A quantum state is the complete description of a quantum system's properties at a given moment. For a single qubit, the quantum state is a vector in a two-dimensional complex Hilbert space, written as |ψ⟩ = α|0⟩ + β|1⟩, where α and β are complex numbers satisfying |α|² + |β|² = 1. This state encodes not just the probabilities of measurement outcomes but also the phase relationships between components, which are crucial for quantum interference and computation.
For multi-qubit systems, the quantum state lives in an exponentially large Hilbert space. An n-qubit system requires 2ⁿ complex amplitudes to fully describe — 50 qubits need over 10¹⁵ amplitudes, each a complex number requiring 16 bytes, totaling petabytes of classical memory. This exponential scaling is both the source of quantum computing's power and the reason classical simulation of quantum systems is intractable for large numbers of qubits.
Quantum states can be pure (described by a single state vector) or mixed (a statistical ensemble of pure states, described by a density matrix). In practice, decoherence causes pure states to evolve into mixed states, losing quantum information to the environment. Quantum state tomography — the experimental reconstruction of a quantum state from repeated measurements — requires exponentially many measurements for full characterization, though techniques like randomized benchmarking and shadow tomography provide efficient partial state information.