Context-Invariant and Local Quasi Hidden Variable (qHV) Modelling Versus Contextual and Nonlocal HV Modelling

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.

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

$$\begin{aligned}&2\nu _{2\times 2}^{(\rho _{d,2})}(x_{1},x_{2},y_{1},y_{2}\mid X_{1},X_{2},Y_{1},Y_{2}) \nonumber \\&\quad = \,\alpha _{X_{1}}^{(+)}(x_{1}|y_{1},y_{2})\alpha _{X_{2}}^{(+)}(x_{2}|y_{1},y_{2})\,\,\mathrm {tr}[\rho _{d,2}\,\, \{ \mathbb {I}_{\mathbb {C}^{d}}\,\,\mathbb {\otimes }\,\{ \mathrm {P}_{Y_{1}}(y_{1})\mathrm {P}_{Y_{2}}(y_{2})\}_{\mathrm {sym}}^{(+)}\}] \nonumber \\&\quad -\alpha _{X_{1}}^{(-)}(x_{1}|y_{1},y_{2})\alpha _{X_{2}}^{(-)}(x_{2}|y_{1},y_{2})\mathrm {tr}[\rho _{d,2}\text { }\{ \mathbb {I} _{\mathbb {C}^{d}}\text { }\mathbb {\otimes } \{\mathrm {P} _{Y_{1}}(y_{1})\mathrm {P}_{Y_{2}}(y_{2})\}_{\mathrm {sym}}^{(-)}\}], \end{aligned}$$

(20)

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

$$\begin{aligned}&\mathrm {tr}[\rho _{d,2}\{ \mathrm {P}_{X_{i}}(x_{i})\otimes \{ \mathrm {P}_{Y_{1}}(y_{1})\mathrm {P}_{Y_{2}}(y_{2})\}_{\mathrm {sym}}^{(\pm )}\}]\nonumber \\&=\alpha _{X_{i}}^{(\pm )}(x_{i}|y_{1},y_{2}))\mathrm {tr}\rho _{d,2}\{ \mathbb {I}_{\mathbb {C}^{d}}\,\,\mathbb {\otimes }\,\,\{ \mathrm {P}_{Y_{1}}(y_{1})\mathrm {P}_{Y_{2}}(y_{2})\}_{\mathrm {sym}}^{(\pm )}\}]. \end{aligned}$$

(21)

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

$$\begin{aligned}&\left\| \nu _{2\times 2}^{(\rho _{d,2})}(\cdot |X_{1},X_{2},Y_{1},Y_{2})\right\| _{var} \nonumber \\&\equiv \sum _{\omega \in \Omega }\left| \nu _{2\times \cdots \times 2}^{(\rho _{d,N})}(x_{1},x_{2},y_{1},y_{2}|X_{1},X_{2},Y_{1},Y_{2})\right| \nonumber \\&\le \frac{1}{2}\left\| \sum \limits _{y_{1},y_{2}}\left| \,\, \{\mathrm {P}_{Y_{1}}(y_{1})\mathrm {P}_{Y_{2}}(y_{2})\}_{\mathrm {sym}}\right| \right\| _{\mathbb {C}^{d}}, \end{aligned}$$

(22)

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

$$\begin{aligned}&\left| \{\mathrm {P}_{Y_{1}}(y_{1})\mathrm {P} _{Y_{2}}(y_{2})\}_{\mathrm {sym}}\right| \nonumber \\&=\{\mathrm {P}_{Y_{1}}(y_{1})\mathrm {P}_{Y_{2}}(y_{2})\}_{\mathrm { sym}}^{(+)}+\{\mathrm {P}_{Y_{1}}(y_{1})\mathrm {P}_{Y_{2}}(y_{2})\}_{ \mathrm {sym}}^{(-)}. \end{aligned}$$

(23)

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

$$\begin{aligned}&\sum \limits _{y_{1},y_{2}}\left| \,\,\{\mathrm {P} _{Y_{1}}(y_{1})\mathrm {P}_{Y_{2}}(y_{2})\}_{\mathrm {sym}}\right| \nonumber \\&=\sum \limits _{k_{1},k_{2}}\left| \alpha _{k_{1},k_{2}}\right| \left( |\phi _{Y_{1}}^{(k_{1}}\rangle \langle \phi _{Y_{1}}^{(k_{1}}| + |\phi _{Y_{2}}^{(k_{2}}\rangle \langle \phi _{Y_{2}}^{(k_{2}}|\right) , \end{aligned}$$

(24)

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

$$\begin{aligned} \left\| \nu _{2\times 2}^{(\rho _{d,2})}(\cdot |X_{1},X_{2},Y_{1},Y_{2})\right\| _{var}\le \sqrt{d}. \end{aligned}$$

(25)

For $N>2,$ the real-valued distribution

$$\begin{aligned} \nu _{2\times \cdots \times 2}^{(\rho _{d,N})}(\omega |X_{1}^{(1)},X_{1}^{(2)},\ldots ,X_{N}^{(1)},X_{N}^{(2)}) \end{aligned}$$

(26)

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,

$$\begin{aligned} \left\| \nu _{2\times \cdots \times 2}^{(\rho _{d,N})}\right\| _{var}\le d^{\frac{N-1}{2}}. \end{aligned}$$

(27)