| quantum state | |
|---|---|
| Name | Quantum state |
| Field | Quantum mechanics |
| Introduced | 20th century |
| Notable examples | qubit, coherent state, Bell state |
quantum state
A quantum state is a mathematical object that encapsulates the complete information about a physical system available within Quantum mechanics. It determines probabilities for outcomes of all possible measurements and underpins technologies from quantum computing to quantum cryptography. Understanding quantum states is central to debates in philosophy of science and to equitable deployment of quantum technologies in society.
A quantum state is represented in formalism either by a normalized vector |ψ⟩ in a Hilbert space ℋ (a pure state) or by a positive, trace‑one operator ρ (a mixed state or density matrix) acting on ℋ. The state encodes expectation values via the Born rule: expectation of an observable A is Tr(ρA) for density operators or ⟨ψ|A|ψ⟩ for state vectors. The mathematical framework relies on linear algebra, functional analysis, and representations of Lie algebras for symmetry operations. Important named constructs include the Schrödinger picture, the Heisenberg picture, and the Dirac notation used to manipulate state vectors and operators. Classic works formalizing these ideas include Dirac's Principles of Quantum Mechanics and von Neumann's formal treatment.
Interpreting what the quantum state "means" has generated competing frameworks: the Copenhagen interpretation emphasizes operational probabilities, the many-worlds interpretation treats |ψ⟩ as universally real, and QBism and relational views focus on information and agents. Representations include wavefunctions ψ(x) in position space, momentum‑space representations via the Fourier transform, occupation number (Fock) states for quantum fields, coherent states for quantum optics, and Bloch sphere visualization for a qubit. The quantum state also appears in algebraic quantum field theory and in the formalism of density functional theory when bridging to many‑body systems.
Measurement maps a quantum state to outcome probabilities; the textbook postulate invokes nonunitary "collapse" of a pure state to an eigenstate of the measured observable. The Born rule quantifies probabilities, while projective measurement and generalized POVMs formalize outcomes. Decoherence explains apparent collapse via entanglement with an environment (described by system–environment density operators and partial traces), giving emergent classicality without invoking physical collapse. Research at institutions like CERN, IBM Quantum, and Quantum Information and Computation groups explores measurement backaction and experimental tests of collapse models (e.g., GRW theory).
Mixed states model statistical ensembles, noise, or ignorance and are described by density operators ρ = Σ p_i |ψ_i⟩⟨ψ_i|. Measures of mixedness include purity Tr(ρ^2) and von Neumann entropy S(ρ) = −Tr(ρ ln ρ), central to quantum thermodynamics and resource theories. Connections exist to Shannon entropy in classical information theory and to operational tasks in quantum information theory, such as quantum channel capacities and entanglement measures. Prominent results include Holevo's theorem and the formulation of quantum statistical mechanics by von Neumann and Ludwig Boltzmann‑inspired approaches.
Closed quantum systems evolve unitarily under the Schrödinger equation with Hamiltonian H: iħ ∂|ψ⟩/∂t = H|ψ⟩. Open systems require completely positive, trace‑preserving maps and master equations (e.g., the Lindblad equation). Time evolution can be studied via propagators, path integrals introduced by Richard Feynman, and numerical methods such as tensor networks and DMRG for many‑body states. Symmetry operations correspond to unitary or antiunitary operators per Wigner's theorem; conservation laws follow from Noether's theorem applied to state evolution.
Entanglement is a property of composite quantum states where joint states cannot be factorized into subsystem states; it fuels protocols like quantum teleportation, superdense coding, and quantum key distribution (e.g., BB84 uses nonorthogonal states). Entangled states lead to violations of Bell's theorem and experimental tests by groups such as those at Caltech and University of Vienna. Quantification uses entropies, entanglement of formation, and operational quantities like entanglement distillation and concurrence. Applications target equitable benefits—deploying quantum sensors and secure communication to underserved communities—while addressing risks of concentration of technological power by corporations or states.
Preparing quantum states uses techniques across platforms: superconducting circuits by Google Quantum AI and Rigetti Computing, trapped ions at IonQ and academic labs (e.g., NIST), photonic states via nonlinear optics and waveguides, and cold atoms in optical lattices developed by groups at MIT and Max Planck Institute for Quantum Optics. Quantum state tomography reconstructs ρ from measurements using maximum likelihood or compressed‑sensing methods. Applications include quantum simulation of materials, quantum metrology with squeezed and NOON states, and nascent quantum computing algorithms (e.g., Shor's algorithm, Grover's algorithm). Ethical deployment requires public engagement, workforce diversity, and policies from bodies like the National Science Foundation to ensure benefits are widely distributed.
Category:Quantum mechanics Category:Quantum information science