Quantum analog of the original Bell inequality for two-qudit states with perfect correlations/anticorrelations

TL;DR

This paper introduces a class of two-qudit states with perfect correlations and proves that their maximal Bell inequality violation is bounded by 3/2, extending understanding of quantum nonlocality bounds beyond qubits.

Contribution

The paper defines a new class of two-qudit states with perfect correlations and establishes an upper bound of 3/2 on their Bell inequality violations, generalizing previous results.

Findings

Maximal violation of Bell inequality by these states is bounded by 3/2.

Two-qudit GHZ states with even dimension exhibit perfect correlations.

The results support the conjecture of a universal 3/2 upper bound for such violations.

Abstract

For an even qudit dimension $d\geq 2,$ we introduce a class of two-qudit states exhibiting perfect correlations/anticorrelations and prove via the generalized Gell-Mann representation that, for each two-qudit state from this class, the maximal violation of the original Bell inequality is bounded from above by the value $3/2$ - the upper bound attained on some two-qubit states. We show that the two-qudit Greenberger-Horne-Zeilinger (GHZ) state with an arbitrary even $d\geq 2$ exhibits perfect correlations/anticorrelations and belongs to the introduced two-qudit state class. These new results are important steps towards proving in general the $\frac{3}{2}$ upper bound on quantum violation of the original Bell inequality. The latter would imply that similarly as the Tsirelson upper bound $2\sqrt{2}$ specifies the quantum analog of the CHSH inequality for all bipartite quantum states, the upper bound $\frac{3}{2}$ specifies the quantum analog of the original Bell inequality for all bipartite quantum states with perfect correlations/ anticorrelations. Possible consequences for the experimental tests on violation of the original Bell inequality are briefly discussed.

Equations

$$\left|\mathrm{tr}[\rho \{A\otimes B\}]-\mathrm{tr}[\rho \{A\otimes \tilde{B}\}]\right|\pm \mathrm{tr}[\rho \{B\otimes \tilde{B}\}]\le 1$$

$$\left|\mathrm{tr}[\rho \{A\otimes B\}]-\mathrm{tr}[\rho \{A\otimes \tilde{B}\}]\right|\pm \mathrm{tr}[\rho \{B\otimes \tilde{B}\}]\le \frac{3}{2}$$

$$⟨{\lambda }_{a_i}{\lambda }_{b_k}⟩:=\sum _{{\lambda }_a,{\lambda }_b\in [-1,1]}{\lambda }_a{\lambda }_b{P}_{(a_i,b_k)}({\lambda }_a,{\lambda }_b)$$

$$\left|⟨{\lambda }_{a_1}{\lambda }_{b_1}⟩-⟨{\lambda }_{a_1}{\lambda }_{b_2}⟩\right|\pm ⟨{\lambda }_{a_2}{\lambda }_{b_2}⟩$$

$$⟨{\lambda }_{a_2}{\lambda }_{b_1}⟩=\pm 1.$$

$$⟨{\lambda }_{a_2}{\lambda }_{b_1}⟩=\pm 1\iff \sum _{{\lambda }_a,{\lambda }_b\in [-1,1]}({\lambda }_a{\lambda }_b\mp 1)P_{(a_i,b_k)}({\lambda }_a,{\lambda }_b)=0.$$

$$P_{(a_i,b_k)}({\lambda }_a,{\lambda }_b)$$

$${\lambda }_a,{\lambda }_b$$

$$\left(\left|⟨{\lambda }_{a_1}{\lambda }_{b_1}⟩-⟨{\lambda }_{a_1}{\lambda }_{b_2}⟩\right|\pm ⟨{\lambda }_{a_2}{\lambda }_{b_2}⟩\right){|}_{\text{perfect}}\le 1$$

$$⟨{\lambda }_{a_i}{\lambda }_{b_k}⟩=\mathrm{tr}[{\rho }_{d\times d}\{A_i\otimes B_k\}],$$

$${\mathcal{B}}_{{\rho }_{d\times d}}^{ob}(A,B^{(\pm )},\tilde{B})=\left|\mathrm{tr}[{\rho }_{d\times d}\{A\otimes B^{(\pm )}\}]-\mathrm{tr}[{\rho }_{d\times d}\{A\otimes \tilde{B}\}]\right|\pm \mathrm{tr}[{\rho }_{d\times d}\{B^{(\pm )}\otimes \tilde{B}\}],$$

$$\mathrm{tr}[{\rho }_{d\times d}\{B^{(\pm )}\otimes B^{(\pm )}\}]=\pm 1,$$

$$\sum _{i,k}({\lambda }_i{\lambda }_k\mp 1)\mathrm{tr}[{\rho }_{d\times d}\{E_{B^{(\pm )}}({\lambda }_i)\otimes E_{B^{(\pm )}}({\lambda }_k)\}]=0,$$

$$\underset{{\rho }_{2\times 2},{\sigma }_a,{\sigma }_{b^{(\pm )}},{\sigma }_{\tilde{b}}}{\max }{\mathcal{B}}_{{\rho }_{2\times 2}}^{ob}({\sigma }_a,{\sigma }_{b^{(\pm )}},{\sigma }_{\tilde{b}}){|}_{\text{perfect}}=\frac{3}{2}.$$

$$\underset{{\rho }_{3\times 3},S_a,S_{b^{(\pm )}},S_{\tilde{b}}}{\max }{\mathcal{B}}_{{\rho }_{3\times 3}}^{ob}(S_a,S_b,S_{\tilde{b}}){|}_{\text{perfect}}\le \frac{3}{2}.$$

$$X$$

$$\mathrm{tr}[X^2]$$

$${\mathcal{L}}_d\leftrightarrow {\mathfrak{R}}_d$$

$${\mathfrak{R}}_d:=\left\{r\in {\mathbb{R}}^{d^2-1}\mid {\Vert r\cdot \Lambda \Vert }_0\le \sqrt{\frac{2}{d}}\right\}$$

$${\mathfrak{R}}_d$$

$$l_d$$

$${\mathcal{L}}_d\supset {\mathcal{L}}_d^{(0)}\leftrightarrow {\mathfrak{R}}_d^{(0)}\subset {\mathfrak{R}}_d$$

$${\mathfrak{R}}_d^{(0)}=\left\{r\in {\mathbb{R}}^{d^2-1}\mid {\Vert r\cdot \Lambda \Vert }_0=\sqrt{\frac{2}{d}},\Vert r\Vert =1\right\}.$$

$$A$$

$$a,b^{(\pm )},\tilde{b}$$

$$\mathrm{tr}[{\rho }_{d\times d}\{A\otimes B\}]=⟨a,T_{{\rho }_{d\times d}}b⟩:=\sum _{n,m}{T}_{{\rho }_{d\times d}}^{(n,m)}{a}_n{b}_m,$$

$$T_{{\rho }_{d\times d}}^{(nm)}:=\mathrm{tr}[{\rho }_{d\times d}\{{\Lambda }_n\otimes {\Lambda }_m\}],n,m=1,...,d^2-1,$$

$${\mathcal{B}}_{{\rho }_{d\times d}}^{ob}(A,B^{(\pm )},\tilde{B})$$

$$⟨b^{(\pm )},T_{{\rho }_{d\times d}}{b}^{(\pm )}⟩$$

$$\underset{A,B^{(\pm )},\tilde{B}\in {\mathcal{L}}_d^{(0)}}{\max }{\mathcal{B}}_{{\rho }_{d\times d}}^{ob}(A,B^{(\pm )},\tilde{B}){|}_{\text{perfect}}$$

$$T_{{\rho }_{d\times d}}{\mathrm{v}}_{{\lambda }_m}^{(j)}={\lambda }_m{\mathrm{v}}_{{\lambda }_m}^{(j)}$$

$$\Vert T_{{\rho }_{d\times d}}\Vert =\underset{m}{\max }\left|{\lambda }_m\right|.$$

$$b^{(\pm )}=\sum _{m,j}{\beta }_{jm}^{(\pm )}{\mathrm{v}}_{{\lambda }_m}^{(j)}\in {\mathfrak{R}}_d^{(0)},\sum _{m,j}{\left({\beta }_{jm}^{(\pm )}\right)}^2=1,$$

Full text

1 Introduction

The Clauser-Horne-Shimony-Holt (CHSH) inequality [1] on local classical correlations was introduced in 1969 and in 11 years Tsirelson [2, 3] proved that, for any bipartite quantum state, possibly, infinite dimensional, the maximal violation of the CHSH inequality cannot exceed $\sqrt{2}$ and that on some two-qubit states this upper bound is attained.

The original Bell inequality for local classical correlations was derived [4, 5] by Bell even earlier – in 1964, however, to our knowledge, until our recent article [6] on quantum violation of the original Bell inequality by general two-qubit and two-qutrit states under spin measurements, the maximal quantum violation of this inequality was considered in the literature only for two-qubit Bell states and, mostly, for the singlet – Pitowsky [7] and Khrennikov&Basieva [8] proved that, for the two-qubit singlet state, the maximal violation is equal to $\frac{3}{2}$ and stressed that this value is more than the Tsirelson [2, 3] upper bound $\sqrt{2}$ on the maximal violation of the CHSH inequality.

There are probably three major reasons for disregarding the problem of finding the maximal violation of the original Bell inequality for a general bipartite quantum state with perfect correlations/anticorrelations.

First of all, only the two-qubit singlet state was known to satisfy the condition on the perfect anticorrelation of outcomes whenever *each *qubit spin observable is measured at both sites – the Bell condition sufficient for the derivation of the original Bell inequality in a local hidden variable (LHV) frame.

Note that, for the derivation of the original Bell inequality in a local hidden variable model, the Bell restriction $\mathrm{tr}[\rho\{B\otimes B\}]=\pm 1$ on a bipartite quantum state $\rho$ on a Hilbert space $\mathcal{H}\otimes\mathcal{H}$ and a quantum observable $B$ on $\mathcal{H}$ is only sufficient but not necessary. In the LHV model, the validity of this condition for only some observable $B$ with eigenvalues in $[-1,1]$ implies the validity for a state $\rho$ of the original Bell inequality

[TABLE]

(in its perfect correlations (plus sign) form or perfect anticorrelations (minus sign) form) for all observables $A,\widetilde{B}$ with eigenvalues in $[-1,1].$ Moreover, as proved in [9], for any dimension of a Hilbert space $\mathcal{H}\otimes\mathcal{H},$ there is the whole class of bipartite quantum states that satisfy the perfect correlation form of inequality (1) for any three quantum observables $A,B,\widetilde{B}$ with eigenvalues in $[-1,1]$ but do not need to exhibit the perfect correlation or anticorrelation of outcomes if some qubit observable is measured at both sites (perfect correlations/anticorrelations, for short).

Secondly, as we have discussed this in Section 2 of [6], the Tsirelson [2, 3] upper bound $2\sqrt{2}$ for the CHSH inequality implies the upper bound $\left(2\sqrt{2}-1\right)$ for the maximal value of the left-hand side of inequality (1) in any bipartite quantum state with perfect correlations/anticorrelations. Therefore, specifically the latter bound was considered to be the least one for violation of the original Bell inequality in a quantum perfect correlation/anticorrelation case. However, as proved in [7, 8], for the singlet, the maximal value of the left hand side of inequality (1) is equal to $\frac{3}{2}$ and this value $\sqrt{2}<\frac{3}{2}<2\sqrt{2}-1$.

Thirdly, for more than 40 years since derivation of the original Bell inequality it was impossible to approach high fidelity in preparation of two-qubit singlet states. Only recently experimenters approached very high levels of fidelity. Therefore, the original Bell inequality was considered as a theoretical statement without any possibility for its experimental verification. The absence of possible experimental applications lowered the interest of theoreticians to this inequality and the main theoretical efforts were put into analysis of the CHSH inequality. We remark that even nowadays the experimental testing of the original Bell inequality is a big challenge, since one has to combine high levels of purity in preparation of the two qubit singlet state and of detection efficiency (see [8] for details).

Analyzing in [6] the maximal violation of the original Bell inequality in a general two-qubit case, we introduced *a necessary and sufficient condition *for a symmetric two-qubit state to exhibit perfect correlations/anticorrelations. We proved [6] that, for each symmetric two-qubit state with perfect correlations/anticorrelations, the maximal violation of the original Bell inequality under spin measurements is upper bounded by the value $\frac{3}{2}$ and specified two-qubit states where this upper bound is attained. We also considered [6] spin measurements on symmetric two-qutrit states with perfect correlations/anticorrelations and found that, in this case, the maximal violation of the original Bell inequality is also upper bounded by $3/2$.

Therefore, in Conclusions of [6], we assumed that, similarly as the Tsirelson upper bound $2\sqrt{2}$ specifies [2, 3] the quantum analog of the CHSH inequality for all bipartite quantum states and all quantum observables with eigenvalues in $[-1,1]$, the upper bound $\frac{3}{2}$ specifies the quantum analog

[TABLE]

of the original Bell inequality for all bipartite quantum states with perfect correlations/ anticorrelations: $\mathrm{tr}[\rho\{B\otimes B\}]=\pm 1.$

In the present paper, for an arbitrary even $d\geq 2,$ we introduce a class of symmetric two-qudit states exhibiting perfect correlations/anticorrelations and show that, for all two-qudit states from this class, the maximal violation of the original Bell inequality over traceless qudit observables with eigenvalues $\pm 1$ is bounded by $3/2$ from above. We prove that, for any even $d\geq 2,$ the two-qudit Greenberger–Horne–Zeilinger (GHZ)* *state exhibits perfect correlations/ anticorrelations and belongs to the introduced state class.

These new results are important steps towards proving that inequality (2) constitutes the quantum analog of the original Bell inequality for all bipartite quantum states with perfect correlations/anticorrelations.

The paper is organized as follows.

In Section 2, we recall the main issues on the derivation of the original Bell inequality in a local hidden variable frame.

In Section 3, we specify shortly our previous results [6] on the maximal violation of the original Bell inequality under spin measurements on arbitrary two-qubit states and two-qutrit states with perfect correlations/anticorrelations.

In Section 4, due to the properties of the generalized Gell-Mann representation for traceless qudit observables proved in [10], we consider violation of the original Bell inequality in a general two-qudit case ($d\geq 2).$ We show (Proposition 1) that, for an arbitrary even $d\geq 2,$ there exists the whole class of states exhibiting perfect correlations/anticorrelations and that the GHZ state belongs (Proposition 2) to this state class. We further prove (Theorem 1) that, for each state from this class, the maximal violation of the original Bell inequality is bounded from above by the value $3/2.$

In Section 5, we summarize the main results of the present paper.

2 Preliminaries: the original Bell inequality

Consider a bipartite correlation scenario111On the general framework for the description of a multipartite correlation scenario, see [11]. where two parties (say, Alice and Bob) perform measurements, indexed by $a_{i},b_{k},$ $i,k=1,2,$ and with outcomes $\lambda_{a},\lambda_{b}\in[-1,1]$ at Alice and Bob sites, respectively. This correlation scenario is described by four joint measurements $(a_{i},b_{k}),$ $i,k=1,2$, and, for a measurement $(a_{i},b_{k}),$ notation $P_{(a_{i},b_{k})}(\lambda_{a},\lambda_{b})$ means the joint probability of the event that Alice observes an outcome $\lambda_{a}$ and Bob – an outcome $\lambda_{b}$, and the expectation value (average) of the product of their outcomes is given by

[TABLE]

For the above correlation scenario, consider the value of the Bell [4, 5]) combination of the product expectations

[TABLE]

under the Bell condition [4, 5] on perfect correlations/anticorrelations:

[TABLE]

In view of (3), this condition takes the form:

[TABLE]

Since $\lambda\in[-1,1],$ this implies that the Bell condition (5) is fulfilled under measurement $(a_{2},b_{1})$ if and only if, for outcomes $\lambda_{a},$ $\lambda_{b}$, with the product $\lambda_{a}\lambda_{b}\neq\pm 1,$ the corresponding joint probability $P_{(a_{2},b_{1})}(\lambda_{a},\lambda_{b})=0$.

Let the probabilistic description of this correlation scenario admits a local hidden variable (LHV) model in the sense that all joint probabilities $P_{(a_{i},b_{k})},$ $i,k=1,2,$ admit the representation

[TABLE]

via a single probability distribution $\nu$ of some variables $\omega\in\Omega$ and conditional probability distributions $P_{a_{i}}(\mathrm{\cdot}$ $|\omega),$ $P_{b_{k}}(\mathrm{\cdot}$ $|\omega)$ of outcomes at Alice’s and Bob’s sites, each distribution depending222Independence of distributions $P_{a_{i}}(\mathrm{\cdot}$ $|\omega),$ $P_{b_{k}}(\mathrm{\cdot}|\omega)$ on setting of other measurements is referred to as Bell locality, see [15] for details. only on a setting of the corresponding measurement at the corresponding site.

Then, under condition (5) on perfect correlations/anticorrelations, the Bell expression (4) for product expectations satisfies the original Bell inequality [4]:

[TABLE]

in its perfect correlation (plus sign) or perfect anticorrelation (minus sign) forms. For the proof of inequality (8) in an LHV model for arbitrary $\lambda_{a},\lambda_{b}\in[-1,1]$ and a more general (than (5)) sufficient condition for its validity, see [9].

3 Quantum violation by two-qubit and two-qutrit states

Let, under a bipartite correlation scenario, two parties perform measurements on a two-qudit state $\rho_{d\times d}$ on $\mathbb{C}^{d}\otimes\mathbb{C}^{d},$ $d\geq 2,$ and measure traceless qudit observables $A_{1},$ $A_{2}=B_{1},B_{2}$ with eigenvalues in $[-1,1]$.

In a quantum case, the product expectations (3) take the form

[TABLE]

and expression (4) (the left hand-side of the original Bell inequality (8)) and the Bell condition (5) reduce to

[TABLE]

and

[TABLE]

respectively, where, for short, we change notations $A_{1}\rightarrow A,$ $B_{1}\rightarrow B^{(\pm)},$ $B_{2}\rightarrow\widetilde{B}$.

Remark 1

Similarly as we have discussed this above in Section 2 (see Eq. (6), let us analyze, for what a quantum observable $B^{(\pm)}$ with eigenvalues in $[-1,1],$ the Bell condition (11) can be fulfilled. Consider the spectral decomposition $B^{(\pm)}=\sum_{i}\lambda_{i}E_{B}(\lambda_{i}),$ of an observable $B^{(\pm)}$ with eigenvalues $\lambda_{i}\in[-1,1]$ and spectral projections $E_{B^{(\pm)}}(\lambda_{i}).$ Substituting this decomposition into (11), we rewrite the Bell condition (11) on perfect correlations/anticorrelations in the form

[TABLE]

*where all joint probabilities $\mathrm{tr}[\rho_{d\times d}\{E_{B^{(\pm)}}(\lambda_{i})\otimes E_{B^{(\pm)}}(\lambda_{k})\}]\geq 0$ and sum up to $1.$This form of condition (11) implies that, for an observable $B^{(\pm)}$ with eigenvalues $\left|\lambda_{i}\right|\leq 1,$ condition (11) is fulfilled *if and only if the operator norm $\left\|B^{(\pm)}\right\|=1$ and, for eigenvalues $\lambda_{i},$ $\lambda_{k}$, for which the product $\lambda_{i}\lambda_{k}\neq\pm 1,$ the corresponding joint probability $\mathrm{tr}[\rho_{d\times d}\{E_{B^{(\pm)}}(\lambda_{i})\otimes E_{B^{(\pm)}}(\lambda_{k})\}]=0.$

For $d=2,$ we specified in Proposition 2 of [6] a necessary and sufficient condition for a symmetric two-qubit state $\rho_{2\times 2}$ to exhibit perfect correlations/anticorrelations if some qubit spin observable $\sigma_{b}$ is measured at both sites. We proved (see Theorem 1 in [6]) that, for each two-qubit state $\rho_{2\times 2}$ with perfect correlations/anticorrelations, the maximal value of the left-hand side (10) of the original Bell inequality (8) over all qubit spin observables $\sigma_{a},\sigma_{b},\sigma_{\widetilde{b}}$ cannot exceed $\frac{3}{2}$ and specified two-qubit states, where this upper bound is attained:

[TABLE]

In Theorem 2 of [6] we also showed that, for each symmetric two-qutrit state $\rho_{3\times 3}$ exhibiting perfect correlations/anticorrelations if some qutrit spin observable $S_{b}$ is measured at both sites, the maximal value of the left- hand side (10) of the original Bell inequality over qutrit spin observables $S_{a},S_{b},S_{\widetilde{b}}$ also admits the bound

[TABLE]

We stressed in [6] that, in (13), (14), the attained upper bound $\frac{3}{2}$ on the maximal violation of the original Bell inequality is less than the upper bound $\left(2\sqrt{2}-1\right)$ for the value of $\mathcal{B}_{\rho_{d\times d}}(A,B^{(\pm)},\widetilde{B}){\LARGE|}_{\text{perfect}}$ following from the Tsirelson upper bound for the CHSH inequality.

In the following section, we proceed to analyse the maximal value of the left-hand side (10) of the original Bell inequality in a general two-qudit case ($d\geq 2)$.

4 Quantum violation in a general two-qudit case

Under the Bell condition (11), let us analyze the maximal value of the Bell expression (10) over traceless qudit observables $A,$ $B,$ $\widetilde{B}$ with eigenvalues in $[-1,1].$ For short, we further denote this set of observables by $\mathcal{L}_{d}$.

For an observable $X\in\mathcal{L}_{d}$, consider the normalized version [10]

[TABLE]

of the generalized Gell-Mann representation333On the generalized Gell-Mann representation for qudit states see [12, 13, 14] and references therein.. Here, notation $\left\|\cdot\right\|$ means the Euclidian norm of a vector $n$ in $\mathbb{R}^{d^{2}-1}$ and $\Lambda_{j},$ $j=1,....,d^{2}-1$  are traceless Hermitian operators on $\mathbb{C}^{d}$ (generators of SU$(d)$ group), satisfying the relation $\mathrm{tr}[\Lambda_{j}\Lambda_{j_{1}}]=2\delta_{jj_{1}},$ and presented in Appendix A. The matrix representations of $\Lambda_{j},$ $j=1,....,d^{2}-1$ constitute the higher dimensional extensions of the Pauli matrices for qubits ($d=2$) and the Gell-Mann matrices for qutrits ($d=3$).

As proved in [10], representation (15) establishes the one-to-one correspondence

[TABLE]

between traceless observables in $\mathcal{L}_{d}$ and $(d^{2}-1)$-dimensional vectors $r\in\mathbb{R}^{d^{2}-1}$ in the set

[TABLE]

which is a subset

[TABLE]

of the ball of radius $l_{d}$ in $\mathbb{R}^{d^{2}-1}$ and also contains the ball $\mathfrak{R}_{d}\supseteq\left\{n\in\mathbb{R}^{d^{2}-1}\mid\left\|n\right\|=\sqrt{\frac{1}{d-1}}\right\}.$ In (17), notation $\left\|\cdot\right\|_{0}$ means the operator norm of observables on $\mathbb{C}^{d}.$

If a qudit dimension $d\geq 2$ is even, then (15) establishes [10] the one-to-one correspondence

[TABLE]

between traceless qudit observables with eigenvalues $\pm 1$ (i. e in subset $\mathcal{L}_{d}^{(0)}\subset\mathcal{L}_{d}$) and vectors in the intersection $\mathfrak{R}_{d}^{(0)}$ of $\mathfrak{R}_{d}$ with the unit sphere:

[TABLE]

For details, see Proposition 1 in Section 2 of [10].

For each of three qudit observables $A,B,\widetilde{B}\in\mathcal{L}_{d}$ standing in expression (10), we specify representation (15) as

[TABLE]

This implies the following expression for the quantum expectation (9):

[TABLE]

where $T_{\rho_{d\times d}}$ is the linear operator on $\mathbb{R}^{d^{2}-1},$ defined in the canonical basis of $\mathbb{R}^{d^{2}-1}$ by the $\left(d^{2}-1\right)\times\left(d^{2}-1\right)$ correlation matrix

[TABLE]

introduced for an arbitrary qudit dimension $d\geq 2$ in [10]. This matrix constitutes a generalization to higher dimensions of the two-qubit correlation matrix considered in [6, 16]. For a symmetric444In the sense that $\rho_{d\times d}$ is invariant under permutation of spaces $\mathbb{C}^{d}$ in the tensor product $\mathbb{C}^{d}\otimes\mathbb{C}^{d}.$ two-qudit state $\rho_{d\times d},$ the operator $T_{\rho_{d\times d}}$ on $\mathbb{R}^{d^{2}-1}$ and its matrix representation (23) are hermitian.

Substituting (22) into relation (10) and condition (11), we, correspondingly, derive:

[TABLE]

In what follows, we consider only symmetric two-qudit states $\rho_{d\times d}$ – in this case, the correlation matrix $\left(T_{\rho_{d\times d}}^{(nm)}\right)$ is hermitian, and also, in view of Remark 1 – only qudit observables $A,B^{(\pm)},\widetilde{B}$ in $\mathcal{L}_{d}^{(0)},$ that is, traceless and with eigenvalues $\pm 1.$ From relation (19) proved in [10] it follows that, under representation (15), these observables are bijectively mapped to vectors in subset $\mathfrak{R}_{d}^{(0)}\subset\mathfrak{R}_{d}$ given by (20).

For the maximal value of the Bell expression $\mathcal{B}_{\rho_{d\times d}}^{ob}(A,B^{(\pm)},\widetilde{B})$ over traceless observables $A,B^{(\pm)},\widetilde{B}\in\mathcal{L}_{d}^{(0)}$ under constraint (11), relation (24) and the one-to-one correspondence (19) imply

[TABLE]

where, for the maximum standing in the second line, the Bell condition on perfect correlations/anticorrelations is given by (25) – the plus sign corresponds to perfect correlations and the minus sign – to perfect anticorrelations.

Let us first analyze when a symmetric two-qudit state may satisfy the Bell condition (25), equivalently, (11).

4.1 Two-qudit states with perfect correlations/anticorrelations

For the hermitian matrix $\left(T_{\rho_{d\times d}}^{nm}\right),$ let $\lambda_{m}$ be an eigenvalue with a multiplicity $k_{\lambda_{m}}$, $\sum_{m}k_{\lambda_{m}}=d^{2}-1,$ and $\mathrm{v}_{\lambda_{m}}^{(j)}\in\mathbb{R}^{d^{2}-1},j=1,...,k_{\lambda_{m}},$ be mutually orthogonal unit eigenvectors

[TABLE]

corresponding to this $\lambda_{m}$. Note that the spectral norm of the correlation matrix (23) satisfies the relation

[TABLE]

Decomposing in (4) a unit vector $b^{(\pm)}\in\mathfrak{R}_{d}^{(0)}$ ($d\geq 2$ is even) via the orthonormal basis $\{\mathrm{v}_{\lambda_{m}}^{(j)}\}:$

[TABLE]

we rewrite the Bell condition (25) in the form

[TABLE]

This form implies the following statement.

Proposition 1** (Sufficient condition)**

Let, for a symmetric two-qudit state $\rho_{d\times d}$ with an even $d\geq 2,$ the correlation matrix $T_{\rho_{d\times d}}$ have the spectral norm

[TABLE]

and, for the eigenvalue $\lambda_{m_{0}}$ of $T_{\rho_{d\times d}}$ with maximal absolute value $\left|\lambda_{m_{{}_{0}}}\right|=\frac{2}{d}$, there exist a unit eigenvector $\mathrm{v}_{\lambda_{m_{0}}}$ belonging to set $\mathfrak{R}_{d}^{(0)}$ given by (20) and satisfying, therefore, the relations:

[TABLE]

Then, for this state $\rho_{d\times d}$, the Bell condition in the form (25) is fulfilled on each vector

[TABLE]

and, correspondingly, the Bell condition in the form (11) – for each qudit observable

[TABLE]

If $\lambda_{m_{{}_{0}}}=\frac{2}{d},$ then the Bell condition on perfect correlations (plus sign) is fulfilled and if $\lambda_{m_{{}_{0}}}=-\frac{2}{d}$, then Bell condition on perfect anticorrelations (minus sign) is fulfilled.

In view of this statement, in what follows, we use the following terminology.

Definition 1

Denote by $\mathfrak{S}_{d\times d}^{(sym)}$ the class of symmetric two-qudit states satisfying relations (31), (32) and, therefore, exhibiting perfect correlations/anticorrelations if at least one of qudit observables in $\mathcal{L}_{d}^{(0)}$ is measured at both sites.

In a two-qubit case, condition (31) reduces to $\left\|T_{\rho_{2\times 2}}\right\|=1$ while relation (32) is fulfilled for each unit eigenvector of $T_{\rho_{2\times 2}}$. Therefore, in a two-qubit case, Proposition 1 of the present article reduces to our Proposition 2 in [6].

Moreover, since, in a two-qubit case relation $\left\|T_{\rho_{2\times 2}}\right\|\leq 1$ holds for each two-qubit state, condition (31) becomes necessary and sufficient for a two-qubit state to exhibit perfect correlations.

As an example of higher dimensional two-qudit states belonging to class $\mathfrak{S}_{d\times d}^{(sym)},$ consider the two-qudit Greenberger–Horne–Zeilinger (GHZ) state

[TABLE]

For $d=2$, this state constitutes one of Bell states and its correlation matrix has the form [6]

[TABLE]

Therefore, $\left\|T_{\rho_{ghz,2}}\right\|=1$. Also, as mentioned above, for $d=2,$ relation (32) is fulfilled for all unit eigenvectors of $T_{\rho_{ghz,2}}.$ Therefore, the two-qubit GHZ state belongs to the class $\mathfrak{S}_{2\times 2}^{(sym)}.$

Consider an even $d>2.$ As it is proved in [10], for the GHZ state $\rho_{ghz,d}$ with an arbitrary $d\geq 2$ (not necessarily even), the correlation matrix $T_{\rho_{ghz,d}}$ has the block diagonal form

[TABLE]

where (i) $T^{(s)}$ is the $\frac{d\left(d-1\right)}{2}\times\frac{d\left(d-1\right)}{2}$ diagonal matrix with all eigenvalues equal to $\frac{2}{d}$; (ii) $T^{(as)}$ is the $\frac{d\left(d-1\right)}{2}\times\frac{d\left(d-1\right)}{2}$ diagonal matrix with all eigenvalues equal to $(-\frac{2}{d})$; and (iii) $T^{(d)}$ is the $(d-1)\times(d-1)$ diagonal matrix with eigenvalues $\frac{2}{d}$.

Therefore, for the two-qudit GHZ state, the spectral norm of its correlation matrix $T_{\rho_{ghz,d}}$ is equal to

[TABLE]

so that condition (31) of Proposition 1 is fulfilled.

From (37) it also follows that, for the GHZ state (35), the hermitian matrix $T_{\rho_{ghz,d}}$ has two eigenvalues $\pm\frac{2}{d}$, hence, two proper subspaces $\mathfrak{J}_{\pm\frac{2}{d}}\subset\mathbb{R}^{d^{2}-1}$, so that each vector $r\in\mathfrak{R}_{d}^{(0)}$ is decomposed as $r=r^{(+)}+r^{(-)},$ where $r^{(\pm)}$ are projections of $r\in\mathfrak{R}_{d}^{(0)}$ on the proper subspaces $\mathfrak{J}_{\pm\frac{2}{d}}$ and constitute eigenvectors (in general, not unit) of $T_{\rho_{ghz,d}}$ corresponding to eigenvalues $(\pm\frac{2}{d}):$

[TABLE]

Let a dimension $d\geq 2$ be even, hence, subset $\mathcal{L}_{d}^{(0)}$ of traceless qudit observables with eigenvalues $\pm 1$ be not empty. Consider a qudit observable $X\in\mathcal{L}_{d}^{(0)}$ of the form

[TABLE]

where (i) $\gamma_{m}$ are arbitrarily chosen positive integers $\gamma_{m}$ guaranteeing tr$[X]=0$; (ii) $|m\rangle$ are mutually orthogonal unit vectors of the computational basis $\{|m\rangle,$ $m=1,....,d\}$ in $\mathbb{C}^{d}.$

Due to the structure (A1) of operators $\Lambda_{jk}^{(s)}$, $\Lambda_{jk}^{(as)}$, $\Lambda_{l}^{(d)}$, under representation (15), to an observable of the form (40) there corresponds the unit vector $r_{X}\in\mathfrak{R}_{d}^{(0)},$ for which the projection $r_{X}^{(-)}=0,$ therefore, $r_{X}=r_{X}^{(+)}\in$ $\mathfrak{R}_{d}^{(0)}\cap\mathfrak{J}_{\frac{2}{d}}.$

Also, if we take a qudit observable $X^{\prime}\in\mathcal{L}_{d}^{(0)}$ of the form

[TABLE]

with arbitrarily chosen positive integers $\gamma_{m},$ then, under representation (15), to an observable (41) there also corresponds the unit vector $r_{X^{\prime}}\in\mathfrak{R}_{d}^{(0)},$ for which the projection $r_{X^{\prime}}^{(-)}=0,$ so that $r_{X^{\prime}}=r_{X^{\prime}}^{(+)}\in$ $\mathfrak{R}_{d}^{(0)}\cap\mathfrak{J}_{\frac{2}{d}}.$

However, to a qudit observable $X^{\prime\prime}\in\mathcal{L}_{d}^{(0)}$ of the form

[TABLE]

with arbitrary positive integers $\gamma_{m},$ under representation (15), there corresponds the unit vector $r_{X^{\prime\prime}}\in\mathfrak{R}_{d}^{(0)}$ with projection $r_{X^{\prime\prime}}^{(+)}=0,$ so that this observable is mapped to the unit eigenvector $r_{X^{\prime\prime}}=r_{X}^{(-)}\in$ $\mathfrak{R}_{d}^{(0)}\cap\mathfrak{J}_{(-\frac{2}{d})}$ of $T_{\rho_{ghz,d}}$ corresponding to eigenvalue $(-\frac{2}{d}).$

Thus: (i) from (38) it follows that, for the GHZ state (35), condition (31) is fulfilled; (ii) from (40)–(42) and (19), (20) it follows that, in case of GHZ state $\rho_{ghz,d}$ with an even $d\geq 2,$ for eigenvalues $\pm\frac{2}{d}$ of the correlation matrix $T_{\rho_{ghz,d}}$, there exist unit eigenvectors which belong to set $\mathfrak{R}_{d}^{(0)}.$

Therefore, by Proposition 1, the GHZ state $\rho_{ghz,d}$ exhibits perfect correlations if any of observables $X,X^{\prime}\in\mathcal{L}_{d}^{(0)}$ of forms (40), (41) is measured at both sites and perfect anticorrelations if at both sites any observable $X^{\prime\prime}\in\mathcal{L}_{d}^{(0)}$ of form (42) is measured.

Summing up, we have proved the following statement.

Proposition 2

For an arbitrary even $d\geq 2,$ the two-qudit GHZ state (35) belongs to the class $\mathfrak{S}_{d\times d}^{(sym)}$ of two-qudit states exhibiting perfect correlations/anticorrelations.

4.2 Quantum analog

For a state $\rho_{d\times d}\in\mathfrak{S}_{d\times d}^{(sym)}$ with an even dimension $d\geq 2$ and a vector $b^{(\pm)}\in\mathfrak{R}_{d}^{(0)}$ specified in Proposition 1, let us now analyze the value of the maximum (4).

Since by (20) $\mathfrak{R}_{d}^{(0)}$ is a subset of the unit sphere in $\mathbb{R}^{d^{2}-1},$ we have the following relation:

[TABLE]

In (4.2), the maximum of the expression

[TABLE]

over unit vectors $\left\|a\right\|=1$ is attained on the unit vector

[TABLE]

and is equal to

[TABLE]

For a vector $b^{(\pm)}\in\mathfrak{R}_{d}^{(0)}$, specified in Proposition 1, decomposition (29) reads

[TABLE]

Expanding also a vector $\widetilde{b}=\sum_{m,j}\widetilde{\beta}_{jm}\mathrm{v}_{\lambda_{m}}^{(j)}$, $\sum_{m,j}\left(\widetilde{\beta}\right)_{jm}^{2}=1$ via the orthonormal basis $\left\{\mathrm{v}_{\lambda_{m}}^{(j)}\right\}$ and substituting this decomposition and decomposition (47) into (46), we derive

[TABLE]

Since, for a state $\rho_{d\times d}\in\mathfrak{S}_{d\times d}^{(sym)}$, all eigenvalues $\lambda_{m}^{2}\leq\frac{4}{d^{2}},$ from the expression in the last line of (4.2) it follows that, for all choices of a vector $b^{(\pm)}\in\mathfrak{R}_{d}^{(0)}$, specified for a state $\rho_{d\times d}\in\mathfrak{S}_{d\times d}^{(sym)}$ in Proposition 1, maximum (4.2) admits the bound

[TABLE]

This implies that, for each state $\rho_{d\times d}\in\mathfrak{S}_{d\times d}^{(sym)}$ with an even $d\geq 2,$ the maximal value of the left-hand side of the original Bell inequality over all traceless qudit observables with eigenvalues $\pm 1$ admits the bound

[TABLE]

As we have proved by in [6] (Theorem 1), for $d=2,$ this upper bound is attained.

Relations (4), (4.2) and (4.2) imply the following statement.

Theorem 1

Let a qudit dimension $d\geq 2$ be even and a symmetric two-qudit state $\rho_{d\times d}$ belong to the class $\mathfrak{S}_{d\times d}^{(sym)}$ of states specified in Proposition 1 and exhibiting perfect correlations/anticorrelations (11) whenever a qudit observable $B^{(\pm)}$ is measured at both sites. Then the maximal value of the left-hand side $\mathcal{B}_{\rho_{d\times d}}(A,B^{(\pm)},\widetilde{B})$ of the original Bell inequality (10) over all traceless qudit observables $A,B^{(\pm)},\widetilde{B}$ with eigenvalues $\pm 1$ admits the bound

[TABLE]

and this upper bound is, for example, attained on two-qubit states specified in [6].

We stress that, in (51), the general upper bound $\frac{3}{2}$ on the maximal violation of the original Bell inequality is less than the upper bound $\left(2\sqrt{2}-1\right),$ which follows for the value of $\mathcal{B}_{\rho_{d\times d}}^{ob}(A,B^{(\pm)},\widetilde{B}){\LARGE|}_{\text{perfect}}$ from the Tsirelson upper bound for the CHSH inequality.

Theorem 1 proves that, for an arbitrary even $d\geq 2$ and all two-qudit states $\rho_{d\times d}\in\mathfrak{S}_{d\times d}^{(sym)}$ with perfect correlations/anticorrelations $\mathrm{tr}[\rho_{d\times d}\{B^{(\pm)}\otimes B^{(\pm)}\}]=\pm 1,$ the quantum analog of the original Bell inequality

[TABLE]

holds for all traceless qudit observables $A,B^{(\pm)},\widetilde{B}$ with eigenvalues $\pm 1.$

5 Conclusions

In our recent article [6], we introduced a *necessary and sufficient condition *for a symmetric two-qubit state to exhibit perfect correlations/ anticorrelations and proved [6] that, for all symmetric two-qubit states exhibiting perfect correlations/ anticorrelations, violation of the original Bell inequality is bounded by $3/2$ from above and that this upper bound is attained.

In the present paper, for an even qudit dimension $d\geq 2,$ we have specified (Proposition 1, Definition 1) a class of two-qudit states exhibiting perfect correlations/anticorrelations and have proved (Theorem 1) that, for each two-qudit state from this class, the maximal violation of the original Bell inequality over all traceless qudit observables with eigenvalues $\pm 1$ is also bounded from above by the value $3/2$. We have shown (Proposition 2) that the two-qudit GHZ state with an arbitrary even $d\geq 2$ belongs to the introduced two-qudit state class.

These our new results are important steps towards proving the $3/2$ upper bound conjecture on the original Bell inequality violation for all bipartite quantum states with perfect correlations/anticorrelations. As we see, the proof of this upper bound for arbitrary higher dimensions is nontrivial and stimulated our application of the generalized Gell-Mann representation.

Although the authors expect that the same technique can lead to the proof of the $3/2$ upper bound in a general case, one cannot exclude that the complexity of calculations would lead to finding other approaches (see, e.g., [17]). Of course, one still cannot exclude that, for the maximal violation of the original Bell inequality by a general bipartite state with perfect correlations/anticorrelations, the upper bound $3/2$ may not be true.

The states specified in this article can also be used in experiments to test violation of the original Bell inequality. For the moment, we cannot guess whether it would be easier to combine high fidelity and detection efficiency for such class of states. But one cannot exclude that these states can play the important role in future tests on violation of the original Bell inequality.

The authors hope that the new results of the present article would attract the interest of the quantum information community to theoretical analysis of the original Bell inequality and its possible experimental testing.

Acknowledgments

E. R. Loubenets was supported within the Academic Fund Program at the National Research University Higher School of Economics (HSE) in 2018-2019 (grant N 18-01-0064) and by the Russian Academic Excellence Project ”5-100”.

6 Appendix A

In (15), the traceless hermitian operators have the following indexation form

[TABLE]

where [12, 13, 14]

[TABLE]

Here, $\{\left|m\right\rangle,$ $m=1,...,d\}$ is the computational basis of $\mathbb{C}^{d}.$ The matrix representations of $\Lambda_{j},$ $j=1,....,d^{2}-1$ constitute the higher–dimensional extensions of the Pauli matrices for qubits ($d=2$) and the Gell-Mann matrices for qutrits ($d=3$).