Robertson–Schrödinger uncertainty relation

E450152

generalization of Heisenberg uncertainty principle quantum mechanical inequality uncertainty relation

The Robertson–Schrödinger uncertainty relation is a generalized quantum mechanical inequality that extends Heisenberg’s uncertainty principle to arbitrary pairs of observables, incorporating both their commutator and statistical correlations.

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Statements (45)

Predicate	Object
instanceOf	generalization of Heisenberg uncertainty principle ⓘ quantum mechanical inequality ⓘ uncertainty relation ⓘ
appliesIn	Hilbert space formalism NERFINISHED ⓘ density operators ⓘ state vectors ⓘ
appliesTo	non-commuting observables ⓘ pairs of observables ⓘ
assumes	normalized quantum state ⓘ
concerns	measurement uncertainty ⓘ preparation uncertainty ⓘ
expresses	lower bound on product of standard deviations ⓘ trade-off between measurement precisions ⓘ
field	quantum mechanics ⓘ
generalizes	Heisenberg uncertainty principle NERFINISHED ⓘ
hasConsequence	correlated observables obey tighter constraints than uncorrelated ones ⓘ no quantum state can have arbitrarily sharp values for all observables ⓘ
hasDomain	theoretical physics ⓘ
hasForm	ΔA² ΔB² ≥ \|⟨[A,B]⟩\|²/4 + \|cov(A,B)\|² ⓘ
hasStrongerFormThan	Heisenberg–Kennard uncertainty relation NERFINISHED ⓘ
holdsFor	any pair of self-adjoint operators with finite variances ⓘ
implies	correlations can increase uncertainty lower bound ⓘ non-commutativity leads to measurement limits ⓘ
includes	commutator term ⓘ covariance term ⓘ statistical correlations between observables ⓘ
mathematicallyBasedOn	inner product spaces ⓘ operator algebra ⓘ
namedAfter	Erwin Schrödinger NERFINISHED ⓘ Howard Percy Robertson NERFINISHED ⓘ
relatedTo	Cauchy–Schwarz inequality NERFINISHED ⓘ variance–covariance matrix positivity ⓘ
specialCase	Heisenberg position–momentum uncertainty relation NERFINISHED ⓘ
usedIn	continuous-variable quantum information ⓘ entanglement criteria ⓘ quantum optics ⓘ quantum state characterization ⓘ
usesConcept	Hermitian operators ⓘ commutator of operators ⓘ covariance matrix ⓘ standard deviation ⓘ
validFor	mixed states ⓘ pure states ⓘ
yearProposed	1929 ⓘ 1930 ⓘ

Referenced by (1)

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uncertainty principle → hasGeneralFormulation → Robertson–Schrödinger uncertainty relation ⓘ