# Non-Deterministic Semantics for Quantum States

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## Abstract

**:**

## 1. Introduction

## 2. The Principle of Truth Functionality and Algebra Homomorphisms

**Definition**

**1.**

**Definition**

**2.**

#### 2.1. Homomorphisms

**Definition**

**3.**

**Definition**

**4.**

#### 2.1.1. Types of Languages and Homomorphisms between Structures

**Definition**

**5.**

**Definition**

**6.**

- 1.
- Individual variables: ${v}_{0},{v}_{1},\dots ,{v}_{n},\dots $.
- 2.
- Auxiliary symbols: left and right parenthesis, and commas.
- 3.
- Propositional connectives: ¬, ∨, ∧, →,…
- 4.
- Equality symbol: =.
- 5.
- Existential quantifier: ∃.
- 6.
- The symbols of τ.

**Definition**

**7.**

**Definition**

**8.**

- $\{{v}_{i}\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}i\ge 0\}\cup C\subseteq X$, where $C\subseteq \tau $, is the set of constants of τ.
- If $f\in {F}_{m}\subseteq \tau $, $1\le m$ and ${t}_{1},\dots ,{t}_{m}\in X$, then $f({t}_{1},\dots ,{t}_{m})\in X$.

**Definition**

**9.**

**Definition**

**10.**

- $\{\alpha :\alpha \phantom{\rule{4.pt}{0ex}}isanatomic\phantom{\rule{4.pt}{0ex}}\tau -formula\}\subseteq X$.
- If $\alpha ,\beta \in X$, then $(\neg \alpha )$, $(\alpha \vee \beta )$, $(\alpha \wedge \beta )$ and $(\alpha \to \beta )\in X$.
- If $\alpha \in X$ and ${v}_{i}$ is a variable, then $(\exists {v}_{i}\alpha )\in X$.

**Definition**

**11.**

- $A\ne \varnothing $.
- $I:\tau \to A\cup \{f:{A}^{m}\to A,1\le m\}\cup (\bigcup \{P\left({A}^{n}\right):1\le n\})$.

**Definition**

**12.**

- h is a function from A to B; $h:A\to B$.
- For each relational n-ary symbol ${r}_{n}$ in τ and each ${a}_{1},\dots ,{a}_{n}\in A$,$$\langle {a}_{1},\dots ,{a}_{n}\rangle \in {r}_{n}^{\mathcal{U}}\phantom{\rule{4.pt}{0ex}}iff\phantom{\rule{4.pt}{0ex}}\langle h\left({a}_{1}\right),\dots ,h\left({a}_{n}\right)\rangle \in {r}_{n}^{\mathcal{V}}.$$
- For each functional n-ary symbol ${f}_{n}$ in τ and each ${a}_{1},\dots ,{a}_{n}\in A$,$$h\left({f}_{n}^{\mathcal{U}}({a}_{1},...,{a}_{n})\right)={f}_{n}^{\mathcal{V}}(h\left({a}_{1}\right),\dots ,h\left({a}_{n}\right)).$$
- For each $c\in \tau ,h\left({c}^{\mathcal{U}}\right)={c}^{\mathcal{V}}.$

## 3. Quantum States and the Gleason and Kochen–Specker Theorems

#### 3.1. Quantum Probabilities and Gleason’S Theorem

- ${M}_{A}(\varnothing )=\mathbf{0}$
- ${M}_{A}\left(\mathbb{R}\right)=\mathbf{1}$
- ${M}_{A}\left({\cup}_{j}\left({B}_{j}\right)\right)={\sum}_{j}{M}_{A}\left({B}_{j}\right)$, for any mutually disjoint family ${B}_{j}$
- ${M}_{A}\left({B}^{c}\right)=\mathbf{1}-{M}_{A}\left(B\right)={\left({M}_{A}\left(B\right)\right)}^{\u27c2}$.

- $\mu \left(\mathbf{0}\right)=0$.
- $\mu \left({P}^{\u27c2}\right)=1-\mu \left(P\right)$
- For any pairwise orthogonal and denumerable family ${\left\{{P}_{j}\right\}}_{j\in \mathbb{N}}$, $\mu \left({\bigvee}_{j}{P}_{j}\right)={\sum}_{j}\mu \left({P}_{j}\right)$.

- $\mu (\varnothing )=0$
- $\mu \left({A}^{c}\right)=1-\mu \left(A\right),\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{where}\phantom{\rule{0.166667em}{0ex}}{(\dots )}^{c}\phantom{\rule{0.166667em}{0ex}}\mathrm{denotes}\mathrm{set}\mathrm{theoretical}\mathrm{complement}$
- for any pairwise disjoint and denumerable family ${\left\{{A}_{i}\right\}}_{i\in \mathbb{N}}$, $\mu \left({\bigcup}_{i}{A}_{i}\right)={\sum}_{i}\mu \left({A}_{i}\right)$.

- $\mu \left(\mathbf{0}\right)=0$
- $\mu \left({a}^{\u27c2}\right)=\mathbf{1}-\mu \left(a\right),\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{where}\phantom{\rule{0.166667em}{0ex}}{(\dots )}^{\u27c2}\phantom{\rule{0.166667em}{0ex}}\mathrm{denotes}\mathrm{the}\mathrm{orthocomplement}$
- for any pairwise orthogonal and denumerable family of elements ${\left\{{a}_{i}\right\}}_{i\in I}$, $\mu \left({\bigvee}_{i}{a}_{i}\right)={\sum}_{i}\mu \left({a}_{i}\right)$.

#### 3.2. The Kochen–Specker Theorem and the Failure of Truth Functionality in Quantum Mechanics

**Definition**

**13.**

**Proposition**

**1.**

- for every P, $v\left(P\right)=1$ iff $v(\neg P)=0$
- for every pair $P,Q\in \mathcal{P}\left(\mathcal{H}\right)$, if $v\left(P\right)=1$ and $P\le Q$, then $v\left(Q\right)=1$.

**Definition**

**14.**

## 4. Non-Deterministic Semantics

#### 4.1. Deterministic Matrices

**Definition**

**15.**

- V is a non-empty set of truth-values.
- D (designated truth-values) is a non-empty proper subset of V.
- For every n-ary connective ⟡ of L, O includes a corresponding function $\tilde{\u27e1}:{V}^{n}\to V$

#### 4.2. Non-Deterministic Matrices (N-Matrices)

**Definition**

**16.**

- V is a non-empty set of truth-values.
- $D\in \mathcal{P}\left(V\right)$ (designated truth-values) is a non-empty proper subset of V.
- For every n-ary connective ⟡ of L, O includes a corresponding function$$\tilde{\u27e1}:{V}^{n}\to \mathcal{P}\left(V\right)\setminus \{\varnothing \}$$

**Definition**

**17.**

- 1.
- A partial dynamic valuation in M (or an M-legal partial dynamic valuation), is a function v from some closed under subformulas subset $\mathcal{W}\subseteq Fr{m}_{L}$ to V, such that for each n-ary connective ⟡ of L, the following holds for all ${\psi}_{1},\dots ,{\psi}_{n}\in \mathcal{W}$:$$v(\u27e1({\psi}_{1},\dots ,{\psi}_{n}))\in \tilde{\u27e1}(v\left({\psi}_{1}\right),\dots ,v\left({\psi}_{n}\right)).$$A partial valuation in M is called a valuation if its domain is $Fr{m}_{L}$.
- 2.
- A (partial) static valuation in M (or an M-legal (partial) static valuation), is a (partial) dynamic valuation (defined in some $\mathcal{W}\subseteq Fr{m}_{L}$) which satisfies also the following composability (or functionality) principle: for each n-ary connective ⟡ of L and for every ${\psi}_{1},\dots ,{\psi}_{n},{\phi}_{1},\dots ,{\phi}_{n}\in \mathcal{W}$, if $v\left({\psi}_{i}\right)=v\left({\phi}_{i}\right)\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}(i=1,\dots ,n)$, then$$v(\u27e1({\psi}_{1},\dots ,{\psi}_{n}))=v(\u27e1({\phi}_{1},\dots ,{\phi}_{n})).$$

**Definition**

**18.**

- 1.
- A (partial) valuation v in M satisfies a formula ψ $(v\vDash \psi $) if ($v\left(\psi \right)$ is defined and) $v\left(\psi \right)\in D$. It is a model of Γ ($v\vDash \mathrm{\Gamma}$) if it satisfies every formula in Γ.
- 2.
- We say that ψ is dynamically (statically) valid in M, in symbols ${\vDash}_{M}^{d}\psi $ (${\vDash}_{M}^{s}\psi $), if $v\vDash \psi $ for each dynamic (static) valuation v in M.
- 3.
- the dynamic (static) consequence relation induced by M is defined as follows: $\mathrm{\Gamma}{\u22a2}_{M}^{d}\Delta $ ($\mathrm{\Gamma}{\u22a2}_{M}^{s}\Delta $) if every dynamic (static) model v in M of Γ satisfies some $\psi \in \Delta $.

**Theorem**

**1.**

**Theorem**

**2.**

## 5. N-Matrices for Probabilistic Theories

#### 5.1. Construction of the N-Matrices for the Quantum Formalism

#### 5.2. The General Case

## 6. Other Logical Aspects of Our Construction

#### 6.1. Quantum N-Matrices and Adequacy

**Definition**

**19.**

- 1.
- $\tilde{\wedge}$:$$If\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}a\in D\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}and\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}b\in D,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}then\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}a\tilde{\wedge}b\subseteq D$$$$\mathit{If}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}a\notin D,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{then}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}a\tilde{\wedge}b\subseteq V\setminus D$$$$\mathit{If}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}b\notin D,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{then}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}a\tilde{\wedge}b\subseteq V\setminus D$$
- 2.
- $\tilde{\vee}$:$$\mathit{If}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}a\in D,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{then}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}a\tilde{\vee}b\subseteq D$$$$\mathit{If}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}b\in D,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{then}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}a\tilde{\vee}b\subseteq D$$$$\mathit{If}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}a\notin D\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{and}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}b\notin D,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{then}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}a\tilde{\vee}b\subseteq V\setminus D$$
- 3.
- $\tilde{\supset}$:$$\mathit{If}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}a\notin D,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{then}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}a\tilde{\supset}b\subseteq D$$$$\mathit{If}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}b\in D,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{then}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}a\tilde{\supset}b\subseteq D$$$$\mathit{If}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}a\in D\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{and}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}b\notin D,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{then}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}a\tilde{\supset}b\subseteq V\setminus D$$

- If $a,b\in D$:

- If $a\in D,$ and $b\notin D$

- If $a,b\notin D$:

- If $a,b\in D$

- If $a\in D,b\notin D\phantom{\rule{4.pt}{0ex}}\mathrm{or}\phantom{\rule{4.pt}{0ex}}a\notin D,b\in D$

- If $a,b\notin D$

#### 6.2. Double Negation

- If $v\left(p\right)=a\in D=[\alpha ,1]$:

- If ${v}_{\left(p\right)}=a\notin D$

#### 6.3. Quantum N-Matrix as a Refinement of an F-Expansion of a Finite N-Matrix

**Definition**

**20.**

- 1.
- ${M}_{1}$ is a refinement of ${M}_{2}$ if ${V}_{1}\subseteq {V}_{2}$, ${D}_{1}={D}_{2}\cap {V}_{1}$, and ${\tilde{\u27e1}}_{{M}_{1}}\left(\overline{x}\right)\subseteq {\tilde{\u27e1}}_{{M}_{2}}\left(\overline{x}\right)$ for every n-ary conective ⟡ of L and every tuple $\overline{x}\in {V}_{1}^{n}$.
- 2.
- Let F be a function that assigns to each $x\in V$ a non-empty set $F\left(x\right)$, such that $F\left({x}_{1}\right)\cap F\left({x}_{2}\right)=\varnothing $ if ${x}_{1}\ne {x}_{2}$. The F-expansion of ${M}_{1}$ is the following N-matrix ${M}_{1}^{F}=\langle {V}_{F},{D}_{F},{O}_{F}\rangle $, with ${V}_{F}={\bigcup}_{x\in V}F\left(x\right)$, ${D}_{F}={\bigcup}_{x\in D}F\left(x\right)$, and ${\tilde{\u27e1}}_{{M}_{1}^{F}}({y}_{1},\dots ,{y}_{n})={\bigcup}_{z\in {\tilde{\u27e1}}_{{M}_{1}}({x}_{1},...,{x}_{n})}F\left(z\right)$ whenever ⟡ is an n-ary connective of L, and ${x}_{i}\in V$, ${y}_{i}\in F\left({x}_{i}\right)$ for every $1\le i\le n$. We say that ${M}_{2}$ is an expansion of ${M}_{1}$ if ${M}_{2}$ is the F-expansion of ${M}_{1}$ for some function F.

**Definition**

**21.**

- 1.
- simple if it is a simple refinement of the F-expansion of ${M}_{1}$.
- 2.
- preserving if $F\left(x\right)\cap {V}_{2}\ne \varnothing $ for every $x\in {V}_{1}$.
- 3.
- strongly preserving if it is preserving, and for every ${x}_{1},\dots ,{x}_{n}\in {V}_{2}$, $\u27e1\in {\u27e1}_{\mathcal{L}}^{n}$ and $y\in {\tilde{\u27e1}}_{1}(\tilde{F}\left[{x}_{1}\right],\dots ,\tilde{F}\left[{x}_{n}\right])$, it holds that the set $F\left(y\right)\cap {\tilde{\u27e1}}_{2}({x}_{1},\dots ,{x}_{n})$ is not empty.

**Proposition**

**2.**

**Proposition**

**3.**

- 1.
- For every $x\in {V}_{2}$, $x\in {D}_{2}$ iff $f\left(x\right)\in {D}_{1}$.
- 2.
- For every ${x}_{1},\dots ,{x}_{n}\in {V}_{2}$ and $y\in {\tilde{\u27e1}}_{2}({x}_{1},\dots ,{x}_{n})$, it holds that $f\left(y\right)\in {\tilde{\u27e1}}_{1}(f\left({x}_{1}\right),\dots ,f\left({x}_{n}\right))$.

**Proposition**

**4.**

## 7. Conclusions

- There are several ways in which one can affirm that the quantum formalism does not obey truth functionality.
- The set of projection operators admits N-matrices, and thus, the N-matrices formalism can be adapted to quantum mechanics.
- Each quantum state can be interpreted as a valuation associated to a non-deterministic semantics. Indeed, the set of quantum states can be characterized as being equivalent to the set of valuations defined by the N-matrices that we propose in Section 5. We have proved that quantum states, considered as valuations, are, in general, dynamic and non-static. We have provided a similar analysis for generalized probabilistic models.
- There exist different candidates for non-deterministic semantics which are compatible with the quantum formalism. We have studied different examples.
- It is possible to give a notion of a logical consequence associated to non-deterministic semantics in the quantum formalism (a study that should be extended, of course, in future work).

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Deterministic Matrices | N-Matrices | |
---|---|---|

Truth values set | V | V |

Designated values set | $D\subset V$ | $D\subset V$ |

Connectives ⟡ | $\tilde{\u27e1}:{V}^{n}\to V$ | $\tilde{\u27e1}:{V}^{n}\to \mathcal{P}\left(V\right)\setminus \{\varnothing \}$ |

Valuations | Non-dynamic | Possibly dynamic and possibly non-static |

Truth-Functional | Yes | Not necessarily |

**Table 2.**Table comparing the different valuations that can be defined on classical vs. quantum propositional systems.

Classical systems | Quantum systems | |
---|---|---|

Lattice | Boolean Algebra | Projections lattice |

Truth-tables | Admit deterministic matrices | Only proper N-matrices |

Truth-Values | Admit valuations in $\{0,1\}$ | Only valuations in $[0,1]$ |

Truth-Functional | Yes (for deterministic states) | No |

Satisfy Adequacy | Yes (for deterministic states) | No |

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Jorge, J.P.; Holik, F.
Non-Deterministic Semantics for Quantum States. *Entropy* **2020**, *22*, 156.
https://doi.org/10.3390/e22020156

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Jorge JP, Holik F.
Non-Deterministic Semantics for Quantum States. *Entropy*. 2020; 22(2):156.
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**Chicago/Turabian Style**

Jorge, Juan Pablo, and Federico Holik.
2020. "Non-Deterministic Semantics for Quantum States" *Entropy* 22, no. 2: 156.
https://doi.org/10.3390/e22020156