## Abstract

For the probabilistic description of all the joint von Neumann measurements on a D-dimensional quantum system, we present the specific example of a context-invariant quasi hidden variable (qHV) model, proved in Loubenets (J Math Phys 56:032201, 2015) to exist for each Hilbert space. In this model, a quantum observable X is represented by a variety of random variables satisfying the functional condition required in quantum foundations but, in contrast to a contextual model, each of these random variables equivalently models X under all joint von Neumann measurements, regardless of their contexts. This, in particular, implies the specific local qHV (LqHV) model for an N-qudit state and allows us to derive the new exact upper bound on the maximal violation of 2\(\times \cdots \times \)2-setting Bell-type inequalities of any type (either on correlation functions or on joint probabilities) under N-partite joint von Neumann measurements on an N-qudit state. For d = 2, this new upper bound coincides with the maximal violation by an N-qubit state of the Mermin–Klyshko inequality. Based on our results, we discuss the conceptual and mathematical advantages of context-invariant and local qHV modelling.

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## Notes

In the mathematical physics literature, this probability model is often named after Kolmogorov. However, in the probability theory literature, the term “Kolmogorov model” is mostly used [2] for the Kolmogorov probability axioms [3]. These axioms hold for a measurement of any nature, in particular, quantum, see also our discussion in [4, 5].

Here, \(\Omega \) is a set and \(\mathcal {F}_{\Omega }\) is an algebra of subsets of \(\Omega .\)

For the general framework on Bell-type inequalities, see [18].

In a contextual model, a quantum observable can be also modelled by a variety of random variables but which of these random variables represents an observable \(X\) under a joint von Neumann measurement depends

*specifically on a context*of this joint measurement, i. e. on other compatible quantum observables measured jointly with \(X\).This notation means [18] that two observables are measured at each of \( N \) sites.

On this notion, see, for example, Sect. 3 in [19].

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## Acknowledgments

I am very grateful to Professor A. Khrennikov for valuable discussions.

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## Appendix

### Appendix

Let us first consider the bipartite case \(N=2.\) For simplicity of notations, denote by \(X_{i}\), \(i=1,2,\) observables measured by Alice and by \(Y_{k},\) \( k=1,2\), measured by Bob. For this case, the values of the real-valued distribution \(\nu _{2\times 2}^{(\rho _{d,2})}(\cdot |X_{1},X_{2},Y_{1},Y_{2}),\) standing in (15), are defined as

where (i) the notation \(Z^{(\pm )}\) means the positive operators, decomposing a self-adjoint operator \(Z=Z^{(+)}-Z^{(-)}\) and satisfying the relation \(Z^{(+)}Z^{(-)}=Z^{(-)}Z^{(+)}=0\), (ii) the probability distributions \(\alpha _{X_{i}}^{(\pm )}(\cdot |y_{1},y_{2}),\) \(i=1,2,\) are defined via the relation

From (20) and (21) it follows that, for the distribution \(\nu _{2\times 2}^{(\rho _{d,2})},\) the total variation norm

where \(\left| \{\mathrm {P}_{Y_{1}}(y_{1})\mathrm {P} _{Y_{2}}(y_{2})\}_{\mathrm {sym}}\right| \) is the absolute value operator

Calculating \(\left| \mathbb {\{}\mathrm {P}_{Y_{1}}(y_{1})\mathrm {P} _{Y_{2}}(y_{2})\}_{\mathrm {sym}}\right| ,\) we find

where \(\phi _{Y}^{(k)}, k=1,\ldots ,d,\) are orthonormal eigenvectors of an observable \(Y\) and \(\alpha _{k_{1},k_{2}}=\langle \phi _{Y_{1}}^{(k_{1})}|\phi _{Y_{2}}^{(k_{2})}\rangle \). Substituting (24) into (22) and taking into account that \(\sum \limits _{k_{i}}\left| \alpha _{k_{1},k_{2}}\right| \le \sqrt{d}, i=1,2,\) we finally derive

For \(N>2,\) the real-valued distribution

in (15) is similar by its construction to distribution (20) with the replacement of \(\{ \mathbb {I}_{\mathbb {C}^{d}}\mathbb {\otimes }\frac{1}{2} \{\mathrm {P}_{Y_{1}}(y_{1})\mathrm {P}_{Y_{2}}(y_{2})_{\mathrm {sym} }^{(\pm )}\} \}\) by the \(N\)-partite tensor product of identity operator \( \mathbb {I}_{\mathbb {C}^{d}}\) and \((N-1)\) factors of the form \(\frac{1}{2} \{\mathrm {P}_{X_{n}^{(1)}}(x_{n}^{(1)})\mathrm {P} _{X_{n}^{(2)}}(x_{n}^{(2)})\}_{\mathrm {sym}}^{(\pm )}\) at each \(n\)th of \( (N-1)\) sites. As a result,

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Loubenets, E.R. Context-Invariant and Local Quasi Hidden Variable (qHV) Modelling Versus Contextual and Nonlocal HV Modelling.
*Found Phys* **45**, 840–850 (2015). https://doi.org/10.1007/s10701-015-9903-8

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DOI: https://doi.org/10.1007/s10701-015-9903-8