# A Logic for Quantum Register Measurements

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

“Is there a propositional logic that has the observational trees as set of models?”

#### 1.1. A Gentle Informal Introduction of Our Proposal

**iff**then each possible measurement (to simplify the treatment, we consider here only the so called PVM-ProjectionValue Measurement [4]) of $\varphi $ returns (probabilistic) a set of new quantum registers in which in turn p is true. Following this intuition, we set that e

**is true in**${\sum}_{{C}_{i}\in {2}^{R}}{a}_{i}\left|{C}_{i}\right.\u232a$ iff e

**is true in**each $\left|{C}_{i}\right.\u232a$ iff $e\in {C}_{i}$.

- S formula $A\vee B$
**is true in**a quantum state $\left|\psi \right.\u232a$ iff after every sequence (eventually the empty sequence) of measurements of $\left|\psi \right.\u232a$ in the resulting state $\left|\psi \right.\u232a$ we have either the truth of A or those of B. - S formula $A\to B$
**is true in**a quantum state $\left|\psi \right.\u232a$ iff after each sequence (eventually the empty sequence) of measurements of $\left|\psi \right.\u232a$, in the resulting state $\left|\psi \right.\u232a$ we have that if A is true then B is true.

#### 1.2. Synopsis

## 2. A Quantum Tree Model for Observations

#### 2.1. Background

#### 2.1.1. Trees

#### 2.1.2. Quantum Registers

- $p\in \left|{b}_{i}\right.\u232a$ to mean that $p\in {b}_{i}$; and
- $p\in {\sum}_{i}{a}_{i}\left|{b}_{i}\right.\u232a$ to mean that $\forall {a}_{j}\ne 0.p\in \left|{b}_{j}\right.\u232a$

#### 2.1.3. Measurement Operators

**Definition**

**1.**

- P is hermitian; and
- $ker(P)\phantom{\rule{4pt}{0ex}}\perp \phantom{\rule{4pt}{0ex}}im(P)$.

**Definition**

**2.**

#### 2.2. Observation Trees

**Definition**

**3**

**.**Let $\mathbb{X}$ be a finite set of propositional symbols. An observational tree is a structure ${\mathbb{T}}_{\mathbb{X}}=\langle \langle T,\le \rangle ,\mathfrak{p},\mathfrak{a},\mathfrak{s}\rangle $ where

- $\mathcal{T}=\langle T,\le \rangle $ is an abstract tree;
- $\mathfrak{p},\mathfrak{a},\mathfrak{s}$ are the following labelling functions:
- -
- $\mathfrak{p}:\mathbb{E}\to {(0,1]}_{\mathbb{R}}$;
- -
- $\mathfrak{a}:T\to \mathfrak{M}$;
- -
- $\mathfrak{s}:T\to {\mathcal{R}}_{\mathbb{X}}\cup \{0\}$

- for which some constraints holds. Let us suppose that $\mathfrak{a}(\alpha )={({P}_{i})}_{i<k}\in \mathfrak{M}$, then:
- -
- $\forall i<k.\phantom{\rule{4pt}{0ex}}({P}_{i}(\alpha )\ne 0\Rightarrow \alpha \ast \langle i\rangle \in T)$;
- -
- if $\forall j\ge K.\phantom{\rule{4pt}{0ex}}\alpha \ast \langle j\rangle \notin T$;
- -
- $\forall i<K$ if $\alpha \ast \langle i\rangle \in T$ then
- -
- $\mathfrak{p}(\alpha ,\alpha \ast \langle i\rangle )=\u2329\mathfrak{s}(\alpha )\mid {P}_{i}\mid \mathfrak{s}(\alpha )\u232a$
- -
- $\mathfrak{s}(\alpha \ast \langle i\rangle )=\frac{{P}_{i}(\mathfrak{s}(\alpha ))}{\sqrt{\mathfrak{p}(\alpha ,\alpha \ast \langle i\rangle )}}$

- $\mathfrak{p}$ assigns a (correct) probability to each edge.
- $\mathfrak{a}$ assigns to each node a sequence of observations (an element in $\mathfrak{M}$), in particular the sequence that generates the current (evaluation of the) state, starting from the root node.
- $\mathfrak{s}$ assigns to each node a quantum register.

**Proposition**

**1**

**.**Let ${\mathbb{T}}_{\mathbb{X}}=\langle \langle T,\le \rangle ,\mathfrak{p},\mathfrak{a},\mathfrak{s}\rangle $ be an observational tree, then

**Remark**

**1.**

## 3. The Logic of Observations

**Definition**

**4**

**.**The language ${\mathfrak{L}}_{\mathbb{P}}$ of ${\mathcal{L}}_{\mathbb{P}}$ is built upon propositional symbols, which we set to $\mathbb{P}$ and connectives $\to ,\wedge ,\vee ,\perp $.

**Definition**

**5**

**.**The semantics of a formula A with respect to to an observational tree ${\mathbb{T}}_{\mathbb{X}}$ with $\mathbb{X}\supseteq \mathbb{P}[A]$ is defined as:

- ${\mathbb{T}}_{\mathbb{X}},\alpha \phantom{\rule{3.33333pt}{0ex}}\u22ab\phantom{\rule{3.33333pt}{0ex}}q$ iff $q\in \mathfrak{s}(\alpha )$;
- ${\mathbb{T}}_{\mathbb{X}},\alpha {\u22af}_{}\perp $
- ${\mathbb{T}}_{\mathbb{X}},\alpha \phantom{\rule{3.33333pt}{0ex}}\u22ab\phantom{\rule{3.33333pt}{0ex}}A\wedge B$ iff ${\mathbb{T}}_{\mathbb{X}},\alpha \phantom{\rule{3.33333pt}{0ex}}\u22ab\phantom{\rule{3.33333pt}{0ex}}A\phantom{\rule{4pt}{0ex}}\&\phantom{\rule{4pt}{0ex}}{\mathbb{T}}_{\mathbb{X}},\alpha \phantom{\rule{3.33333pt}{0ex}}\u22ab\phantom{\rule{3.33333pt}{0ex}}B$
- ${\mathbb{T}}_{\mathbb{X}},\alpha \phantom{\rule{3.33333pt}{0ex}}\u22ab\phantom{\rule{3.33333pt}{0ex}}A\vee B$ iff ${\mathbb{T}}_{\mathbb{X}},\alpha \phantom{\rule{3.33333pt}{0ex}}\u22ab\phantom{\rule{3.33333pt}{0ex}}A\phantom{\rule{4pt}{0ex}}\mathsf{OR}\phantom{\rule{4pt}{0ex}}{\mathbb{T}}_{\mathbb{X}},\alpha \phantom{\rule{3.33333pt}{0ex}}\u22ab\phantom{\rule{3.33333pt}{0ex}}B$
- ${\mathbb{T}}_{\mathbb{X}},\alpha \phantom{\rule{3.33333pt}{0ex}}\u22ab\phantom{\rule{3.33333pt}{0ex}}A\to B$ iff $\forall \beta \le \alpha \phantom{\rule{4pt}{0ex}}{\mathbb{T}}_{\mathbb{X}},\beta \phantom{\rule{3.33333pt}{0ex}}\u22ab\phantom{\rule{3.33333pt}{0ex}}A\phantom{\rule{4pt}{0ex}}\Rightarrow \phantom{\rule{4pt}{0ex}}{\mathbb{T}}_{\mathbb{X}},\beta \phantom{\rule{3.33333pt}{0ex}}\u22ab\phantom{\rule{3.33333pt}{0ex}}B$

**Proposition**

**2.**

- 1.
- ${\mathbb{T}}_{\mathbb{X}},\alpha \phantom{\rule{3.33333pt}{0ex}}\u22ab\phantom{\rule{3.33333pt}{0ex}}A\iff \forall \beta \le \alpha \Rightarrow {\mathbb{T}}_{\mathbb{X}},\beta \phantom{\rule{3.33333pt}{0ex}}\u22ab\phantom{\rule{3.33333pt}{0ex}}A$;
- 2.
- ${\mathbb{T}}_{\mathbb{X}},\langle \phantom{\rule{4pt}{0ex}}\rangle \phantom{\rule{3.33333pt}{0ex}}\u22ab\phantom{\rule{3.33333pt}{0ex}}A\iff \forall \alpha \in {T}_{\mathbb{X}}.{\mathbb{T}}_{\mathbb{X}},\alpha \phantom{\rule{3.33333pt}{0ex}}\u22ab\phantom{\rule{3.33333pt}{0ex}}A$.

**Proof.**

**Proposition**

**3.**

#### 3.1. From Observational Trees to Kripke Models

- ${T}_{\mathbb{T}}=T$;
- $\alpha \u2291\beta \iff \beta \le \alpha $;
- ${V}_{\mathbb{T}}:{T}_{\mathbb{T}}\to {2}^{\mathbb{P}}$ is s.t. $q\in {V}_{\mathbb{T}}(\alpha )\iff q\in \mathfrak{s}(\alpha )$.

**Proposition**

**4.**

**intuitionistic**Kripke model.

**Definition**

**6**

**.**The semantics of a formula A with respect to to an Kripke Model ${\mathfrak{K}}_{{\mathbb{T}}_{\mathbb{X}}}$ with $\mathbb{X}\supseteq \mathbb{P}[A]$ is defined as:

- ${\mathfrak{K}}_{{\mathbb{T}}_{\mathbb{X}}},\alpha \u22a9q$ iff $q\in {V}_{\mathbb{T}}(\alpha )$;
- ${\mathfrak{K}}_{{\mathbb{T}}_{\mathbb{X}}},\alpha \neg \u22a9\perp $;
- ${\mathfrak{K}}_{{\mathbb{T}}_{\mathbb{X}}},\alpha \u22a9A\wedge B$ iff ${\mathfrak{K}}_{{\mathbb{T}}_{\mathbb{X}}},\alpha \u22a9A\phantom{\rule{4pt}{0ex}}\&\phantom{\rule{4pt}{0ex}}{T}_{\mathbb{T}},\alpha \u22a9B$;
- ${\mathfrak{K}}_{{\mathbb{T}}_{\mathbb{X}}},\alpha \u22a9A\vee B$ iff ${\mathfrak{K}}_{{\mathbb{T}}_{\mathbb{X}}},\alpha \u22a9A\phantom{\rule{4pt}{0ex}}\mathsf{OR}\phantom{\rule{4pt}{0ex}}{\mathfrak{K}}_{{\mathbb{T}}_{\mathbb{X}}},\alpha \u22a9B$;
- ${\mathfrak{K}}_{{\mathbb{T}}_{\mathbb{X}}},\alpha \u22a9A\to B$ iff $\forall \beta ,\alpha {\u2291}_{\mathbb{T}}\beta \phantom{\rule{4pt}{0ex}}\Rightarrow ({\mathfrak{K}}_{{\mathbb{T}}_{\mathbb{X}}},\beta \u22a9A\phantom{\rule{4pt}{0ex}}\Rightarrow \phantom{\rule{4pt}{0ex}}{\mathfrak{K}}_{{\mathbb{T}}_{\mathbb{X}}},\beta \u22a9B)$.

**Proposition**

**5.**

**Proof.**

**Corollary**

**1.**

**Theorem**

**1.**

**Proof.**

#### 3.2. From Kripke Models to Observational Trees

**Theorem**

**2.**

**Proof.**

**Step****1**- Choose a set of distinguishable propositional symbols $PN=\{{p}_{t}:t\in N\}$ s.t. ${\mathbb{P}}_{T}\cap N=\varnothing $ and build the Hilbert Space is ${\mathcal{H}}_{PN\cup {\mathbb{P}}_{T}}$.
**Step****2**- Define ⊑ as ${\le}^{-}1$ ($t\u2291u\iff u\le t$).
**Step****3**- Let $\mathfrak{a}(t)$ be the set of projectors ${\mathcal{O}}_{t}=\{{P}_{{i}_{1}},\dots {P}_{{i}_{m}}\}$ defined as:$${\mathcal{O}}_{t}=\left\{\begin{array}{c}\varnothing \phantom{\rule{4pt}{0ex}}\mathrm{if}\phantom{\rule{4pt}{0ex}}t\phantom{\rule{4pt}{0ex}}\mathrm{is}\mathrm{a}\mathrm{leaf}\hfill \\ \{{P}_{{i}_{1}},\dots {P}_{{i}_{m}}\}\phantom{\rule{4pt}{0ex}}\mathrm{s}.\mathrm{t}.\phantom{\rule{4pt}{0ex}}\forall j\in [1,m].{P}_{{i}_{j}}\phantom{\rule{4pt}{0ex}}\mathrm{is}\mathrm{the}\mathrm{projector}\mathrm{in}\mathrm{the}\mathrm{subspace}\mathrm{of}\mathrm{registers}\hfill \\ \phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\beta \phantom{\rule{4pt}{0ex}}\mathrm{s}.\mathrm{t}.\phantom{\rule{4pt}{0ex}}t\ast \langle {i}_{j}\rangle \in \beta \phantom{\rule{4pt}{0ex}}\mathrm{and}\phantom{\rule{4pt}{0ex}}t\ast \langle {i}_{j}\rangle \u2291t,\phantom{\rule{4pt}{0ex}}\mathrm{otherwise}.\hfill \end{array}\right.$$
**Step****4**- The functions $\mathfrak{p},\mathfrak{s}$ are univocally defined by the following labeling $\mathfrak{s}\langle \phantom{\rule{4pt}{0ex}}\rangle $ of the root.Let us consider the set of L of leaves of K, and consider for each $u\in :$ the set ${C}_{u}=\{t:t\in N\phantom{\rule{4pt}{0ex}}\&\phantom{\rule{4pt}{0ex}}u\u2291t\phantom{\rule{4pt}{0ex}}\&\phantom{\rule{4pt}{0ex}}t\in N\}$ and the set $P{r}_{u}={\bigcup}_{t\in {C}_{u}}V(t)$. We define $\mathfrak{s}\langle \phantom{\rule{4pt}{0ex}}\rangle ={\sum}_{u\in L}\frac{1}{\sqrt{\left|L\right|}}\left|{C}_{u}\cup P{r}_{u}\right.\u232a$

**Example**

**1.**

**Corollary**

**2.**

**Theorem**

**3.**

## 4. Possible Developments

- We have shown that intuitionistic logic is “the” logic of observational tree. This means that we could think to move from the model theoretic approach to a proof theoretical one. It is well known that, via the so called Curry–Howard isomorphism, it is possible to associate a lambda calculus to the intuitionistic proofs. Is it possible to give a quantum interpretation of such a calculus? Our idea is to start again with the BHK interpretation of intuitionistic logic. For example, according to this interpretation, a proof of $A\to B$ could be seen as a measurement process that transforms each measurement process A into one of B.
- We think also to extend the model theoretic approach in order to deal with unitary transformations. One possibility we have in mind is to add a temporal (possibly classical or intuitionistic) dimension to intuitionistic logic, so that we can move in two different directions: an intuitionist one linked to the measurement process, and an linear temporal one that is linked to unitary evolution of the quantum system. The studies of Finger and Gabbay on the temporalization of logical system could help (see, e.g., [19].)

## Author Contributions

## Funding

## Conflicts of Interest

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Masini, A.; Zorzi, M. A Logic for Quantum Register Measurements. *Axioms* **2019**, *8*, 25.
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Masini A, Zorzi M. A Logic for Quantum Register Measurements. *Axioms*. 2019; 8(1):25.
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Masini, Andrea, and Margherita Zorzi. 2019. "A Logic for Quantum Register Measurements" *Axioms* 8, no. 1: 25.
https://doi.org/10.3390/axioms8010025