The generalized Gell-Mann representation and violation of the CHSH inequality by a general two-qudit state

TL;DR

This paper introduces a generalized Gell-Mann representation for qudit observables, analyzes CHSH inequality violations in two-qudit states, and derives new bounds for the maximal CHSH expectation, with explicit results for GHZ states.

Contribution

It develops a generalized Gell-Mann framework and establishes new bounds for CHSH violation in two-qudit states, including explicit maximum values for GHZ states across dimensions.

Findings

New bounds for CHSH expectation in two-qudit states.

Exact maximum CHSH expectation for two-qudit GHZ states.

For two-qubit states, bounds coincide with known Tsirelson results.

Abstract

We formulate and prove the main properties of the generalized Gell-Mann representation for traceless qudit observables with eigenvalues in $[-1,1]$ and analyze via this representation violation of the CHSH inequality by a general two-qudit state. For the maximal value of the CHSH expectation in a two-qudit state with an arbitrary qudit dimension $d\geq2$, this allows us to find two new bounds, lower and upper, expressed via the spectral properties of the correlation matrix for a two-qudit state. We have not yet been able to specify if the new upper bound improves the Tsirelson upper bound for each two-qudit state. However, this is the case for all two-qubit states, where the new lower bound and the new upper bound coincide and reduce to the precise two-qubit CHSH\ result of Horodeckis, and also, for the Greenberger-Horne-Zeilinger (GHZ) state with an odd $d\geq2,$ where the new upper bound is less than the upper bound of Tsirelson. Moreover, we explicitly find the correlation matrix for the two-qudit GHZ state and prove that, for this state, the new upper bound is attained for each dimension $d\geq2$ and this specifies the following new result: for the two-qudit GHZ state, the maximum of the CHSH expectation over traceless qudit observables with eigenvalues in $[-1,1]$ is equal to $2\sqrt{2}$ if $d\geq2$ is even and to $\frac{2(d-1)}{d}\sqrt{2}$ if $d>2$ is odd.

Equations

$$Y$$

$$\alpha$$

$$\mathrm{tr}[{\Lambda }_j{\Lambda }_{j_1}]=2{\delta }_{j{j}_1}.$$

$$({\Lambda }_{12}^{(s)},...,{\Lambda }_{1d}^{(s)},...,{\Lambda }_{d-1,d}^{(s)},{\Lambda }_{12}^{(as)},...,{\Lambda }_{1d}^{(as)},...,{\Lambda }_{d-1,d}^{(as)},{\Lambda }_1^{(d)},...,{\Lambda }_{d-1}^{(d)}),$$

$${\Lambda }_{mk}^{(s)}$$

$${\Lambda }_{mk}^{(as)}$$

$${\Lambda }_l^{(d)}$$

$$X=n\cdot \Lambda ,n_j=\frac{1}{2}\mathrm{tr}[X{\Lambda }_j],$$

$$\mathrm{tr}[X^2]=2{\Vert n\Vert }^2.$$

$$\sqrt{\frac{2}{d}}\le \frac{{\Vert n\cdot \Lambda \Vert }_0}{\Vert n\Vert }\le \sqrt{\frac{2(d-1)}{d}}\le \sqrt{\frac{2}{1+{\delta }_{d2}}},$$

$$\frac{{\Vert n\cdot \sigma \Vert }_0}{\Vert n\Vert }{\mid }_{d=2}=1,n\in {\mathbb{R}}^3,$$

$$X$$

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

$${\Vert X\Vert }_0\le 1\iff {\Vert n_X\cdot \Lambda \Vert }_0\le \sqrt{\frac{2}{d}}.$$

$$\Phi :{\mathcal{L}}_d\to {\mathfrak{R}}_d$$

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

$$\mathrm{tr}[X^2]=(d-s)=d{\Vert n_X\Vert }^2.$$

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

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

$$X_n=\sqrt{\frac{d}{2}}\left(n\cdot \Lambda \right),\mathrm{tr}[X_n^2]=d{\Vert n\Vert }^2,$$

$${\Vert X_r\Vert }_0=1\text{, }\mathrm{tr}[X_r^2]=d.$$

$${\mathfrak{R}}_2=\{n\in {\mathbb{R}}^3\mid \Vert n\Vert \le 1\}$$

$$\sqrt{\frac{2}{d}}\frac{1}{{\Vert {\Lambda }_j\Vert }_{op}},j=1,...,d^2-1,$$

$$l_d:=\underset{n\in {\mathfrak{R}}_d}{\max }\Vert n\Vert =\sqrt{\frac{1}{d}\underset{X\in {\mathcal{L}}_d}{\max }\mathrm{tr}[X^2]}=\sqrt{\frac{1}{d}\underset{{\lambda }_m\in D}{\max }\sum _m{\lambda }_m^2,}$$

$$l_d$$

$$\Vert n\Vert \le l_d\le 1,$$

$${\mathfrak{R}}_d\supseteq \left\{n\in {\mathbb{R}}^{d^2-1}\mid \Vert n\Vert =\sqrt{\frac{1}{d-1}}\right\},$$

$$\tilde{n}=\sqrt{\frac{2}{d}}\frac{n}{{\Vert n\cdot \Lambda \Vert }_0}\in {\mathfrak{R}}_d.$$

$$X=\sqrt{\frac{d}{2}}\left(n\cdot \Lambda \right)$$

$${\mathcal{L}}_d^{(0)}\leftrightarrow {\mathfrak{R}}_d^{(0)}=\left\{n\in {\mathfrak{R}}_d\mid \Vert n\Vert =1\right\}$$

$${\mathcal{B}}_{chsh}(A_1,A_2;B_1,B_2)=A_1\otimes (B_1+B_2)+A_2\otimes (B_1-B_2)$$

$$\left|\mathrm{tr}[{\rho }_{d\times d}{\mathcal{B}}_{chsh}(A_1,A_2;B_1,B_2)]\right|$$

Full text

1 Introduction

Different aspects of quantum violation of the Clauser–Horne–Shimony–Holt (CHSH) inequality [1] were studied since 1969 in a huge number of papers (see [2, 3, 4, 5, 6, 7, 8] and references therein).

It is well known due to Tsirelson [3, 4] that, for all bipartite quantum states, the maximal value of the CHSH expectation cannot exceed the (Tsirelson) upper bound $2\sqrt{2}$ and that this upper bound is attained on the two-qubit Bell states.

It is also well known that, for the maximum of the CHSH expectation in an arbitrary two-qubit state over traceless qubit observables with eigenvalues $\pm 1$, Horodeckis [6] found the precise value specified via the correlation properties of this two-qubit state.

However, for a general two-qudit state with an arbitrary qudit dimension $d>2,$ bounds on the maximal value of the CHSH expectation in terms of the correlation properties of this state have not been analyzed.

In the present paper, we formulate and prove the main properties of the generalized Gell-Mann representation for traceless qudit observables with eigenvalues in $[-1,1]$ and study via this representation violation of the CHSH inequality by a general two-qudit state.

For the maximal value of the CHSH expectation in a two-qudit state with an arbitrary qudit dimension $d\geq 2$, we find two *new bounds, lower and upper, *expressed via the spectral properties of the correlation matrix of this two-qudit state.

We have not yet been able to specify if the new upper bound improves the Tsirelson upper bound for each two-qudit state. However, this is the case for every two-qubit state, where the new lower bound and the new upper bound coincide and reduce to the precise two-qubit result of Horodeckis in [6], and also, for the two-qudit Greenberger–Horne–Zeilinger (GHZ) state with an arbitrary odd $d\geq 2$, where the new upper bound is less* *than the upper bound of Tsirelson [3, 4].

Applying our new general results for the two-qudit Greenberger–Horne–Zeilinger (GHZ) state with an arbitrary $d\geq 2$, we explicitly find the correlation matrix for this state and prove that, for the two-qudit GHZ state, the new upper bound is attained for each $d\geq 2$ and specifies the precise value of the CHSH expectation maximum in this state: $2\sqrt{2}$ if $d\geq 2$ is even and $\frac{2(d-1)}{d}\sqrt{2}$ if $d>2$ is odd.

The paper is organized as follows.

In Section 2, we formulate and prove (Theorem 1) the main properties of the generalized Gell-Mann representation for traceless qudit observables with eigenvalues in $[-1,1].$

In Section 3, via the properties of the generalized Gell-Mann representation proved in Section 2, we study the maximal value of the CHSH expectation for a general two-qudit state and derive new bounds (Theorem 2) on the maximal value of the CHSH expectation for a two-qudit state with an arbitrary $d\geq 2.$

In Section 4, we show that, for the GHZ state with an odd $d\geq 2$, the new upper bound is less (Proposition 2) than the upper bound of Tsirelson [3, 4] and prove (Theorem 3) that, for the two-qudit GHZ state with an arbitrary $d\geq 2$, the new upper bound is attained.

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

2 Representation of traceless qudit observables

For an arbitrary observable $Y$ on $\mathbb{C}^{d}$, $d\geq 2,$ the generalized Gell-Mann representation (see e. g. in [9, 10, 11]) reads:

[TABLE]

where $n$ is a vector 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

[TABLE]

The tuple $\Lambda:=(\Lambda_{1},...,\Lambda_{d^{2}-1})$ of these traceless hermitian operators has the following form

[TABLE]

where

[TABLE]

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

Representation (1) constitutes decomposition of a qudit observable $Y$ in the orthonormal basis $\{\mathbb{I},\Lambda_{j},j=1,....,d^{2}-1\}$ of the Hilbert-Schmidt space $\mathcal{H}_{sch}$ where qudit observables are vectors and the scalar product is defined via $\langle Y,Y^{\prime}\rangle_{\mathcal{H}_{sch}}:=\mathrm{tr}[YY^{\prime}].$

Note that, for derivation of our new results in Sections 2, 3, a choice in (1) of a basis of $\mathcal{H}_{sch}$ is not essential – we could take any basis of $\mathcal{H}_{sch}.$ Though the new result presented in Section 4 also does not itself depend on a basis choice in representation (1), our proof of this result is based specifically on basis (3).

For a traceless qudit observable $X$, representation (1) takes the form

[TABLE]

and implies

[TABLE]

The following statement is proved in Appendix A.

Lemma 1

For each $n\in\mathbb{R}^{d^{2}-1}$ and each $d\geq 2,$

[TABLE]

where (i) notation $\left\|\cdot\right\|_{0}$ means the operator norm of an observable on $\mathbb{C}^{d}$ and $\left\|\cdot\right\|$ – the Euclidian norm of a vector $n$ in $\mathbb{R}^{d^{2}-1};$ (ii) $\delta_{d2}=1$ for $d=2$ and $\delta_{d2}=0,$ otherwise.

From the lower bound in (9) it follows that, for each $n\in\mathbb{R}^{d^{2}-1},$ relation $\left\|n\cdot\Lambda\right\|_{0}\leq\sqrt{\frac{2}{d}}$ implies $\left\|n\right\|\leq 1,$ and if $\left\|n\cdot\Lambda\right\|_{0}\leq\sqrt{\frac{2}{d}}\left\|n\right\|,$ then  $\left\|n\cdot\Lambda\right\|_{0}=\sqrt{\frac{2}{d}}\left\|n\right\|$.

Remark 1

In the qubit case ($d=2)$, operators $\Lambda_{j},$ $j=1,2,3,$ constitute the Pauli operators $\sigma_{j},j=1,2,3$, and bounds (9) reduce to the well-known relation

[TABLE]

valid for any traceless qubit observable $n\cdot\sigma$ – projection of the qubit spin on a direction $n$ in $\mathbb{R}^{3}.$

In what follows, we use the following notations, for short.

Definition 1

Denote by $\mathcal{L}_{d}$ the set of all traceless qudit observables on $\mathbb{C}^{d}$ with eigenvalues in $[-1,1]$ and by $\mathcal{L}_{d}^{(s)}\subset\mathcal{L}_{d}$ the subset of all traceless qudit observables $X_{s}$ with a dimension $s\geq 0$ of their kernels and all their eigenvalues in set $\{-1,0,1\}$. Subset $\mathcal{L}_{d}^{(0)}\neq\varnothing$ iff a dimension $d\geq 2$ is even.

If $s>0$, then it constitutes the multiplicity of the zero eigenvalue of an observable in $\mathcal{L}_{d}^{(s)}$. If $s=0,$ then an observable $X_{0}\in\mathcal{L}_{d}^{(0)}$ does not have the zero eigenvalue.

Normalizing a vector $n$ in decomposition (7) in view of the lower bound in (9), we represent each traceless qudit observable $X\in\mathcal{L}_{d}$ in the form

[TABLE]

so that, under representation (11),

[TABLE]

Therefore, representation (11) establishes the mapping

[TABLE]

of all traceless observables in $\mathcal{L}_{d}$ to vectors $n\in\mathbb{R}^{d^{2}-1}$ in the set

[TABLE]

and since (11) constitutes the decomposition via the basis $\{\mathbb{I},$ $\Lambda_{j},$ $j=1,....,d^{2}-1\}$ of the Hilbert-Schmidt space $\mathcal{H}_{sch}$, the mapping $\Phi$ is injective.

Under representation (11), for all observables $X\in\mathcal{L}_{d}^{(s)}$,

[TABLE]

Therefore, all traceless observables in $\mathcal{L}_{d}^{(s)}\subset\mathcal{L}_{d},$ $s\geq 0,$ are mapped to vectors $n$ in

[TABLE]

In particular, for an even $d\geq 2,$ all traceless qudit observables with eigenvalues $\pm 1$ (i. e. in subset $\mathcal{L}_{d}^{(0)}\subset\mathcal{L}_{d}),$ are mapped to vectors in

[TABLE]

Conversely, let $n$ be an arbitrary vector in set $\mathfrak{R}_{d},$ defined by (15). For the traceless qudit observable, corresponding to each $n\in\mathfrak{R}_{d}$ via the representation

[TABLE]

the operator norm $\left\|X_{n}\right\|_{0}\leq 1$ and, therefore, $X_{n}\in\mathcal{L}_{d}$.  Hence, the injective mapping (14) is surjective, therefore, bijective.

Furthermore, let subset $\mathfrak{R}_{d}^{(0)}\subset\mathfrak{R}_{d}$, given by (18), be not empty and $r\in\mathfrak{R}_{d}^{(0)}$ be a unit vector in this subset. For the observable $X_{r}=\sqrt{\frac{d}{2}}\left(r\cdot\Lambda\right)$ in $\mathcal{L}_{d}$ corresponding to a unit vector $r\in\mathfrak{R}_{d}^{(0)}$ via (19), we have

[TABLE]

But this is possible iff $X_{r}\in\mathcal{L}_{d}^{(0)}$. Therefore, representation (19) establishes the one-to-one correspondence between observables in the 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. Since $\mathcal{L}_{d}^{(0)}\neq\varnothing$ iff a dimension $d\geq 2$ is even, also, subset $\mathfrak{R}_{d}^{(0)}\neq\varnothing$ iff a dimension $d\geq 2$ is even.

Let us now analyze *geometry *of the set $\mathfrak{R}_{d}$, $d\geq 2,$ defined by (15).

As specified in Remark 1, in the qubit ($d=2$) case, for each $n\in\mathbb{R}^{3},$ the norm $\left\|n\cdot\sigma\right\|_{0}=\left\|n\right\|$. Therefore, in (15), relation $\left\|n\cdot\Lambda\right\|_{0}\leq 1$ is equivalent to $\left\|n\right\|\leq 1,$ the set

[TABLE]

constitutes the unit ball in $\mathbb{R}^{3}$.

Due to (15), for each vector $n\in\mathfrak{R}_{d}$ along $j$-$th$ coordinate axis in $\mathbb{R}^{d^{2}-1}$, the norm cannot exceed

[TABLE]

where the operator norms of all observables (4) and (5) are equal to $1$, whereas the operator norms of observables (6) vary from $1$ to $\sqrt{\frac{2(d-1)}{d}}.$ Therefore, the lengths (22) of vectors along different coordinate axes in $\mathbb{R}^{d^{2}-1}$ vary from $\sqrt{\frac{2}{d}}$ to $\sqrt{\frac{1}{d-1}}$ and are equal to each other if only $d=2.$ Thus, for $d>2,$ the form of the bounded set $\mathfrak{R}_{d}$ with respect to different axes is asymmetric.

The maximal norm $l_{d}:=\max_{n\in\mathfrak{R}_{d}}\left\|n\right\|$ of a vector $n\in\mathfrak{R}_{d}$ is calculated via (12):

[TABLE]

and is equal to

[TABLE]

In (23), $\{\lambda_{m}\in[-1,1],$ $m=1,...,d\}$ are eigenvalues of a traceless observable $X=\sqrt{\frac{d}{2}}\left(n\cdot\Lambda\right)\in\mathcal{L}_{d}$ and $D=\left\{\left|\lambda_{m}\right|\leq 1,\text{ }m=1,...,d,\text{ }\sum_{m}\lambda_{m}=0\right\}.$

For an even d\geq 2,\the maximal norm $l_{d}$ is attained on vectors $n\in\mathfrak{R}_{d}^{(0)},$ corresponding to observables in $\mathcal{L}_{d}^{(0)},$ for an odd $d\geq 2\ -$ on vectors $n\in\mathfrak{R}_{d}^{(1)}$, corresponding to observables in $\mathcal{L}_{d}^{(1)}$ (see Definition 1).

From (23) it follows, that, for all $n\in\mathfrak{R}_{d},$ the norms

[TABLE]

so that $\mathfrak{R}_{d}$ is a subset of the ball in $\mathbb{R}^{d^{2}-1}$of radius $l_{d}\leq 1.$ Relations (9), (15) imply that

[TABLE]

therefore, $\mathfrak{R}_{d}$ contains the ball of radius $\sqrt{\frac{1}{d-1}}.$ Note also that, for each vector $n\in\mathbb{R}^{d^{2}-1},$ the vector

[TABLE]

Summing up, we have proved the following statement.

Theorem 1

Representation

[TABLE]

establishes the one-to-one correspondence $\mathcal{L}_{d}\leftrightarrow\mathfrak{R}_{d}$ between traceless qudit observables $X$ with eigenvalues in $[-1,1]$ and vectors $n$ in the set $\mathfrak{R}_{d},$ defined by (15) and having, in general, the complicated form specified above by (21)–( 27). The maximal norm $l_{d}$ of a vector in $\mathfrak{R}_{d}$ is equal to $l_{d}$ $=1$ if a qudit dimension $d\geq 2$ is even and to $l_{d}$ $=\sqrt{\frac{d-1}{d}}$ if a qudit dimension $d\geq 2$ is odd. Under the one-to-one correspondence $\mathcal{L}_{d}\leftrightarrow\mathfrak{R}_{d}$ established by (28), sets

[TABLE]

and are not empty iff a dimension $d\geq 2$ is even.

In the next section, we use Theorem 1 for analyzing violation of the CHSH inequality by an arbitrary two-qudit state.

3 The maximal value of the CHSH expectation for a two-qudit state

Let $\rho_{d\times d}$ be a state on $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ and $A_{i},B_{k},$ $i,k=1,2$, be traceless observables on $\mathbb{C}^{d}$ with eigenvalues in $[-1,1],$ that is, $A_{i},B_{k}\in\mathcal{L}_{d}$ (see Definition 1).

Consider a bipartite correlation scenario111On the general framework for the probabilistic description of a multipartite correlation scenario with arbitrary numbers of settings and outcomes per site, see [13]. where each of two parties, say, Alice and Bob, performs measurements of two qudit observables in a state $\rho_{d\times d}$. Let Alice measure observables $A_{1},A_{2}\in\mathcal{L}_{d}$ and Bob – observables $B_{1},B_{2}\in\mathcal{L}_{d}$. For this quantum bipartite scenario, the Bell operator associated with the CHSH inequality has the form

[TABLE]

and, for a state $\rho_{d\times d},$ the left-hand side of the CHSH inequality222In the local hidden variable (LHV) model, the expectation $\left|\left\langle\mathcal{B}_{chsh}\right\rangle_{lhv}\right|\leq 2$ – the CHSH inequality [1]. reads:

[TABLE]

Remark 2

For a bipartite correlation scenario with two observables per site, the CHSH inequality was originally [1] introduced and proved in case of dichotomic observables with outcomes $\pm 1$ at each site. It was, however, further proved that, in the LHV frame, the CHSH inequality holds for a bipartite correlation scenario with outcomes in $[-1,1]$ of any spectral type. For the proof, see, for example, Proposition 1.4.3 in [14] and also, Section 3.2 in [15]. Note also that by Corollaries 1, 2 in [15] each of known correlation Bell inequalities, introduced originally for scenarios with outcomes $\pm 1,$ holds also for scenarios with outcomes in $[-1,1]$ of any spectral type.

For the expectation (31) of the Bell operator (30) in a state $\rho_{d\times d},$ let us analyze the least upper bound

[TABLE]

over observables $A_{i},B_{k}\in\mathcal{L}_{d}.$

For these observables, representation (28) reads

[TABLE]

where set $\mathfrak{R}_{d}$ is given by (15) and specified in Theorem

1. By (33) each expectation $\mathrm{tr}[\rho_{d\times d}\left\{A_{i}\otimes B_{k}\right\}]$ in (31) takes the form

[TABLE]

where $T_{\rho_{d\times d}}$ is the linear operator on $\mathbb{R}^{d^{2}-1}$, defined in the standard basis of $\mathbb{R}^{d^{2}-1}$ via the two-qudit correlation matrix with real elements:

[TABLE]

This $\left(d^{2}-1\right)\times\left(d^{2}-1\right)$ matrix constitutes a generalization to higher dimensions of the two-qubit correlation matrix, considered in [6, 11]. If a state $\rho_{d\times d}$ is invariant under permutation of spaces $\mathbb{C}^{d}$ in the tensor product $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$, permutationally invariant, for short, then $T_{\rho_{d\times d}}$ is hermitian. Note that since $\left|\text{ }\mathrm{tr}[\rho_{d\times d}\left\{A\otimes B\right\}]\text{ }\right|\leq 1$ for any state $\rho_{d\times d}$ and all observables $A,B\in\mathcal{L}_{d},$ relation (34) implies

[TABLE]

Substituting (34) into (31), we have

[TABLE]

Due to the one-to-one correspondence $\mathcal{L}_{d}\leftrightarrow\mathfrak{R}_{d}$ (see Theorem 1) between traceless qudit observables with eigenvalues in $[-1,1]$ and vectors in the subset $\mathfrak{R}_{d}$ of $\mathbb{R}^{d^{2}-1}$ given by (15), relation (37) implies

[TABLE]

Taking into account continuity in $a_{i},b_{k}$ of scalar products on $\mathbb{R}^{d^{2}-1},$ boundedness and closure of set $\mathfrak{R}_{d}$ and the one-to-one correspondence $\mathcal{L}_{d}\leftrightarrow\mathfrak{R}_{d},$ we have

[TABLE]

In the last line of (39), the maximum over $a_{i}\in\mathfrak{R}_{d},$ $i=1,2,$ is attained on vectors

[TABLE]

which are not necessarily unit, and is equal to

[TABLE]

Noting further that by (15), for arbitrary $b_{1},b_{2}\in\mathfrak{R}_{d},$ vectors $\frac{b_{1}\pm b_{2}}{2}$ are also in $\mathfrak{R}_{d}$, we generalize the calculation method used by Horodeckis in [6] and introduce two vectors $r_{1},r_{2}\in\mathfrak{R}_{d}$, not necessarily mutually orthogonal and satisfying the relations

[TABLE]

Substituting (42) into (41) and (41) into (39), we have

[TABLE]

where the maximum over $\theta\in[0,\pi/2]$ is attained at

[TABLE]

and is given by

[TABLE]

From (39)–(45) it follows

[TABLE]

Let us first show that the derived new expression (46) for maximum (39) leads at once to the Tsirelson upper bound. Namely, due to property (27) and relation (36), specified with vectors

[TABLE]

we have the bound

[TABLE]

Substituting (48) into (46), we have

[TABLE]

that is, the Tsirelson upper bound derived originally [3] in the other way.

Furthermore, the derived expression (46) for the maximal value of (31) allows us to introduce the following two new general bounds, lower and upper, see Appendix B for the proof.* *

Theorem 2

For an arbitrary two-qudit state $\rho_{d\times d},$ $d\geq 2,$ with the correlation matrix $T_{\rho_{d\times d}}$ defined by (35), the maximal value of the CHSH expectation (31) over all traceless qudit observables with eigenvalues in $[-1,1]$ admits the bounds

[TABLE]

where $\lambda_{\rho_{d\times d}}\geq$ $\widetilde{\lambda}_{\rho_{d\times d}}\geq 0$* *are two greater eigenvalues, corresponding to two linear independent eigenvectors of the positive hermitian matrix $T_{\rho_{d\times d}}^{{}^{{\dagger}}}T_{\rho_{d\times d}},$ and $l_{d}=1$ if a qudit dimension $d\geq 2$ is even and $l_{d}=\sqrt{\frac{d-1}{d}}$ if a qudit dimension $d\geq 2$ is odd.

For each two-qubit state, the lower and upper bounds in (50) coincide and (50) reduces to

[TABLE]

i. e. to the precise two-qubit result found by Horodeckis in [6].

As we prove in Section 4, for the two-qudit GHZ state, the upper bound (50) is also attained and gives the precise value for the maximum of the CHSH expectation (31) in this state.

4 The two-qudit GHZ state

Let us now specify the upper bound in (50) for the two-qudit GHZ state

[TABLE]

For this state, the correlation matrix $T_{\rho_{ghz,d}}$ is hermitian. Calculating its elements due to relation (35) and expressions (3), (4)–(6), we come for this matrix to the following diagonal-block form

[TABLE]

where

(i) $T^{(s)}$ is the $\frac{d\left(d-1\right)}{2}\times\frac{d\left(d-1\right)}{2}$ matrix with elements

[TABLE]

which are equal to $\frac{2}{d}$ on the diagonal and to zero, otherwise;

(ii) $T^{(as)}$ is the $\frac{d\left(d-1\right)}{2}\times\frac{d\left(d-1\right)}{2}$ matrix with elements

[TABLE]

which are equal to $(-\frac{2}{d})$ on the diagonal and to $0,$ otherwise;

(iii) $T^{(d)}$ is the $(d-1)\times(d-1)$ matrix with elements

[TABLE]

which are equal to $\frac{2}{d}$ on the diagonal and to zero, otherwise.

Thus, for the two-qudit GHZ state (52),

[TABLE]

and each vector $n\in\mathbb{R}^{d^{2}-1}$ is an eigenvector of $T_{\rho_{ghz,d}}^{2}$ with eigenvalue $\frac{4}{d^{2}}.$ From (57) it follows that, for the GHZ state, the eigenvalues in (50) are given by

[TABLE]

so that, for the GHZ state, the general upper bound in (50) reduces to

[TABLE]

where $l_{d}=1$ if a dimension $d\geq 2$ is even and $l_{d}=\sqrt{\frac{d-1}{d}}$ if a dimension $d\geq 2$ is odd. This proves the following statement.

Proposition 1

For the two-qudit GHZ state (52) and traceless qudit observables with eigenvalues in $[-1,1]$, the new upper bound introduced in Theorem 1 is equal to $2l_{d}^{2}\sqrt{2}$ and, if a qudit dimension $d\geq 2$ is odd, then this upper bound is less than the general upper bound $2\sqrt{2}$ of Tsirelson [3, 4].

Furthermore, let us prove that, for the GHZ state, the upper bound in (50) is attained.

If $d=2$, then the two-qubit GHZ state constitutes one of the four states in the Bell basis and the upper bound (59) is attained.

Consider an arbitrary $d>2.$ From (53)–(56) it follows that the hermitian matrix $T_{\rho_{ghz,d}}$ has two proper subspaces $\mathfrak{J}_{\pm\frac{2}{d}}\subset\mathbb{R}^{d^{2}-1},$ corresponding to eigenvalues $(\pm\frac{2}{d})$ and each vector $r\in\mathfrak{R}_{d}$ admits decomposition $r=r^{(+)}+r^{(-)},$ where $r^{(\pm)}$ are projections of $r\in\mathfrak{R}_{d}$ onto the proper subspaces $\mathfrak{J}_{\pm\frac{2}{d}}$, respectively, and

[TABLE]

If projection $r^{(\pm)}\neq 0,$ then it constitutes an eigenvector of $T_{\rho_{ghz,d}}$ (not necessarily unit) corresponding to the eigenvalue $\pm\frac{2}{d},$ respectively.

Let $X$ be a traceless qudit observable with: (i) $d$ mutually orthogonal unit eigenvectors given by the unit vectors in the computational basis $\{|j\rangle,$ $j=1,....,d\}$ of $\mathbb{C}^{d}$; (ii) eigenvalues $\pm 1$ if a dimension $d>2$ is even and eigenvalues $0,\pm 1$ with multiplicity $s=1$ of the zero eigenvalue – if a dimension $d>2$ is odd.

By Definition 1, a traceless observable $X$ belongs to subset $\mathcal{L}_{d}^{(s)}\subset\mathcal{L}_{d}^{(s)},$ where $s=0$ if $d>2$ is even and $s=1$ if $d>2$ is odd. The operator norm of this observable is equal to $\left\|X\right\|_{0}=1.$Under representation (11) to this observable $X$ there corresponds vector $r_{X}\in\mathfrak{R}_{d}^{(s)},$ $s=0,1,$ where $\mathfrak{R}_{d}^{(s)}$ is given by (17).

From (11) it follows that components of vector $r_{X}$ are given by $r_{X}^{(j)}=\frac{1}{\sqrt{2d}}\mathrm{tr}[X\Lambda_{j}],$ $j=1,....,(d^{2}-1).$ In view of (53), (55), components of $r_{X}$, corresponding to projection $r_{X}^{(-)}$ onto the proper subspace $\mathfrak{J}_{-\frac{2}{d}}\subset\mathbb{R}^{d^{2}-1}$ of $T_{\rho_{ghz,d}}$ corresponding to the eigenvalue ($-\frac{2}{d}),$ are defined by traces $\frac{1}{\sqrt{2d}}\mathrm{tr}[X\Lambda_{jk}^{(as)}]$. Due to the structure (5) of operators $\Lambda_{jk}^{(as)}$ and the above specified structure of an observable $X\in\mathcal{L}_{d}^{(s)},$ all these traces are equal to zero. Therefore, $r_{X}^{(-)}=0$ and $r_{X}=r_{X}^{(+)}.$

Further, since $r_{X}=r_{X}^{(+)}\in\mathfrak{R}_{d}^{(s)},$ $s=0,1,$ we have by (17): $\left\|r_{X}^{(+)}\cdot\Lambda\right\|_{0}=\sqrt{\frac{2}{d}}$ and $\left\|r_{X}^{(+)}\right\|=l_{d}$. Moreover, for $d>2,$ we can always find at least two such observables $X_{1}$ and $X_{2},$ for which $r_{X_{1}}^{(+)}\neq-r_{X_{2}}^{(+)}.$

Taking all this into account for the maximum in the second line of (46), we derive:

[TABLE]

Therefore, Eqs. (46), (61) imply

[TABLE]

Comparing (59) and (62), we come to the following new result.

Theorem 3

For the two-qudit GHZ state $\rho_{ghz,d}$   with an arbitrary $d\geq 2,$ the upper bound in Theorem 1 is attained for each $d\geq 2$ and specifies the maximal value of the CHSH expectation (31) in this state

[TABLE]

Here, $l_{d}^{2}=1$ if a qudit dimension $d\geq 2$ is even and $l_{d}=\sqrt{\frac{d-1}{d}}$ if a qudit dimension $d\geq 2$ is odd.

This result for the maximal value of the CHSH expectation in the GHZ state can be also derived if to substitute the correlation matrix (53) directly into maximum (39).

5 Conclusions

In the present paper, we have formulated and proved the properties (Theorem 1) of the generalized Gell-Mann representation for traceless qudit observables with eigenvalues in $[-1,1]$ and studied via this representation the maximal value of the CHSH expectation in a general two-qudit state with an arbitrary qudit dimension $d\geq 2.$

For the maximal value

[TABLE]

of the CHSH expectation in a general two-qudit state $\rho_{d\times d},$ $d\geq 2,$ and traceless qudit observables with eigenvalues in $[-1,1],$ we have derived the precise expression (46) via the correlation matrix (35) for this two-qudit state. This expression explicitly leads to the upper bound of Tsirelson [3, 4], and to *two new bounds (50), lower and upper, expressed *(Theorem 2) via the spectral properties of the correlation matrix for a two-qudit state $\rho_{d\times d},$ $d\geq 2$.

We have not yet been able to specify if the new upper bound in (50) improves the Tsirelson upper bound for each two-qudit state. However, this is the case:

(i) for each two-qubit state, where the new lower bound and the new upper bound coincide and reduce to the precise value of (64) found by Horodeckis [6]; 

(ii) for the two-qudit GHZ state (52) with an arbitrary odd $d\geq 2,$ where the new upper bound is less (Proposition 1) than the upper bound of Tsirelson [3, 4].

Moreover, for the two-qudit GHZ state (52), we have explicitly found its correlation matrix (53) and proved (Theorem 3) that, for the two-qudit GHZ state with an arbitrary qudit dimension $d\geq 2,$ the new upper bound in (50) is attained and this specifies the following *new result: for the GHZ state (52), *the maximum of the CHSH expectation over traceless qudit observables with eigenvalues in $[-1,1]$ is equal to $2\sqrt{2}$ if $d\geq 2$ is even and to $\frac{2(d-1)}{d}\sqrt{2}$ if $d>2$ is odd.

6 Appendix A

Consider the proof of Lemma 1. The operator norm of a qudit observable $n\cdot\Lambda$ is given by

[TABLE]

For a pure state $\left|\psi\right\rangle\left\langle\psi\right|,$ the normalized version of decomposition (1) reads

[TABLE]

where $r_{\psi}\in\mathbb{R}^{d^{2}-1}.$ Since tr$[\Lambda_{j}]=0,$ it follows from (A2) and (2) that

[TABLE]

Substituting (A2) into (A1) and taking into account (2), (A3), we have

[TABLE]

This proves the upper bound in (9). To prove the lower bound and the last upper bound in (9), we use (8) and relations

[TABLE]

which imply

[TABLE]

Eqs. (A4), (A6) prove the statement of Lemma 1.

7 Appendix B

Consider the proof of Theorem 1. According to (27) and (25)

[TABLE]

Also, by the upper bound in (9)

[TABLE]

Relations (B1), (B2) imply

[TABLE]

Substituting this into the maximum in the second line of (46), we derive

[TABLE]

Taking further into account that, in view of (25), (26), $\mathfrak{R}_{d}$ is a subset of the ball of radius $l_{d}$ and also contains the ball of radius $\frac{1}{\sqrt{d-1}},$ we have

[TABLE]

and

[TABLE]

Therefore, from (B4)–(B6) it follows

[TABLE]

Note also that, for each radius $R_{0}$ of the sphere in $\mathbb{R}^{d^{2}-1},$

[TABLE]

where $\lambda_{\rho_{d\times d}}\geq$ $\widetilde{\lambda}_{\rho_{d\times d}}\geq 0$** **are two greater eigenvalues, corresponding to two linear independent eigenvectors of the positive hermitian matrix $T_{\rho_{d\times d}}^{{}^{\ast}}T_{\rho_{d\times d}}$.

Substituting (B8) into (B7), we derive

[TABLE]

In view of (46), this proves the statement of Theorem 1.