Uncertainty Relations Associated with Correlations in Mixed Quantum States

  • Sergey A. Ponomarenko
  • Emil Wolf
Conference paper

Abstract

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:
$$ \langle {(\Delta \hat{A})^2}\rangle \langle {(\Delta \hat{B})^2}\rangle \geqslant \frac{1}{4}|\langle \psi |[\hat{A}\hat{B}] - |\psi \rangle {|^2} $$
(1)
$$ P\{ \phi (t) = \psi (t)\} = 1 $$
(1.2)
Inline 1\( \geqslant \frac{{p - 1}}{m} \)

Here [Â,B]−≡ÂB−B 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].

Keywords

Generalize Variance Mixed State Uncertainty Relation Density Operator Open Quantum System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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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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