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Showing 211–240 of 1709 reference entries

PhysicsRC charging and discharge

A simple resistor–capacitor circuit has an exponential timescale.

ModelPhysicsSeries RLC impedance magnitude

Sinusoidal response combines resistance and net reactance.

ModelPhysicsFourier heat conduction

A temperature gradient drives conductive heat flux.

ModelPhysicsLatent heat of a phase transition

Energy changes phase at the specified coexistence conditions.

ModelPhysicsApproximate vapor-pressure relation

Vapor pressure changes with temperature through latent enthalpy.

ModelPhysicsBlackbody total emission

Thermal equilibrium fixes an ideal absorber's total emitted power.

ModelPhysicsCanonical thermal ensemble

Fixed-temperature equilibrium weights states by energy.

ModelPhysics · InformationGibbs entropy

Microstate probabilities quantify statistical entropy.

DefinitionPhysicsQuantum observable expectations

A state and observable determine the mean of repeated outcomes.

ModelPhysicsRobertson uncertainty relation

A commutator constrains two outcome spreads in the same state.

ModelPhysicsSpin one-half and Pauli operators

A spin one-half state has two outcomes along a chosen axis.

ModelPhysics · InformationPure and mixed density operators

A density operator represents quantum statistical states.

DefinitionPhysicsIdeal projective measurement

Projectors determine outcome probabilities and conditional states.

ModelPhysics · InformationComposite states and entanglement

A joint pure state can fail to factor into subsystem states.

DefinitionPhysicsExchange symmetry of identical particles

Boson and fermion states respond differently to exchange.

ModelPhysics · Mathematics & logicGeodesic motion

Free test motion follows the metric connection.

ModelPhysicsEinstein gravitational field equations

Spacetime curvature responds to stress and energy.

ModelPhysicsSchwarzschild static-observer redshift

Clocks at different radii measure different light frequencies.

ModelPhysics · Earth & cosmosFriedmann cosmological evolution

Homogeneity and isotropy reduce gravitational expansion to two balances.

ModelMeasurementCalibration

Relate indications to standards, then infer measurement results.

DefinitionMeasurementType A and Type B uncertainty evaluation

The classification describes how a component is evaluated.

MethodMeasurementExpanded uncertainty and coverage

A coverage factor scales standard uncertainty to a reported interval.

MethodMeasurement · Mathematics & logicCovariance in uncertainty models

Shared input variation belongs in the joint uncertainty description.

MethodMeasurementMeasurement models

Specify how input quantities determine the measurand.

MethodMeasurement · InformationMonte Carlo uncertainty propagation

Sample the joint input model to approximate the output distribution.

MethodMeasurementRepeatability and reproducibility

Precision depends on which measurement conditions are held or changed.

DefinitionMeasurementResolution and sensitivity

Detectable change and response scale are different instrument properties.

DefinitionMeasurementDynamic instrument response

Response depends on timescale, gain and phase.

MethodPhysicsHydrogen Coulomb energy levels

The ideal one-electron atom has discrete bound energies.

ModelPhysics · ChemistryAtomic state labels and Pauli occupancy

Quantum labels specify orbitals and spin states.

Model
DefinitionSource checked

Composite states and entanglement

PhysicsInformation

A joint pure state can fail to factor into subsystem states.

Key knowledge

For specified finite-dimensional subsystems, the joint space is H_A⊗H_B. A pure state is entangled when it cannot be written as ψ_A⊗ψ_B; its subsystem state is obtained by partial trace.

Values and equations

H = H_A⊗H_B; ρ_A = Tr_B(|ψ⟩⟨ψ|)

H, H_A, H_B
Joint and subsystem Hilbert spaces
ψ
Normalized joint pure state
ρ_A
Reduced density operator
Tr_B
Partial trace over subsystem B

Conditions and scope

  • The subsystem partition is specified.
  • Nonfactorization of a mixed density operator alone does not establish entanglement.
Sources · 1

References

Sources retain their recorded scope. A source link alone does not establish verification of every statement.

  1. Quantum Matter 2 (2024), sections 1, 2 and 4F. H. L. Essler / University of OxfordCourse notes · PDFwww-thphys.physics.ox.ac.ukRead source

Evidence and review

Section 4.1: tensor space, product-state exception and reduced density matrix.

2026-10-10 · Bipartite pure-state criterion only; mixed-state separability tests excluded.

Related reference entries

Explore relationships within this area of study and its connections to other areas.

Showing 1–6 of 6 relationships

Scientific relationships · 6 / 6

The relationship type describes the connection. Its linked statements, sources and conditions define the scope.

  1. Uses definition
    Relationship sources · 1

    Linked statements

    • For specified finite-dimensional subsystems, the joint space is H_A⊗H_B. A pure state is entangled when it cannot be written as ψ_A⊗ψ_B; its subsystem state is obtained by partial trace.

    A specified pure bipartite state yields a reduced density operator by partial trace.

  2. Related to
    Relationship sources · 1

    Linked statements

    • Ordinary bosonic wave functions are symmetric and fermionic wave functions antisymmetric under simultaneous exchange of the complete coordinates and spin labels of two identical particles.

    Multiparticle states must respect exchange of complete identical-particle labels; an arbitrary distinguishable-factor model alone does not impose this.

  3. Related to
    Relationship sources · 1

    Linked statements

    • Independent commuting angular momenta combine as J=J₁+J₂. At fixed j₁,j₂, allowed total j runs in unit steps from |j₁−j₂| to j₁+j₂; coupled states expand in product states with Clebsch–Gordan coefficients that vanish unless m=m₁+m₂.

    Angular momenta act on distinct tensor factors; the coupled basis need not factor into separate component eigenstates.

  4. Related to
    Relationship sources · 1

    Linked statements

    • A physical channel on a subsystem is modeled by a completely positive trace-preserving map. In finite dimensions it has a Kraus representation sum K-a rho K-a† with sum K-a† K-a=I. Complete positivity requires positivity even when the map acts on part of an entangled reference system; mere positivity is insufficient. The Kraus representation need not be unique.

    A reference system makes complete positivity operationally necessary.

  5. Related to
    Relationship sources · 1

    Linked statements

    • A quantum channel Φ(ρ)=Σ_kK_kρK_k† with Σ_kK_k†K_k=I remains positive when extended by an identity map on an arbitrary ancilla. This complete positivity is stronger than positivity on isolated inputs. A fixed unitary interaction with an initially uncorrelated environment and subsequent partial trace supplies such a channel.

    Ancilla extension and environment tracing require composite-system structure.

  6. Related to
    Relationship sources · 2

    Linked statements

    • The 2017 Marletto–Vedral proposal links entanglement between two masses to nonclassical mediation under explicit subsystem and locality assumptions. Di Biagio's September 2026 analysis distinguishes circuit/subsystem locality from relativistic no-signalling and argues that entanglement alone need not exclude every classical-gravity/quantum-matter model. Compare quantitative entanglement and decoherence predictions while excluding nongravitational interactions.

    An entanglement witness depends on the specified subsystem partition.

Entry detailsPermanent link to this entry ↗
Record identity
physics-tensor-product-entanglement
Revision
university-reference.1
Structured reference record
University reference · University core expansion