# Quantum Models of Classical World

## Abstract

**:**

**PACS**03.65.Ta; 03.65.-w; 03.65.Ca

Contents | |

1 Introduction | 791 |

I Corrected language of quantum mechanics | 807 |

2 States, observables and symmetries | 808 |

2.1 States | 808 |

2.2 Observables | 817 |

2.3 Galilean group | 829 |

3 Composition of quantum systems | 837 |

3.1 Composition of heterogeneous systems | 838 |

3.2 Composition of identical systems | 845 |

3.3 State reduction | 864 |

II The models | 867 |

4 Quantum models of classical properties | 867 |

4.1 Modified correspondence principle | 869 |

4.2 Maximum entropy assumption in classical mechanics | 871 |

4.3 Classical ME packets mechanics | 873 |

4.4 Quantum ME packets | 878 |

4.5 Classical limit | 887 |

4.6 A model of classical rigid body | 889 |

4.7 Joint measurement of position nd momentum | 893 |

5 Quantum models of preparation and registration | 896 |

5.1 Old theory of measurement | 897 |

5.2 New theory of measurement | 901 |

5.3 Models of direct registrations | 903 |

5.4 Comparison with other changes of separation status | 915 |

6 Conclusions | 917 |

Acknowledgements | 920 |

References | 920 |

## 1. Introduction

Similarly, Ref. [21], p. 17 describes Stern–Gerlach experiment:We can consider the centre of mass [of a microscopic system] as a ’special’ measuring apparatus...

Paradoxically, this notion of measurement apparatus (mostly called “meter”) is quite useful for the very precise modern quantum experiments, such as non-demolition measurements or weak measurements, see, e.g., Refs. [22,23]. Quite generally, these experiments utilize auxiliary microscopic systems called ancillas. If the pointer is interpreted as an ancilla, then the old theory works well at least for the interaction of the measured system with the ancilla. However, the fact that ancillas themselves must be registered by macroscopic detectors is “deemed irrelevant”. A more detailed analysis is given in Section 5.1.The microscopic object under investigation is the magnetic moment μ of an atom.. . . The macroscopic degree of freedom to which it is coupled in this model is the centre of mass position $\mathbf{r}$.. . . I call this degree of freedom macroscopic because different final values of $\mathbf{r}$ can be directly distinguished by macroscopic means, such as the detector.. . . From here on, the situation is simple and unambiguous, because we have entered the macroscopic world: The type of detectors and the detail of their functioning are deemed irrelevant.

#### 1.0.1. Examples of quantum systems

#### 1.0.2. Examples of quantum experiments

- Electrons. One possible source (called field emission, see e.g. Ref. [25], p. 38) consists of a cold cathode in the form of a sharp tip and a flat anode with an aperture in the middle at some distance from the cathode, in a vacuum tube. The electrostatic field of, say, few kV will enable electrons to tunnel from the metal and form an electron beam of about ${10}^{7}$ electrons per second through the aperture, with a relatively well-defined average energy.
- Neutrons can be obtained through nuclear reaction. This can be initiated by charged particles or gamma rays that can be furnished by an accelerator or a radioactive substance. For example the so-called Ra-Be source consists of finely divided RaCl${}_{2}$ mixed with powdered Be, contained in a small capsule. Decaying Ra provides alpha particles that react with Be. The yield for 1 mg Ra is about ${10}^{4}$ neutrons per second with broad energy spectrum from small energies to about 13 MeV. The emission of neutrons is roughly spherically symmetric centred at the capsule.
- Atoms and molecules. A macroscopic specimen of the required substance in gaseous phase at certain temperature can be produced, e.g., by an oven. The gas is in a vessel with an aperture from which a beam of the atoms or molecules emerges.

#### 1.0.3. Realist Model Approach to quantum mechanics

The electron is viewed here as a real object that is extended over the whole width of the biprism apparatus. After this intuitive introduction, we give now a general and systematic account of our realist interpretation.…An electron object runs through between the wire and both the left and right plate simultaneously and interferes with itself afterwards …

- Their values may be arbitrary mathematical entities (sets, maps between sets, etc.). For example, the Hamiltonian of a closed quantum system involves a relation between energy and some other observables of the system. This relation is an example of such a complex property.
- Their values do not need to be directly obtained by individual registrations. For example, to measure a cross-section a whole series of scattering experiments must be done. Thus, their values do not necessarily possess probability distribution but may be equivalent to, or derivable from, probability distributions.

**Basic Ontological Hypothesis of Quantum Mechanics**A sufficient condition for a property to be objective is that its value is uniquely determined by a preparation according to the rules of standard quantum mechanics. The “value” is the value of the mathematical expression that describes the property and it may be more general than just a real number. To observe an objective property, many registrations of one or more observables are necessary.

This can be contrasted with the usual cautious characterization of the subject, as e.g. [21], p. 13:Quantum mechanics studies objective properties of existing microscopic objects.

...quantum theory is a set of rules allowing the computation of probabilities for the outcomes of tests [registrations] which follow specific preparations.

#### 1.0.4. Probability and information

- $\mathcal{C}={\mathcal{C}}_{1}\vee \dots \vee {\mathcal{C}}_{N}$, where ∨ is the logical union (disjunction),
- each ${\mathcal{C}}_{i}$ be still recognizable and reproducible,
- each outcome allowed by $\mathcal{C}$ is uniquely determined by one of ${\mathcal{C}}_{i}$’s.

**Definition**

**1**

**Part I**

**Corrected language of quantum mechanics**

## 2. States, observables and symmetries

#### 2.1. States

#### 2.1.1. Mathematical preliminaries

**Theorem 1**Trace defines the norm ${\parallel \mathsf{A}\parallel}_{s}$ on ${\mathbf{L}}_{r}\left(\mathbf{H}\right)$ by

**Definition 2**The norm (4) is called trace norm and all elements of ${\mathbf{L}}_{r}\left(\mathbf{H}\right)$ with finite trace norm are called trace-class. The set of all trace-class operators is denoted by $\mathbf{T}\left(\mathbf{H}\right)$.

**Theorem 2**$\mathbf{T}\left(\mathbf{H}\right)$ with the operation of linear combination of operators on $\mathbf{H}$, partial ordering ≥ defined above and completed with respect to the norm (4) is an ordered Banach space. A trace-class operator is bounded, its trace is finite and its spectrum is discrete.

**Definition 3**Face $\mathbf{W}$ is a (norm) closed subset of $\mathbf{T}{\left(\mathbf{H}\right)}_{1}^{+}$ that is invariant with respect to convex combinations and contains all convex components of any $\mathbf{T}\in \mathbf{W}$.

**Theorem 3**Every face $\mathbf{W}\subset \mathbf{T}{\left(\mathbf{H}\right)}_{1}^{+}$ can be written as $\mathbf{W}\left(\mathsf{T}\right)$ for a suitably chosen $\mathsf{T}\in \mathbf{W}\left(\mathsf{T}\right)$ where $\mathbf{W}\left(\mathsf{T}\right)$ is the smallest face that contains $\mathsf{T}$.

**Theorem 4**To each face $\mathbf{W}$ of $\mathbf{T}{\left(\mathbf{H}\right)}_{1}^{+}$ there is a unique projection $\mathsf{P}:\mathbf{H}\mapsto {\mathbf{H}}^{\prime}$, where ${\mathbf{H}}^{\prime}$ is a closed subspace of $\mathbf{H}$, for which $\mathsf{T}\subset \mathbf{W}$ is equivalent to

**Theorem 5**Let $\mathsf{P}\left(\mathbf{H}\right)$ be infinite-dimensional and let ${\mathsf{T}}_{1},{\mathsf{T}}_{2}\in {\mathbf{W}}_{\mathsf{P}}$ be positive definite on $\mathsf{P}\left(\mathbf{H}\right)$. Then

**Proof**Suppose that ${\mathsf{T}}_{1}$ is a component of ${\mathsf{T}}_{2}$. Then, there is ${\mathsf{T}}_{3}$ and $w\in (0,1)$ such that

**Definition 4**An element $\mathsf{T}$ is called extremal element of $\mathbf{T}{\left(\mathbf{H}\right)}_{1}^{+}$ if $\mathbf{W}\left(\mathsf{T}\right)$ is zero-dimensional, i.e., if the condition

**Theorem 6**$\mathsf{T}$ is extremal iff $\mathsf{T}=|\psi \rangle \langle \psi |$, where ψ is a unit vector of $\mathbf{H}$.

#### 2.1.2. General rules

**Rule 1**With each quantum system $\mathcal{S}$ of type τ, a complex separable Hilbert space ${\mathbf{H}}_{\tau}$ is associated. ${\mathbf{H}}_{\tau}$ is a representation space of certain group associated with Galilean group and τ determines the representation (see Section 2.3).

**Rule 2**The state space of $\mathcal{S}$ is $\mathbf{T}{\left({\mathbf{H}}_{\tau}\right)}_{1}^{+}$. For each $\mathsf{T}\in \mathbf{T}{\left({\mathbf{H}}_{\tau}\right)}_{1}^{+}$, there is a preparation $\mathcal{P}$ that prepares a system $\mathcal{S}$ of type τ in state $\mathsf{T}$. $\mathsf{T}$ is then an objective property of $\mathcal{S}$.

**Definition 5**Let ${\mathcal{P}}_{1}$ and ${\mathcal{P}}_{2}$ be two preparation of $\mathcal{S}$ and $w\in [0,1]$. Statistical mixture

**Rule 3**Let ${\mathcal{P}}_{1}$ and ${\mathcal{P}}_{2}$ be two preparation of $\mathcal{S}$ and let the corresponding states be ${\mathsf{T}}_{1}$ and ${\mathsf{T}}_{2}$. Then the statistical mixture (10) prepares state

**Definition 6**Let $\mathcal{P}$ of the form (10) prepare $\mathcal{S}$ in state $\mathsf{T}$. Then we write (11) and call the right-hand side of Equation (11) the statistical decomposition of $\mathsf{T}$. States that have a non-trivial statistical decomposition ($w\in (0,1)$) will be called decomposable, otherwise indecomposable.

- objectivity: a state of a system is an objective property of the system,
- universality: any system is always in some state,
- exclusivity: a system cannot be in two different states simultaneously,
- completeness: any state of a system contains maximum information that can exist about the system.
- locality the state of a system determines the position of the system.

**Definition 7**Let $\mathcal{S}$ be system of type τ and $f:\mathbf{T}{\left({\mathbf{H}}_{\tau}\right)}_{1}^{+}\mapsto \mathbb{R}$ a function. Then the proposition “$f=a$” is a simple property. Simple properties form a Boolean lattice with the logical operation union and intersection. Each simple property defines a subset $\{\mathsf{T}\in \mathbf{T}{\left({\mathbf{H}}_{\tau}\right)}_{1}^{+}\phantom{\rule{4pt}{0ex}}|\phantom{\rule{4pt}{0ex}}f\left(\mathsf{T}\right)=a\}$. The Boolean lattice of simple properties is isomorphic to the Boolean lattice of the subsets.

#### 2.2. Observables

#### 2.2.1. Mathematical preliminaries

**Definition 8**Let $\mathbf{F}$ be the Boolean lattice of all Borel subsets of ${\mathbb{R}}^{n}$. A positive operator valued (POV) measure

- 1.
- positivity: $\mathsf{E}\left(X\right)\ge 0$ for all $X\in \mathbf{F}\phantom{\rule{4pt}{0ex}},$
- 2.
- σ-additivity: if $\left\{{X}_{k}\right\}$ is a countable collection of disjoint sets in $\mathbf{F}$ then$$\mathsf{E}\left({\cup}_{k}{X}_{k}\right)=\sum _{k}\mathsf{E}\left({X}_{k}\right)\phantom{\rule{4pt}{0ex}},$$
- 3.
- normalization:$$\mathsf{E}\left({\mathbb{R}}^{n}\right)=\mathsf{1}\phantom{\rule{4pt}{0ex}}.$$

**Theorem 7**${\mathbf{L}}_{r}{\left(\mathbf{H}\right)}_{\le 1}^{+}$ is the set of elements of ${\mathbf{L}}_{r}\left(\mathbf{H}\right)$ satisfying the inequality

**Definition 9**Let $\mathsf{E}:\mathbf{F}\mapsto \mathbf{L}\left(\mathbf{H}\right)$ and ${\mathsf{E}}^{\prime}:{\mathbf{F}}^{\prime}\mapsto \mathbf{L}\left(\mathbf{H}\right)$ be two POV measures that satisfy

**Theorem 8**For any POV measure $\mathsf{E}:\mathbf{F}\mapsto \mathbf{L}\left(\mathbf{H}\right)$ the following two conditions are equivalent

**Theorem 9**If ${\mathsf{T}}_{1}$ and ${\mathsf{T}}_{2}$ from $\mathbf{T}{\left({\mathbf{H}}_{\tau}\right)}_{1}^{+}$ satisfy

**Theorem 10**For each $\mathsf{T}\in \mathbf{T}{\left({\mathbf{H}}_{\tau}\right)}_{1}^{+}$ and $\mathsf{E}\in {\mathbf{L}}_{r}{\left({\mathbf{H}}_{\tau}\right)}_{\le 1}^{+}$, the condition $tr\left[\mathsf{T}\mathsf{E}\right]=1$ is equivalent to

**Theorem 11**For any POV measure $\mathsf{E}:\mathbf{F}\mapsto \mathbf{L}\left(\mathbf{H}\right)$ and any $\mathsf{T}\in \mathbf{T}{\left(\mathbf{H}\right)}_{1}^{+}$, the mapping

#### 2.2.2. General rules

**Rule 4**Any quantum mechanical observable for system $\mathcal{S}$ of type τ is mathematically described by some POV measure $\mathsf{E}:\mathbf{F}\mapsto {\mathbf{L}}_{r}\left({\mathbf{H}}_{\tau}\right)$. Each outcome of an individual registration of the observable $\mathsf{E}\left(X\right)$ performed on quantum object $\mathcal{S}$ yields an element of $\mathbf{F}$. Each registration apparatus that interacts with $\mathcal{S}$ determines a unique observable of $\mathcal{S}$.

**Definition 10**If a PV measure is an observable, the observable is called sharp.

**Rule 5**The number ${p}_{\mathsf{T}}^{\mathsf{E}}\left(X\right)$ defined by Equation (14) is the probability that a registration of the observable $\mathsf{E}\left(X\right)$ performed on object $\mathcal{S}$ in the state $\mathsf{T}$ leads to a result in the set X.

**Definition 11**Let $\mathsf{A}$ be a sharp observable and $\mathsf{T}$ be a state. Then

**Definition 12**The normalized correlation in a state $\mathsf{T}$ of any two commuting sharp observables $\mathsf{A}$ and $\mathsf{B}$ is defined by

**Proposition 1**The average ${\langle \mathsf{A}\rangle}_{\mathsf{T}}$ and variance ${\Delta}_{\mathsf{T}}\mathsf{A}$ of any sharp observable $\mathsf{A}$ and correlation $C(\mathsf{A},\mathsf{B},\mathsf{T})$ of any two commuting sharp observables $\mathsf{A}$ and $\mathsf{B}$ in state $\mathsf{T}$ of object $\mathcal{S}$ are an objective (dynamical) property of $\mathcal{S}$ that has been prepared in state $\mathsf{T}$.

#### 2.2.3. Joint measurability

**Proposition 2**Let $\mathsf{A}$ and $\mathsf{B}$ be two s.a. operators with a common invariant domain, let $i[\mathsf{A},\mathsf{B}]$ have a s.a. extension and let $\mathsf{T}$ be an arbitrary state. Then

**Definition 13**Two elements ${\mathsf{E}}_{1}$ and ${\mathsf{E}}_{2}$ of ${\mathbf{L}}_{r}{\left(\mathbf{H}\right)}_{\le 1}^{+}$ are jointly measurable if there is a POV measure $\mathsf{E}:\mathbf{F}\mapsto {\mathbf{L}}_{r}{\left(\mathbf{H}\right)}_{\le 1}^{+}$ such that ${\mathsf{E}}_{1}=\mathsf{E}\left({X}_{1}\right)$ and ${\mathsf{E}}_{2}=\mathsf{E}\left({X}_{2}\right)$ for some ${X}_{1}$ and ${X}_{2}$ in $\mathbf{F}$.

**Proposition 3**Two effects ${\mathsf{E}}_{1}$ and ${\mathsf{E}}_{2}$ are jointly measurable if and only if there are three elements ${\mathsf{E}}_{1}^{\prime}$, ${\mathsf{E}}_{2}^{\prime}$ and ${\mathsf{E}}_{12}^{\prime}$ in ${\mathbf{L}}_{r}{\left(\mathbf{H}\right)}_{\le 1}^{+}$ such that

#### 2.2.4. Contextuality

#### 2.2.5. Superselection rules

**Rule 6**All effects that can be registered on system $\mathcal{S}$ mixing ${\mathcal{S}}_{1}$ and ${\mathcal{S}}_{2}$ have the form ${\mathsf{A}}_{1}\oplus {\mathsf{A}}_{2}$ where ${\mathsf{A}}_{1}\in {\mathbf{L}}_{r}{\left({\mathbf{H}}_{1}\right)}_{\le 1}^{+}$ and ${\mathsf{A}}_{2}\in {\mathbf{L}}_{r}{\left({\mathbf{H}}_{2}\right)}_{\le 1}^{+}$.

**Definition 14**A discrete sharp observable $\mathsf{Z}$ of a system $\mathcal{S}$ with Hilbert space $\mathbf{H}$ is called superselection observable if $\mathsf{Z}$ commutes with all observables of $\mathcal{S}$, sharp or not sharp. The existence of such an observable is called superselection rule. The eigenspaces of $\mathsf{Z}$ are called superselection sectors. All superselection observables form the centre $\mathbf{Z}$ of the algebra of all sharp observables of $\mathcal{S}$.

**Definition 15**Two states ${\mathsf{T}}_{1}$ and ${\mathsf{T}}_{2}$ are equivalent with respect to a set $\mathbf{O}$ of observables if these states assign the same probability measures to each observable of $\mathbf{O}$, that is , ${p}_{{\mathsf{T}}_{1}}^{\mathsf{E}}={p}_{{\mathsf{T}}_{2}}^{\mathsf{E}}$ for each $\mathsf{E}\in \mathbf{O}$. In that case, we write ${\mathsf{T}}_{1}{\cong}_{\mathbf{O}}{\mathsf{T}}_{2}$.

**Proposition 4**Given a system $\mathcal{S}$ with the sets $\mathbf{O}$ and ${\mathbf{T}}_{\mathbf{O}}{\left(\mathbf{H}\right)}_{1}^{+}$ of observables and state classes, respectively. Then the following statements are equivalent:

**A**- $\mathsf{Z}$ is a superselection observable.
**B**- Each state is equivalent to a unique convex combination of eigenstates of $\mathsf{Z}$.

**Rule 7**The states of system $\mathcal{S}$ mixing ${\mathcal{S}}_{1}$ and ${\mathcal{S}}_{2}$ are the equivalence classes with respect to the set of observables $\mathbf{O}$ of the form ${\mathsf{A}}_{1}\oplus {\mathsf{A}}_{2}$. The unique convex combination ${\mathsf{C}}_{\mathsf{Z}}\left(\mathsf{T}\right)$ of eigenstates of $\mathsf{Z}$ to which each element $\mathsf{T}$ of $\mathbf{T}{\left(\mathbf{H}\right)}_{1}^{+}$ is equivalent describes the physical meaning of the class.

#### 2.3. Galilean group

**Rule 8**The same experiments performed in two different Newtonian inertial frames have the same results, i.e., give the frequencies of observable values.

**Proposition 5**State ${\mathsf{T}}_{g,g\left(f\right)}$ and effect ${\mathsf{E}}_{g,g\left(f\right)}\left(X\right)$ satisfy

#### 2.3.1. Closed systems

**Rule 9**Let $\mathcal{S}$ be an closed system of type τ. Then there is a unique linear map $\mathsf{U}\left(g\right):{\mathbf{H}}_{\tau}\mapsto {\mathbf{H}}_{\tau}$ for each element $g\in {\overline{\mathbf{G}}}^{+}$ so that

**Proposition 6**$g\mapsto \mathsf{U}\left(g\right)$ is a unitary ray representation of ${\overline{\mathbf{G}}}^{+}$ on ${\mathbf{H}}_{\tau}$

**Definition 16**The three s.a. operators ${\mathsf{P}}^{k}$ are components of total momentum , ${\mathsf{J}}^{k}$ are components of total angular momentum and ${\mathsf{Q}}^{k}=(1/M){\mathsf{K}}^{k}$ are components of position (centre of mass) of $\mathcal{S}$. M is the total mass of $\mathcal{S}$ and

**Rule 10**The non-zero commutators of the generators are

#### 2.3.2. Time translations

**Rule 11**Let $\mathcal{S}$ be a system of type τ and let external fields f be given. Then, time translation from ${t}_{1}$ to ${t}_{2}$ is represented by unitary operator $\mathsf{U}(f,{t}_{2},{t}_{1})$ on ${\mathbf{H}}_{\tau}$ satisfying

**Definition 17**The operator $\mathsf{H}\left(t\right)$ defined by

**Proposition 7**Let quantum system $\mathcal{S}$ have a Hamiltonian $\mathsf{H}\left(t\right)$. Then, the dynamical evolution of $\mathcal{S}$ in Schrödinger picture obeys von Neumann–Liouville equation of motion

**Proposition 8**Let $\mathsf{T}$ be an extremal state $|\varphi \rangle \langle \varphi |$. Then

**Rule 12**Let $\mathsf{T}$ be the state with statistical decomposition (11) Then, its time evolution is a state operator with statistical decomposition

**Definition 18**Each unitary transformation $\mathsf{U}:{\mathbf{H}}_{\tau}\mapsto {\mathbf{H}}_{\tau}$ that leaves the Hamiltonian $\mathsf{H}$ of $\mathcal{S}$ invariant,

## 3. Composition of quantum systems

#### 3.1. Composition of heterogeneous systems

#### 3.1.1. Tensor product of Hilbert spaces

**Rule 13**Let ${\mathcal{S}}_{1}$ and ${\mathcal{S}}_{2}$ be two heterogeneous quantum systems and their Hilbert spaces be ${\mathbf{H}}_{1}$ and ${\mathbf{H}}_{2}$, respectively. Then, the system $\mathcal{S}$ composed of ${\mathcal{S}}_{1}$ and ${\mathcal{S}}_{2}$ has the Hilbert space $\mathbf{H}={\mathbf{H}}_{1}\otimes {\mathbf{H}}_{2}$, its states are elements of $\mathbf{T}{({\mathbf{H}}_{1}\otimes {\mathbf{H}}_{2})}_{1}^{+}$ and its effects are elements of ${\mathbf{L}}_{r}{({\mathbf{H}}_{1}\otimes {\mathbf{H}}_{2})}_{\le 1}^{+}$.

**Rule 14**Let system $\mathcal{S}$ composed of heterogeneous systems ${\mathcal{S}}_{1}$ and ${\mathcal{S}}_{2}$ be prepared in state $\mathsf{T}\in \mathbf{T}{({\mathbf{H}}_{1}\otimes {\mathbf{H}}_{2})}_{1}^{+}$. Then ${\mathcal{S}}_{1}$ is simultaneously prepared in state ${\Pi}_{2}\left(\mathsf{T}\right)$ and ${\mathcal{S}}_{1}$ in state ${\Pi}_{2}\left(\mathsf{T}\right)$. The observables of ${\mathcal{S}}_{1}$ and ${\mathcal{S}}_{2}$ can be identified with observables ${\mathsf{E}}_{1}\otimes {\mathsf{1}}_{2}$ and ${\mathsf{1}}_{1}\otimes {\mathsf{E}}_{2}$ respectively, of the composite. ${\mathcal{S}}_{1}$ and ${\mathcal{S}}_{2}$ are called subsystems of $\mathcal{S}$.

**Definition 19**Let heterogeneous systems ${\mathcal{S}}_{1}$ and ${\mathcal{S}}_{2}$ have Hilbert spaces ${\mathbf{H}}_{1}$ and ${\mathbf{H}}_{2}$. Let the representative of $(g,\varphi )\in {\overline{\mathbf{G}}}_{c}^{+}$ on ${\mathbf{H}}_{k}$ be denoted by ${\tilde{\mathsf{U}}}_{k}(g,\varphi )$ and that on ${\mathbf{H}}_{1}\otimes {\mathbf{H}}_{2}$ by $\tilde{\mathsf{U}}(g,\varphi )$. If

**Proposition 9**Given a one parameter subgroup of ${\overline{\mathbf{G}}}_{c}^{+}$ with generator $\mathsf{G}$ of its representation on ${\mathbf{H}}_{1}\otimes {\mathbf{H}}_{2}$, ${\mathsf{G}}_{1}$ of its representations on ${\mathbf{H}}_{1}$ and ${\mathsf{G}}_{2}$ of its representation on ${\mathbf{H}}_{2}$. Then, in the case of non-interacting subsystems,

**Rule 15**Suppose that the state of the system composed of heterogeneous systems ${\mathcal{S}}_{1}+{\mathcal{S}}_{2}$ is $\mathsf{T}$. The necessary and sufficient condition for the statistical decomposition of the state of ${\mathcal{S}}_{1}$ to be

#### 3.1.2. Entanglement

**Definition 20**If object $\mathcal{S}$ composed of two heterogeneous quantum objects ${\mathcal{S}}_{1}$ and ${\mathcal{S}}_{2}$ is in an indecomposable state $\mathsf{W}$ that satisfies condition (40), one says that ${\mathcal{S}}_{1}$ and ${\mathcal{S}}_{2}$ are entangled or that state $\mathsf{W}$ is entangled.

**Proposition 10**Let $\mathcal{S}$ be composed of heterogeneous systems ${\mathcal{S}}_{1}$ and ${\mathcal{S}}_{2}$ of different types. Let $\mathsf{T}$ be the state of $\mathcal{S}$ and let

#### 3.2. Composition of identical systems

#### 3.2.1. Identical subsystems

**Rule 16**Let ${\mathcal{S}}^{N}$ be a quantum system composed of N subsystems $\mathcal{S}$, each of type τ with Hilbert space ${\mathbf{H}}_{\tau}$. Then, the Hilbert space of ${\mathcal{S}}^{N}$ is ${\mathbf{H}}_{\tau s}^{N}$ for subsystems with integer spin and ${\mathbf{H}}_{\tau a}^{N}$ for those with half-integer spin. If systems $\mathcal{S}$ are closed and do not interact, then the representation of group ${\overline{\mathbf{G}}}_{c}^{+}$ on ${\mathbf{H}}_{\tau s}^{N}$ or ${\mathbf{H}}_{\tau a}^{N}$ is the tensor product of its N representations on ${\mathbf{H}}_{\tau}$.

**Definition 21**Systems with integer spin are called bosons and those with half-integer spin are called fermions. The symmetry properties of states lead to Bose–Einstein statistics for bosons and Fermi–Dirac one for fermions.

**Proposition 11**Let $\mathsf{G}$ be the generator of subgroup $g\left(t\right)$ of ${\overline{\mathbf{G}}}_{c}^{+}$ on ${\mathbf{H}}_{\tau}$. Then, the generator $\tilde{\mathsf{G}}$ of $g\left(t\right)$ on ${\mathbf{H}}_{\mathcal{R}\left(\tau \right)}^{N}$ is given by

**Proposition 12**Possible states of system ${\mathcal{S}}^{N}$ composed of N systems of type τ are elements of $\mathbf{T}{({\mathbf{H}}_{\mathcal{R}\left(\tau \right)}^{N})}_{1}^{+}$ and the effects of ${\mathcal{S}}^{N}$ are elements of ${\mathbf{L}}_{r}{({\mathbf{H}}_{\mathcal{R}\left(\tau \right)}^{N})}_{\le 1}^{+}$.

**Experiment I**: State $\mathsf{P}\left[\psi \right]$ of particle ${\mathcal{S}}_{1}$ of type τ is prepared in our laboratory.

**Experiment II**: State $\mathsf{P}\left[\psi \right]$ is prepared as in Experiment I and state $\mathsf{P}\left[\varphi \right]$ of particle ${\mathcal{S}}_{2}$ of the same type τ is prepared simultaneously in a remote laboratory.

#### 3.2.2. Cluster separability

**Cluster Separability II**No quantum experiment with a system in a local laboratory is affected by the mere presence of an identical system in remote parts of the universe.

... a state $\mathsf{w}$ is called remote if $\parallel \mathsf{A}\mathsf{w}\parallel $ is vanishingly small, for any operator $\mathsf{A}$ which corresponds to a quantum test in a nearby location. ... We can now show that the entanglement of a local quantum system with another system in a remote state (as defined above) has no observable effect.

**Rule 17**Let $\mathcal{S}$ be a quantum system of type τ with Hilbert space ${\mathbf{H}}_{\tau}$. Any preparation of $\mathcal{S}$ must give it separation status D satisfying $D\ne \varnothing $. Then the prepared state of $\mathcal{S}$ is an element of $\mathbf{T}{\left({\mathbf{H}}_{\tau}\right)}_{1}^{+}$ and D-local effects of ${\mathbf{L}}_{r}{\left({\mathbf{H}}_{\tau}\right)}_{\le 1}^{+}$ are individually registrable on $\mathcal{S}$ but only these are.

#### 3.2.3. Mathematical theory of D-local observables

**Definition 22**Let $D\subset {\mathbb{R}}^{3}$ be open, let $\mathsf{A}$ be an operator on $\mathbf{H}$ and let the following conditions hold:

- 1.
- $A({\overrightarrow{x}}_{1},\cdots ,{\overrightarrow{x}}_{N};{\overrightarrow{x}}_{1}^{\prime},\cdots ,{\overrightarrow{x}}_{N}^{\prime})$ is the zero distribution for any ${\overrightarrow{x}}_{1},\cdots ,{\overrightarrow{x}}_{N}\in {\mathbb{R}}^{3N}\backslash {D}^{N}$.
- 2.
- $$\int {d}^{3N}{x}^{\prime}A({\overrightarrow{x}}_{1},\cdots ,{\overrightarrow{x}}_{N};{\overrightarrow{x}}_{1}^{\prime},\cdots ,{\overrightarrow{x}}_{N}^{\prime})f({\overrightarrow{x}}_{1}^{\prime},\cdots ,{\overrightarrow{x}}_{N}^{\prime})=0$$$$\text{supp}f({\overrightarrow{x}}_{1}^{\prime},\cdots ,{\overrightarrow{x}}_{N}^{\prime})\subset {\mathbb{R}}^{3N}\backslash {D}^{N}\phantom{\rule{4pt}{0ex}}.$$

**Example 1**$N=1$, $A(\overrightarrow{x};{\overrightarrow{x}}^{\prime})={x}^{1}\delta \left(\overrightarrow{x}-{\overrightarrow{x}}^{\prime}\right)$. $\mathsf{A}$ is ${\mathbb{R}}^{3}$-local and the filter is ${\mathbb{R}}^{3}$.

**Example 2**$N=1$, $\psi \left(\overrightarrow{x}\right)$ is a wave function with support D. Then $|\psi \left(\overrightarrow{x}\right)\rangle \langle \psi \left(\overrightarrow{x}\right)|$ is ${D}^{\prime}$-local, where ${D}^{\prime}$ is the interior of D, and the filter is the family of all open sets containing ${D}^{\prime}$.

**Definition 23**A POV measure $$