#
A Different Angle on Quantum Uncertainty (Measure Angle)^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Effective-Number Theory

#### 2.1. Additivity

#### 2.2. Monotonicity

^{−}) monotonicity is designed to identify functions respecting cumulation. To place its meaning in a more conventional context, we point out that imposing it in conjunction with symmetry (S) (see below) results in a well-known property of Schur concavity [6].

#### 2.3. Effective-Number Functions

^{−}), the axioms defining effective-number functions $\mathcal{N}=\mathcal{N}\left[C\right]$ incorporate intuitive and easily formulated features such as symmetry, continuity, and previously mentioned “boundary values” associated with uniform and $\delta $-function distributions. The complete list of these additional requirements is formally specified below.

_{−}), (S), (B2) and (C).

#### 2.4. Minimal Amount

**Theorem**

**1.**

## 3. Measure Aspect of Quantum Uncertainty

- [U${}_{0}$]
- The μ-uncertainty of $\mid \phantom{\rule{-0.166667em}{0ex}}\psi \phantom{\rule{0.166667em}{0ex}}\rangle $ with respect to $\left\{\phantom{\rule{0.166667em}{0ex}}\mid \phantom{\rule{-0.166667em}{0ex}}i\phantom{\rule{0.166667em}{0ex}}\rangle \right\}$ is at least ${\mathcal{N}}_{\u2605}[\phantom{\rule{0.166667em}{0ex}}\mid \phantom{\rule{-0.166667em}{0ex}}\psi \phantom{\rule{0.166667em}{0ex}}\rangle ,\left\{\phantom{\rule{0.166667em}{0ex}}\mid \phantom{\rule{-0.166667em}{0ex}}i\phantom{\rule{0.166667em}{0ex}}\rangle \right\}]$ states (Equations (7) and (10)).

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Effective-number assignments for (

**left**) uniform and (

**middle**) $\delta $-function probability distributions. Is there a well-founded prescription(s) for generic distributions (

**right**)?

**Figure 2.**Composing probability distributions generated by lattice Schrödinger particle. Parts characterized by N, P (red), and M, Q (blue) combine into a total described by $N+M$, . See the discussion in the text.

**Figure 3.**(left panel) Distribution C is more cumulated than distribution B. (right panel) Distributions cannot be readily compared by their cumulation. Note that discrete dependencies were replaced with continuous ones for better clarity.

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Horváth, I.; Mendris, R.
A Different Angle on Quantum Uncertainty (Measure Angle). *Proceedings* **2019**, *13*, 8.
https://doi.org/10.3390/proceedings2019013008

**AMA Style**

Horváth I, Mendris R.
A Different Angle on Quantum Uncertainty (Measure Angle). *Proceedings*. 2019; 13(1):8.
https://doi.org/10.3390/proceedings2019013008

**Chicago/Turabian Style**

Horváth, Ivan, and Robert Mendris.
2019. "A Different Angle on Quantum Uncertainty (Measure Angle)" *Proceedings* 13, no. 1: 8.
https://doi.org/10.3390/proceedings2019013008