Coherence and Quantum Optics VIII pp 331-332 | Cite as

# Uncertainty Relations Associated with Correlations in Mixed Quantum States

## Abstract

*B*on a system in a state |;ψ〉. In essence, the uncertainty relation indicates that due to intrinsic indeterminacy of the quantum state, the product of the variances of two noncommuting observables Â and

*B*cannot be less than a certain value, which is expressed mathematically as:

Here [Â,*B*]−≡Â*B*−*B*Â is the commutator of a pair of operators, and 〈(△Â)^{2}〉=〈Â^{2}〉−〈Â^{2}〉 is the variance of the operator *Â*. It should be noted that the pertains to a *pure* state. Uncertainty relations of this type were extensively studied as early as in the 193O’s [3]. There has also been a considerable interest in uncertainty relations associated with joint measurements of noncommuting observables [5–8] as well as in the generalized parameter-based UR’s that do not explicitly depend on the expectation value of the commutator [8, 9]. Further, a generalization of the Heisenberg-type UR (1) to open quantum systems was obtained [10, 11], and the nature of the states that minimize such a generalized UR was examined. These studies have shown that, at least when the observables Â and *B* are the coordinate and the momentum, the most general minimum-uncertainty state must be a pure state [10].

## Keywords

Generalize Variance Mixed State Uncertainty Relation Density Operator Open Quantum System## References and links

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