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arXiv:1303.5238

Purity dependent uncertainty relation and possible enhancement of quantum tunneling phenomenon

Vladimir N. Chernega

Published Mar 21, 2013• 2 citations
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Abstract

The position-momentum uncertainty relations containing the dependence of their quantum bounds on state purity parameter $μ$ are discussed in context of possibilities to influence on the potential barrier transparency by means of decoherence processes. The behavior of barrier transparency $D$ is shown to satisfy the condition $μ^{-1}\ln D=const$. The particular case of thermal state with temperature $T$ where the purity parameter is a function of temperature is considered. For large temperature the condition for the barrier transparency is shown to be $T\ln D=const$.

DOI
10.48550/arxiv.1303.5238

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Uncertainty relations for mixed quantum states (precisely, purity-bounded position–momentum relations, developed by Bastiaans and then by Man'ko and Dodonov) are studied in the general multidimensional case. An expression for a family of mixed states at the lower bound of uncertainty relation is obtained. It is shown that in the case of entropy-bounded uncertainty relations, the lower-bound state is thermal, and a transition from the one-dimensional problem to the multidimensional one is trivial. Results of numerical calculation of the relation of the lower bound for different types of generalized purity are presented. Analytical expressions for general purity-bounded relations for highly mixed states are obtained.
For a tunneling system we consider the coupling to the dissipative environment of the coordinate q, the momentum p and the potential function U(q). In the frame of the quantum theory of open systems we derive a quantum master equation, fundamental constraints and uncertainty relations. We obtain an analytical expression of the tunneling spectrum and show that at thermal equilibrium its shape entirely depends on the barrier characteristics while its amplitude is proportional to the strength of the dissipative coupling.
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).
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).
A concise review of various mathematical formulations of the uncertainty relations in quantum mechanics discovered since 1927 is given. Besides the traditional Heisenberg inequality, the modifications made by Schrödinger and Robertson, as well as generalizations to sets of several noncommuting operators, are considered. The "entropic" inequalities and "local" uncertainty relations, together with inequalities which connect the so-called total width and the mean peak width of a wave function, are discussed. Inequalities for the products of higher order moments of the coordinate and momentum are presented. Inequalities making the uncertainty relations more accurate when the "purity" of a quantum state is fixed are demonstrated. Diverse formulations of the energy-time uncertainty relations are considered.
We explore the effects of Markovian bath coupling and initial position-momentum correlations on the coherence and purity of Gaussian quantum states. Our analysis focuses on the roles these factors play in the dynamics of quantum coherence, coherence lengths, and state purity. Our results reveal that initial position-momentum correlations have a remarkable impact on the quantum properties of the mixed state. These correlations lead to opposing behaviors in coherence and purity: as quantum coherence increases in response to stronger correlations, purity diminishes, and vice versa. This inverse relationship illustrates the phenomenon where, governed by these initial correlations, a state with greater mixing can display enhanced quantum coherence compared to a less mixed state. We also observe an unanticipated coherence freezing phenomenon, quantified by the relative entropy of coherence. Notably, this freezing is driven by initial position-momentum correlations, although the final frozen value is independent of these correlations.