| Quantum states | |
|---|---|
| Name | Quantum state |
| Field | Quantum physics |
| Introduced | 1920s |
| Notable people | Paul Dirac, John von Neumann, Erwin Schrödinger, Max Born |
Quantum states A quantum state is a mathematical object that fully describes the properties and statistical behavior of a physical system in quantum mechanics. Quantum states determine the probabilities of outcomes for all possible measurements, underpinning technologies such as quantum computing and quantum cryptography. Understanding quantum states is central to foundations, experiments at institutions like CERN and IBM Quantum, and theoretical frameworks developed by scientists such as Paul Dirac and John von Neumann.
A quantum state assigns expectation values and probability distributions for observables represented by operators on a Hilbert space. In the canonical formalism developed by Paul Dirac and John von Neumann, pure quantum states are represented by rays (equivalence classes of vectors) in a complex separable Hilbert space or by normalized state vectors |ψ⟩ in Dirac notation. Statistical mixtures are described by density operators ρ, positive trace-class operators with trace one. The Born rule, introduced by Max Born, connects the mathematical representation to experimental probabilities: for an observable represented by a self-adjoint operator Â, the probability of outcome in a spectral projection P is Tr(ρP). The formalism accommodates continuous variables via wave functions ψ(x) in position representation and discrete bases such as the spin basis for two-level systems or qubits.
Pure states correspond to maximal information about a system and are represented by one-dimensional projectors |ψ⟩⟨ψ|. Mixed states model classical uncertainty or entanglement with inaccessible subsystems and are convex combinations of pure states. The distinction is operational: a state ρ is pure iff Tr(ρ^2)=1; mixed states have Tr(ρ^2)<1. Historical work by John Bell and later developments in quantum information theory formalized properties like purity, entropy measures such as the von Neumann entropy, and resource theories. Experiments at facilities such as National Institute of Standards and Technology (NIST) and platforms like trapped-ion and superconducting qubit architectures routinely prepare and characterize both pure and mixed states.
Closed quantum systems evolve unitarily according to the Schrödinger equation (or equivalently via the unitary operator U(t)=e^{-iHt/ħ} generated by the Hamiltonian H). Open systems interacting with environments are described by quantum dynamical semigroups and master equations such as the Lindblad equation, developed in the context of quantum optics and decoherence studies. Techniques from quantum control and adiabatic theorems (e.g., Born–Oppenheimer approximation) guide state manipulation in experiments by groups like D-Wave Systems and research labs at MIT and Harvard University. Time evolution preserves purity for unitary dynamics but generally increases entropy under irreversible interactions with reservoirs modeled by completely positive, trace-preserving (CPTP) maps, also called quantum channels.
Observables are represented by self-adjoint operators; measurement outcomes follow the Born rule. Ideal projective measurements are formalized by PVMs while generalized measurements use POVMs and Kraus operators to represent realistic detectors. Measurement can induce state update rules often called "collapse"; in von Neumann's measurement model collapse maps a pre-measurement pure state to an eigenstate corresponding to the observed eigenvalue. Alternatives to collapse include decoherence theory (explored by e.g. Wojciech Zurek) and measurement models in foundations of quantum mechanics. Experimental tests using Bell tests, weak measurement, and quantum non-demolition measurement probe the boundary between unitary evolution and collapse.
Composite systems are described by the tensor product of subsystem Hilbert spaces. Entanglement, first highlighted in the Einstein–Podolsky–Rosen paradox and formalized by Erwin Schrödinger, is a nonclassical correlation of quantum states with no classical analogue. Entangled states such as Bell states and the GHZ state are resources for protocols including quantum teleportation and superdense coding. Measures of entanglement include entanglement entropy, concurrence, and entanglement of formation. Multipartite entanglement is exploited in quantum error correction codes like the Shor code and in proposals for quantum metrology and quantum simulation by groups at Caltech and Google Quantum AI.
Quantum states are the carriers of quantum information. The qubit is the basic unit; protocols such as quantum key distribution (e.g., BB84) use specific state ensembles for security. Characterizing unknown states uses quantum state tomography, which reconstructs ρ from measurement data via techniques including maximum likelihood estimation and compressed sensing. Recent advances include randomized benchmarking, direct fidelity estimation, and machine-learning-assisted tomography, implemented on platforms by IBM Quantum and academic groups at University of Oxford and University of Toronto. Quantum coding theorems, such as the Holevo bound and quantum noiseless coding theorem, connect state ensembles to information capacities of quantum channels.
Quantum states play a central role in interpretational debates: are they epistemic (states of knowledge) or ontic (states of reality)? Competing views include the Copenhagen interpretation, many-worlds interpretation (Everettian), de Broglie–Bohm theory, and epistemic approaches such as QBism. No-go theorems like the Pusey–Barrett–Rudolph theorem constrain epistemic models by showing under certain assumptions that quantum states cannot be interpreted merely as statistical information about underlying hidden variables. Foundational research connects to experiments testing contextuality (e.g., Kochen–Specker theorem) and locality (e.g., Bell's theorem), shaping how communities at Perimeter Institute and Institute for Quantum Optics and Quantum Information think about the ontological status and operational utility of quantum states.
Category:Quantum mechanics Category:Quantum information theory