Acta Physica Sinica

The uncertainty relations and squeezing effects for mixed states*

Deng Wen-Ji, Liu Ping, Xu Xiao, (1)华南理工大学材料科学研究所,广州　510640; (2)华南理工大学物理系,广州　510640

Published Jan 1, 2004• 1 citation

Abstract

Based upon the Heisenberger's uncertainty relations for mixed states, the minimu muncertainty and squeezing effects are studies. Although all minimum uncertainty states must be pure states, the nucertainty of one dynamical variable can be squeezed in some pure states or mixed states, even if they are not minimum uncert ainty states for the pair of dynamical variables. In addition, we present the ge neral uncertainty relation of statistics, which has the similar mathmatical stru cture of Heisenberg's indeterminacy principle, but has no relation with quantum interfernce.

DOI: 10.7498/aps.53.3668

Related papers

The uncertainty relations and minimum uncertainty states

Wenji Deng, Xu Yun-Hua, Liu Ping +1 more

Based upon some basic properties of Hermitian operators and Hilbert state vector s, the Heisenberg's uncertainty relations are analyzed in details. The necessary and sufficient condition for the minimum uncertainty states is re-obtained dire ctly from our proof. We propose also the problem of squeezed states for any coup le of dynamical variables, and solve it by using the bosonic creation and annihi lation operators.

Acta Physica SinicaJan 1, 20031 citation

Heisenberg uncertainty relation for mixed states

Shunlong Luo

The Heisenberg uncertainty relation sets a fundamental limit for quantum measurement of incompatible observables. Its standard form derived by Weyl and Robertson is of purely quantum nature when the state is pure. However, for mixed states, because the variance involving a mixed state is a hybrid of classical and quantum uncertainty, the conventional uncertainty relation is of a ``mixed'' flavor. It is desirable to seek some decomposition of variance into classical and quantum parts and to cast the Heisenberg uncertainty relation for mixed states in a more quantum form. By use of the skew information introduced by Wigner and Yanase in 1963, we make such an attempt and establish a different uncertainty relation which is stronger than the conventional one.

Physical Review AOct 20, 2005192 citations

Uncertainty Relations for Mixed Quantum States

V. V. Dodonov, Vladimir I. Man’ko

We give a review of different forms of uncertainty relations for mixed quantum states obtained over the last two decades and present many new results. The nonclassical properties of mixed states minimizing the purity-bounded uncertainty relations (a possibility of sub-Poissonian statistics, squeezing etc) are considered. The normalized Hilbert–Schmidt distance between the minimizing states and the ‘most classical ’ thermal states is used as a ‘measure of nonclassicality ’ together with the Mandel parameter. For highly mixed minimizing states (whose ‘purities ’ are very small), the normalized Hilbert–Schmidt distance tends to a finite limit, which depends on the nature of the state (15 % of the maximal possible distance if the deviation from pure states is characterized by the ‘standard purity ’ Tr ρ̂2 and 37 % if the ‘superpurity ’ limr→0[Tr(ρ̂1+1/r)]r is chosen as a measure of deviation).

Jan 1, 19881 citation

Correlations in open quantum systems and associated uncertainty relations

Sergey A. Ponomarenko, E. Wolf

We study how correlations in a state of an open quantum system affect the intrinsic uncertainties of the expectation values of an arbitrary pair of noncommuting observables. We show that for such observables, there exist Heisenberg-type uncertainty relations that take fully into account correlations in the state of the system. If the quantum system is in a pure state, such uncertainty relations reduce to the conventional one. We obtain an equation for the density operator of a general state that minimizes the new uncertainty relations, and demonstrate that in the important case of coordinate and momentum operators, the minimum-uncertainty states are displaced squeezed thermal states.

Physical Review AMay 9, 200117 citations

Improvement of uncertainty relations for mixed states

Yong‐Moon Park

We study a possible improvement of uncertainty relations. The Heisenberg uncertainty relation employs commutator of a pair of conjugate observables to set the limit of quantum measurement of the observables. The Schrödinger uncertainty relation improves the Heisenberg uncertainty relation by adding the correlation in terms of anti-commutator. However both relations are insensitive whether the state used is pure or mixed. We improve the uncertainty relations by introducing additional terms which measure the mixtureness of the state. For the momentum and position operators as conjugate observables and for the thermal state of quantum harmonic oscillator, it turns out that the equalities in the improved uncertainty relations hold.

arXiv:math-ph/0409008Journal of Mathematical PhysicsMar 28, 200518 citations

Purity- and entropy-bounded uncertainty relations for mixed quantum states

V. V. Dodonov

We give a review of different forms of uncertainty relations for mixed quantum states obtained over the last two decades and present many new results. The nonclassical properties of mixed states minimizing the purity-bounded uncertainty relations (a possibility of sub-Poissonian statistics, squeezing etc) are considered. The normalized Hilbert-Schmidt distance between the minimizing states and the `most classical' thermal states is used as a `measure of nonclassicality' together with the Mandel parameter. For highly mixed minimizing states (whose `purities' are very small), the normalized Hilbert-Schmidt distance tends to a finite limit, which depends on the nature of the state (15% of the maximal possible distance if the deviation from pure states is characterized by the `standard purity' Tr 2 and 37% if the `superpurity' lim r →0 [Tr ( 1+1/ r )] r is chosen as a measure of deviation).

Journal of Optics B Quantum and Semiclassical OpticsApr 5, 200268 citations