# A New Logic, a New Information Measure, and a New Information-Based Approach to Interpreting Quantum Mechanics

## Abstract

**:**

## 1. Introduction: Where Does QM Math Come from?

## 2. Materials and Methods

- Distinctions versus indistinctions,
- Definiteness versus indefiniteness,
- Distinguishability versus indistinguishability

Thus, superposition, with the attendant riddles of entanglement and reduction, remains the central and generic interpretative problem of quantum theory [6] [p. 27]

From these two basic ideas alone—indefiniteness and the superposition principle—it should be clear already that quantum mechanics conflicts sharply with common sense. If the quantum state of a system is a complete description of the system, then a quantity that has an indefinite value in that quantum state is objectively indefinite; its value is not merely unknown by the scientist who seeks to describe the system. Furthermore, since the outcome of a measurement of an objectively indefinite quantity is not determined by the quantum state, and yet the quantum state is the complete bearer of information about the system, the outcome is strictly a matter of objective chance—not just a matter of chance in the sense of unpredictability by the scientist. Finally, the probability of each possible outcome of the measurement is an objective probability. Classical physics did not conflict with common sense in these fundamental ways [7] [p. 47].

These statements… may collectively be called “the Literal Interpretation” of quantum mechanics. This is the interpretation resulting from taking the formalism of quantum mechanics literally, as giving a representation of physical properties themselves, rather than of human knowledge of them, and by taking this representation to be complete [8] [pp. 6–7].

## 3. Results

#### 3.1. The Lattice of Partitions

#### 3.2. The Logic of Partitions

#### 3.3. Logical Information Theory: Logical Entropy

#### 3.4. Skeletonized Quantum States as Partitions

Every thing, however, as to its possibility, further stands under the principle of thoroughgoing determination; according to which, among all possible predicates of things, insofar as they are compared with their opposites, one must apply to it. [19] [p. B600]

In quantum mechanics, however, identical particles are truly indistinguishable. This is because we cannot specify more than a complete set of commuting observables for each of the particles; in particular, we cannot label the particle by coloring it blue [20] [p. 446].

## 4. Superposition as Indefiniteness in the Quantum ‘Underworld’

The wave formalism offers a convenient mathematical representation of this latency, for not only can the mathematics of wave effects, like interference and diffraction, be expressed in terms of the addition of vectors (that is, their linear superposition; see [23] [Chap. 29.5]), but the converse, also holds [24] [p. 303]

It follows from the linearity of the operators which represent observables of quantum mechanical systems that any measurable physical property which happens to be shared by all of the individual mathematical terms of some particular superposition (written down in any particular basis) will necessarily also be shared by the full superposition, considered as a single quantum-mechanical state, as well [25] [p. 234].

Heisenberg [35] [p. 53]… used the term “potentiality” to characterize a property which is objectively indefinite, whose value when actualized is a matter of objective chance, and which is assigned a definite probability by an algorithm presupposing a definite mathematical structure of states and properties. Potentiality is a modality that is somehow intermediate between actuality and mere logical possibility. That properties can have this modality, and that states of physical systems are characterized partially by the potentialities they determine and not just by the catalogue of properties to which they assign definite values, are profound discoveries about the world, rather than about human knowledge [8] [p. 6].

The historical reference should perhaps be dismissed, since quantum mechanical potentiality is completely devoid of teleological significance, which is central to Aristotle’s conception. What it has in common with Aristotle’s conception is the indefinite character of certain properties of the system. [45] [pp. 313–314]

- Where the classical notion of fully definite reality is replaced by the idea of indefiniteness at the quantum level (not definite all-the-way-down),
- Where the state-reduction jump from an indefinite state to a more definite state is due to making distinctions (“more-definite its from dits”),
- Where change with no distinctions stays at the same level of indefiniteness (unitary evolution).

#### 4.1. Superposition and the Born Rule

#### 4.2. Quantum States

The off-diagonal terms of a density matrix … are often called quantum coherences because they are responsible for the interference effects typical of quantum mechanics that are absent in classical dynamics [51] [p. 177].

#### 4.3. Quantum Observables

#### 4.4. Quantum Measurement

**Theorem**

**1.**

**Example**

**1.**

**Theorem**

**2**

**.**The increase in logical entropy from $h\left(\rho \right(\pi \left)\right)$ to $h\left(\widehat{\rho}\left(\pi \right)\right)$ in the Lüders mixture operation $\widehat{\rho}\left(\pi \right)={\sum}_{r\in g\left(U\right)}{P}_{{g}^{-1}\left(r\right)}\rho \left(\pi \right){P}_{{g}^{-1}\left(r\right)}$ is the sum of the squares of the off-diagonal non-zero entries in $\rho \left(\pi \right)$ that were zeroed in the measurement operation $\rho \left(\pi \right)\u21dd\widehat{\rho}\left(\pi \right)$.

**Theorem**

**3**

**.**In the Lüders mixture operation $\widehat{\rho}={\sum}_{r\in g\left(U\right)}{P}_{{V}_{r}}\rho {P}_{{V}_{r}}$, the increase in quantum logical entropy from $h\left(\rho \right)$ to $h\left(\widehat{\rho}\right)$ is the sum of the absolute squares of the off-diagonal entries in ρ that were zeroed in the measurement operation $\rho \u21dd\widehat{\rho}$.

#### 4.5. Other Aspects of QM Mathematics

#### 4.5.1. Commuting, Non-Commuting, and Conjugate Operators

**Theorem**

**4**

- F and G are commuting if $\mathcal{SE}=V$;
- F and G are incompatible if $\mathcal{SE}\ne V$;
- F and G are conjugate if $\mathcal{SE}=0$ (zero space).

**Partition math**: A set of compatible partitions $\pi ,\sigma ,\dots ,\gamma $ defined by $f,g,\dots ,h:U\to \mathbb{R}$ is said to be complete, i.e., a Complete Set of Compatible Attributes or CSCA, if their join is the partition whose blocks are of cardinality one (i.e., ${\mathbf{1}}_{U}$). Then, the elements $u\in U$ are uniquely characterized by the ordered set of values $\left(f\right(u),g(u),\dots ,h(u)$.

**QM math**: A set of commuting observables $F,G,\dots ,H$ is said to be complete, i.e., a Complete Set of Commuting Observables or CSCO [60] [p. 57], if the join of their eigenspace DSDs is the DSD whose subspaces are of dimension one. Then, the simultaneous eigenvectors of the operators are uniquely characterized by the ordered set of their eigenvalues.

#### 4.5.2. Feynman’s Treatment of Measurement

If you could, in principle, distinguish the alternative final states (even though you do not bother to do so), the total, final probability is obtained by calculating the probability for each state (not the amplitude) and then adding them together. If you cannot distinguish the final states even in principle, then the probability amplitudes must be summed before taking the absolute square to find the actual probability [63] [pp. 3–9].

Feynman’s approach is based on the contrast between processes that are distinguishable within a given physical context and those that are indistinguishable within that context. A process is distinguishable if some record of whether or not it has been realized results from the process in question; if no record results, the process is indistinguishable from alternative processes leading to the same end result [64] [p. 314].

In other words, the superposition of amplitudes … is only valid if there is no way to know, even in principle, which path the particle took. It is important to realize that this does not imply that an observer actually takes note of what happens. It is sufficient to destroy the interference pattern, if the path information is accessible in principle from the experiment or even if it is dispersed in the environment and beyond any technical possibility to be recovered, but in principle still “out there”. The absence of any such information is the essential criterion for quantum interference to appear [65] [p. 484].

**Theorem**

**5**

**.**Measurement is described in the set case by the Lüders mixture operation $\widehat{\rho}\left(\pi \right)={\sum}_{r\in g\left(U\right)}{P}_{{g}^{-1}\left(r\right)}\rho \left(\pi \right){P}_{{g}^{-1}\left(r\right)}$. The State Reduction Principle then states that if an off-diagonal entry $\rho {\left(\pi \right)}_{ik}\ne 0$ (i.e., ${u}_{i}$ and ${u}_{k}$ are in a same-block superposition), then, if $g\left({u}_{i}\right)\ne g\left({u}_{k}\right)$ (i.e., the interaction distinguishes ${u}_{i}$ and ${u}_{k}$), then $\widehat{\rho}{\left(\pi \right)}_{ik}=0$ (i.e., the ‘coherence’ between ${u}_{i}$ and ${u}_{k}$ is decohered and a reduction is made).

**Theorem**

**6**

**.**Measurement is described in QM by the Lüders mixture operation $\widehat{\rho}={\sum}_{r\in g\left(U\right)}{P}_{{V}_{r}}\rho {P}_{{V}_{r}}$ (measuring ρ by G). The State Reduction Principle then states that if an off-diagonal entry ${\rho}_{ik}\ne 0$ (i.e., the G-eigenvectors $|{u}_{i}\rangle $ and $|{u}_{k}\rangle $ are in a superposition in ρ), then, if $|{u}_{i}\rangle $ and $|{u}_{k}\rangle $ have different G-eigenvalues (i.e., the vectors are distinguished by G), then ${\widehat{\rho}}_{ik}=0$ (i.e., the vectors are decohered—this is not the Zeh/Zurek (for all practical purposes) “decoherence” [68], but the old-fashioned change from the coherence of a superposition pure state into a decohered mixture of states—and a reduction is made).

#### 4.5.3. Von Neumann’s Type I and Type II Processes

- Type I process of measurement and state reduction
- Type II process obeying the Schrödinger equation.

It seems unbelievable that there is a fundamental distinction between “measurement” and “non-measurement” processes. Somehow, the true fundamental theory should treat all processes in a consistent, uniform fashion [71] [p. 245].

#### 4.5.4. Hermann Weyl’s Imagery for Measurement

In Einstein’s theory of relativity the observer is a man who sets out in quest of truth armed with a measuring-rod. In quantum theory he sets out armed with a sieve [72] [p. 267].

#### 4.6. A Skeletal Analysis of the Double-Slit Experiment

**Case 1**: There are detectors at the slits to distinguish between the two superposed states, so the state reduces to the half–half mixture of $\left\{a\right\}$ and $\left\{c\right\}$. The same mixture results if one or the other slit is simply closed. Then, $\left\{a\right\}$ evolves by the non-singular dynamics to $\{a,b\}$, which hits the wall and reduces to $\left\{a\right\}$ or $\left\{b\right\}$ with half–half probability. Similarly, $\left\{c\right\}$ evolves to $\{b,c\}$, which hits the wall and reduces to $\left\{b\right\}$ or $\left\{c\right\}$ with half–half probability. Since this is the case of distinctions between the alternative paths to $\left\{a\right\}$, $\left\{b\right\}$, or $\left\{c\right\}$, we add the probabilities to obtain:

**Case 2**: There are no detectors to distinguish between the (open) slits in the superposition $\{a,c\}$, so it linearly evolves by the dynamics $\{a,c\}=\left\{a\right\}+\left\{c\right\}\u21dd\{a,b\}+\{b,c\}=\{a,c\}$. Hence, the probabilities at the wall are:

## 5. Discussion

The conceptual elements of quantum theory that now underlie our picture of the physical world include objective chance, quantum interference, and the objective indefiniteness of dynamical quantities. Quantum interference, which is directly observable, was readily absorbed by the physics community. Objective chance and indefiniteness, being of more philosophical significance, gained acceptance only after much debate and conceptual analysis, when it was recognized that observed phenomena are better understood through these notions than through older ones or hidden variables [41] [p. vii].

- Superposition is the key non-classical notion of state in QM math;
- Objective indefiniteness is its ontological counterpart;
- The quantum notion of becoming (vN Type I) is the jump from an indefinite state to a more definite state;
- The quantum notion of evolution (vN Type II) is between states at the same level of indefiniteness.

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CSCA | Complete Set of Compatible Attributes |

CSCO | Complete Set of Commuting Observables |

PII | Principle of Identity of Indistinguishables |

QM | Quantum Mechanics |

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**Figure 5.**Partition lattice with join $\pi \vee \sigma =\left\{\right\{a\},\{b,c\left\}\right\}\vee \left\{\right\{a,b\},\{c\left\}\right\}=\left\{\right\{a\},\{b\},\{c\left\}\right\}$.

Algebra of subsets $\wp \left(U\right)$ of U | Algebra of partitions $\mathsf{\Pi}\left(U\right)$ on U |

Its = elements of subsets | Dits = distinctions of partitions |

P. O. inclusion of subsets $S\subseteq T$ | P. O. inclusion of ditsets $dit\left(\sigma \right)\subseteq dit\left(\pi \right)$ |

Join: $S\vee T=S\cup T$ | Join: $dit(\sigma \vee \pi )=dit\left(\sigma \right)\cup dit\left(\pi \right)$ |

Top: subset U with all elements | Top: partition ${\mathbf{1}}_{U}$ with all possible dits |

Bottom: subset ∅ with no elements | Bottom: partition ${\mathbf{0}}_{U}$ with no dits |

Choice Function | Quantum Measurement (Maximal) |
---|---|

$f\left(\right\{a,b,\dots ,c\left\}\right)=b$ | $\alpha |a\rangle +\beta |b\rangle +\cdots +\gamma |c\rangle \stackrel{\mathrm{indeterminate}}{\u21dd}|b\rangle $ |

$f\left(\right\{b\left\}\right)=b$ | $|b\rangle \stackrel{\mathrm{determinate}}{\u21dd}|b\rangle $ |

Partition Concept | Corresponding Quantum Concept |
---|---|

Non-singleton block, e.g., $\{a,b\}$ | Superposition pure state |

Indiscrete partition ${\mathbf{0}}_{U}=\left\{\{a,b,c,d\}\right\}$ | Largest pure state |

Singleton block, e.g., $\left\{d\right\}$ | Classical state (no superposition) |

Discrete partition ${\mathbf{1}}_{U}=\{\left\{a\right\},\left\{b\right\},\left\{c\right\},\left\{d\right\}\}$ | Mixture of classical states |

Partition Math | Hilbert Space Math |
---|---|

Density matrix: $\rho \left(\pi \right)$ | $\rho $ |

Orthonormal vectors: $\langle {b}_{{j}^{\prime}}|{b}_{j}\rangle ={\delta}_{j{j}^{\prime}}$ | $\langle {u}_{{i}^{\prime}}|{u}_{i}\rangle ={\delta}_{i{i}^{\prime}}$ |

Eigenvalues: $Pr\left({B}_{1}\right),\dots ,Pr\left({B}_{m}\right),0,\dots ,0$ | ${\lambda}_{1},\dots ,{\lambda}_{n}$ |

Spectral decomposition: $\rho \left(\pi \right)={\sum}_{j=1}^{m}Pr\left({B}_{j}\right)|{b}_{j}\rangle \langle {b}_{j}|$ | $\rho ={\sum}_{i=1}^{n}{\lambda}_{i}|{u}_{i}\rangle \langle {u}_{i}|$ |

Non-zero off-diagonals: indistinction in a block ${B}_{j}$ | Coherence in a superposition |

Set Concept | Vector-Space Concept |
---|---|

Partition ${\left\{{f}^{-1}\left(r\right)\right\}}_{r\in f\left(U\right)}$ | DSD ${\left\{{V}_{r}\right\}}_{r\in f\left(U\right)}$ |

$U={\uplus}_{r\in f\left(U\right)}{f}^{-1}\left(r\right)$ | $V={\oplus}_{r\in f\left(U\right)}{V}_{r}$ |

Numerical attribute $f:U\to \mathbb{R}$ | Observable $F{u}_{i}=f\left({u}_{i}\right){u}_{i}$ |

$f\upharpoonright S=rS$ | $F{u}_{i}=r{u}_{i}$ |

Constant set S of f | Eigenvector ${u}_{i}$ of F |

Value r on constant set S | Eigenvalue r of eigenvector ${u}_{i}$ |

Characteristic fcn. ${\chi}_{S}:U\to \{0,1\}$ | Projection operator ${P}_{\left[S\right]}{u}_{i}={\chi}_{S}\left({u}_{i}\right){u}_{i}$ |

${\cup}_{r\in f\left(U\right)}({f}^{-1}\left(r\right)\cap \left(\right))=I:\wp \left(U\right)\to \wp \left(U\right)$ | ${\sum}_{r\in f\left(U\right)}{P}_{{V}_{r}}=I:V\to V$ |

Spectral Decomp. $f={\sum}_{r\in f\left(U\right)}r{\chi}_{{f}^{-1}\left(r\right)}$ | Spectral Decomp. $F={\sum}_{r\in f\left(U\right)}r{P}_{{V}_{r}}$ |

Set of r-constant sets $\wp \left({f}^{-1}\left(r\right)\right)$ | Eigenspace ${V}_{r}$ of r-eigenvectors |

Cartesian product $U\times {U}^{\prime}$ | Tensor product $V\otimes {V}^{\prime}$ |

Partition Math | Hilbert Space Math |
---|---|

State: $\rho \left(\pi \right)={\sum}_{j=1}^{m}Pr\left({B}_{j}\right)|{b}_{j}\rangle \langle {b}_{j}|$ | $\rho ={\sum}_{i=1}^{n}{\lambda}_{i}|{u}_{i}\rangle \langle {u}_{i}|$ |

Observable: $g={\sum}_{r\in g\left(U\right)}r{\chi}_{{g}^{-1}\left(r\right)}:U\to \mathbb{R}$ | $G={\sum}_{r\in g\left(U\right)}r{P}_{{V}_{r}}$ |

Measurement: $\widehat{\rho}\left(\pi \right)={\sum}_{r\in g\left(U\right)}{P}_{{g}^{-1}\left(r\right)}\rho \left(\pi \right){P}_{{g}^{-1}\left(r\right)}$ | $\widehat{\rho}={\sum}_{r\in g\left(U\right)}{P}_{{V}_{r}}\rho {P}_{{V}_{r}}$ |

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## Share and Cite

**MDPI and ACS Style**

Ellerman, D.
A New Logic, a New Information Measure, and a New Information-Based Approach to Interpreting Quantum Mechanics. *Entropy* **2024**, *26*, 169.
https://doi.org/10.3390/e26020169

**AMA Style**

Ellerman D.
A New Logic, a New Information Measure, and a New Information-Based Approach to Interpreting Quantum Mechanics. *Entropy*. 2024; 26(2):169.
https://doi.org/10.3390/e26020169

**Chicago/Turabian Style**

Ellerman, David.
2024. "A New Logic, a New Information Measure, and a New Information-Based Approach to Interpreting Quantum Mechanics" *Entropy* 26, no. 2: 169.
https://doi.org/10.3390/e26020169