In quantum computing, the state of a qubit under superposition is described using a mathematical construct called a quantum state vector or simply a state vector. The state vector represents the quantum state of the qubit and contains information about the probabilities associated with different possible measurement outcomes.
Mathematically, a qubit under superposition can be expressed as a linear combination of its basis states, typically denoted as |0⟩ and |1⟩. These basis states represent the two possible classical states of the qubit: |0⟩ corresponds to the state "0" and |1⟩ corresponds to the state "1". The state vector of a qubit in superposition is given by:
|ψ⟩ = α|0⟩ + β|1⟩
Here, α and β are complex numbers, known as probability amplitudes, and represent the coefficients of the basis states. The probability of measuring the qubit in state |0⟩ is given by |α|^2, and the probability of measuring it in state |1⟩ is given by |β|^2. The total probability of obtaining one of the measurement outcomes must be equal to 1, so |α|^2 + |β|^2 = 1.
The complex probability amplitudes α and β encode both the magnitude and phase information of the quantum state. The phase information allows for the interference and superposition effects that are unique to quantum systems.
It's important to note that the state vector represents the qubit's quantum state in an abstract mathematical sense and doesn't directly correspond to physical properties like the position or momentum of a particle. Instead, it describes the probabilities associated with measurement outcomes when the qubit is observed.
The state vector evolves over time according to the rules of quantum mechanics, with operations like gates and measurements modifying the probability amplitudes and affecting the subsequent behavior of the qubit. By applying quantum gates and manipulating the state vector, quantum computations can be performed exploiting the properties of superposition and entanglement to achieve computational advantages over classical systems.