Uncertainty Relations Associated with Correlations in Mixed Quantum States

  • Sergey A. Ponomarenko
  • Emil Wolf
Conference paper


The conventional uncertainty relation (UR) is a cornerstone of the modern quantum theory of measurement. [1], and consequently is treated in most quantum-mechanics textbooks. However, contrary to the widespread view, the usual “textbook” UR pertains to separate rather than joint measurements [2] of observables  or 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  and B cannot be less than a certain value, which is expressed mathematically as:
$$ \langle {(\Delta \hat{A})^2}\rangle \langle {(\Delta \hat{B})^2}\rangle \geqslant \frac{1}{4}|\langle \psi |[\hat{A}\hat{B}] - |\psi \rangle {|^2} $$
$$ P\{ \phi (t) = \psi (t)\} = 1 $$
Inline 1\( \geqslant \frac{{p - 1}}{m} \)

Here [Â,B]−≡ÂBB is the commutator of a pair of operators, and 〈(△Â)2〉=〈Â2〉−〈Â2〉 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  and B are the coordinate and the momentum, the most general minimum-uncertainty state must be a pure state [10].


Generalize Variance Mixed State Uncertainty Relation Density Operator Open Quantum System 
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Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Sergey A. Ponomarenko
    • 1
  • Emil Wolf
    • 1
  1. 1.Department of Physics and AstronomyUniversity of RochesterRochester

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