In quantum computing, the qubit is typically viewed as a discrete two-level system, much like a classical bit. It can represent either a 0 or a 1, just as a classical bit can represent either a 0 or a 1. This discrete nature of qubits forms the basis for quantum computation and quantum algorithms.
However, qubits also have the unique property of superposition, which sets them apart from classical bits. Superposition allows a qubit to exist in a linear combination of the 0 and 1 states. This means that a qubit can be in a state that is partially 0 and partially 1 simultaneously. Mathematically, we represent this superposition as a linear combination of the basis states |0⟩ and |1⟩, such as α|0⟩ + β|1⟩, where α and β are complex numbers that describe the probability amplitudes of the respective states.
The continuous-valued aspect of qubits might be related to the fact that the probability amplitudes α and β can take on any complex value, which includes both real and imaginary components. This allows for a continuous range of possible values for the probability amplitudes, unlike classical bits where the probability is either 0 or 1.
However, it's important to note that even though the probability amplitudes can take on a continuous range of values, the measurement outcomes of qubits are still discrete, yielding either a 0 or a 1 with certain probabilities. The continuous-valued nature of qubits primarily refers to the range of values that the probability amplitudes can assume, not to the measurement outcomes themselves.
In summary, qubits are fundamentally discrete two-level systems, but their ability to exist in superposition allows them to have a continuous range of probability amplitudes. This combination of discrete measurement outcomes and continuous-valued probability amplitudes forms the foundation of quantum computation and quantum algorithms.