Bell inequalities for two qudits

Correlator-based Bell inequalities

We combine all of the correlator operators introduced in the first specification for the state |Ψi,

Cˆ_Ψ1^(d) =X

r,t,k

Cˆ_k^(rt), (2.55)

and use its expectation value C_Ψ1^(d) = h ˆC_Ψ1^(d)i to be a single identification of the correlation properties of entangled qudits. Then it is interesting to investigate what is the maximal values of C_Ψ1^(d) that can be provided by classical correlations under local-realistic theories.

The expectation value C_Ψ1^(d) for any physical systems can be represented as:

where P (·) denotes a joint probability for getting a set of result (v1^(r), v₂^(t)) which satisfies a condition shown in the bracket. In order to have a compact form of C_Ψ1^(d) for a convenient discussion, we define the following variables:

χ11 = v₁⁽¹⁾+ v⁽¹⁾₂ + ˙d11, χ12 = −v1⁽¹⁾− v2⁽²⁾+ ˙d12, χ22 = v₁⁽²⁾+ v⁽²⁾₂ + ˙d22,

χ21 = −v1⁽²⁾− v2⁽¹⁾− 1 + ˙d²¹, (2.57)

where ˙d_rt denotes a multiple of d and χ_rt∈ {−1, 0} for r, t = 1, 2. In particular, the sum of the variables satisfies the constrain:

X2 r,t=1

χrt .

= −1. (2.58)

With the defined variables, C_Ψ1^(d) is written as

C_Ψ1^(d) = X2 r,t=1

P (χrt= 0) − P (χ^rt= −1). (2.59)

Next, we proceed to consider the extreme values of C_Ψ1^(d) under the local-realistic theories.

The all possible sets of (χ11, χ12, χ22, χ21) which fulfill the constraint of the sum of the variables are as the following: (i) three of the variables are 0 and the rest is −1. (ii) all of the variables are −1. The first class can be applied to arbitrary d, whereas the

second one is only applicable for d = 3 . Thus, we have C_Ψ1,LR^(d) = 2 for the class (i) and C_Ψ1,LR⁽³⁾ = −4 for (ii), which mean that for all the generators of the convex polytope for C_Ψ1,LR^(d) the value of C_Ψ1,LR^(d) is equal or less than 2. Therefore, in the regime governed by local-realistic theories, the value of C_Ψ1,LR^(d) is bounded by 2, i.e., C_Ψ1,LR^(d) ≤ 2.

For a generalized Bell state, the summation of all of the correlators can be calculated analytically and we have results, we realize that h ˆC_Ψ1^(d)i > CΨ1,LR^(d) . Therefore, the quantum correlations are stronger than the ones predicted by the local-realistic theories. With this fact, the derived kernel C_Ψ1^(d) can be utilized to tell quantum correlations from classical ones.

From a geometric point of view, we have examined our Bell-type inequality by the work of Masanes about tightness of Bell inequalities [115]. The result shows that the inequality is non-tight, i.e., it is not an optimal detector of non-local-realistic correlation.

The detailed proof and discussions are given in Appendix A.

We proceed to consider another Bell inequality which consists of only the sets of cor-relators C_kα^(rt) = h ˆC_kα^(rt)i for α = 0 presented in the second specification for the generalized Bell states. The kernel of our Bell inequality is of the form

C_Ψ2^′(d) = X

Using the following substitutions,

χ11 = v₁⁽¹⁾− v2⁽¹⁾+ ˙d11, χ12 = −v1⁽¹⁾+ v⁽²⁾₂ + ˙d12, χ₂₂ = v₁⁽²⁾− v2⁽²⁾+ ˙d₂₂,

χ21 = −v1⁽²⁾+ v⁽¹⁾₂ − 1 + ˙d²¹, (2.62)

C_Ψ2^′(d) is expressed by

C_Ψ2^′(d) = X2 r,t=1

P (χ_rt= 0) − P (χrt= −1), (2.63)

with the constraint P2

r,t=1χrt .

= −1. Then, by the same method as the approach for determining the extreme values of C_Ψ1,LR^(d) , one has C_Ψ2,LR^′(d) ≤ 2. Whereas, for a generalized Bell state, the expectation values of ˆC_Ψ2^′(d) are h ˆC_Ψ2^′(d)i = [csc²(_4d^π) − csc²(^3π_4d)]/(2d²) and are greater than the local-realistic upper bound for arbitrary d.

The above Bell inequality is non-tight. The proof for showing its tightness is similar to the one for C_Ψ1,LR^(d) ≤ 2, refer to Appendix A. In addition, although the values of maximal quantum violation are slightly smaller than the CGLMP inequalities [17], the total number of joint probabilities required by each of the presented correlation functions C_k0^(rt) is only 2d, which is much smaller than that in Fu’s general correlation function [109], which is about O(d²) (refer to the following discussions). Moreover, the factors for violations of Bell inequalities are larger than the ones for SLK inequalities [18] for d > 2 (see below).

Another feature of the sum of all correlators is its robustness to noise. If a generalized Bell state is suffered from white noise and turns into a mixed one, say ρ, with a noise fraction pnoise, the value of h ˆC_Ψ2^′(d)i for the state ρ becomes h ˆC_Ψ2^′(d)i^ρ = (1 − p^noise)h ˆC_Ψ2^′(d)i. If the criterion, h ˆC_Ψ2^′(d)i^ρ > 2, i.e., pnoise < 1 − 2/h ˆC_Ψ2^′(d)i, is imposed on the system, one ensures that the mixed state still exhibits quantum correlations in outcomes of measurements.

For instance, to maintain the quantum correlation for the limit of large d, the system

must have pnoise < 0.30604.

CGLMP inequality

In the second specification of the state |Ψi, we have proposed 4⌊d/2⌋ sets of correlators to describe the structure of correlation. We use a linear combination of all of these correlators to detect quantum correlations. Since each correlator is a function of α, a combination of correlator operators could be of the form:

Cˆ_Ψ2^(d) = X

α,r,t,k

f (α) ˆC_kα^(rt), (2.64)

where f (α) denotes a coefficient of combination which is function of α. If we let f (α) be

f (α) = 1 − 2α

d − 1, (2.65)

the summation of all of the correlators C_kα^(rt) becomes the kernel of the CGLMP inequality [17]:

C_Ψ2^(d) = C_Ψ2^′(d)+

⌊d/2−1⌋X

α=1

X2 r,t=1

Xd−1 k=0

(1 − 2α

d − 1)C_kα^(rt), (2.66)

where C_Ψ2^′(d) is the kernel of correlator-based Bell inequality defined by Eq. (2.61). Es-pecially, note that for d = 2, 3 the CGLMP inequalities are the correlator-based Bell inequalities. The local realistic constraint proposed by Collins et al. [17] specifies that correlations have to satisfy the condition: C_Ψ2,LR^(d) ≤ 2. On the other hand, by Eq. (2.42), quantum correlations of a generalized Bell state gives a violation of the CGLMP inequality for arbitrary high-dimensional systems. Thus, through the related discussions in the sec-ond specification for |Ψi, we realize that the CGLMP inequality is composed of correlators for correlations.

It is worth representing Eq, (2.66) in the following form: same simple form as the kernel of the CHSH inequality [13], and the linear combinations of correlators, C^(rt), are just the general correlation functions of the CHSH inequality for arbitrarily high-dimensional systems introduced by Fu [109]. Each C^(rt) provides

⌊d/2⌋ sets of correlators for identifying correlations and then consists of 2d⌊d/2⌋ joint probabilities.

SLK inequality

Following a way similar to the one for constructing the kernel of the CGLMP inequality, we take a linear combination of the operators of correlators proposed in the third specification for the generalized Bell state and give an operator of the form

Cˆ_Ψ3^(d) = X

η,µ,r,t,k

f^(rt)(η, µ) ˆC_kηµ^(rt), (2.69)

where f^(rt)(η, µ) is a coefficient of combination and depends on a local measurement chosen and a set of variables (η, µ). Let us give a concrete example to show above formulation by the following summation of correlators:

Xd−1

where zero and implies that one can always have the following relation:

Xd−1

If we choose the same measurement settings as the ones mentioned in the third speci-fication and let n⁽¹⁾_1,2 = 0, n⁽²⁾_1,2 = 1/2 (refer to Eqs. (2.35) and (2.45)), ν = ν11= 0, ν22 = 1, and ν12 = ν21 = 1/2, one obtains P_d−1

η=1g(η)^(hh)= (1 − d)/2 and g(0)^(hh) = (d − 1)/2 for h = 1, 2. Thus, one arrives at the exact forms of f^(rt)(η, µ) for r = t = h that are given by f^(hh)(0, µ) = 1/2. Furthermore, with Eq. (2.71) we have g(η)⁽¹²⁾ = g(η)⁽²¹⁾ = 0, which means that there are only two local measurement settings involved in the kernel. Thus h ˆC_Ψ3^(d)i = CΨ3^(d) is of the following explicit form:

For a generalized Bell state, the values of correlators h ˆC_kηµ^(hh)i = P (v^(h)1

= −k, v. 2^(h) = k) − P (v1^(h)

= µ − k, v. ^(h)2 = k) strictly satisfy the criteria (2.47) by the facts h ˆC_kηµ^(hh)i = 1/d for all parameters considered.

In the operator representation, C_Ψ3^(d) can be represented in the form

Cˆ_Ψ3^(d) = 1 show ˆC_Ψ3^(d) can be used to construct entanglement witness operators to detect states close to |Ψi.

One also can utilize ˆC_Ψ3^(d) to detect a state under a local unitary transformation of one of the qudits of |Ψi. For example, the state |Ψ^νi = (I ⊗ ˆSν) |Ψi, where ˆSν is a phase shift operator such that ˆSν|vi = ω^νv|vi, is detected by the following operator:

Cˆ_Ψ3ν^(d) = (I ⊗ ˆS_ν) ˆC_Ψ3^(d)(I ⊗ ˆS_ν^†)

Furthermore, if ν = 1/4 is chosen and other parameters involved in g^(rt)(η) are set as the previous ones, the expectation value h ˆC_Ψ3ν^(d) i = CΨ3ν^(d) can be represented in terms of correlators:

where in the last line the summation of local measurement setting does not include

(r, t) = (1, 1). Since g^(rt)(µ) 6= 0 for all r’s and t’s, four measurement settings are required for realizing ˆC_Ψ3ν^(d) . For generalized GHZ states, the sum of correlators is C_Ψ3^(d) = C_Ψ3ν^(d) = d − 1. Son et al. [18] have shown that local-realistic theories predict the value of CΨ3ν^(d) by

C_Ψ3ν,LR^(d) ≤ 1

4[3 cot( π

4d) − cot(π

3d)] − 1, (2.79)

for arbitrarily high-dimensional systems, which is called the SLK inequality. The SLK inequality is shown to be violated by the generalized Bell state by a factor:

d→∞lim C_Ψ3^(d)

C_Ψ3ν,LR^(d) = 3π

8 ≃ 1.1781, (2.80)

for large limit of d, which is smaller than the ones for the CGLMP [17] (≃ 1.4849) and the correlator-based (≃ 1.4410) Bell inequalities.

2.5 Correlators imbedded in entanglement witness