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

Entropy Cost of ‘Erasure’ in Physically Irreversible Processes

Ruth E. Kastner, Andreas Schlatter

A restricted form of Landauer’s principle, independent of computational considerations, is shown to hold for thermal systems by reference to the joint entropy associated with conjugate observables. It is shown that the source of the compensating entropy for irreversible physical processes is due to the ontological uncertainty attending values of such mutually incompatible observables, rather than due to epistemic uncertainty, as traditionally assumed in the information-theoretic approach. In particular, it is explicitly shown that erasure of logical (epistemic) information via reset operations is not equivalent to erasure of thermodynamic entropy, so that the traditional, information-theoretic form of Landauer’s principle is not supported by the physics. A further implication of the analysis is that, in principle, there can be no Maxwell’s Demon in the real world.A restricted form of Landauer’s principle, independent of computational considerations, is shown to hold for thermal systems by reference to the joint entropy associated with conjugate observables. It is shown that the source of the compensating entropy for irreversible physical processes is due to the ontological uncertainty attending values of such mutually incompatible observables, rather than due to epistemic uncertainty, as traditionally assumed in the information-theoretic approach. In particular, it is explicitly shown that erasure of logical (epistemic) information via reset operations is not equivalent to erasure of thermodynamic entropy, so that the traditional, information-theoretic form of Landauer’s principle is not supported by the physics. A further implication of the analysis is that, in principle, there can be no Maxwell’s Demon in the real world.

arXiv:2307.02643MathematicsJan 8, 20245 citations

On Derivations of the Uncertainty Principle

Ernest R. Davidson

The Journal of Chemical PhysicsFeb 15, 196521 citations

Entropic uncertainty relations – A survey

Stephanie Wehner, Andreas Winter

Uncertainty relations play a central role in quantum mechanics. Entropic uncertainty relations in particular have gained significant importance within quantum information, providing the foundation for the security of many quantum cryptographic protocols. Yet, rather little is known about entropic uncertainty relations with more than two measurement settings. In this note we review known results and open questions. The uncertainty principle is one of the fundamental ideas of quantum mechanics. Since Heisenberg’s uncertainty relations for canonically conjugate variables, they have been one of the most prominent examples of how quantum mechanics differs from the classical world (Heisenberg, 1927). Uncertainty relations today are probably best known in the form given by (Robertson, 1929), who extended Heisenberg’s result to two arbitrary observables A and B. Robertson’s relation states that if we prepare many copies of the state |ψ〉, and measure each copy individually using either A or B, we haveUncertainty relations play a central role in quantum mechanics. Entropic uncertainty relations in particular have gained significant importance within quantum information, providing the foundation for the security of many quantum cryptographic protocols. Yet, rather little is known about entropic uncertainty relations with more than two measurement settings. In this note we review known results and open questions. The uncertainty principle is one of the fundamental ideas of quantum mechanics. Since Heisenberg’s uncertainty relations for canonically conjugate variables, they have been one of the most prominent examples of how quantum mechanics differs from the classical world (Heisenberg, 1927). Uncertainty relations today are probably best known in the form given by (Robertson, 1929), who extended Heisenberg’s result to two arbitrary observables A and B. Robertson’s relation states that if we prepare many copies of the state |ψ〉, and measure each copy individually using either A or B, we have

Jan 1, 2009239 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.