This paper is generally concerned with understanding how the uncertainty principle arises in formulations of quantum mechanics, such as the decoherent histories approach, whose central goal is the assignment of probabilities to histories. We first consider histories characterized by position or momentum projections at two moments of time. Both exact and approximate (Gaussian) projections are studied. Shannon's information is used as a measure of the uncertainty expressed in the probabilities for these histories. We derive a number of inequalities in which the uncertainty principle is expressed as a lower bound on the information of phase space distributions derived from the probabilities for two-time histories. We go on to consider histories characterized by position samplings at $n$ moments of time. We derive a lower bound on the information of the joint probability for $n$ position samplings. Similar bounds are derived for histories characterized by samplings of other variables. All lower bounds on the information of histories have the general form $\mathrm{ln}(\frac{{V}_{H}}{{V}_{S}})$ , where ${V}_{H}$ is a volume element of history space, which we define, and ${V}_{S}$ is the volume of that space probed by the projections. We thus obtain a concise and general form of the uncertainty principle referring directly to the histories description of the system, and making no reference to notions of phase space.This paper is generally concerned with understanding how the uncertainty principle arises in formulations of quantum mechanics, such as the decoherent histories approach, whose central goal is the assignment of probabilities to histories. We first consider histories characterized by position or momentum projections at two moments of time. Both exact and approximate (Gaussian) projections are studied. Shannon's information is used as a measure of the uncertainty expressed in the probabilities for these histories. We derive a number of inequalities in which the uncertainty principle is expressed as a lower bound on the information of phase space distributions derived from the probabilities for two-time histories. We go on to consider histories characterized by position samplings at $n$ moments of time. We derive a lower bound on the information of the joint probability for $n$ position samplings. Similar bounds are derived for histories characterized by samplings of other variables. All lower bounds on the information of histories have the general form $\mathrm{ln}(\frac{{V}_{H}}{{V}_{S}})$, where ${V}_{H}$ is a volume element of history space, which we define, and ${V}_{S}$ is the volume of that space probed by the projections. We thus obtain a concise and general form of the uncertainty principle referring directly to the histories description of the system, and making no reference to notions of phase space.

arXiv:gr-qc/9304039Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsSep 15, 199342 citations

A GENERALIZATION OF THE UNCERTAINTY PRINCIPLE

A. Gamba

Jun 1, 19500 citations

Role of information theoretic uncertainty relations in quantum theory

Petr Jizba B, Jacob Dunningham, Jaewoo Joo

Uncertainty relations based on information theory for both discrete and continuous distribution functions are briefly reviewed. We extend these results to account for (differential) Rényi entropy and its related entropy power. This allows us to find a new class of information-theoretic uncertainty relations (ITURs). The potency of such uncertainty relations in quantum mechanics is illustrated with a simple two-energy-level model where they outperform both the usual Robertson–Schrödinger uncertainty relation and Shannon entropy based uncertainty relation. In the continuous case the ensuing entropy power uncertainty relations are discussed in the context of heavy tailed wave functions and Schrödinger cat states. Again, improvement over both the Robertson–Schrödinger uncertainty principle and Shannon ITUR is demonstrated in these cases. Further salient issues such as the proof of a generalized entropy power inequality and a geometric picture of information-theoretic uncertainty relations are also discussed.Uncertainty relations based on information theory for both discrete and continuous distribution functions are briefly reviewed. We extend these results to account for (differential) Rényi entropy and its related entropy power. This allows us to find a new class of information-theoretic uncertainty relations (ITURs). The potency of such uncertainty relations in quantum mechanics is illustrated with a simple two-energy-level model where they outperform both the usual Robertson–Schrödinger uncertainty relation and Shannon entropy based uncertainty relation. In the continuous case the ensuing entropy power uncertainty relations are discussed in the context of heavy tailed wave functions and Schrödinger cat states. Again, improvement over both the Robertson–Schrödinger uncertainty principle and Shannon ITUR is demonstrated in these cases. Further salient issues such as the proof of a generalized entropy power inequality and a geometric picture of information-theoretic uncertainty relations are also discussed.

Apr 1, 201535 citations

Some extensions of the uncertainty principle

Steeve Zozor, M. Portesi, Christophe Vignat

arXiv:0709.3011Physica A Statistical Mechanics and its ApplicationsApr 7, 200881 citations

Decoherence limit of quantum systems obeying generalized uncertainty principle: New paradigm for Tsallis thermostatistics

Petr Jizba B, Gaetano Lambiase, Giuseppe Gaetano Luciano +1 more

The generalized uncertainty principle (GUP) is a phenomenological model whose purpose is to account for a minimal length scale (e.g., Planck scale or characteristic inverse-mass scale in effective quantum description) in quantum systems. In this paper, we study possible observational effects of GUP systems in their decoherence domain. We first derive coherent states associated to GUP and unveil that in the momentum representation they coincide with Tsallis probability amplitudes, whose nonextensivity parameter $q$ monotonically increases with the GUP deformation parameter $\ensuremath{\beta}$. Second, for $\ensuremath{\beta}<0$ (i.e., $q<1$ ), we show that, due to Bekner-Babenko inequality, the GUP is fully equivalent to information-theoretic uncertainty relations based on Tsallis-entropy-power. Finally, we invoke the maximal entropy principle known from estimation theory to reveal connection between the quasiclassical (decoherence) limit of GUP-related quantum theory and nonextensive thermostatistics of Tsallis. This might provide an exciting paradigm in a range of fields from quantum theory to analog gravity. For instance, in some quantum gravity theories, such as conformal gravity, aforementioned quasiclassical regime has relevant observational consequences. We discuss some of the implications.The generalized uncertainty principle (GUP) is a phenomenological model whose purpose is to account for a minimal length scale (e.g., Planck scale or characteristic inverse-mass scale in effective quantum description) in quantum systems. In this paper, we study possible observational effects of GUP systems in their decoherence domain. We first derive coherent states associated to GUP and unveil that in the momentum representation they coincide with Tsallis probability amplitudes, whose nonextensivity parameter $q$ monotonically increases with the GUP deformation parameter $\ensuremath{\beta}$. Second, for $\ensuremath{\beta}<0$ (i.e., $q<1$), we show that, due to Bekner-Babenko inequality, the GUP is fully equivalent to information-theoretic uncertainty relations based on Tsallis-entropy-power. Finally, we invoke the maximal entropy principle known from estimation theory to reveal connection between the quasiclassical (decoherence) limit of GUP-related quantum theory and nonextensive thermostatistics of Tsallis. This might provide an exciting paradigm in a range of fields from quantum theory to analog gravity. For instance, in some quantum gravity theories, such as conformal gravity, aforementioned quasiclassical regime has relevant observational consequences. We discuss some of the implications.

arXiv:2201.07919Physical review. D/Physical review. D.Jun 15, 202245 citations

n

moments of time. We derive a lower bound on the information of the joint probability for

n

position samplings. Similar bounds are derived for histories characterized by samplings of other variables. All lower bounds on the information of histories have the general form

\mathrm{ln}(\frac{{V}_{H}}{{V}_{S}})

, where

{V}_{H}

is a volume element of history space, which we define, and

{V}_{S}

is the volume of that space probed by the projections. We thus obtain a concise and general form of the uncertainty principle referring directly to the histories description of the system, and making no reference to notions of phase space.

q

monotonically increases with the GUP deformation parameter $\ensuremath{\beta}$. Second, for $\ensuremath{\beta}<0$ (i.e.,

q<1

), we show that, due to Bekner-Babenko inequality, the GUP is fully equivalent to information-theoretic uncertainty relations based on Tsallis-entropy-power. Finally, we invoke the maximal entropy principle known from estimation theory to reveal connection between the quasiclassical (decoherence) limit of GUP-related quantum theory and nonextensive thermostatistics of Tsallis. This might provide an exciting paradigm in a range of fields from quantum theory to analog gravity. For instance, in some quantum gravity theories, such as conformal gravity, aforementioned quasiclassical regime has relevant observational consequences. We discuss some of the implications.