Nonsignaling as the consistency condition for local quasi-classical probability modeling of a general multipartite correlation scenario

A possibility of the description of quantum measurements in terms of the classical probability model has been a point of intensive discussion ever since the seminal publications of von Neumann [1], Kolmogorov [2], Einstein, Podolsky and Rosen (EPR) [3] and Bell [4, 5].

Although, in the quantum physics literature, one can still find the misleading¹ claims on a peculiarity of 'quantum probabilities' and 'quantum events', the probabilistic description of each quantum measurement satisfies the Kolmogorov axioms [2] for the theory of probability.

Namely, each measurement on a quantum system represented initially by a state ρ on a complex separable Hilbert space $\mathcal {H}$ is described by the probability space² $(\Lambda , \mathcal {F}_{\Lambda },\mathrm{tr}[\rho \mathrm{M}(\cdot )]),$ where Λ is a set of measurement outcomes, $\mathcal {F}_{\Lambda }$ is a σ-algebra of observed events F⊆Λ and $\mathrm{tr} [\rho \mathrm{M}(\cdot )]:\mathcal {F}_{\Lambda }\rightarrow [0,1]$ is the probability measure with values tr[ρM(F)], $F\in \mathcal {F}_{\Lambda },$ each defining the probability that, under this quantum measurement, an outcome λ belongs to a set $F\in \mathcal {F} _{\Lambda }.$ Here, M is a normalized ($\mathrm{M}(\Lambda )= \mathbb {I}_{\mathcal {H}})$ measure with values M(F), $F\in \mathcal {F}_{\Lambda },$ that are positive operators on $\mathcal {H}$—that is, a normalized positive operator-valued (POV) measure³ on $(\Lambda ,\mathcal {F} _{\Lambda })$.

The measure-theoretic structure of the Kolmogorov axioms [2] is crucial and the probabilistic description of each measurement in every application field satisfies these probability axioms.

However, the classical probability model, which is also often named⁴ after Kolmogorov in the mathematical physics literature and where system observables and states are represented by random variables and probability measures on a single measurable space $(\Omega ,\mathcal {F}_{\Omega }),$ describes correctly randomness in the classical statistical mechanics and many other application fields, but fails either to reproduce noncontextually [10] the statistical properties of all quantum observables on a Hilbert space of a dimension$\ \dim \mathcal {H}\ge 3$ or to simulate via random variables, each depending only on a setting of the corresponding measurement at the corresponding site, the probabilistic description of a quantum correlation scenario on an arbitrary N-partite quantum state. For details and references, see section 1.4 in [11] and the introduction in [12].

The probabilistic description⁵ of an arbitrary multipartite correlation scenario also cannot be reproduced via the classical probability model.

Note that, in the quantum theory literature, the interpretation of quantum measurements in the classical probability terms is generally referred to as a hidden variable (HV) model.

In [13], we have introduced for a general correlation scenario the notion of a local quasi hidden variable (LqHV) model, where locality and the measure-theoretic structure inherent to a local hidden variable (LHV) model are preserved, but the positivity of a simulation measure is dropped. We have proved [13] that every quantum S₁ × ⋅⋅⋅ × S_N-setting correlation scenario admits LqHV modeling and specified the state parameter determining quantitatively a possibility of an S₁ × ⋅⋅⋅ × S_N-setting LHV description of an N-partite quantum state.

In this paper, we further develop the LqHV approach introduced in [13]. The paper is organized as follows.

In section 2, we specify for a general multipartite correlation scenario the notion of a deterministic LqHV model,where all joint probability distributions of a correlation scenario are simulated via a single measure space with a normalized bounded real-valued measure and random variables, each depending only on a setting of the corresponding measurement at the corresponding site. We show that the existence for a general correlation scenario of some LqHV model implies the existence for this scenario of a deterministic LqHV model.

In section 3, we prove that an arbitrary multipartite correlation scenario admits a deterministic LqHV model if an only if all its joint probability distributions satisfy the consistency condition, constituting the general nonsignaling condition formulated by equation (10) in [12].

In section 4, we summarize the main mathematical results of this paper and discuss their conceptual implication.

Consider an N-partite correlation scenario, where each nth of N ⩾ 2 parties (players) performs S_n ⩾ 1 measurements with outcomes λ_n ∈ Λ_n of an arbitrary type and $\mathcal {F}_{\Lambda _{n}}$ is a σ-algebra of events F_n⊆ Λ_n observed at the nth site. We label each measurement at the nth site by a positive integer s_n = 1, ..., S_n and each of N-partite joint measurements, induced by this correlation scenario and with outcomes (λ₁, ..., λ_N) ∈ Λ₁ × ⋅⋅⋅ × Λ_N, by an N-tuple (s₁, ..., s_N), where the nth component refers to a measurement at the nth site.

For concreteness, we further specify an S₁ × ⋅⋅⋅ × S_N -setting correlation scenario by the symbol $\mathcal {E}_{S},$ where S := S₁ × ⋅⋅⋅ × S_N, and denote by $P_{(s_{1},\ldots ,s_{N})}^{( \mathcal {E}_{S})}$ a probability measure, defined on the direct product ⁶ (Λ₁ × ⋅⋅⋅ × Λ_N, $\mathcal {F}_{\Lambda _{1}}\otimes \cdots \otimes \mathcal {F}_{\Lambda _{N}})$ of measurable spaces $(\Lambda _{n},\mathcal {F}_{\Lambda _{n}}),$ n = 1, ..., N, and describing an N-partite joint measurement (s₁, ..., s_N) under a scenario $\mathcal {E}_{S}$.

Remark 1. The superscript $\mathcal {E}_{S}$ in notation $P_{(s_{1},\ldots ,s_{N})}^{( \mathcal {E}_{S})}$ indicates that, in contrast to a correlation scenario represented by the so-called nonsignaling boxes [15, 16] and described by joint probability distributions $P_{(s_{1},\ldots ,s_{N})}^{( \mathcal {E}_{S})}\equiv P_{(s_{1},\ldots ,s_{N})}$, s₁ = 1, ..., S₁, ..., s_N = 1, ..., S_N, each depending only on settings of the corresponding measurements at the corresponding sites, for a general correlation scenario $\mathcal {E}_{S},$ each distribution $P_{(s_{1},\ldots ,s_{N})}^{(\mathcal {E}_{S})}$ may also depend on settings of all (or some) other measurements. The latter is, for example, the case under a classical correlation scenario with 'one-sided' or 'two-sided' memory [17].

If under an N-partite joint measurement (s₁, ..., s_N) of a scenario $\mathcal {E}_{S}$ only outcomes of M < N parties 1 ⩽ n₁ < ⋅⋅⋅ < n_M ⩽ N are taken into account while outcomes of all other parties are ignored, then the joint probability distribution of outcomes observed at these M sites is described by the marginal probability distribution:

$$\begin{eqnarray} &&\fl P_{(s_{1},\ldots ,s_{N})}^{(\mathcal {E}_{S})}(\Lambda _{1}\times \cdots \times \Lambda _{n_{1}-1}\times \mathrm{d}\lambda _{_{n_{1}}}\times \Lambda _{n_{1}+1}\times \cdots \times \Lambda _{n_{_{M}}-1}\times \mathrm{d}\lambda _{_{n_{_{_{M}}}}}\times \Lambda _{n_{_{M}}+1}\times \cdots \times \Lambda _{_{N}}).\nonumber\\ \end{eqnarray} \tag{ 1 }$$

In particular,

$$\begin{equation} P_{(s_{1},\ldots ,s_{N})}^{(\mathcal {E}_{S})}(\Lambda _{1}\times \cdots \times \Lambda _{n-1}\times \mathrm{d}\lambda _{n}\times \Lambda _{n+1}\times \cdots \times \Lambda _{_{N}}) \end{equation} \tag{ 2 }$$

is the probability distribution of outcomes observed at the nth site under a joint measurement (s₁, ..., s_N) of the scenario $\mathcal {E}_{S}.$

Remark 2. Throughout this paper, for a measure τ on the direct product (Λ × ⋅⋅⋅ × Λ', $\mathcal {F}_{\Lambda }\otimes \cdots \otimes \mathcal {F}_{\Lambda ^{\prime }})$ of some measurable spaces, we often use notation τ(dλ × ⋅⋅⋅ × dλ') outside an integral. This allows us to easily specify the structure of different marginals of τ.

For the probabilistic description of a general correlation scenario, consider the following simulation model introduced in [13].

Definition 1 ([13]). An S₁ × ⋅⋅⋅ × S_N-setting correlation scenario $\mathcal {E}_{S},$ with joint probability distributions $P_{(s_{1},\ldots ,s_{N})}^{(\mathcal {E}_{S})},$ s₁ = 1, ..., S₁, ..., s_N = 1, ..., S_N, and outcomes (λ₁, ..., λ_N) ∈ Λ₁ × ⋅⋅⋅ × Λ_N of an arbitrary type admits a local quasi hidden variable (LqHV) model if all of its joint probability distributions admit the representation

$$\begin{eqnarray} &&\fl P_{(s_{1},\ldots ,s_{N})}^{(\mathcal {E}_{S})}(F_{1}\times \cdots \times F_{N}) =\int _{\Omega }P_{1}^{(s_{_{1}})}(F_{1}\hbox{ }|\hbox{ }\omega )\cdot \ldots \cdot P_{N}^{(s_{_{N}})}(F_{N}\hbox{ }|\hbox{ }\omega )\hbox{ } \nu _{\mathcal {E}_{S}}(\mathrm{d}\omega ), \\ &&F_{1} \in \mathcal {F}_{\Lambda _{1}},\ldots ,F_{N}\in \mathcal {F}_{\Lambda _{N}}, \nonumber \end{eqnarray} \tag{ 3 }$$

in terms of a single measure space $( \Omega ,\mathcal {F}_{\Omega },\nu _{\mathcal {E}_{S}})$ with a normalized bounded real-valued measure $\nu _{\mathcal {E}_{S}}$ and conditional probability measures P^(s_n)_n( · | $\omega ):\mathcal {F}_{\Lambda _{n}}\rightarrow [0,1],$ defined $\nu _{_{\mathcal {E}_{S}}}$-a.e. (almost everywhere) on Ω and such that, for each s_n = 1, ..., S_n and every n = 1, ..., N, the function P^(s_n)_n(F_n | · ): Ω → [0, 1] is measurable for each $F_{n}\in \mathcal { F}_{\Lambda _{n}}.$

In a triple $\left( \Omega ,\mathcal {F}_{\Omega },\nu \right)$ representing a measure space, Ω is a non-empty set, $\mathcal {F}_{\Omega }$ is a σ-algebra of subsets of Ω and ν is a measure on a measurable space $\left( \Omega ,\mathcal {F}_{\Omega }\right) .$ A real-valued measure ν is called normalized if ν(Ω) = 1 and bounded [14] if |ν(F)| ⩽ M < ∞ for all $F\in \mathcal {F}_{\Omega }.$ Note that each bounded real-valued measure ν admits [14] the Jordan decomposition ν = ν⁺ − ν⁻ via positive measures

$$\begin{equation} \fl \nu ^{+}(F):=\sup _{F^{\prime }\in \mathcal {F}_{\Omega },F^{\prime }\subseteq F}\nu (F^{\prime }),\quad \nu ^{-}(F):=-\inf _{F^{\prime }\in \mathcal {F}_{\Omega },F^{\prime }\subseteq F}\nu (F^{\prime }),\quad \forall F\in \mathcal {F}_{\Omega }, \end{equation} \tag{ 4 }$$

with disjoint supports.

We stress that, in an LqHV model (3), a normalized bounded real-valued measure $\nu _{\mathcal {E}_{S}}$ has a simulation character and may, in general, depend (via the subscript $\mathcal {E}_{S}$) on measurement settings at all (or some) sites, as an example, see measure (39) in [13].

The structure of each LqHV model is such that though some values of a simulation measure $\nu _{\mathcal {E}_{S}}$ may be negative, the integral standing on the right-hand side of representation (3) is non-negative for all $F_{1}\in \mathcal {F}_{\Lambda _{1}},\ldots ,F_{N}\in \mathcal {F}_{\Lambda _{N}}.$

If, for a correlation scenario $\mathcal {E}_{S}$, there exists representation (3), where a normalized bounded real-valued measure $\nu _{\mathcal {E}_{S}}$ is positive (hence, is a probability measure), then this scenario admits an LHV model formulated for a general case by equation (26) in [12].

As is discussed in detail in [13], the concept of an LqHV model incorporates as particular cases and generalizes in one whole both types of simulation models known in the literature—an LHV model and an affine model [18]. Note that the latter model, where all distributions $P_{(s_{1},\ldots ,s_{N})}^{(\mathcal {E}_{S})}$ of the scenario $\mathcal {E}_{S}$ are expressed via the affine sum of some LHV distributions, is in principle built up on the concept of an LHV model.

Introduce now the following special type of LqHV model.

Definition 2. An LqHV model (3) is called deterministic if there exist $\mathcal {F} _{\Omega }/\mathcal {F}_{\Lambda _{n}}$-measurable functions (random variables) $f_{n,s_{n}}:$ Ω → Λ_n, such that, in representation (3), all conditional probability measures P^(s_n)_n( · |ω), s_n = 1, ..., S_n, n = 1, ..., N, have the special form:

$$\begin{equation} P_{n}^{(s_{n})}(F_{n}\hbox{ }|\hbox{ }\omega )=\chi _{f_{n,s_{n}}^{-1}(F_{n})}(\omega ),\quad \forall F_{n}\in \mathcal {F} _{\Lambda _{n}}, \end{equation} \tag{ 5 }$$

$\nu _{\mathcal {E}_{S}}$-a.e. on Ω.

Here, f⁻¹(F) = {ω ∈ Ω∣f(ω) ∈ F} is the pre-image of a set $F\in \mathcal {F}_{\Lambda }$ under a mapping f: Ω → Λ and χ_D( · ) is the indicator function of a subset D⊆Ω, that is, χ_D(ω) = 1 for ω ∈ D and χ_D(ω) = 0 for ω∉D.

The notion of a deterministic LqHV model generalizes the concept of a deterministic⁷ LHV model formulated for a general multipartite correlation scenario in section 4 of [12].

From (3) and (5) it follows that if an S₁ × ⋅⋅⋅ × S_N-setting correlation scenario $\mathcal {E}_{S}$ admits a deterministic LqHV model, then all its joint probability distributions $P_{(s_{1},\ldots ,s_{N})}^{(\mathcal {E}_{S})}$ admit the representation

$$\begin{eqnarray} P_{(s_{1},\ldots ,s_{N})}^{(\mathcal {E}_{S})}(F_{1}\times \cdots \times F_{N})& =\int _{\Omega }\chi _{f_{1,s_{1}}^{-1}(F_{1})}(\omega )\cdot \ldots \cdot \chi _{f_{N,s_{N}}^{-1}(F_{N})}(\omega )\hbox{ }\nu _{\mathcal {E}_{S}}( \mathrm{d}\omega ) \nonumber\\ & =\nu _{\mathcal {E}_{S}}\big( f_{1,s_{1}}^{-1}(F_{1})\cap \cdots \cap f_{N,s_{_{N}}}^{-1}(F_{N})\big) , \nonumber \end{eqnarray}$$

$$\begin{equation} F_{1} \in \mathcal {F}_{\Lambda _{1}},\ldots ,F_{N}\in \mathcal {F}_{\Lambda _{N}}, \nonumber \end{equation} \tag{ 6 }$$

via a normalized bounded real-valued measure $\nu _{\mathcal {E}_{S}}$ on some measurable space $(\Omega ,\mathcal {F}_{\Omega })$ and random variables $f_{n,s_{n}}:\Omega \rightarrow \Lambda _{n},$ each depending only on a setting of the s_nth measurement at the nth site.

We stress that, in a deterministic LqHV model, the relation between a simulation measure $\nu _{\mathcal {E}_{S}}$ and random variables $f_{n,s_{n}},$ s_n = 1, ..., S_n, n = 1, ..., N, modeling scenario measurements is such that the joint probabilities of scenario events are reproduced due to (6) only via non-negative values of $\nu _{ \mathcal {E}_{S}}$.

Representation (6), in turn, implies that for arbitrary bounded measurable real-valued functions $\varphi _{n}:\Lambda _{n}\rightarrow \mathbb {R},$ n = 1, ..., N, the product expectation

$$\begin{equation} \fl \langle \varphi _{_{1}}(\lambda _{1})\cdot \ldots \cdot \varphi _{_{N}}(\lambda _{N})\rangle _{(s_{1},\ldots ,s_{N})}^{(\mathcal {E} _{S})}:=\int _{\Lambda }\varphi _{_{1}}(\lambda _{1})\cdot \ldots \cdot \varphi _{_{N}}(\lambda _{N})P_{(s_{1},\ldots ,s_{N})}^{(\mathcal {E}_{S})}( \mathrm{d}\lambda _{1}\times \cdots \times \mathrm{d}\lambda _{N}) \end{equation} \tag{ 7 }$$

takes the form

$$\begin{equation} \fl \langle \varphi _{_{1}}(\lambda _{1})\cdot \ldots \cdot \varphi _{_{N}}(\lambda _{N})\rangle _{(s_{1},\ldots ,s_{N})}^{(\mathcal {E} _{S})}=\int (\varphi _{_{1}}\circ f_{1,s_{1}})(\omega )\cdot \ldots \cdot (\varphi _{_{N}}\circ f_{N,s_{N}})(\omega )\hbox{ }\nu _{\mathcal {E}_{S}}( \mathrm{d}\omega ), \end{equation} \tag{ 8 }$$

which differs from the form of the product expectations in a deterministic LHV model (see equation (31) in [12]) only by the fact that a normalized bounded real-valued measure $\nu _{\mathcal {E}_{S}}$ in (8) does not need to be positive.

Recall that, for a given correlation scenario, a deterministic LHV model constitutes the version of the local classical probability model, where only the observed joint probability distributions are reproduced [12].

Therefore, a deterministic LqHV model (6) corresponds to the local quasi-classical probability model, where, in contrast to the local classical probability model, an 'underlying' probability space is replaced by a measure space $(\Omega ,\mathcal {F}_{\Omega },\nu )$ with a normalized bounded real-valued measure ν not necessarily positive and where

• (i)  
  observables with a value space $(\Lambda ,\mathcal {F}_{\Lambda })$ are represented only by such random variables f: Ω → Λ for which ν(f⁻¹(F)) ⩾ 0, $\forall F\in \mathcal {F} _{\Lambda };$
• (ii)  
  a joint measurement of two observables f₁ and f₂, each with a value space $(\Lambda _{n},\mathcal {F}_{\Lambda _{n}}),$ is possibleif and only if ν(f⁻¹₁(F₁)∩f⁻¹₂(F₂)) ⩾ 0 for all $F_{n}\in \mathcal {F}_{\Lambda _{n}}.$

The following statement is proved in appendix A.

Proposition 1. If an S₁ × ⋅⋅⋅ × S_N-setting correlation scenario $\mathcal {E} _{S}$ admits some LqHV model (3), then it also admits a deterministic LqHV model (6).

This proposition and theorem 1 of [13] imply.

Proposition 2. An S1 × ⋅⋅⋅ × SN-setting correlation scenario $\mathcal {E} _{S}$ admits a deterministic LqHV model (6) if and only if, on the direct product space (ΛS11 × ⋅⋅⋅ × ΛSNN, $\mathcal {F}_{\Lambda _{1}}^{\otimes S_{1}}\otimes \cdots \otimes \mathcal {F}_{\Lambda _{N}}^{\otimes S_{N}})$, there exists a normalized bounded real-valued measure8

$$\begin{eqnarray} &&\fl \mu _{\mathcal {E}_{S}}\big( \mathrm{d}\lambda _{1}^{(1)}\times \cdots \times \mathrm{d}\lambda _{1}^{(S_{1})}\times \cdots \times \mathrm{d} \lambda _{N}^{(1)}\times \cdots \times \mathrm{d}\lambda _{N}^{(S_{N})}\big) , \nonumber\\&&\ms \lambda _{n}^{(s_{n})} \in \Lambda _{n},\quad s_{n}=1,\ldots ,S_{n},\quad n=1,\ldots ,N, \end{eqnarray} \tag{ 9 }$$

returning all joint probability distributions $P_{(s_{1},\ldots ,s_{N})}^{( \mathcal {E}_{S})}$ of a scenario $\mathcal {E}_{S}$ as the corresponding marginals.

Let us now analyze under what condition on joint probability distributions an arbitrary multipartite correlation scenario admits a deterministic LqHV model.

Suppose that under an S₁ × ⋅⋅⋅ × S_N-setting correlation scenario $\mathcal {E}_{S},$ for all joint measurements $(s_{1},\ldots ,s_{_{N}}),$ (s'₁, ..., s_N') with 1 ⩽ M < N common settings $s_{n_{1}},\ldots ,s_{n_{_{M}}}$ at arbitrary sites 1 ⩽ n₁ < ⋅⋅⋅ < n_M ⩽ N, the marginal probability distributions (1) of outcomes observed at these sites coincide, that is,

$$\begin{eqnarray} &&\fl P_{(s_{1},\ldots ,s_{N})}^{(\mathcal {E}_{S})}(\Lambda _{1}\times \cdots \times \Lambda _{n_{1}-1}\times \mathrm{d}\lambda _{_{n_{1}}}\times \Lambda _{n_{1}+1}\times \cdots \times \Lambda _{n_{_{M}}-1}\times \mathrm{d}\lambda _{_{n_{_{_{M}}}}}\times \Lambda _{n_{_{M}}+1}\times \cdots \times \Lambda _{_{N}}) \nonumber\\ &&=P_{(s_{1}^{\prime },..,s_{_{N}}^{\prime })}^{(\mathcal {E}_{S})}(\Lambda _{1}\times \cdots \times \Lambda _{n_{1}-1}\times \mathrm{d}\lambda _{_{n_{1}}}\times \Lambda _{n_{1}+1}\times \cdots \times \Lambda _{n_{_{M}}-1}\times \mathrm{d}\lambda _{_{n_{_{_{M}}}}}\nonumber\\ &&\;\;\;\times \,\Lambda _{n_{_{M}}+1}\times \cdots \times \Lambda _{_{N}}). \end{eqnarray} \tag{ 10 }$$

As we discuss this in section 3 of [12], for a general correlation scenario $\mathcal {E}_{S}$ with a finite number of measurement settings at each site, condition (10) does not automatically imply that the coinciding marginals, standing on the left-hand and right-hand sides of equation (10), depend only on settings of measurements $s_{n_{1}},\ldots ,s_{n_{_{M}}}$ at sites 1 ⩽ n₁ < ⋅⋅⋅ <n_M ⩽ N.

This means that condition (10) should be distinguished from the condition

$$\begin{eqnarray} &&\fl P_{(s_{1},\ldots ,s_{_{N}})}^{(\mathcal {E}_{S})}(\Lambda _{1}\times \cdots \times \Lambda _{n_{1}-1}\times \mathrm{d}\lambda _{_{n_{1}}}\times \Lambda _{n_{1}+1}\times \cdots \times \Lambda _{n_{_{M}}-1}\times \mathrm{d}\lambda _{_{n_{_{_{M}}}}}\times \Lambda _{n_{_{M}}+1}\times \cdots \times \Lambda _{_{N}}) \nonumber\\ &&\equiv P_{(s_{n_{1}},\ldots ,s_{n_{_{M}}})}(\mathrm{d}\lambda _{n_{_{1}}}\times \cdots \times \mathrm{d}\lambda _{n_{_{M}}}), \nonumber\\ &&s_{1} =1,\ldots ,S_{1},\ldots ,s_{N}=1,\ldots ,S_{N},\quad M=1,\ldots ,N, \end{eqnarray} \tag{ 11 }$$

usually argued in the literature to follow if M < N from condition (10).

Although condition (11) implies condition (10), the converse is not, in general, true; see proposition 1 in [12].

In view of their physical interpretations discussed in detail in [12], we call conditions (10) and (11) the nonsignaling condition and the EPR locality condition, respectively. Moreover, since in the literature on quantum information⁹ specifically the joint combination of conditions (10) and (11) is often called nonsignaling, in order to exclude a possible misunderstanding, we further refer to the consistency condition (10) as the general nonsignaling condition.

We stress that the nonsignaling condition in the sense of [15, 16] implies the general nonsignaling condition (10), but the converse of this statement is not, in general, true.

The following theorem is proved in appendix B.

Theorem 1. An S₁ × ⋅⋅⋅ × S_N-setting correlation scenario $\mathcal {E }_{S}$ admits a deterministic LqHV model (6) if and only if all its joint probability distributions $P_{(s_{1},\ldots ,s_{N})}^{(\mathcal {E}_{S})},$ s₁ = 1, ..., S₁, ..., s_N = 1, ..., S_N, satisfy the consistency condition (10), constituting the general nonsignaling condition formulated in [12].

Consider, in particular, an S₁ × ⋅⋅⋅ × S_N-setting correlation scenario performed on an N-partite quantum state ρ on a complex separable Hilbert space $\mathcal {H}_{1}\otimes \cdots \otimes \mathcal {H}_{N}$ and described by the joint probability distributions

$$\begin{eqnarray} &&\mathrm{tr}\big[\rho \big\lbrace \mathrm{M}_{1}^{(s_{1})}(F_{1})\otimes \cdots \otimes \mathrm{M}_{N}^{(s_{_{N}})}(F_{N})\big\rbrace \big], \nonumber\\ &&F_{n} \in \mathcal {F}_{n},\quad s_{n}=1,\ldots ,S_{n},\quad n=1,\ldots ,N, \end{eqnarray} \tag{ 12 }$$

where M^(s_n)_n is a POV¹⁰ measure on $(\Lambda _{n},\mathcal {F} _{\Lambda _{n}})$ representing on a Hilbert space $\mathcal {H}_{n}$ a quantum measurement s_n at the nth site.

Since each quantum correlation scenario (12) satisfies condition (10) (as well as condition (11)), theorem 1 implies

Corollary 1. For every quantum state ρ on a complex separable Hilbert space $\mathcal {H}_{1}\otimes \cdots \otimes \mathcal {H}_{N}$ and arbitrary positive integers S₁, ..., S_N ⩾ 1, the probabilistic description of each quantum S₁ × ⋅⋅⋅ × S_N -setting correlation scenario (12) admits a deterministic LqHV model.

In view of the above proposition 1, the statement of corollary 1 agrees with the statement of theorem 2 in [13].

In this paper, we have introduced (definition 2) the notion of a deterministic LqHV model, where all joint probability distributions of a multipartite correlation scenario are simulated via a single measure space $(\Omega ,\mathcal {F}_{\Omega },\nu ),$ with a normalized bounded real-valued measure ν not necessarily positive, and random variables which are local in the sense that each of these random variables depends only on a setting of the corresponding measurement at the corresponding site.

We have proved (theorem 1) that a general S₁ × ⋅⋅⋅ × S_N -setting correlation scenario admits a deterministic LqHV model if and only if all its joint probability distributions satisfy the consistency condition (10), constituting the general nonsignaling condition formulated in [12].

This general result, in particular, implies (corollary 1) that the probabilistic description of each S₁ × ⋅⋅⋅ × S_N-setting correlation scenario (12) on an N-partite quantum state admits modelling in local quasi-classical probability terms.

From the conceptual point of view, these mathematical results specify a new probability model that has the measure-theoretic structure $(\Omega , \mathcal {F}_{\Omega },\nu ),$ resembling the structure of the classical probability model but reduces to the latter iff a normalized bounded real-valued measure ν is positive. In the frame of this quasi-classical probability model:

• (i)  
  observables with a value space $(\Lambda ,\mathcal {F}_{\Lambda })$ are represented only by such random variables f: Ω → Λ for which ν(f⁻¹(F)) ⩾ 0, $\forall F\in \mathcal {F} _{\Lambda };$
• (ii)  
  a joint measurement of two observables f₁ and f₂, each with a value space $(\Lambda _{n},\mathcal {F}_{\Lambda _{n}}),$ is possible if and only if ν(f⁻¹₁(F₁)∩f⁻¹₂(F₂)) ⩾ 0 for all $F_{n}\in \mathcal {F}_{\Lambda _{n}},$ n = 1, 2.

In the quasi-classical probability model, the relation between a simulation measure ν and random variables modeling observables is such that probabilities of the observed events are reproduced only via positive values of a normalized bounded real-valued measure ν.

The local version of the quasi-classical probability model covers (theorem 1) the probabilistic description of each nonsignaling multipartite correlation scenario, in particular, each multipartite correlation scenario (corollary 1) on an N-partite quantum state.

Proof of proposition 1. Let a scenario $\mathcal {E}_{S}\mathrm{\ }$ admit an LqHV model (3). Introduce the normalized real-valued measure

$$\begin{eqnarray} &&\fl \mu _{\mathcal {E}_{S}}\big( \mathrm{d}\lambda _{1}^{(1)}\times \cdots \times \mathrm{d}\lambda _{1}^{(S_{1})}\times \cdots \times \mathrm{d} \lambda _{N}^{(1)}\times \cdots \times \mathrm{d}\lambda _{N}^{(S_{N})}\big)\nonumber \\ &&:=\int _{\Omega }\hbox{ }\left\lbrace \prod \limits _{s_{n}=1,\ldots ,S_{n},\hbox{ } n=1,\ldots ,N}P_{n}^{(s_{n})}(\mathrm{d}\lambda _{n}^{(s_{n})}\mid \omega )\hbox{ }\right\rbrace \hbox{ }\nu _{\mathcal {E}_{S}}(\mathrm{d}\omega ). \end{eqnarray} \tag{ A.1 }$$

This measure is bounded (see the proof of theorem 1 in [13]) and returns all distributions $P_{(s_{1},\ldots ,s_{N})}^{(\mathcal {E}_{S})}$ of the scenario $\mathcal {E}_{S}$ as the corresponding marginals. The latter means the factorizable representation

$$\begin{eqnarray} &&\fl P_{(s_{1},\ldots ,s_{N})}^{(\mathcal {E}_{S}\mathcal {)}}(F_{1}\times \cdots \times F_{N}) =\int \chi _{F_{1}}\big(\lambda _{1}^{(s_{1})}\big)\cdot \ldots \cdot \chi _{F_{N}}\big(\lambda _{N}^{(s_{N})}\big)\hbox{ }\mu _{\mathcal {E}_{S}}\big(\mathrm{d }\lambda _{1}^{(1)} \nonumber\\ &&\times \cdots \times \mathrm{d}\lambda _{1}^{(S_{1})}\times \cdots \times \mathrm{d}\lambda _{N}^{(1)}\times \cdots \times \mathrm{d}\lambda _{N}^{(S_{N})}\big), \nonumber \\ &&F_{1} \in \mathcal {F}_{\Lambda _{1}},\ldots ,F_{N}\in \mathcal {F}_{\Lambda _{N}}, \end{eqnarray} \tag{ A.2 }$$

for all s_n = 1, ..., S_n, n = 1, ..., N. Denote

$$\begin{eqnarray} &&\widetilde{\omega } :=\big(\lambda _{1}^{(1)},\ldots ,\lambda _{1}^{(S_{1})},\ldots ,\lambda _{N}^{(1)},\ldots ,\lambda _{N}^{(S_{N})}\big),\nonumber \\ &&\widetilde{\Omega } :=\Lambda _{1}^{S_{1}}\times \cdots \times \Lambda _{N}^{S_{N}},\quad \mathcal {F}_{\widetilde{\Omega }}:=\mathcal {F} _{\Lambda _{1}}^{\otimes S_{1}}\otimes \cdots \otimes \mathcal {F}_{\Lambda _{N}}^{\otimes S_{N}},\nonumber \\ &&\widetilde{\nu }_{\mathcal {E}_{S}}(\mathrm{d}\widetilde{\omega }) :=\mu _{ \mathcal {E}_{S}}\big(\mathrm{d}\lambda _{1}^{(1)}\times \cdots \times \mathrm{d} \lambda _{1}^{(S_{1})}\times \cdots \times \mathrm{d}\lambda _{N}^{(1)}\times \cdots \times \mathrm{d}\lambda _{N}^{(S_{N})}\big) \end{eqnarray} \tag{ A.3 }$$

and introduce the $\mathcal {F}_{\widetilde{\Omega }}/\mathcal {F}_{\Lambda _{n}}$-measurable functions $f_{n,s_{n}}:\widetilde{\Omega }\rightarrow \Lambda _{n},$ each defined by the relation $f_{n,s_{n}}(\widetilde{\omega } )=\lambda _{n}^{(s_{n})}.$ Then

$$\begin{equation} \chi _{_{F_{n}}}\big(\lambda _{n}^{(s_{n})}\big)\equiv \chi _{f_{n,s_{n}}^{-1}(F_{n})}(\widetilde{\omega }),\quad \forall F_{n}\in \mathcal {F}_{\Lambda _{n}}, \end{equation} \tag{ A.4 }$$

and, in view of (A.3), (A.4), representation (A.2) takes the form

$$\begin{eqnarray} P_{(s_{1},\ldots ,s_{N})}^{(\mathcal {E}_{S}\mathcal {)}}(F_{1}\times \cdots \times F_{N})& =\int _{\Omega }\chi _{f_{1,s_{1}}^{-1}(F_{1})}( \widetilde{\omega })\cdot \ldots \cdot \chi _{f_{N,s_{_{N}}}^{-1}(F_{N})}( \widetilde{\omega })\hbox{ }\widetilde{\nu }_{\mathcal {E}_{S}}(\mathrm{d} \widetilde{\omega }) \\ & =\widetilde{\nu }_{\mathcal {E}_{S}}( f_{1,s_{1}}^{-1}(F_{1})\cap \cdots \cap f_{N,s_{_{N}}}^{-1}(F_{N})) . \nonumber \end{eqnarray} \tag{ A.5 }$$

This proves the statement of proposition 1.

Proof of theorem 1. If an N-partite correlation scenario $\mathcal {E }_{S}$, with a setting S = S₁ × ⋅⋅⋅ × S_N, admits a deterministic LqHV model (6), then, clearly, the consistency condition (10) is fulfilled.

Conversely, let the scenario $\mathcal {E}_{S}$ satisfy the consistency condition (10). Consider first a bipartite (N = 2) scenario $\mathcal {E}_{S}$ with a setting S = S₁ × S₂ and joint probability distributions $P_{(s_{1},s_{2})}^{(\mathcal {E}_{S})}$ satisfying condition (10). Since, under condition (10), marginals $P_{(s_{1},1)}^{(\mathcal {E} _{S})}(F_{1}\times \Lambda _{2}),\ldots ,$ $P_{(s_{1},S_{2})}^{(\mathcal {E} _{S})}(F_{1}\times \Lambda _{2})$ at site '1' coincide for all s₂ = 1, ..., S₂, for the simplicity of notation, we denote these coinciding marginals as

$$\begin{equation} P_{(s_{1},1)}^{(\mathcal {E}_{S})}(F_{1}\times \Lambda _{2})=\cdots =P_{(s_{1},S_{2})}^{(\mathcal {E}_{S})}(F_{1}\times \Lambda _{2}):=P_{s_{1}}^{(\mathcal {E}_{S})}(F_{1}), \end{equation} \tag{ B.1 }$$

for all $F_{1}\in \mathcal {F}_{\Lambda _{1}}$ and each s₁ = 1, ..., S₁ at site '1'. The superscript $\mathcal {E}_{S}$ in notation $P_{s_{1}}^{( \mathcal {E}_{S})}$ indicates that, for a general correlation scenario, this marginal does not need to depend only on a setting of measurement s₁ at site '1' (see remark 1).

Quite similarly,

$$\begin{equation} P_{(1,s_{2})}^{(\mathcal {E}_{S})}(\Lambda _{1}\times F_{2})=\cdots =P_{(S_{1},s_{2})}^{(\mathcal {E}_{S})}(\Lambda _{1}\times F_{2}):=P_{s_{2}}^{(\mathcal {E}_{S})}(F_{2}), \end{equation} \tag{ B.2 }$$

for all $F_{2}\in \mathcal {F}_{\Lambda _{2}}$ and each s₂ = 1, ..., S₂ at site '2'.

Introduce the normalized bounded real-valued bounded measure $\mu _{\mathcal { E}_{S}}$ on (ΛS11 × ΛS22, $\mathcal {F} _{\Lambda _{1}}^{\otimes S_{1}}\otimes \mathcal {F}_{\Lambda _{2}}^{\otimes S_{2}})$ with values

$$\begin{eqnarray} &&\fl \mu _{\mathcal {E}_{S}}\hbox{ }\big(F_{1}^{(1)}\times \cdots \times F_{1}^{(S_{1})}\times F_{2}^{(1)}\times \cdots \times F_{2}^{(S_{2})}\big)\nonumber\\ &&:=\sum _{s_{1},s_{2}}\left\lbrace P_{(s_{1},s_{2})}^{(\mathcal {E} _{S})}\big(F_{1}^{(s_{1})}\times F_{2}^{(s_{2})}\big)\prod \limits _{\widetilde{s} _{1}\ne s_{1}}P_{\widetilde{s}_{1}}^{(\mathcal {E}_{S})}\big(F_{1}^{(\widetilde{s }_{1})}\big)\prod \limits _{\widetilde{s}_{2}\ne s_{2}}P_{\widetilde{s}_{2}}^{( \mathcal {E}_{S})}\big(F_{2}^{(\widetilde{s}_{2})}\big)\right\rbrace \nonumber \\ &&\;\;\;-\hbox{ }(S_{1}S_{2}-1)\prod \limits _{s_{1}}P_{s_{1}}^{(\mathcal {E} _{S})}\big(F_{1}^{(s_{1})}\big)\prod \limits _{s_{2}}P_{s_{2}}^{(\mathcal {E} _{S})}\big(F_{2}^{(s_{2})}\big), \end{eqnarray} \tag{ B.3 }$$

for all $F_{n}^{(s_{n})}\in \mathcal {F}_{\Lambda _{n}},$ s_n = 1, ..., S_n, n = 1, 2. It is easy to verify that this measure returns all joint probability distributions $P_{(s_{1},s_{2})}^{(\mathcal {E}_{S})}$ of a bipartite nonsignaling scenario $\mathcal {E}_{S}$ as the corresponding marginals. By proposition 2, this implies that a bipartite correlation scenario satisfying condition (10) admits a deterministic LqHV model.

Let N = 3. Consider a tripartite correlation scenario $\mathcal {E}_{S}$ with a setting S = S₁ × S₂ × S₃ and joint probability distributions $P_{(s_{1},s_{2},s_{3})}^{(\mathcal {E})}$ satisfying condition (10). In addition to the one-party marginals denoted similar to notation (B.2), we denote by $P_{(s_{1},s_{2})}^{(\mathcal {E}_{S})},$ $P_{(s_{1},s_{3})}^{(\mathcal {E}_{S})}$ and $P_{(s_{2},s_{3})}^{(\mathcal {E})}$ the coinciding two-party marginals at the corresponding sites, that is,

$$\begin{eqnarray} \fl P_{(s_{1},s_{2},1)}^{(\mathcal {E}_{S})}(F_{1}\times F_{2}\times \Lambda _{3})& =\cdots =P_{(s_{1},s_{2},S_{3})}^{(\mathcal {E}_{S})}(F_{1}\times F_{2}\times \Lambda _{3}):=P_{(s_{1},s_{2})}^{(\mathcal {E}_{S})}(F_{1}\times F_{2}),\nonumber \\ \fl P_{(s_{1},1,s_{3})}^{(\mathcal {E}_{S})}(F_{1}\times \Lambda _{2}\times F_{3})& =\cdots =P_{(s_{1},S_{2},s_{3})}^{(\mathcal {E}_{S})}(F_{1}\times \Lambda _{2}\times F_{3}):=P_{(s_{1},s_{3})}^{(\mathcal {E}_{S})}(F_{1}\times F_{3}),\nonumber \\ \fl P_{(1,s_{2},s_{3})}^{(\mathcal {E}_{S})}(\Lambda _{1}\times F_{2}\times F_{3})& =\cdots =P_{(S_{1},s_{2},s_{3})}^{(\mathcal {E}_{S})}(\Lambda _{1}\times F_{2}\times F_{3}):=P_{(s_{2},s_{3})}^{(\mathcal {E})}(F_{2}\times F_{3}), \end{eqnarray} \tag{ B.4 }$$

for all $F_{n}\in \mathcal {F}_{\Lambda _{n}},$ s_n = 1, ..., S_n, n = 1, 2, 3.

Similar to our construction of measure (B.3) for a bipartite case, introduce the normalized bounded real-valued measure $\widetilde{\mu }_{ \mathcal {E}_{S}}$ on (ΛS11 × ΛS22 × ΛS33, $\mathcal {F}_{\Lambda _{1}}^{\otimes S_{1}}\otimes \mathcal {F}_{\Lambda _{2}}^{\otimes S_{2}}\otimes \mathcal {F}_{\Lambda _{3}}^{\otimes S_{3}})$ with values

$$\begin{eqnarray} &&\fl \widetilde{\mu }_{\mathcal {E}_{S}}\hbox{ }\big(F_{1}^{(1)}\times \cdots \times F_{1}^{(S_{1})}\times F_{2}^{(1)}\times \cdots \times F_{2}^{(S_{2})}\times F_{3}^{(1)}\times \cdots \times F_{3}^{(S_{3})}\big) \nonumber\\ &&\fl :=\!\sum _{s_{1},s_{2},s_{_{3}}}\!\!\left\lbrace P_{(s_{1},s_{2},s_{_{3}})}^{(\mathcal { E}_{S})}\big(F_{1}^{(s_{1})}\times F_{2}^{(s_{2})}\times F_{3}^{(s_{3})}\big)\!\prod \limits _{\widetilde{s}_{1}\ne s_{1}}P_{\widetilde{ s_{1}}}^{(\mathcal {E}_{S})}\big(F_{1}^{(\widetilde{s}_{1})}\big)\!\prod \limits _{ \widetilde{s}_{2}\ne s_{2}}P_{\widetilde{s}_{2}}^{(\mathcal {E} _{S})}\big(F_{2}^{(\widetilde{s}_{2})}\big)\prod \limits _{\widetilde{s}_{3}\ne s_{3}}P_{\widetilde{s}_{3}}^{(\mathcal {E}_{S})}\big(F_{3}^{(\widetilde{s} _{3})}\big)\right\rbrace \nonumber \\ &&\fl \;\;\;\;-\,(S_{1}-1)\prod \limits _{s_{1}}P_{s_{1}}^{(\mathcal {E} _{S})}\big(F_{1}^{(s_{1})}\big)\sum _{s_{2},s_{3}}\hbox{ }\left\lbrace P_{(s_{2},s_{3})}^{( \mathcal {E}_{S})}\big(F_{2}^{(s_{2})}\times F_{3}^{(s_{3})}\big)\prod \limits _{ \widetilde{s}_{2}\ne s_{2}}P_{\widetilde{s}_{2}}^{(\mathcal {E} _{S})}\big(F_{2}^{(\widetilde{s}_{2})}\big)\prod \limits _{\widetilde{s}_{3}\ne s_{3}}P_{\widetilde{s}_{3}}^{(\mathcal {E}_{S})}\big(F_{3}^{(\widetilde{s} _{3})}\big)\right\rbrace \nonumber \\ &&\fl \;\;\;\;-\,(S_{2}-1)\prod \limits _{s_{2}}P_{s_{2}}^{(\mathcal {E} _{S})}\big(F_{2}^{(s_{2})}\big)\sum _{s_{1},s_{3}}\left\lbrace P_{(s_{1},s_{3})}^{( \mathcal {E}_{S})}\big(F_{1}^{(s_{1})}\times F_{3}^{(s_{3})}\big)\prod \limits _{ \widetilde{s}_{1}\ne s_{1}}P_{\widetilde{s}_{1}}^{(\mathcal {E} _{S})}\big(F_{1}^{(\widetilde{s}_{1})}\big)\prod \limits _{\widetilde{s}_{3}\ne s_{3}}P_{\widetilde{s}_{3}}^{(\mathcal {E}_{S})}\big(F_{3}^{(\widetilde{s} _{3})}\big)\right\rbrace \nonumber \\ &&\fl \;\;\;\;-\,(S_{3}-1)\prod \limits _{s_{3}}P_{s_{3}}^{(\mathcal {E} _{S})}\big(F_{3}^{(s_{3})}\big)\sum _{s_{1},s_{2}}\left\lbrace P_{(s_{1},s_{2})}^{( \mathcal {E}_{S})}\big(F_{1}^{(s_{1})}\times F_{2}^{(s_{2})}\big)\prod \limits _{ \widetilde{s}_{1}\ne s_{1}}P_{\widetilde{s}_{1}}^{(\mathcal {E} _{S})}\big(F_{1}^{(\widetilde{s}_{1})}\big)\prod \limits _{\widetilde{s}_{2}\ne s_{2}}P_{\widetilde{s}_{2}}^{(\mathcal {E}_{S})}\big(F_{2}^{(\widetilde{s} _{2})}\big)\right\rbrace \nonumber \\ &&\fl \;\;\;\;+\,(2S_{1}S_{2}S_{3}-S_{1}S_{2}-S_{2}S_{3}-S_{1}S_{3}+1)\prod \limits _{s_{1}}P_{s_{1}}^{(\mathcal {E}_{S})}\big(F_{1}^{(s_{1})}\big)\prod \limits _{s_{2}}P_{s_{2}}^{(\mathcal {E}_{S})}\big(F_{2}^{(s_{2})}\big)\prod \limits _{s_{3}}P_{s_{3}}^{(\mathcal {E}_{S})}\big(F_{3}^{(s_{3})}\big), \nonumber\\ \end{eqnarray} \tag{ B.5 }$$

for all sets $F_{n}^{(s_{n})}\in \mathcal {F}_{\Lambda _{n}},$ s_n = 1, ..., S_n, n = 1, 2, 3. Measure $\widetilde{\mu }_{\mathcal {E}_{S}}$ returns all joint probability distributions $P_{(s_{1},s_{2},s_{_{3}})}^{( \mathcal {E}_{S})}$ of a tripartite nonsignaling scenario $\mathcal {E}_{S}$ as the corresponding marginals. By proposition 2, the latter implies that a correlation scenario $\mathcal {E}_{S}$ satisfying condition (10) admits a deterministic LqHV model.

The obvious generalization to an arbitrary N-partite case of the measure constructions used in (B.3) and (B.5) proves the sufficiency part of theorem 1.