of possible states growing exponentially with the number of qubits.
In summary, a single qubit in a two-level quantum system can represent a superposition of two states, whereas the number of possible states that can be encoded grows exponentially with the number of qubits in higher-dimensional quantum systems.