A quantum state is the mathematical and physical description of the condition of a quantum system that contains the information required by the theory to predict the probabilities of possible measurements. For a pure state, it can be represented by a state vector in a Hilbert space, commonly written using Dirac ket notation. For systems described statistically or entangled with an unobserved environment, a density operator or density matrix provides the more general description.
The quantum state is not simply a list of classical properties such as exact position, velocity and orientation. Instead, it encodes probability amplitudes, correlations and the possible outcomes of measurements. A quantum state can describe an electron, photon, atom, molecule, spin system, superconducting circuit or many particle system. The appropriate mathematical representation depends on the physical system and the observables of interest.
A pure quantum state is represented by a vector up to an overall physically irrelevant complex phase. If a state vector is multiplied by a global phase factor of unit magnitude, all ordinary measurement probabilities remain unchanged. Quantum states are normally normalised so that the total probability of all possible outcomes is one. A general state can be expressed as a superposition of basis states, with complex coefficients whose squared magnitudes contribute to measurement probabilities in that basis.
The choice of basis is a mathematical representation rather than a claim that the system physically contains separate classical components corresponding to every basis vector. Different bases can reveal different aspects of the same state. This basis dependence is important in quantum mechanics because a state that is definite with respect to one observable can be a superposition with respect to another incompatible observable.
The wave function is one important representation of a quantum state. For a nonrelativistic particle in position space, the wave function psi of x and t contains the probability amplitude associated with position. The Born rule states that the squared magnitude of the wave function gives the probability density for finding the particle at a particular location under the relevant measurement. The wave function can also be represented in momentum space through a Fourier transform, producing a momentum space amplitude.
These different representations contain equivalent information when transformed correctly. A wave function is generally complex valued, so its phase carries physical information even though the overall global phase is not observable. Relative phase between components of a superposition can affect interference and therefore measurable probabilities.
Quantum states evolve according to quantum dynamical laws. In ordinary nonrelativistic quantum mechanics, the Schrodinger equation governs the time evolution of a closed system. If the Hamiltonian is time independent, energy eigenstates evolve by phase factors determined by their energies. A superposition of different energy eigenstates develops changing relative phases, which can produce time dependent observable behaviour. More generally, time evolution of a closed quantum system is unitary, preserving the inner products and total probability structure of the state space. Open systems interacting with environments require more general descriptions because their reduced states can evolve nonunitarily when environmental degrees of freedom are ignored. Master equations, quantum channels and density matrices are used to describe such processes.
A quantum state determines the probabilities of measurement outcomes through the Born rule. If an observable has eigenstates forming an appropriate basis, the probability of obtaining a particular eigenvalue depends on the projection of the system state onto the corresponding eigenspace. Measurement can therefore be understood as an operation that extracts information from the state while generally changing the state as well.
In ideal projective measurement, an outcome projects the state into the associated eigenspace, although more general measurements are represented using positive operator valued measures and quantum instruments. These more general mathematical tools describe realistic measurements, noisy detectors and measurements involving ancillary systems. The relationship between state, observable and measurement is one of the defining features of quantum theory and distinguishes a quantum state from a classical hidden list of predetermined measurement results.
Composite quantum systems require special attention because their joint state can be entangled. If two systems A and B are independent, a product state can be written as the tensor product of a state for A and a state for B. More general joint states cannot necessarily be separated this way. Such states are entangled and contain correlations that are properties of the joint system rather than of either subsystem alone.
When only one part of an entangled system is considered, its reduced state is generally mixed even when the combined system is in a pure state. This illustrates why a complete description of a composite quantum system cannot always be constructed by assigning independent complete states to each component. Tensor products and density matrices are therefore essential tools for quantum information theory and many particle quantum mechanics.
Quantum states can differ according to the physical degrees of freedom they encode. A spin one half particle can have spin states relative to a chosen measurement axis. A photon can have polarisation states, spatial modes, frequency modes or combinations of these degrees of freedom. An atom can have electronic, vibrational and hyperfine states. A molecule can have rotational and vibrational quantum states.
A qubit is a two dimensional quantum system whose state can be represented as a superposition of two basis states. Larger quantum systems have exponentially growing Hilbert spaces in the number of idealised qubits, which is one reason simulating general quantum systems with classical computers can become computationally difficult. The physical state space nevertheless depends on the actual system and constraints, so not every mathematically imaginable state is necessarily physically accessible.
A quantum state should not be confused with a classical state, a probability distribution or a wave in ordinary space. Classical probability describes uncertainty about values that can, in principle within the model, coexist as definite alternatives, whereas a quantum state contains amplitudes whose phases can generate interference. A quantum state also does not necessarily correspond to a directly observable object.
It is a theoretical description used to calculate measurement statistics, and interpretations of what the state represents physically differ among interpretations of quantum mechanics. Despite these interpretive differences, the formal state concept is extremely precise and experimentally powerful. Quantum states provide the central language connecting superposition, uncertainty, entanglement, interference, measurement and quantum information. Understanding the quantum state is therefore essential to understanding essentially every other major concept in quantum mechanics.
