1 Introduction↩︎

Violation of Bell-like inequalities represents a crucial test of the character of the fundamental laws of nature, as it is incompatible with local-realism and in particular with local hidden-variables theories. The most popular variant of these inequalities is the CHSH version [1]. Given a two-qubit system, where Alice (Bob) can measure two observables, \(A, A'\) (\(B, B'\)) which can take values \(\{+1, -1\}\), the distribution of measurements are compatible with local realism if and only if \(|\langle {\cal O}_{\rm Bell}\rangle|\leq 2\) with \({\cal O}_{\rm Bell}=AB+AB'+A'B-A'B'\). As it is well known, quantum mechanics can violate this CHSH inequality for certain entangled states. More precisely, if the state is pure there is always a choice of \(A, A', B, B'\) which violates CHSH [2]. If it is a mixture, that is not guaranteed [3].

Of course, given a quantum state, the amount of potential Bell violation depends on the choice of the four \(A, A', B, B'\) observables. It was shown in ref.[3] that for a generic qubit-qubit state, \(\rho\), expressed as \[\begin{align} \rho=\frac{1}{4}\left( \mathbb{1}_2\otimes \mathbb{1}_2 +\sum_i(B_i^+ \sigma_i\otimes \mathbb{1}_2 + B_i^- \mathbb{1}_2\otimes \sigma_i) + \sum_{ij}C_{ij} \sigma_i\otimes \sigma_j \right) \label{generalrho2qu} \end{align}\tag{1}\] (with \(B_i^\pm, C_{ij}\) real coefficients), the maximum value of \(|\langle {\cal O}_{\rm Bell}\rangle|\) is given by \[\begin{align} \max_{{A},{A'},{B},{B'}}\left|\langle {\cal O}_{\rm Bell}\rangle\right|=\max_{{A},{A'},{B},{B'}}\left|\langle AB\rangle+\langle AB'\rangle+\langle A'B\rangle-\langle A'B'\rangle\right|=2\sqrt{\kappa_1+\kappa_2}\;, \label{CHSH-abC95max} \end{align}\tag{2}\] where \(\kappa_1,\kappa_2\) are the largest eigenvalues of \(C^T C\). The authors provided also the explicit choice of \(A, A', B, B'\) leading to this maximum value. In this way one can easily check whether a (qubit-qubit) state generates probability distributions incompatible with local realism. Complementary results are derived when only fixing spectral properties of the density matrix \(\rho\), such as the eigenvalues or the purity. For these cases, the maximum value \(|\langle {\cal O}_{\rm Bell}\rangle|\) has been analytically obtained [4]–[6].

Beyond the qubit-qubit case things get much more involved. As a matter of fact, there is not even a general description of the region (polytope) of probability distribution of \(A, A', B, B'\) which is compatible with local realism. The celebrated CGLMP inequalities [7] represent some facets of such polytope, but in general they do not provide a complete description of it. On the other hand, for qubit-qudit states it was shown by Pironio [8] that all the facets defining the “classical" polytope are given by CHSH-type inequalities. Nevertheless, this does not solve the problem of determining the maximum Bell violation for a given \(\rho\), and thus whether the probabilities of physical observables of the system can be described by a classical (local-realistic) theory. In particular, Eq.(2 ) cannot be extrapolated to higher dimension. Then, in principle one should explore all possibilities for the \(A, A', B, B'\) observables, a very expensive computational task, as it involves a large number of parameters, which grows rapidly as the dimension of the qudit increases. On the other hand, \({\cal H}_2 \otimes {\cal H}_d\) states are of high physical interest. E.g. in the context of high-energy physics, it would be interesting to show that systems like \({\rm top}-W\)-boson, produced at the LHC, can exhibit the same Bell non-locality as systems of two spin-1/2 particles (like top pairs). In other context, qubit-qudit systems also play an important role in quantum information processing [9]–[12].

In this paper we address the task of evaluating the maximal Bell violation for a generic qubit-qudit system, obtaining easily computable expressions. This also allows us to examine other issues, for example the possibility of enhancing the Bell violation by embedding the qudit Hilbert space in one of larger dimension.

In section 2 we present our approach to the problem and the general result for maximal Bell-violation in qubit-qudit systems, which is the main result of this paper. In section 3 we apply our general results to address a number of issues. In section 3.1 we give simple lower and upper bounds on \(\langle {\cal O}_{\rm Bell} \rangle_{\rm max}\), which respectively represent sufficient and necessary conditions for violation of local realism. In section 3.2 we use our general results to obtain a bound on the degree of purity required for nonlocality. Section 3.3 is devoted to study the statistics of nonlocality in random density matrices. In section 3.4 we examine the possibility of improving the amount of Bell-violation by embedding Bob’s Hilbert space in one of larger dimension. In section 3.5 we discuss how the results derived for projective measurements are generalized to positive operator-valued measurements (POVMs). In section 3.6 we illustrate our results by studying a family of density matrices in a qubit-qutrit system. Finally, in section 4 we summarize our results and conclusions. We include also an appendix to present the derivations of section 3.5 in higher detail.

2 Main result↩︎

Let us consider a general qubit-qudit system, with Hilbert space \({\cal H}_2 \otimes {\cal H}_d\). Any \(2d\times 2d\) density matrix in this space can be unambiguously expressed as \[\rho=\frac{1}{2}\left[\mathbb{1}_2\otimes\beta_0+\sigma_1\otimes\beta_1+\sigma_2\otimes\beta_2+\sigma_3\otimes\beta_3\right], \label{rho}\tag{3}\] where \(\sigma_i\) are the standard Pauli matrices and \(\{\beta_0,\beta_i\}\) are \(d\times d\) Hermitian matrices. In particular, the \(\beta_0\) matrix coincides with Bob’s reduced density matrix, \(\require{physics} \rho_B=\Tr_A \rho=\beta_0\), and therefore verifies \(\require{physics} \Tr \beta_0 = 1\).¹ Besides, the \(\beta-\)matrices must lead to a positive semidefinite \(\rho\) matrix; other than that they are arbitrary. The previous expression is a kind of Schmidt decomposition of the \({\cal H}_2 \otimes {\cal H}_d\) density matrix.

Following the result obtained by Pironio [8], we know that all the facets defining the polytope of Local Hidden Variables (LHV) are given by CHSH-type inequalities over the probability distributions of the system. In other words, all the “tight" Bell-like inequalities (those whose violation is a sufficient and necessary condition to violate local realism), involving two observables for both Alice and Bob, can be written as CHSH-type inequalities, \(|\langle O_{\rm Bell}\rangle|\leq 2\), with \[{\cal O}_{\rm Bell} = A \otimes (B+B') + A'\otimes (B-B') \;.\] Here \(A\) and \(A'\) (\(B\) and \(B'\)) are \(2\times2\) (\(d\times d\)) linear Hermitian observables with eigenvalues \(\{+1, -1\}\) (\(\{+1, -1\}\) with some degeneracy). Its expectation value is given by \[\require{physics} \langle {\cal O}_{\rm Bell} \rangle = \Tr \left( \rho {\cal O}_{\rm Bell} \right) = \frac{1}{2}\sum_{i=1}^3 \left\{\Tr(\sigma_i A) \, \Tr \left( \beta_i (B+B') \right) \; + \; \Tr(\sigma_i A') \, \Tr\left(\beta_i (B-B')\right) \right\}. \label{60OB62}\tag{4}\] As it is well known, for local realistic theories \(\langle {\cal O}_{\rm Bell} \rangle \leq 2\), while in quantum theories it can reach \(2 \sqrt{2}\). Our goal is to find its maximal value: \[\langle {\cal O}_{\rm Bell} \rangle_{\rm max} = \max_{A,A',B,B'} \langle {\cal O}_{\rm Bell} \rangle\;. \label{maxAB}\tag{5}\] For the qubit-qubit case, the cross-terms \(\sigma_i\otimes\beta_i\) in (3 ) can be expressed as \(\frac{1}{2}C_{ij}\sigma_i\otimes\sigma_j\), where \(C_{ij}\) is a real matrix. Then, it was shown in [3] that the maximum value of \(\langle {\cal O}_{\rm Bell} \rangle\) is given by \(2\sqrt{\kappa_1+\kappa_2}\), where \(\kappa_1,\kappa_2\) are the largest eigenvalues of the \(C^TC\) matrix. Such nice result cannot be extrapolated to the qubit-qudit case for various reasons. In particular, the freedom for the choice of the \(B, B'\) observables is dramatically greater, as they live in the space of \(d\times d\) Hermitian matrices. Finally, for \(d>2\) there is an increasing number of CHSH inequalities to be examined, corresponding to the distribution of \(+1\)s and \(-1\)s of the \(B\) and \(B'\) eigenvalues.

For this analysis it is convenient to define \(\vec{r}_A\), \(\vec{r}_{A'}\) and \(\vec{r}_B\), \(\vec{r}_{B'}\) vectors as \[\require{physics} \begin{align} \vec{r}_A & = & \left( \Tr (\sigma_1 A ), \Tr (\sigma_2 A ),\Tr (\sigma_3 A ) \right ), \nonumber \\ \vec{r}_B & = & \left(\Tr (\beta_1 B ), \Tr (\beta_2 B ),\Tr (\beta_3 B )\right ) \label{rArB} \end{align}\tag{6}\] and similar expressions for \(\vec{r}_{A'}\) and \(\vec{r}_{B'}\). Note that these are real vectors from the Hermiticity of the involved matrices. Then Eq.(4 ) reads \[\langle {\cal O}_{\rm Bell} \rangle = \frac{1}{2} \vec{r}_A (\vec{r}_B+ \vec{r}_{B^\prime}) + \frac{1}{2} \vec{r}_{A^\prime} (\vec{r}_B- \vec{r}_{B^\prime})\;. \label{Bell95r}\tag{7}\] Incidentally, this expression is explicitly invariant under simultaneous rotations in the 3-spaces of the \(\sigma_i\), \(\beta_i\) matrices, which is in turn a consequence of the invariance of \(\rho\), Eq.(3 ), under that operation. Now notice that \(\frac{1}{2}\vec{r}_A\), \(\frac{1}{2}\vec{r}_{A'}\) are unit vectors. This comes from \(A, A'\) having eigenvalues \(\{+1,-1\}\) and thus vanishing trace², so they can be expressed as \[\require{physics} \begin{align} A= \sum\frac{1}{2}\Tr (\sigma_i A )\sigma_i= \frac{1}{2}\vec{r}_A \vec{\sigma} \label{A} \end{align}\tag{8}\] and similarly for \(A'\). Now, since \(\require{physics} \Tr A^2 ={\rm Tr}\, {A^\prime}^2 =\Tr \mathbb{1}_2=2\), we get \(\|\vec{r}_{A}\|^2=\|\vec{r}_{A^\prime}\|^2=4\). Apart from that, the \(\vec{r}_{A}, \vec{r}_{A^\prime}\) vectors are arbitrary since, for any choice of them, the corresponding \(A, A'\) observables are given by (8 ). Therefore, for a given pair \((B, B^\prime)\), the optimal choice of \((A, A^\prime)\) is \(\vec{r}_A \parallel ( \vec{r}_B + \vec{r}_{B^\prime})\) and \(\vec{r}_{A^\prime} \parallel ( \vec{r}_B - \vec{r}_{B^\prime})\), so that \[\langle {\cal O}_{\rm Bell} \rangle_{\rm max} =\, \max_{B,B'} \Bigl\{ \| \vec{r}_B + \vec{r}_{B^\prime}\| + \| \vec{r}_B- \vec{r}_{B^\prime}\| \Bigl\}. \label{maxBB}\tag{9}\] As expected, this expression is also invariant under 3-rotations. Unfortunately, the \(\beta-\)matrices do not follow the Pauli algebra, so a similar argument cannot be done for the \(\vec{r}_{B}, \vec{r}_{B^\prime}\) vectors, in particular they do not have a fixed normalization. As already mentioned, this is part of the extra intricacy of the qubit-qudit case compared to the qubit-qubit one. In order to solve (9 ) it is useful the following lemma:

• Lemma I

  Let \(\vec{v} ,\vec{w}\) be two arbitrary vectors in a plane. Consider a simultaneous rotation of angle \(\varphi\) of both vectors and call the new vectors \(\vec{v} (\varphi) , \vec{w} (\varphi)\), so that \(\vec{v}= \vec{v} (0) ,\;\vec{w}= \vec{w}(0)\). Then the following identity takes place: \[\bigg( \| \vec{v}+ \vec{w}\| + \| \vec{v} - \vec{w}\| \bigg)^2 = 4 \max_{ \varphi } \bigg\{v_1(\varphi)^2 +w_2(\varphi)^2\bigg\}, \label{lemma}\tag{10}\] where the subscripts 1, 2 denote the components of the vectors. This equation can be easily checked by choosing an initial reference frame for which the longest vector, say \(\vec{v}\), has \(v_2 = 0\). Then the expression within curl brackets reads \(v_1^2 \cos^2 \varphi + (w_1 \sin \varphi +w_2 \cos \varphi)^2\), which is maximal at \[\varphi = \displaystyle{ \frac{1}{2} \arctan \frac{2 w_1 w_2 }{v_1^2 + w_2^2 - w_1^2 } }\] and the value at the maximum coincides with the l.h.s. of (10 ). Clearly, the lemma holds when we allow for rotations in 3-space, \(\vec{v} ,\vec{w}\rightarrow {\cal R}\vec{v} ,{\cal R}\vec{w}\), i.e. \[\Bigl( \| \vec{v}+ \vec{w}\| + \| \vec{v} - \vec{w}\| \Bigl)^2 = 4\, \max_{ {\cal R} } \Bigl\{({\cal R}\vec{v})_1^2 +({\cal R}\vec{w})_2^2 \Bigl\} \label{lemma3D}\tag{11}\] where \({\cal R}\) is an arbitrary rotation in 3D, characterized by the three Euler angles. This becomes obvious by taking into account that the r.h.s. of this equation reaches its maximum when the two vectors have vanishing third component, \(({\cal R}\vec{v})_3 =({\cal R}\vec{w})_3=0\), so that the problem reduces to the above rotation in the plane.

Applying the previous lemma, Eq.(11 ), to the Bell expectation value, Eq.(9 ) we get \[\begin{align} \langle {\cal O}_{\rm Bell} \rangle_{\rm max} = 2 \, \max_{B,B',\, {\cal R}} \sqrt{ \left| ({\cal R} \vec{r}_B )_1 \right |^2 + \left| ({\cal R} \vec{r}_{B^\prime} )_2 \right|^2}\;. \label{r95squared} \end{align}\tag{12}\] From the definition of \(\vec{r}_B\), Eq.(6 ), \(\require{physics} ({\cal R} \vec{r}_B )_i= \Tr [({\cal R}\vec{\beta})_i\cdot B]\), and an analogous expression for \(({\cal R} \vec{r}_{B'} )_i\). Hence \[\require{physics} \begin{align} \langle {\cal O}_{\rm Bell} \rangle_{\rm max} =2\, \max_{B,B',\, {\cal R}} \sqrt{ \left| \Tr [({\cal R} \vec{\beta})_1 \cdot B] \right |^2 + \left| \Tr [({\cal R} \vec{\beta})_2 \cdot B'] \right |^2}\;. \label{r95squared2} \end{align}\tag{13}\] Now we take into account the following: for a generic Hermitian matrix, \(M\), with eigenvalues \(\lambda_i\), and an arbitrary Hermitian, involutory matrix \(B\) (i.e. \(B^2=\mathbb{1}_d\)), it happens that \(\require{physics} \max_{B} \Tr[M\cdot B]=\sum_i |\lambda_i|\). ³ This maximum is achieved when \(B\) is aligned with \(M\), i.e. they are diagonalized by the same unitary matrix, and the signs of the \(B\) eigenvalues (\(1\) or \(-1\)) are chosen equal to the signs of the corresponding \(\lambda_i\). This is precisely our case, since \(B, B'\) are Hermitian involutory matrices, but other than that arbitrary. Here we allow \(B, B'\) to be \(\pm \mathbb{1}_d\), which is the optimal choice when all \(\lambda_i\) have the same sign (we comment below on the meaning of this case). Consequently, the maximum value of \(\langle {\cal O}_{\rm Bell} \rangle\) is given by \[\boxed{ \begin{array}{rcl} &&\\[-3mm] \langle {\cal O}_{\rm Bell} \rangle_{\rm max} &=& \displaystyle{2\, \max_{{\cal R}} \sqrt{\| ({\cal R} \vec{\beta})_1 \|_1 ^2 + \| ({\cal R}\vec{\beta})_2 \|_1 ^2 } \, } \\[3mm] & = & \displaystyle{2\, \max_{{\cal R}} \left[ \left(\sum_{i=1}^{d} |\lambda_i^{(1)}({\cal R})| \right) ^2 + \left( \sum_{i=1}^{d} |\lambda_i^{(2)}({\cal R})| \right) ^2 \right]^{1/2}}\; \\[-3mm] \label{generalresult} \end{array} }\tag{14}\] where \(\lambda_i^{(1,2)} ({\cal R})\) stand for the eigenvalues of the \(SO(3)\)-rotated \(\beta\) matrices, \(({\cal R}\vec{\beta})_1, ({\cal R} \vec{\beta})_2\). This is the main result of our paper. It follows that a necessary and sufficient condition for nonlocality is that the above expression (14 ) is larger than \(2\).

Let us briefly comment on some aspects of this result.

• Note that in principle one should consider all the possibilities for the distribution of \(1\)s and \(-1\)s of \(B\) and \(B'\) eigenvalues. Each possibility corresponds to a different CHSH inequality, which represents \(\sim d^2\) CHSH inequalities. However the above result automatically selects the optimal choice for the \(1s\) and \(-1s\) of the \(B,B'\) observables; in other words, the CHSH inequality which gives the maximal violation of the given density matrix.

• When all the \(\lambda_i^{(1)}({\cal R})\) and/or \(\lambda_i^{(2)}({\cal R})\) at the maximum of Eq.(14 ) have the same sign, this entails setting either \(B=\pm\mathbb{1}_d\) and/or \(B'=\pm\mathbb{1}_d\), which is known to give no violation for CHSH-type inequalities [13].

• To see the computational advantage of the above expression, note the following. In this procedure, given a \(\rho\) matrix, once it is expressed in the form (3 ), we have to perform a (usually numerical) maximization of Eq.(14 ). This implies to scan the three Euler angles of the \({\cal R}\) rotation, which is a very cheap computation. It should be compared with the \(4+2d(d-1)\) parameters for each CHSH inequality in the initial expression (5 ). Even in the simplest qubit-qutrit case this represents 16 parameters.

• The \(A, A', B, B'\) observables that realize the maximum Bell-violation are straightforward to obtain. Once we have determined the matrices \(({\cal R}\vec{\beta})_1\), \(({\cal R}\vec{\beta})_2\) that maximize (14 ) we simply set \[\begin{align} B=U_1 D_1 U_1^\dagger,\;\;\; B'=U_2 D_2 U_2^\dagger, \end{align}\] where \(U_{1,2}\) are the diagonalizing unitary matrices, i.e. \(U_a ({\cal R}\vec{\beta})_a U_a^\dagger={\rm diag}\,(\lambda_i^{(a)})\), and \(D_a={\rm diag}({\rm sign}\,[\lambda_i^{(a)}])\). The corresponding \(A,A'\) observables are given by Eq.(8 ), with \(\vec{r_A}\), \(\vec{r_{A'}}\) the unit vectors aligned along \(( \vec{r}_B + \vec{r}_{B^\prime})\),  \(( \vec{r}_B - \vec{r}_{B^\prime})\) (see discussion after Eq.(8 )), and \[\require{physics} \begin{align} \vec{r}_B = \left(\Tr \left[({\cal R}\vec{\beta})_1 B \right], \Tr\left[({\cal R}\vec{\beta})_2 B\right], \Tr\left[({\cal R}\vec{\beta})_3 B\right] \right ) \end{align}\]

  and similarly for \(B'\).

• Let us finally see that expression (14 ) is consistent with the qubit-qubit result (2 ) obtained in ref.[3].

  In such scenario, comparing expressions (1 ) and (3 ) for \(\rho\), the \(\beta\) matrices read \(\beta_0=\frac{1}{2}(\mathbb{1}_2+\sum_i B_i^-\sigma_i)\) and \(\beta_i=\frac{1}{2}(B_i^+\mathbb{1}_2+\sum_j C_{ij}\sigma_j)\). On the other hand, assuming that the state violates a CHSH inequality, the corresponding observables \(A, A', B, B'\) must have eigenvalues \(\{+1,-1\}\). (The other inequivalent possibility, namely one or more observables proportional to the identity, leads to no CHSH-violation [13].) In that case, the terms involving \(B_i^\pm\) are irrelevant as they cancel in \(\require{physics} \Tr{\rho {\cal O}_{\rm Bell}}\), Eq. (4 ). Now, the (real) matrix \(C\) can be diagonalized by two orthogonal transformations, \({\cal R}_A, {\cal R}_B\in O(3)\): \[\begin{align} C= {\cal R}_A \Sigma {\cal R}_B^T,\;\;\;\Sigma={\rm diag}\{\mu_1, \mu_2, \mu_3\}, \end{align}\] ordered as \(\mu_1\geq\mu_2\geq\mu_3\geq0\). This is equivalent to perform appropriate changes of basis in the Alice and Bob Hilbert spaces. Hence, in this new basis \[\begin{align} \rho=\frac{1}{2}\left( \mathbb{1}_2\otimes \mathbb{1}_2 + \sum_{i} \sigma_i\otimes \beta_i \;+\;\cdots \right)\;, \end{align}\] where the dots denote terms which are irrelevant for the previous reasons, and \(\beta_i=\frac{1}{2}\mu_i\sigma_i\) up to a sign⁴. Now, from Eq.(14 ) we have to maximize \(\| ({\cal R}\vec{\beta})_1 \|_1 ^2 + \| ({\cal R}\vec{\beta})_2 \|_1 ^2\) where \({\cal R}\) is an arbitrary \(SO(3)\) rotation. Using the fact that the eigenvalues of \(\vec{v}\cdot\vec{\sigma}\) are \(\pm \|\vec{v}\|\) we get \[\begin{align} \| ({\cal R} \vec{\beta})_i \|_1 ^2= \sum_j \mu_j^2 |{\cal R}_{ij}|^2\;, \end{align}\] so the maximum value of \(\| ({\cal R} \vec{\beta})_1 \|_1 ^2 + \| ({\cal R}\vec{\beta})_{2} \|_1 ^2\) occurs for \({\cal R}_{13}={\cal R}_{23}=0\). Then \[\begin{align} \max_{{\cal R}}\{\| ({\cal R} \vec{\beta})_1 \|_1 ^2 + \| ({\cal R}\vec{\beta})_2 \|_1 ^2\}= \sum_{i=1,2}\sum_j \mu_j^2 |{\cal R}_{ij}|^2=\mu_1^2 +\mu_2^2 \end{align}\] (independent of \({\cal R}_{ij}\)). Plugging this result in (14 ) we recover Eq.(2 ).

3 Applications↩︎

3.1 Lower and upper bounds on Bell violation↩︎

From the general expression for the maximal Bell violation, Eq.(14 ), we can easily extract simple lower and upper bounds on \(\langle {\cal O}_{\rm Bell} \rangle_{\rm max}\). They are useful as a quick exploration of nonlocality.

The lower bound comes from simply taking \({\cal R}=\mathbb{1}_3\). In other words, once the density matrix has been expressed as in Eq.(3 ), we can assure that \[\begin{align} \langle {\cal O}_{\rm Bell} \rangle_{\rm max}\;\geq\;2\; \sqrt{\| \beta_1 \|_1 ^2 + \| \beta_2 \|_1 ^2 } \; =\; 2\, \left[ \left(\sum_{i=1}^{d} |\lambda_i^{(1)}| \right) ^2 + \left( \sum_{i=1}^{d} |\lambda_i^{(2)}| \right) ^2 \right]^{1/2}, \label{lowerbound} \end{align}\tag{15}\] where in this case \(\lambda_i^{(1,2)}\) stand for the eigenvalues of the initial \(\beta_1, \beta_2\) matrices (no rotation applied). More precisely, \(\beta_1, \beta_2\) correspond to the beta matrices with larger trace-norm. We find that usually this bound is close to the actual value and becomes closer for higher qudit dimension.

In order to get an upper bound on \(\langle {\cal O}_{\rm Bell} \rangle_{\rm max}\) from Eq.(14 ), we use the inequality \(\norm{\,\cdot\,}_1^2\leq d\, \norm{\,\cdot\,}_2^2\) involving the 1 and 2-norm over \(d\times d\) matrices ⁵, so that \[\sum_{a=1}^2 \| ({\cal R} \vec{\beta})_a \|_1^2 \;\leq\; \sum_{a=1}^3 \| ({\cal R} \vec{\beta})_a \|_1^2 \;\leq\; d\,\sum_{a=1}^3 \| ({\cal R} \vec{\beta})_a \|_2^2 \;=\; d\,\sum_{a=1}^3 \| \beta_a \|_2^2. \label{chain}\tag{16}\] The equality comes from the fact that the last expression is invariant under \(O(3)\) rotations, so we can take the initial \(\beta\)-matrices to evaluate the upper bound. Hence, \[\langle {\cal O}_{\rm Bell} \rangle_{\rm max}\;\leq\; 2\sqrt{d} \left[ \sum_{a=1}^3 \left(\sum_{i=1}^{d} |\lambda_i^{(a)}|^2 \right) \, \right]^{1/2}.\] In summary, \[2 \left[ \sum_{a=1}^2 \left(\sum_{i=1}^{d} |\lambda_i^{(a)}| \right) ^2 \, \right]^{1/2}\;\leq\; \langle {\cal O}_{\rm Bell} \rangle_{\rm max}\;\leq\;2\sqrt{d} \left[ \sum_{a=1}^3 \left(\sum_{i=1}^{d} |\lambda_i^{(a)}|^2 \right) \, \right]^{1/2} \label{bounds}\tag{17}\] where, in all the equations of this subsection, (15 )-(17 ), \(\lambda_i^{(a)}\) stand for the eigenvalues of the unrotated \(\beta_a\) matrices in Eq.(3 ).

3.2 Purity and non-locality↩︎

Purity is simply defined as \(\require{physics} \Tr \rho^2\), and provides a simple measure of how pure a state is [14], in particular \(\require{physics} \Tr \rho^2=1\) (\(\require{physics} \Tr \rho^2=1/D\)) for a pure (completely mixed) state, where \(D\) is the dimension of the Hilbert space. Clearly, a maximally-mixed bipartite state is separable (and thus local in the Bell sense), since the density matrix is proportional to the identity. In contrast, a pure state, if entangled, is non-local as well [2], [15]. Then, one can wonder what is the minimum degree of purity a state should have in order to present non-local correlations.

This question can be adressed for qubit-qudit systems, using our previous results. From the expression of \(\rho\) in Eq.(3 ) it follows that \[\require{physics} \begin{align} \Tr \rho^2=\frac{1}{2}\left[\Tr \beta_0^2\;+ \sum_{i=1,2,3}\Tr \beta_i^2\right]\;. \end{align}\] Here \(\require{physics} \beta_0=\rho_B=\Tr_A \rho\) is the reduced density matrix in Bob’s side, note that \(\require{physics} \Tr \rho_B^2\geq1/d\). On the other hand, \(\require{physics} \sum_{i=1,2,3}\Tr \beta_i^2=\; d\,\sum_{i=1}^3 \| \beta_i \|_2^2\). Then, using relations (16 ) and (14 ) we get \[\require{physics} \begin{align} \Tr \rho^2\geq\frac{1}{2}\left[\Tr \rho_B^2 + \frac{1}{4d}\langle {\cal O}_{\rm Bell} \rangle_{\rm max}^2\right]\geq \frac{1}{2d}\left[1 + \frac{1}{4}\langle {\cal O}_{\rm Bell} \rangle_{\rm max}^2\right]\;, \end{align}\] and thus a bound on the purity of \(\rho\) in order to have Bell violation (\(\langle {\cal O}_{\rm Bell} \rangle_{\rm max}>2\)):

\[\require{physics} \begin{align} \text{Bell-Violation}\;\Rightarrow\;\Tr \rho^2>\frac{1}{2}\left[\Tr \rho_B^2+\frac{1}{d}\right] \geq \frac{1}{d}\;. \end{align}\] Notice here that generically \(\require{physics} \Tr \rho^2\geq1/(2d)\).

3.3 Non-locality of \(2\times d\) Random Matrices↩︎

As it is well known, for a bipartite quantum system most pure states are entangled and thus violate Bell-inequalities. The situation is more complicated to analyze (and actually unknown) for general mixed states.

Our main result on Bell-violation, Eq.(14 ), allows to explore the statistics of non-locality in different sets of (generically mixed) qubit-qudit states, in particular in random mixed states. As a matter of fact, there is not a unique way to define a "random mixed state", since it depends on the the probability measure used (the following results are however similar in all cases). A well motivated one is the Random Bures set. It can be generated from the Ginibre ensemble and the random unitary matrix ensemble – distributed according to the Haar measure –. Let \(G\) and \(U\) be two \(D \times D\) random matrices of these ensembles. Then \[\require{physics} \rho_{\rm Bures} = \frac{ (\mathbb{1} + U) G G^\dagger (\mathbb{1} + U^\dagger)}{\Tr [ (\mathbb{1} + U) G G^\dagger (\mathbb{1} + U^\dagger)]}\] provides a random \(D \times D\) Bures state [16].

In order to study the statistics of non-locality of these states for qubit-qudit (where \(D=2d\)) we have generated sets of \(\sim 10,000\) density matrices for different \(d\) values, and evaluated the optimal Bell operator value for each state. The corresponding probability distributions of \(\langle {\cal O}_{\rm Bell} \rangle_{\rm max}\) are displayed in Fig 1. They are very close to normal distributions with means and variances given in table 1. Clearly, the higher \(d\), the smaller the mean value and the variance of \(\langle {\cal O}_{\rm Bell} \rangle_{\rm max}\); and thus the less likely to have non-local correlations. More precisely, for \(d=2, 4\) the \(p-\)value of non-locality is \(9.1\%, 1.4\%\) respectively. For \(d\geq 10\), such probability drops below \(10^{-4}\). In the \(d\rightarrow\infty\) limit we find numerically that \(\require{physics} \|\Tr \beta_{1,2,3}\|_1 \sim 0.494\). Then \(\langle {\cal O}_{\rm Bell} \rangle_{\rm max}^{d\rightarrow\infty} \sim 2 \sqrt{2} \; 0.494 \sim 1.397\). This result can be obtained in a few seconds using a standard laptop, by just generating a small set of random matrices of \({\cal O}(1000)\) dimension.

3.4 Embeddings↩︎

Having a recipe for the optimal Bell-violation for \(2 \times d\) systems allows us to address the following question: if we embed Bob’s Hilbert space in one of larger dimension, is it possible to improve the amount of Bell-violation by a suitable choice of the new (higher dimensional) \(\tilde{B}, \tilde{B}'\) observables? One may even think of the possibility of starting with an (entangled) state which does not violate Bell inequalities, but it does it in the extended Hilbert space. As we are about to see, the answer is negative for both questions.

Let us start with a generic state in a \({\cal H}_2 \otimes {\cal H}_{d_1}\) Hilbert space, characterized by a density matrix

\[\rho=\frac{1}{2}\left[\mathbb{1}_2\otimes\beta_0+\sigma_1\otimes\beta_1+\sigma_2\otimes\beta_2+\sigma_3\otimes\beta_3\right],\] where \(\{\beta_0,\beta_i\}\) are \(d_1\times d_1\) matrices. Now let us consider Bob’s Hilbert space as part of a higher dimensional one, \({\cal H}_{d_1}\subset {\cal H}_{d_2}\) with \(d_2>d_1\). Thus we embed the above state in the new Hilbert space by considering the \(\{\beta_0,\beta_i\}\) matrices as the upper-left block of a block diagonal \(d_2\times d_2\) matrix: \[ \beta_0\rightarrow\tilde{\beta}_0= \left(\begin{array}{cc} \beta_0 & \mathbb{0}_{d_1\times (d_2-d_1)} \\ \mathbb{0}_{(d_2-d_1)\times d_1} & \mathbb{0}_{(d_2-d_1)\times (d_2-d_1)} \end{array}\right),\;\;\beta_i\rightarrow\tilde{\beta}_i= \left(\begin{array}{cc} \beta_i & \mathbb{0}_{d_1\times (d_2-d_1)} \\ \mathbb{0}_{(d_2-d_1)\times d_1} & \mathbb{0}_{(d_2-d_1)\times (d_2-d_1)} \end{array}\right), \label{betas}\tag{18}\] where the \(\mathbb{0}\) matrices have all entries vanishing. In terms of the higher-dimension observables and \(\beta-\)matrices, Eq.(13 ) reads: \[\require{physics} \begin{align} \langle {\cal O}_{\rm Bell} \rangle_{\rm max} =2\, \max_{\tilde{B},\tilde{B}',\, {\cal R}} \sqrt{ \left| \Tr [({\cal R} \vec{\tilde{\beta}})_1 \cdot \tilde{B}] \right |^2 + \left| \Tr [({\cal R} \vec{\tilde{\beta}})_2 \cdot \tilde{B}'] \right |^2}\;. \label{r95squared3} \end{align}\tag{19}\] Note that the \({\cal R} \vec{\tilde{\beta}}\) matrices are block-diagonal, with the same texture of zeroes as matrices (18 ). Hence, they have the same \(d_1\) eigenvalues as \({\cal R} \vec{\beta}\) plus \(d_2-d_1\) zeroes. Therefore, for a given rotation \({\cal R}\), the optimal choice for \(\tilde{B}, \tilde{B}'\) in Eq.(19 ) yields the same result as the optimal choice in the \({\cal H}_2 \otimes {\cal H}_{d_1}\) system, namely \[\require{physics} \begin{align} \Tr \left[ ({\cal R} \vec{\tilde{\beta}})_1 \cdot \tilde{B}\right] = \sum_{i=1}^{{d_1}} |\lambda_i^{(1)}({\cal R})| \;,\;\;\; \Tr \left[({\cal R} \vec{\tilde{\beta}})_2 \cdot \tilde{B}'\right] = \sum_{i=1}^{{d_1}} |\lambda_i^{(2)}({\cal R})|\;, \label{traces} \end{align}\tag{20}\] where \(\lambda_i^{(1,2)}({\cal R})\) stand for the eigenvalues of the \(({\cal R} \vec{\beta})_1, ({\cal R} \vec{\beta})_2\) matrices. Hence we recover the same result as for the initial system, Eq.(14 ). Note that in this case there are many choices of \(\tilde{B}, \tilde{B}'\) which yield the same result (20 ).

3.5 Extension to POVMs↩︎

Let us briefly comment on how the results we have derived for projection-valued measures (PVMs) generalize to the case of positive operator-valued measures (POVMs).

A POVM is mathematically characterized by a set of positive semi-definite Hermitian operators, \(\{F_k\}_k\), fulfilling \(\sum_{k} F_k=\mathbb{1}\). The general CHSH inequality for a bipartite scenario, Alice and Bob, with binary POVM inputs, labeled by \(a,a'\) for Alice and \(b,b'\) for Bob, and binary outcomes, \({\pm 1}\), is given by \[E (a,b)+ E (a,b')+E (a',b)-E (a',b') \leq 2. \label{CHSHExp}\tag{21}\] Here, \(E(a,b)= P(1,1|a,b)-P(1,-1|a,b)-P(-1,1|a,b)+P(-1,-1|a,b)\) and \(P(i,j|a,b)\) is the probability of inputs \(a\) and \(b\) to output outcomes \(i\) and \(j\) respectively. This holds similarly with all of the other terms. It is worth-mentioning that in this work we are considering a bipartite scenario with 2 inputs each and with 2 and \(d\) outcomes respectively. Therefore, in the CHSH inequality in hand, we have collapsed the \(d\) different outcomes of the POVMs acting on Bob’s Hilbert space into the binary outcomes \(j=-1,1\).

Inequality 21 can be written as well in terms of Hermitian operators (see Appendix A for the details of the derivation) as: \[|\langle {\cal O}_{\rm Bell}\rangle|\leq 2, \quad \text{with}\quad {\cal O}_{\rm Bell}=AB+AB'+A'B-A'B'. \label{CHSHPOVMs}\tag{22}\] The only difference with respect to the PVM case comes from the spectral decomposition of the operators \(A,A',B,B'\). In particular, \(A\) and \(A'\) (\(B\) and \(B'\)) are \(2\times2\) (\(d\times d\)) Hermitian operators with eigenvalues contained in the interval \([-1,1]\).

If we further consider the operators \(A\) and \(A'\) to be traceless, then \[A=\frac{\cal S}{2}\vec{r}_A \vec{\sigma}, \quad A'=\frac{{\cal S}'}{2}\vec{r}_{A'} \vec{\sigma},\] with \({\cal S},{\cal S}'\in (0,1]\) and \(\|\vec{r}_{A}\|^2=\|\vec{r}_{A^\prime}\|^2=4\). Then, the derivation of the optimal value \(\langle {\cal O}_{\rm Bell} \rangle_{\rm max}\) can be reproduced in exactly the same way as in section 2. In particular, one gets \[\langle {\cal O}_{\rm Bell} \rangle_{\rm max} =\, \max_{{\cal S},{\cal S}',B,B'} \Bigl\{ {\cal S}\| \vec{r}_B + \vec{r}_{B^\prime}\| + {\cal S}' \| \vec{r}_B- \vec{r}_{B^\prime}\| \Bigl\},\] instead of the previous Eq.(9 ). The optimization over \({\cal S}\) and \({\cal S}'\) is trivial, namely \({\cal S}={\cal S}'=1\). Now, by using Lemma I, this becomes \[\require{physics} \begin{align} \langle {\cal O}_{\rm Bell} \rangle_{\rm max} =2\, \max_{B,B',\, {\cal R}} \sqrt{ \left| \Tr [({\cal R} \vec{\beta})_1 \cdot B] \right |^2 + \left| \Tr [({\cal R} \vec{\beta})_2 \cdot B'] \right |^2}, \end{align}\] where now \(B\) and \(B'\) are Hermitian operators with eigenvalues \(\mu\in[-1,1]\). However, this feature does not increase the complexity of the maximization. Actually, it holds that \(\require{physics} \max_{B\in \text{POVM}} \Tr[M\cdot B]=\norm{M}_1\). I.e., the maximum value over POVMs is attained when considering PVMs, and for that choice we still get \[\langle {\cal O}_{\rm Bell} \rangle_{\rm max}= 2\, \max_{{\cal R}} \sqrt{\| ({\cal R} \vec{\beta})_1 \|_1 ^2 + \| ({\cal R}\vec{\beta})_2 \|_1 ^2 }\;.\]

The last result is in agreement with the one obtained for the qubit-qubit case [17]. On the contrary, if we let \(A\) and \(A'\) have a non-vanishing trace, we expect that the previous value is no longer optimal. This is not surprising, as the same happens even in the qubit-qubit case, for which the presence of a non-vanishing trace leads to larger optimal values [18].

3.6 A qubit-qutrit case study↩︎

To illustrate the use of the general result on the maximal Bell-violation (14 ) let us consider an example in the context of the qubit-qutrit system. As it is well known, for mixed states entanglement does not necessarily leads to violation of quantum realism (i.e. Bell-violation). A popular example of this fact in the qubit-qubit case are the Werner states, \(\rho=\frac{1}{4}( \mathbb{1}_2\otimes \mathbb{1}_2 - \eta \sum_{i} \sigma_i\otimes \sigma_i)\), which for \(1/3<\eta\leq 1/\sqrt{2}\) are entangled but do not violate any CHSH inequality. For a qubit-qutrit system we can perform a similar analysis, using both our result (14 ) and the fact that in this case the Peres-Horodecki [19], [20] criterion, i.e. the existence of some negative eigenvalue of the partially transposed matrix \(\rho^{T_2}\), provides a necessary and sufficient condition for entanglement. For the sake of concreteness, let us consider the qubit-qutrit state \[\begin{align} \rho=x|\psi_1\rangle\langle\psi_1| \;+\; y|\psi_2\rangle\langle\psi_2| \;+\; z|\psi_3\rangle\langle\psi_3|, \end{align}\] where \(0\leq (x,y,z)\leq 1\) with \(x+y+z=1\), and (in an obvious notation) \[\begin{align} |\psi_1\rangle=\frac{1}{\sqrt{2}}\left(|00\rangle+|11\rangle\right),\;\; |\psi_2\rangle=\frac{1}{\sqrt{2}}\left(|01\rangle+|12\rangle\right),\;\; |\psi_3\rangle=\frac{1}{\sqrt{2}}\left(|02\rangle+|10\rangle\right). \end{align}\] Explicitly, \[\rho = \frac{1}{2} \left( \! \begin{array}{cccccc} x & 0 & 0 & 0 & x & 0 \\ 0 & y & 0 & 0 & 0 & y \\ 0 & 0 & 1-x-y &1-x-y & 0 & 0 \\ 0 & 0 & 1-x-y & 1-x-y & 0 & 0 \\ x & 0 & 0 & 0 & x & 0 \\ 0 & y & 0 & 0 & 0 & y \end{array} \! \right) \,. \label{rho322x3}\tag{23}\] The physical region, where \(\rho\) is positive definite, corresponds to \(x+y\leq 1\) (triangle in Fig.2). Using the Peres-Horodecki criterion, it is easy to check that \(\rho\) is entangled for any value of \(x,y\), except for \(x=y=1/3\). Fig.2, left panel, shows the value of the logarithmic negativity \(E=\log_2\left(\|\rho^{T_2}\|_1\right)\) in the \(x-y\) plane. The logarithmic negativity, which provides a sound measurement of entanglement [21], [22], is greater than 0 in the whole physical region except at that particular point.

For the analysis of the Bell-violation, we first express \(\rho\) in the form (3 ), which amounts to the following \(\beta-\)matrices: \[\beta_0 =\frac{1}{2} \left( \begin{array}{ccc} 1-y&0&0\\ 0&x+y&0\\ 0&0&1-x \end{array} \right),\; \beta_1 =\frac{1}{2} \left( \begin{array}{ccc} 0&x&-x-y+1\\ x&0&y\\ -x-y+1&y&0 \end{array} \right),\; \nonumber \label{betas01}\tag{24}\] \[\beta_2 =\frac{1}{2} \left( \begin{array}{ccc} 0&i x&i (x+y-1) \\ -i x&0&i y\\ -i (x+y-1)&-i y&0 \end{array} \right),\; \beta_3 =\frac{1}{2} \left( \begin{array}{ccc} 2 x+y-1&0&0\\0&y-x&0\\ 0&0&-x-2 y+1 \end{array} \right).\; \label{betas23}\tag{25}\] Plugging these expressions in Eq.(14 ) and performing a simple numerical optimization we can obtain the maximal Bell-violation in the \(x-y\) plane, which is shown in Fig.2, right panel. Similarly to the Werner qubit-qubit states, there is a sizeable region in which the state is entangled but \(|\langle O_{\rm Bell}\rangle|\leq 2\), so local realism is not violated.

4 Summary and conclusions↩︎

We have considered the violation of Bell-like inequalities in the context of a qubit-qudit system with arbitrary dimension. These inequalities represent a crucial test of local realism, i.e the possibility that the outputs of physical measurements on the system could be reproduced by a (classical) theory of hidden variables. The violation of such inequalities requires that the state is entangled, but (for mixed states) the opposite is not necessarily true. In previous literature [8] it was shown that for these systems, the “classical" polytope, i.e. the region of probability distribution of observables \(A, A', B, B'\) which is compatible with local realism, is bounded by CHSH-type [1] inequalities. However, given a state \(\rho\), this does not solve the problem of determining the maximum Bell violation and thus whether the system can be described by a classical (local-realistic) theory. The usual recipes for a qubit-qubit system [3] cannot be applied beyond the lowest dimensionality. Hence, in principle one should explore all possibilities for the \(A, A', B, B'\) observables involved in a CHSH inequality, an expensive computational task, which entails to optimize \(\sim 2d^2\) parameters and thus increases quickly with the dimension of the qudit.

In this paper we have addressed the task of evaluating the maximal Bell violation for a generic qubit-qudit system, obtaining easily computable expressions. Our central result, given in Eq.(14 ), generically amounts to a simple optimization in three angles, independently of the qudit dimension, and it automatically selects the strongest CHSH inequality among all the possible ones. Moreover, this result also holds when considering a larger number of observables acting on the qudit space, since for that scenario the “classical” polytope is still bounded by CHSH-type inequalities [8]. We also give lower (and usually close to optimal) bounds and upper bounds on the Bell-violation, which can be immediately computed. We have applied our general results to address a number of issues. Namely, we have obtained a bound on the degree of purity required in a system to exhibit nonlocality. Also, we have studied the statistics of nonlocality in random density matrices. Besides, we have shown that it is not possible to improve the amount of Bell-violation by embedding Bob’s Hilbert space in one of larger dimension. Likewise, we discuss how the previous results, which were derived for projective measurements, are generalized to positive operator-valued measurements (POVMs). Finally, as an example of the use of our results we have considered a 2-parameter family of density matrices in the context of a qubit-qutrit system and determined the region of such parameter space in which the state is entangled and the region where local realism is violated, showing that both are correlated but the former is broader than the latter.

The results presented here can be used for any qubit-qudit system, independently of its physical nature; e.g. in the analysis of non-local correlations in top-\(W\) [23], [24] or photon-\(Z\) production [25], [26] at the LHC, or even (in the large-\(d\) limit) hybrid discrete-continuous systems such as a cavity atom-light system[27], [28].

Acknowledgements↩︎

We are grateful to J.A. Aguilar-Saavedra for useful discussions. The authors acknowledge the support of the Spanish Agencia Estatal de Investigacion through the grants “IFT Centro de Excelencia Severo Ochoa CEX2020-001007-S" and PID2022-142545NB-C22 funded by MCIN/AEI/10.13039/501100011033 and by ERDF. The work of A.B. is supported through the FPI grant PRE2020-095867 funded by MCIN/AEI/10.13039/501100011033.

Appendix A↩︎

In this Appendix we show the equivalence between the CHSH inequality as a function of probabilities 21 , with that of the CHSH inequality as an expectation value of an Hermitian operator 22 .

Let us recall the CHSH inequality in terms of the probabilities for the 2 inputs for both Alice (\(a,a'\)) and Bob (\(b,b'\)), giving outcomes \({\pm 1}\): \[E (a,b)+ E (a,b')+E (a',b)-E (a',b') \leq 2. \label{CHSHExp2}\tag{26}\] Here, \(E(a,b)= P(1,1|a,b)-P(1,-1|a,b)-P(-1,1|a,b)+P(-1,-1|a,b)\) and \(P(i,j|a,b)\) is the probability of inputs \(a\) and \(b\) to output outcomes \(i\) and \(j\) respectively. This holds similarly with all of the other terms. Each probability is computed by the Born rule via \[\require{physics} P(i,j|a,b)=\Tr{\rho\left( F_i^{a}\otimes F_j^{b}\right)},\quad i,j=-1,1, \label{BornRule}\tag{27}\] where \(F_i^{a}\) and \(F_j^{b}\) are respectively the positive semi-definite Hermitian operators (not necessarily projectors) associated with outcomes \(i\) and \(j\) of POVMs labeled by inputs \(a\) and \(b\), that in addition verify the identities \[\begin{align} F_1^{a}+F_{-1}^{a}=\mathbb{1}, \;\;\;F_1^{b}+F_{-1}^{b}=\mathbb{1}. \end{align}\] Plugging the expression of the probabilities 27 into Eq. 26 yields the inequality \[\langle A\otimes B\rangle+\langle A\otimes B'\rangle+\langle A'\otimes B\rangle -\langle A'\otimes B'\rangle\leq 2, \label{CHSHPOVMs2}\tag{28}\] with \(\require{physics} \langle A\otimes B\rangle=\Tr{\rho \left(A\otimes B\right)}\) and in which we have defined \[A=F_1^{a}-F_{-1}^{a}=\mathbb{1}-2F_{-1}^{a},\quad B=F_1^{b}-F_{-1}^{b}=\mathbb{1}-2F_{-1}^{b}, \label{identity}\tag{29}\] where in the second step of each equality we have used (29 ). Analogous relations define \(A'\) and \(B'\). Since all \(F^{a}_{i}\) and \(F^{b}_{j}\) are positive semi-definite operators satisfying (29 ), their eigenvalues are all non-negative and bounded from above by 1. In consequence, from the definition of the operators \(A\) and \(B\) (and, in the same fashion, for \(A'\) and \(B'\)), we deduce that their eigenvalues have to be contained in the \([-1,1]\) interval. For instance, when dealing with PVMs the \(F\) operators are projectors and the eigenvalues of \(A,A',B,B'\) are exactly \(\pm 1\).

It is worth-mentioning that in this work we are considering a bipartite scenario, Alice and Bob, with 2 inputs each and with 2 and \(d\) outcomes respectively. Therefore, in the CHSH inequality in hand, we have collapsed the \(d\) different outcomes of the POVMs acting on Bob’s Hilbert space into the binary outcomes \(j=-1,1\). This has no impact on the above reasoning, since this condition only implies that \(F^{b}_j=\sum_{k_j} F^{b}_{k_j}\), where the index \(k_j\) in the sum ranges over all the collapsed outcomes corresponding to \(j\).

Author Contributions↩︎

All authors have contributed equally

References↩︎