This paper was converted on www.awesomepapers.org from LaTeX by an anonymous user.
Want to know more? Visit the Converter page.

Activation of hidden nonlocality using local filtering operations based on CGLMP inequality

Asmita Kumari [email protected] S. N. Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700106, India
Abstract

Entanglement is necessary but not sufficient to demonstrate nonlocality as there exist local entangled states which do not violate any Bell inequality. In recent years, the activation of nonlocality (known as hidden nonlocality) by using local filtering operations has gained considerable interest. In the original proposal of Popescu [Phys. Rev. Lett. 74, 2619 (1995)] the hidden nonlocality was demonstrated for the Werner class of states in d≥5d\geq 5. In this paper, we demonstrate the hidden nonlocality for a class of mixed entangled states (convex mixture of a pure state and color noise) in an arbitrary dd-dimensional system using suitable local filtering operations. For our demonstration, we consider the quantum violation of Collins-Linden-Gisin-Masser-Popescu (CGLMP) inequality which has hitherto not been considered for this purpose. We show that when the pure state in the aforementioned mixed entangled state is a maximally entangled state, the range of the mixing parameter for revealing hidden nonlocality increases with increasing the dimension of the system. Importantly, we find that for d≥8d\geq 8, hidden non-locality can be revealed for the whole range of mixing parameter. Further, by considering another pure state, the maximally CGLMP-violating state, we demonstrate the activation of nonlocality by using the same local filtering operation.

I Introduction

Bell’s theorem Bell1964 is one of the most remarkable discoveries of quantum theory. This no-go theorem elegantly discriminates the quantum theory from the local classical theories by demonstrating that all the predictions of quantum theory cannot be reproduced by local realist models. Such a feature widely known as quantum nonlocality and is commonly demonstrated through the quantum violation of suitable Bell inequalities. Note that, the entanglement Schrödinger1935 ; epr1935 ; horodecki09 between specially separated quantum systems is necessary for demonstrating nonlocality. However, entanglement is not sufficient to generate a nonlocal quantum correlation. There exist entangled states which admit local realist models and hence do not violate any Bell inequality. It remains a challenging problem in higher dimensional systems to find the connection between entanglement and nonlocality.

In the last few decades, much effort has been put forward to activate nonlocality (commonly known as hidden nonlocality) for various kinds of entangled states admitting local models. In his pioneering paper, Popescu Popescu1995 first demonstrated the quantum violation of Bell-Clauser-Horne-Shimony-Holt (CHSH) inequality by applying local filters on the Werner states Werner1989 admitting local model. Such an activation of nonlocality by applying filtering operation on local entangled state is valid in d≥5d\geq 5. By using different class of state Gisin GISIN1996 , demonstrated a nonlocality activation protocol for a suitably chosen two-qubit local entangled state through the quantum violation of CHSH inequality HORODECKI1995340 ; clause1969 . Later, local filtering with projective measurement was generalized for positive operator value measures in Brunner2013 . An alternative route of activating nonlocality is also introduced in peres1996 ; aditi2005 ; pala2012 ; bancal2011 by using multiple copies of the entangled state while each of the copies admits a local model.

The present papers also concerns the activation of nonlocality (or revealing hidden nonlocality) of dd-dimensional quantum systems using local filtering operations. Most works towards this direction are limited to 22-dimensional local quantum systems and demonstrated through the violation of CHSH inequality Popescu1995 ; GISIN1996 ; Brunner2013 ; frank2002 ; Pramanik2019 ; ming2017 as it provides the necessary and sufficient conditions. The original work of Popescu Popescu1995 was for high-dimensional Werner state in d≥5d\geq 5. But, activation of nonlocality for other class of high-dimension local entangled states is less explored. Hirsch et. al. Brunner2013 strengthen the argument in Popescu1995 for POVMs by considering qutrit-qubit and qutrit-qutrit entangled state admitting local models. However, the generalized argument to demonstrate hidden nonlocality using suitable Bell inequalities for any arbitrary dimensional system remains unexplored. We note here that, for higher outcome systems, the well-known Collins-Gisin-Linden-Massar-Popescu (CGLMP) Collins2002 ; pin2016 inequalities provide necessary and sufficient conditions to demonstrate nonlocality. Hence, it could be an interesting line of study to explore the hidden nonlocality for an arbitrary high-dimensional system using suitable local filtering based on the quantum violation of CGLMP inequality.

In this work, we demonstrate the hidden nonlocality for arbitrary high-dimensional bipartite local entangled states using local filtering operation through the quantum violation of CGLMP inequality. We consider a class of states for our study is given by cabello05

ρd=q​|ψd⟩​⟨ψd|+(1−q)|0⟩​⟨0|⊗𝕀dd\displaystyle\rho_{d}=q|\psi_{d}\rangle\langle\psi_{d}|+(1-q)|0\rangle\langle 0|\otimes\frac{\mathbb{I}_{d}}{d} (1)

where qq (0<q≤10<q\leq 1) is the mixing parameter and |ψd⟩∈𝒞d⊗𝒞d|\psi_{d}\rangle\in\mathcal{C}^{d}\otimes\mathcal{C}^{d} is a pure state. By first taking the mixed entangled state where |ψd⟩|\psi_{d}\rangle is a maximally entangled state, we show that the range of mixing parameter showing hidden non-locality increases with increasing the dimension of the system. Importantly, we demonstrate that for d≥8d\geq 8, the hidden non-locality can be revealed for any non-zero value of the mixing parameter. We note that, except for d=2d=2, CGLMP inequality is maximally violated by a non-maximally entangled state (known as maximally CGLMP-violating state) Acin2002 . Further, we explore the hidden nonlocality when |ψd⟩|\psi_{d}\rangle in Eq. (1) is a maximally CGLMP violating state. We observe that the hidden nonlocality can be demonstrated for a comparatively wider range of mixing parameter compared to the former case while |ψd⟩|\psi_{d}\rangle is a maximally entangled state.

This paper is organized as follows. In Sec. II, we focus on preliminaries which includes a brief discussion on the quantum violations of CHSH inequalities and CGLMP inequalities. We discuss the effect of local filtering on 22-dimensional mixed entangled state in Sec. III. In Sec. IV, we generalize the application of local filtering operation for dd-dimensional bipartite quantum system using CGLMP inequality. Finally, in Sec V, we discuss our results.

II Preliminaries

Before presenting our results on activating nonlocality by using local filtering operations, let us first discuss the range of mixing parameter for the state in Eq. (1) for which the quantum violation of CHSH and CGLMP inequalities is obtained.

II.1 Quantum violation of CHSH inequality

The CHSH inequality is the simplest Bell’s inequality defined in two-party, two-measurement, and two-outcome per measurement scenario clause1969 . In the CHSH scenario, two space-like separated parties, Alice and Bob perform measurements of two dichotomic observable (A1,A2)(A_{1},A_{2}) and (B1,B2)(B_{1},B_{2}) respectively. The CHSH inequality valid for any local theory can be written as

⟨A1​B1⟩+⟨A1​B2⟩+⟨A2​B1⟩−⟨A2​B2⟩≤2\displaystyle\langle A_{1}B_{1}\rangle+\langle A_{1}B_{2}\rangle+\langle A_{2}B_{1}\rangle-\langle A_{2}B_{2}\rangle\leq 2 (2)

By considering that Alice and Bob share a two-qubit state (putting d=2d=2 in Eq. (1)) is of the form

ρ2=q​|ψ2⟩​⟨ψ2|+(1−q)|0⟩​⟨0|⊗𝕀22\displaystyle\rho_{2}=q|\psi_{2}\rangle\langle\psi_{2}|+(1-q)|0\rangle\langle 0|\otimes\frac{\mathbb{I}_{2}}{2} (3)

where |ψ2⟩=12(|00⟩+|11⟩|\psi_{2}\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle is maximally entangled two-qubit state.

For a suitable choice of observable, the maximum quantum value of CHSH inequality for the state in Eq. (3) is obtained to be 2​2​q2\sqrt{2}q. It puts restriction on qq (q>12=0.707q>\frac{1}{\sqrt{2}}=0.707) for the violation of CHSH inequality. In the range of 0<q≤0.7070<q\leq 0.707 even if the state ρ2\rho_{2} is entangled, it does not violate CHSH inequality. Following Popencu’s Popescu1995 idea, GisinGISIN1996 demonstrated that using local filtering the above range can be made narrower, i.e., for a lower value of mixing parameter CHSH inequality can be violated thereby revealing the hidden nonlocality.

II.2 Quantum violation of CGLMP inequality

In CGLMP scenario, Alice performs the measurement of two observable A1A_{1} and A2A_{2} and Bob performs B1B_{1} and B2B_{2}. Each of the measurements produces dd outcome (0,1,2,….,d−10,1,2,....,d-1). The CGLMP inequality was derived as Collins2002 ; pin2016

(Id)L\displaystyle(I_{d})_{L} =\displaystyle= |∑k=0⌊d2⌋−1(1−2​kd−1)[P(A1=B1+k)+P(B1=A2+k+1)+P(A2=B2+k)+P(B2=A1+k)\displaystyle\bigg{|}\sum^{\left\lfloor\frac{d}{2}\right\rfloor-1}_{k=0}\bigg{(}1-\frac{2k}{d-1}\bigg{)}\bigg{[}P(A_{1}=B_{1}+k)+P(B_{1}=A_{2}+k+1)+P(A_{2}=B_{2}+k)+P(B_{2}=A_{1}+k)
−(P(A1=B1−k−1)+P(B1=A2−k)+P(A2=B2−k−1)+P(B2=A1−k−1))]|≤2\displaystyle-\bigg{(}P(A_{1}=B_{1}-k-1)+P(B_{1}=A_{2}-k)+P(A_{2}=B_{2}-k-1)+P(B_{2}=A_{1}-k-1)\bigg{)}\bigg{]}\bigg{|}\leq 2

which is valid for any local theory. Here, the subscript LL denotes the local and P​(Aa=Bb+k)P(A_{a}=B_{b}+k) denotes the probabilities of the outcomes of Alice’s measurement AaA_{a} and Bob’s measurement BbB_{b} (a,b=1,2a,b=1,2) that differ by kk mod dd as

P​(Aa=Bb+k)=∑j=0d−1P​(Aa=j,Bb=j+km​o​dd)\displaystyle P(A_{a}=B_{b}+k)=\sum^{d-1}_{j=0}P(A_{a}=j,B_{b}=j+k\ \ mod\ \ d) (5)

If the shared state between Alice and Bob is

|ψd⟩=1d​∑j=0d−1|j⟩A⊗|j⟩B\displaystyle|\psi_{d}\rangle=\frac{1}{\sqrt{d}}\sum^{d-1}_{j=0}|j\rangle_{A}\otimes|j\rangle_{B} (6)

and the non-degenerate eigenvectors of measuring operators AaA_{a} (a=1,2a=1,2) of Alice and BbB_{b} (b=1,2b=1,2) of Bob of the form

|k⟩Aa=1d​∑j=0d−1exp⁡(i​2​πd​j​(k+αa))​|j⟩A\displaystyle|k\rangle_{A_{a}}=\frac{1}{\sqrt{d}}\sum^{d-1}_{j=0}\exp(i\frac{2\pi}{d}j(k+\alpha_{a}))|j\rangle_{A} (7)

and

|l⟩Bb=1d​∑j=0d−1exp⁡(i​2​πd​j​(−l+αa))​|j⟩B\displaystyle|l\rangle_{B_{b}}=\frac{1}{\sqrt{d}}\sum^{d-1}_{j=0}\exp(i\frac{2\pi}{d}j(-l+\alpha_{a}))|j\rangle_{B} (8)

respectively, with

α1=0,α2=1/2,β1=1/4a​n​dβ2=−1/4.\displaystyle\alpha_{1}=0,\ \ \ \alpha_{2}=1/2,\ \ \ \beta_{1}=1/4\ \ \ and\ \ \ \beta_{2}=-1/4. (9)

Then, the joint probability can be obtained by using quantum Fourier transformation Nielsen2010QuantumCA is given by

P​(Aa=k,Bb=l)=1d3​[12​sin2⁡[π​(k−l+αa+βb)/d]]\displaystyle P(A_{a}=k,B_{b}=l)=\frac{1}{d^{3}}\bigg{[}\frac{1}{2\sin^{2}{[\pi(k-l+\alpha_{a}+\beta_{b})/d]}}\bigg{]} (10)

By using Eq. (10) and by putting the values of αa\alpha_{a} and βb\beta_{b} the optimal quantum value of CGLMP functional is obtained as

(Id)Qo​p​t\displaystyle(I_{d})_{Q}^{opt} =\displaystyle= 4​d​∑k=0⌊d2⌋−1(1−2​kd−1)​(pk−p−(k+1))\displaystyle 4d\sum^{\left\lfloor\frac{d}{2}\right\rfloor-1}_{k=0}\bigg{(}1-\frac{2k}{d-1}\bigg{)}(p_{k}-p_{-(k+1)}) (11)

where,

pc=PQ​(A1=B1+c)=12​sin2⁡[π​(c+1/4)/d]\displaystyle p_{c}=P_{Q}(A_{1}=B_{1}+c)=\frac{1}{2\sin^{2}{[\pi(c+1/4)/d]}} (12)

where c∈(k,k+1)c\in(k,k+1) is an integer which denotes that the probability of AaA_{a} and BbB_{b} differ by constant integer cc.

Let us now consider that Alice and Bob share a mixed entangled state ρd\rho_{d} in Eq.(1) where |ψd⟩|\psi_{d}\rangle is a maximally entangled state. In such a case, the quantum value of the CGLMP functional gives

Id,Q′=q​Id,Q\displaystyle I^{\prime}_{d,Q}=qI_{d,Q} (13)

which in turn imposes a lower bound on the mixing parameter qq for the violation of CGLMP inequality is given by

q>2(Id,Q)o​p​t\displaystyle q>\frac{2}{(I_{d,Q})^{opt}} (14)

where (Id,Q)o​p​t(I_{d,Q})^{opt} is the optimal value of Id,QI_{d,Q}. It is demonstrated that for d→∞d\rightarrow\infty, nonlocality can be observed when q>0.673q>0.673 Collins2002 . Hence, within the range 0<q≤0.6730<q\leq 0.673 the state ρd\rho_{d} admits the local model as there is no violation of CGLMP inequality.

Instead, if one takes the pure state |ψd⟩|\psi_{d}\rangle in ρd\rho_{d} to be maximally CGLMP violating state the range of mixing parameter is derived fonseca18 ; roy20 as 0.637<q≤10.637<q\leq 1 for d=10d=10.

We demonstrate that for both the cases where |ψd⟩|\psi_{d}\rangle in Eq.(1) is maximally entangled state and maximally CGLMP violating state, the local filtering can reveal the hidden nonlocality within that range of mixing parameters admitting the local model. However, the ranges are different in those two cases, as derived in our work.

III Hidden non-locality in 22-dimensional mixed entangled state

Before proceeding to demonstrate our results, let us briefly discuss the activating the nonlocality using local filtering operation for a two-qubit mixed entangled state in Eq. (3). For this, we take a specific form Brunner2013 of local filtering operators for Alice and Bob are the following.

FA=ξ​|0⟩​⟨0|+|1⟩​⟨1|;FB=δ​|0⟩​⟨0|+|1⟩​⟨1|\displaystyle F_{A}=\xi|0\rangle\langle 0|+|1\rangle\langle 1|;\ \ \ \ \ \ F_{B}=\delta|0\rangle\langle 0|+|1\rangle\langle 1| (15)

Alice and Bob apply these local filtering operations on the respective subsystems of the shared state. Here, 0≤ξ,δ≤10\leq\xi,\delta\leq 1 with δ=ξq\delta=\frac{\xi}{\sqrt{q}}. After the operation of local filters FAF_{A} and FBF_{B}, the shared state ρ2\rho_{2} in Eq.(3) transforms as

ρ2F\displaystyle\rho^{F}_{2} =\displaystyle= 1N2​(FA⊗FB)​ρ2​(FA⊗FB)†\displaystyle\frac{1}{N_{2}}(F_{A}\otimes F_{B})\rho_{2}(F_{A}\otimes F_{B})^{\dagger}
=\displaystyle= 1N2[q|ψ2⟩⟨ψ2|+q2(ξ2−q)(|ψ2⟩⟨00|+|00⟩⟨ψ2|)\displaystyle\frac{1}{N_{2}}\bigg{[}q|\psi_{2}\rangle\langle\psi_{2}|+\sqrt{\frac{q}{2}}(\xi^{2}-\sqrt{q})\bigg{(}|\psi_{2}\rangle\langle 00|+|00\rangle\langle\psi_{2}|\bigg{)}
+(1−q)​ξ22|01⟩⟨01|+ξ4−2​ξ2​q3/2+q22​q|00⟩⟨00|]\displaystyle+\frac{(1-q)\xi^{2}}{2}|01\rangle\langle 01|+\frac{\xi^{4}-2\xi^{2}q^{3/2}+q^{2}}{2q}|00\rangle\langle 00|\bigg{]}

where, N2=12​(q+(1−q)​ξ2+ξ4q)N_{2}=\frac{1}{2}\left(q+(1-q)\xi^{2}+\frac{\xi^{4}}{q}\right) is the normalization constant. This filtered state ρ2F\rho^{F}_{2} violates CHSH inequality for a wider range than the unfiltered state. The CHSH violation is obtained for the range of 0.665<q≤10.665<q\leq 1 at ξ=0.79\xi=0.79 compared to the range 0.707<q≤10.707<q\leq 1 obtained for the unfiltered state. Hence, the action of the local filtering operator turns the local state into a nonlocal state within the range 0.665<q<0.7070.665<q<0.707. Similar results of revealing hidden nonlocality using local filtering operations in 22-dimensional quantum systems are discussed in GISIN1996 ; Brunner2013 ; frank2002 ; Pramanik2019 ; ming2017 . We now proceed to reveal hidden nonlocality for the dd-dimensional system through the quantum violation of CGLMP inequality which has not hitherto been explored.

IV Hidden non-locality in dd-dimensional mixed entangled state

We take the mixed entangled state as in Eq. (1) and the local filtering operators of Alice and Bob are of the form

FA=ξ​|0⟩​⟨0|+∑j=1d−1|j⟩​⟨j|;FB=δ​|0⟩​⟨0|+∑j=1d−1|j⟩​⟨j|\displaystyle F_{A}=\xi|0\rangle\langle 0|+\sum^{d-1}_{j=1}|j\rangle\langle j|;\ \ \ F_{B}=\delta|0\rangle\langle 0|+\sum^{d-1}_{j=1}|j\rangle\langle j| (17)

respectively Mat2020 . Alice and Bob apply the above local filtering operations on their respective subsystems. As mentioned, we consider two different forms of |ψd⟩∈𝒞d⊗𝒞d|\psi_{d}\rangle\in\mathcal{C}^{d}\otimes\mathcal{C}^{d} in Eq. (1), viz., the maximally entangled state and the maximally CGLMP-violating state.

IV.1 When |ψd⟩|\psi_{d}\rangle is maximally entangled state

For the mixed state ρd\rho_{d} with |ψd⟩|\psi_{d}\rangle is the maximally entangled state as in Eq. (6), the application of local filtering operation (in Eq.(17)) transforms ρd\rho_{d} as

ρdF\displaystyle\rho^{F}_{d} =\displaystyle= 1Nd​(FA⊗FB)​ρd​(FA⊗FB)†\displaystyle\frac{1}{N_{d}}(F_{A}\otimes F_{B})\rho_{d}(F_{A}\otimes F_{B})^{\dagger}
=\displaystyle= 1d​Nd[q(∑j=0d−1|j⟩|j⟩)(∑j=0d−1⟨j|⟨j|)+q(ξ2−q)[(∑j=0d−1|j⟩|j⟩)⟨00|+|00⟩(∑j=0d−1⟨j|⟨j|)]\displaystyle\frac{1}{dN_{d}}\bigg{[}q\bigg{(}\sum^{d-1}_{j=0}|j\rangle|j\rangle\bigg{)}\bigg{(}\sum^{d-1}_{j=0}\langle j|\langle j|\bigg{)}+\sqrt{q}(\xi^{2}-\sqrt{q})\bigg{[}\bigg{(}\sum^{d-1}_{j=0}|j\rangle|j\rangle\bigg{)}\langle 00|+|00\rangle\bigg{(}\sum^{d-1}_{j=0}\langle j|\langle j|\bigg{)}\bigg{]}
+(1−q)ξ2|0⟩⟨0|⊗∑j=1d−1|j⟩⟨j|+((1−q)​ξ4q+(ξ2−q)2)|00⟩⟨00|]\displaystyle+(1-q)\xi^{2}|0\rangle\langle 0|\otimes\sum^{d-1}_{j=1}|j\rangle\langle j|+\bigg{(}\frac{(1-q)\xi^{4}}{q}+(\xi^{2}-\sqrt{q})^{2}\bigg{)}|00\rangle\langle 00|\bigg{]}

where, Nd=[q+(1−q)​ξ2]​(1−1d)+ξ4q​dN_{d}=[q+(1-q)\xi^{2}](1-\frac{1}{d})+\frac{\xi^{4}}{qd} is the normalization factor. Using quantum Fourier transformation Nielsen2010QuantumCA for |k⟩Aa|k\rangle_{A_{a}} and |l⟩Bb|l\rangle_{B_{b}} in Eq.(7) and Eq.(8) for this filtered state, we derive the joint probabilities as

PQF​(Aa=k,Bb=l)\displaystyle P^{F}_{Q}(A_{a}=k,B_{b}=l) (19)
=1d3​Nd[q2​sin2⁡[π​(k−l+αa+βb)/d]\displaystyle=\frac{1}{d^{3}N_{d}}\bigg{[}\frac{q}{2\sin^{2}{[\pi(k-l+\alpha_{a}+\beta_{b})/d]}}
+q​(ξ2−q)sin⁡[π​(k−l+αa+βb)/d]+S1]\displaystyle+\frac{\sqrt{q}(\xi^{2}-\sqrt{q})}{\sin{[\pi(k-l+\alpha_{a}+\beta_{b})/d]}}+S_{1}\bigg{]}

where, S1=(1−q)​(d−1)​ξ2+(1−q)​ξ4q+(ξ2−q)2S_{1}=(1-q)(d-1)\xi^{2}+\frac{(1-q)\xi^{4}}{q}+(\xi^{2}-\sqrt{q})^{2}. The detailed derivations of Eq. (IV.1) and Eq. (19) are placed in Appendix A. Since S1S_{1} is independent of kk, all terms in the joint probabilities follow the symmetric relation given by

PQF​(Aa=k,Bb=l)=PQF​(Aa=k+c,Bb=l+c)\displaystyle P^{F}_{Q}(A_{a}=k,B_{b}=l)=P^{F}_{Q}(A_{a}=k+c,B_{b}=l+c) (20)

where cc is an integer. Using the symmetry in Eq. (20), the optimal quantum value of CGLMP functional is derived as

Id,QF\displaystyle I^{F}_{d,Q} =\displaystyle= 4​d​∑k=0⌊d2⌋−1(1−2​kd−1)​(pkF−p−(k+1)F)\displaystyle 4d\sum^{\left\lfloor\frac{d}{2}\right\rfloor-1}_{k=0}\bigg{(}1-\frac{2k}{d-1}\bigg{)}(p^{F}_{k}-p^{F}_{-(k+1)}) (21)
Refer to caption
Figure 1: Shaded region formed by Id,QF>2I^{F}_{d,Q}>2 shows the violation of CGLMP inequality for d=3,4,5,6,7,8,9,10d=3,4,5,6,7,8,9,10 and d=100d=100 after local filtering operation.
Refer to caption
Figure 2: Maximum quantum violation of CGLMP inequality for d=6,7,8,9d=6,7,8,9 and d=10d=10 are plotted with respect to ξ\xi and qq.

where,

pcF=PQF​(A1=B1+c)\displaystyle p^{F}_{c}=P^{F}_{Q}(A_{1}=B_{1}+c) =\displaystyle= 1d3​Nd[q2​sin2⁡[π​(c+1/4)/d]\displaystyle\frac{1}{d^{3}N_{d}}\bigg{[}\frac{q}{2\sin^{2}{[\pi(c+1/4)/d]}} (22)
+\displaystyle+ q​(ξ2−q)sin⁡[π​(c+1/4)/d]+S1]\displaystyle\frac{\sqrt{q}(\xi^{2}-\sqrt{q})}{\sin{[\pi(c+1/4)/d]}}+S_{1}\Bigg{]}
dd Nonlocal region before filtering Nonlocal region after filtering Region of hidden nonlocality     ξ\xi
33 0.696<q≤10.696<q\leq 1 0.664<q≤10.664<q\leq 1 0.664−0.6960.664-0.696     0.850.85
44 0.690<q≤10.690<q\leq 1 0.648<q≤10.648<q\leq 1 0.648−0.6900.648-0.690     0.810.81
55 0.687<q≤10.687<q\leq 1 0.627<q≤10.627<q\leq 1 0.627−0.6870.627-0.687     0.710.71
66 0.684<q≤10.684<q\leq 1 0.610<q≤10.610<q\leq 1 0.610−0.6840.610-0.684     0.600.60
77 0.683<q≤10.683<q\leq 1 0.524<q≤10.524<q\leq 1 0.524−0.6830.524-0.683     0.250.25
88 0.682<q≤10.682<q\leq 1 0<q≤10<q\leq 1     0−0.6820-0.682     →0\rightarrow 0
99 0.681<q≤10.681<q\leq 1 0<q≤10<q\leq 1     0−0.6810-0.681     →0\rightarrow 0
1010 0.680<q≤10.680<q\leq 1 0<q≤10<q\leq 1     0−0.6800-0.680     →0\rightarrow 0
100100 0.674<q≤10.674<q\leq 1 0<q≤10<q\leq 1     0−0.6740-0.674     →0\rightarrow 0
Table 1: Range of mixing parameter qq violating CGLMP inequality before and after local filtering operation when pure state |ψd⟩|\psi_{d}\rangle in Eq. (1) is a maximally entangled state.

Note that for ξ=δ=1\xi=\delta=1, the local filters in Eq.(17) becomes 𝕀d\mathbb{I}_{d}, and Id,QFI^{F}_{d,Q} reduce to Id,Q′I^{\prime}_{d,Q} given in Eq.(13).

It is known from Eq. (14) that in obtaining the nonlocality for the state ρd\rho_{d} in Eq. (1), the range of mixing parameters slowly increases with the increment of dimension dd. For example, for d=3d=3 the range 0.696<q≤10.696<q\leq 1 becomes 0.674<q≤10.674<q\leq 1 for d=100d=100. We demonstrate that by the action of local filtering operations the range of qq gradually increases with the increment of dd and the decrement of ξ\xi. Interestingly, the nonlocality can be revealed for any nonzero value of mixing parameter qq for d≥8d\geq 8 when ξ→0\xi\rightarrow 0. The ranges of mixing parameter qq revealing hidden nonlocality for the dimensions for d=3,4,5,6,7,8,9,10d=3,4,5,6,7,8,9,10 and d=100d=100 are given in Table 1.

In Fig. 1, we exhibit the nonlocal region i.e., the region satisfies Id,QF>2I^{F}_{d,Q}>2 (CGLMP inequality violation) for various dimensions against the mixing parameter qq and the parameter ξ\xi involved in local filter operators. We note that the violation of CGLMP inequality is not obtained for all values of ξ\xi. In Fig.2, we show that the quantum violation of CGLMP inequality after local filtering operation for d=6,7,8,9d=6,7,8,9 and d=10d=10 with respect to ξ\xi and qq. It shows that local filtering operation does not increase the maximum quantum value of CGLMP inequality but only increases the range of mixing parameter revealing the quantum violation. The range of mixing parameter increases for lower value of quantum violation of CGLMP inequality.

IV.2 When |ψd⟩|\psi_{d}\rangle is maximally violating state

Note that CGLMP inequality is maximally violated by a non-maximally entangled state (known as maximally CGLMP-violating state) if d>2d>2. We consider the same local filtering operations as the previous case. We discuss the range of mixing parameter of ρd\rho_{d} in Eq.(1) for d=3,4d=3,4 and d=5d=5 both before and after local filtering operation.

The CGLMP inequality in Eq.(II.2) for d=3d=3, reduces to

(I3)L\displaystyle(I_{3})_{L} =\displaystyle= P​(A1=B1)+P​(B1=A2+1)+P​(A2=B2)\displaystyle P(A_{1}=B_{1})+P(B_{1}=A_{2}+1)+P(A_{2}=B_{2})
+\displaystyle+ P(B2=A1)−(P(A1=B1−1)+P(B1=A2)\displaystyle P(B_{2}=A_{1})-\big{(}P(A_{1}=B_{1}-1)+P(B_{1}=A_{2})
+\displaystyle+ P(A2=B2−1)+P(B2=A1−1))≤2\displaystyle P(A_{2}=B_{2}-1)+P(B_{2}=A_{1}-1)\big{)}\leq 2

If the mixed entangled state shared between Alice and Bob is of the form

ρ3=q​|ψ3⟩​⟨ψ3|+(1−q)|0⟩​⟨0|⊗𝕀33\displaystyle\rho_{3}=q|\psi_{3}\rangle\langle\psi_{3}|+(1-q)|0\rangle\langle 0|\otimes\frac{\mathbb{I}_{3}}{3} (24)

where,

|ψ3⟩=γ1​|00⟩+γ2​|11⟩+γ3​|22⟩\displaystyle|\psi_{3}\rangle=\gamma_{1}|00\rangle+\gamma_{2}|11\rangle+\gamma_{3}|22\rangle (25)

is maximally violating state for suitable value of γ1,γ2\gamma_{1},\gamma_{2} and γ3=1−(γ12+γ22)\gamma_{3}=\sqrt{1-(\gamma_{1}^{2}+\gamma_{2}^{2})}. For this shared state and operators AaA_{a} (a=1,2a=1,2) and BbB_{b} (b=1,2b=1,2) with eigenvectors given in Eq.(7) and Eq.(8), the maximum quantum value of CGLMP inequality is given by

I3,Q=2.915​q\displaystyle I_{3,Q}=2.915q (26)

obtained at γ1=0.6169,γ2=0.4888\gamma_{1}=0.6169,\gamma_{2}=0.4888 and γ3=0.6169\gamma_{3}=0.6169 fonseca18 ; roy20 . The violation of the inequality in Eq. (IV.2) is obtained within the range 0.686<q≤10.686<q\leq 1.

To reveal hidden nonlocality outside 0.686<q≤10.686<q\leq 1, we apply local filtering operations defined by

FA=ξ​|0⟩​⟨0|+|1⟩​⟨1|+|2⟩​⟨2|\displaystyle F_{A}=\xi|0\rangle\langle 0|+|1\rangle\langle 1|+|2\rangle\langle 2| (27)
FB=δ​|0⟩​⟨0|+|1⟩​⟨1|+|2⟩​⟨2|\displaystyle F_{B}=\delta|0\rangle\langle 0|+|1\rangle\langle 1|+|2\rangle\langle 2|

on the respective local part of the shared state where δ=ξq\delta=\frac{\xi}{\sqrt{q}}. The quantum value of (I3)L(I_{3})_{L} for the filtered state

ρ3F=(FA⊗FB)​ρ3​(FA⊗FB)†T​r​[(FA⊗FB)​ρ3​(FA⊗FB)†]\displaystyle\rho^{F}_{3}=\frac{(F_{A}\otimes F_{B})\rho_{3}(F_{A}\otimes F_{B})^{\dagger}}{Tr[(F_{A}\otimes F_{B})\rho_{3}(F_{A}\otimes F_{B})^{\dagger}]} (28)

and measuring operators AaA_{a} (a=1,2a=1,2) and BbB_{b} (b=1,2b=1,2) with eigenvectors given in Eq.(7) and Eq.(8) is derived as

(I3F)Q=2.218​ξ2​q+0.696​qξ4​(0.047​q+0.333)q−0.666ξ2(q−1.)+0.619q\displaystyle(I^{F}_{3})_{Q}=\frac{2.218\xi^{2}\sqrt{q}+0.696q}{\frac{\xi^{4}(0.047q+0.333)}{q}-0.666\xi^{2}(q-1.)+0.619q} (29)
dd Non-local region before filtering Non-local region after filtering Region of hidden non-locality     ξ\xi
33 0.686<q≤10.686<q\leq 1 0.625<q≤10.625<q\leq 1 0.625−0.6860.625-0.686     0.730.73
44 0.672<q≤10.672<q\leq 1 0.585<q≤10.585<q\leq 1 0.585−0.6720.585-0.672     0.640.64
55 0.663<q≤10.663<q\leq 1 0.539<q≤10.539<q\leq 1 0.539−0.6630.539-0.663     0.540.54
Table 2: Range of mixing parameter qq violating CGLMP inequality before and after filtering operation when pure state |ψd⟩|\psi_{d}\rangle in Eq. (1) is a maximally CGLMP violating state.

We obtain the quantum violation of CGLMP ((I3F)Q>2(I^{F}_{3})_{Q}>2) for the range of 0.625<q≤10.625<q\leq 1 at ξ=0.73\xi=0.73. This range is wider than the range 0.664<q≤10.664<q\leq 1 obtained for ρd\rho_{d} with |ψd⟩|\psi_{d}\rangle as a maximally entangled state. Hence, the range of mixing parameter showing hidden nonlocality using a maximally violating state is 0.625<q<0.6860.625<q<0.686.

Further, we derive the range of the mixing parameter for d=4d=4 and 55 as given in Table 2. It is seen that the hidden nonlocality can be demonstrated for the lower value of the mixing parameter if the dimension of the system is increased. The detailed derivation of the range of qq revealing hidden nonlocality for d=4d=4 and 55 is placed in Appendix B.

V Discussion

In summary, we have demonstrated the hidden nonlocality for a class of local entangled states in an arbitrary dd-dimensional system by using local filtering operations. Popescu’s original proposal Popescu1995 of revealing hidden nonlocality based on the quantum violation of CHSH inequality was demonstrated for Werner state in d≥5d\geq 5. We have demonstrated the activation of nonlocality for a class of local entangled state in arbitrary dd dimensions using local filtering operation through the quantum violation of CGLMP inequality which has hitherto not been studied. For this, we have considered a class of local mixed entangled states which is a convex mixture of a pure entangled state and the color noise.

We considered two different cases, viz., when the pure state |ψd⟩|\psi_{d}\rangle in Eq. (1) is a maximally entangled state and the maximally CGLMP-violating state. In the former case, we demonstrated that by suitable local filtering operations, the range of mixing parameter qq revealing the nonlocality increases with the increment of the dimension of the system. Importantly, for d≥8d\geq 8, the quantum violation of CGLMP inequality is obtained for any nonzero value of the mixing parameter qq. In the latter case when the pure state |ψd⟩|\psi_{d}\rangle in Eq. (1) is the maximally CGLMP-violating state, we showed that the hidden nonlocality can be demonstrated for a comparatively wider range of qq compared to the maximally entangled state for d=3,4,5d=3,4,5. However, we have shown this case up to d=5d=5 and we conjecture that for the dimension d≥6,7d\geq 6,7 the range will be wider compared to the former case which can be further studied.

Finally, we propose a few problems as a follow-up of our study. Note that, our work is restricted to projective measurement and hence generalizing it for POVMs could be an interesting line of future research. We have considered a specific class of mixed entangled state admitting local models. It will then be interesting to study the activation of nonlocality for other class of mixed entangled states through the quantum violation of CGLMP inequality. This could also be an interesting avenue for future research which calls for further study.

Acknowledgments

Asmita Kumari acknowledges the Research Associateship from S. N. Bose National Centre for Basic Sciences, Kolkata, India.

References

Appendix A Hidden nonlocality in dd-dimensional mixed entangled state.

To investigate hidden nonlocality for the state ρd\rho_{d} in Eq.(1) with |ψd⟩|\psi_{d}\rangle as maximally entangled state, let us assume that Alice and Bob perform local filter operations defined in Eq.(17) on their respective local part of the shared state. The operation of local filters transform the shared state ρd\rho_{d} to un-normalized state given by

ρ~dF\displaystyle\tilde{\rho}^{F}_{d} =\displaystyle= (FA⊗FB)​ρd​(FA⊗FB)†\displaystyle(F_{A}\otimes F_{B})\rho_{d}(F_{A}\otimes F_{B})^{\dagger}
=\displaystyle= q​(FA⊗FB)​|ψd⟩​⟨ψd|(FA⊗FB)†+1−qd​(FA⊗FB)|0⟩​⟨0|⊗𝕀d​(FA⊗FB)†\displaystyle q(F_{A}\otimes F_{B})|\psi_{d}\rangle\langle\psi_{d}|(F_{A}\otimes F_{B})^{\dagger}+\frac{1-q}{d}(F_{A}\otimes F_{B})|0\rangle\langle 0|\otimes\mathbb{I}_{d}(F_{A}\otimes F_{B})^{\dagger}
=\displaystyle= qd​[δ​ξ​|00⟩+∑j=1d−1|j⟩A​|j⟩B]​[δ​ξ​⟨00|+∑j=1d−1⟨j|​⟨j|]+(1−qd)​ξ2​|0⟩​⟨0|⊗(δ2​|0⟩​⟨0|+∑j=1d−1|j⟩​⟨j|)\displaystyle\frac{q}{d}\bigg{[}\delta\xi|00\rangle+\sum^{d-1}_{j=1}|j\rangle_{A}|j\rangle_{B}\bigg{]}\bigg{[}\delta\xi\langle 00|+\sum^{d-1}_{j=1}\langle j|\langle j|\bigg{]}+\bigg{(}\frac{1-q}{d}\bigg{)}\xi^{2}|0\rangle\langle 0|\otimes\bigg{(}\delta^{2}|0\rangle\langle 0|+\sum^{d-1}_{j=1}|j\rangle\langle j|\bigg{)}
=\displaystyle= qd​[(δ​ξ−1)​|00⟩+∑j=0d−1|j⟩​|j⟩]​[(δ​ξ−1)​⟨00|+∑j=0d−1⟨j|​⟨j|]+(1−qd)​ξ2​|0⟩​⟨0|⊗(δ2​|0⟩​⟨0|+∑j=1d−1|j⟩​⟨j|)\displaystyle\frac{q}{d}\bigg{[}(\delta\xi-1)|00\rangle+\sum^{d-1}_{j=0}|j\rangle|j\rangle\bigg{]}\bigg{[}(\delta\xi-1)\langle 00|+\sum^{d-1}_{j=0}\langle j|\langle j|\bigg{]}+\bigg{(}\frac{1-q}{d}\bigg{)}\xi^{2}|0\rangle\langle 0|\otimes\bigg{(}\delta^{2}|0\rangle\langle 0|+\sum^{d-1}_{j=1}|j\rangle\langle j|\bigg{)}
=\displaystyle= 1d[q(δξ−1)2|00⟩⟨00|+q(δξ−1)(∑j=0d−1|j⟩|j⟩)⟨00|+q(δξ−1)|00⟩(∑j=0d−1⟨j|⟨j|)\displaystyle\frac{1}{d}\bigg{[}q(\delta\xi-1)^{2}|00\rangle\langle 00|+q(\delta\xi-1)\bigg{(}\sum^{d-1}_{j=0}|j\rangle|j\rangle\bigg{)}\langle 00|+q(\delta\xi-1)|00\rangle\bigg{(}\sum^{d-1}_{j=0}\langle j|\langle j|\bigg{)}
+q(∑j=0d−1|j⟩|j⟩)(∑j=0d−1⟨j|⟨j|)+(1−q)δ2ξ2|0⟩⟨0|⊗|0⟩⟨0|+(1−q)ξ2|0⟩⟨0|⊗∑j=1d−1|j⟩⟨j|]\displaystyle+q\bigg{(}\sum^{d-1}_{j=0}|j\rangle|j\rangle\bigg{)}\bigg{(}\sum^{d-1}_{j=0}\langle j|\langle j|\bigg{)}+(1-q)\delta^{2}\xi^{2}|0\rangle\langle 0|\otimes|0\rangle\langle 0|+(1-q)\xi^{2}|0\rangle\langle 0|\otimes\sum^{d-1}_{j=1}|j\rangle\langle j|\bigg{]}

The normalized filtered state with δ=ξq\delta=\frac{\xi}{\sqrt{q}} can be re-written as

ρdF\displaystyle\rho^{F}_{d} =\displaystyle= 1d​Nd[q(∑j=0d−1|j⟩|j⟩)(∑j=0d−1⟨j|⟨j|)+q(ξ2−q)[(∑j=0d−1|j⟩|j⟩)⟨00|+|00⟩(∑j=0d−1⟨j|⟨j|)]\displaystyle\frac{1}{dN_{d}}\bigg{[}q\bigg{(}\sum^{d-1}_{j=0}|j\rangle|j\rangle\bigg{)}\bigg{(}\sum^{d-1}_{j=0}\langle j|\langle j|\bigg{)}+\sqrt{q}(\xi^{2}-\sqrt{q})\bigg{[}\bigg{(}\sum^{d-1}_{j=0}|j\rangle|j\rangle\bigg{)}\langle 00|+|00\rangle\bigg{(}\sum^{d-1}_{j=0}\langle j|\langle j|\bigg{)}\bigg{]}
+(1−q)ξ2|0⟩⟨0|⊗∑j=1d−1|j⟩⟨j|+((1−q)​ξ4q+(ξ2−q)2)|00⟩⟨00|]\displaystyle+(1-q)\xi^{2}|0\rangle\langle 0|\otimes\sum^{d-1}_{j=1}|j\rangle\langle j|+\bigg{(}\frac{(1-q)\xi^{4}}{q}+(\xi^{2}-\sqrt{q})^{2}\bigg{)}|00\rangle\langle 00|\bigg{]}

Where, Nd=[q+(1−q)​ξ2]​(1−1d)+ξ4q​dN_{d}=[q+(1-q)\xi^{2}](1-\frac{1}{d})+\frac{\xi^{4}}{qd} is the normalization constant. The next task of Alice and Bob is to perform local measurements using operators AaA_{a} (a=1,2a=1,2) and BbB_{b} (b=1,2b=1,2) with eigenvectors |k⟩Aa|k\rangle_{A_{a}} (Eq.(7)) and |l⟩Bb|l\rangle_{B_{b}} (Eq.(8)) on their respective part of the shared filtered state ρdF\rho^{F}_{d}. The joint correlation function for this filtered state can be calculated as given below.

PQ​MF​(Aa=k,Bb=l)\displaystyle P^{F}_{QM}(A_{a}=k,B_{b}=l) =\displaystyle= T​r​[(|k⟩Aa​⟨k|⊗|l⟩Bb​⟨l|)​ρdF]\displaystyle Tr\bigg{[}(|k\rangle_{A_{a}}\langle k|\otimes|l\rangle_{B_{b}}\langle l|)\rho^{F}_{d}\bigg{]}
=\displaystyle= 1d​Nd[qTr[(|k⟩Aa⟨k|⊗|l⟩Bb⟨l|)(∑j=0d−1|j⟩|j⟩)(∑j=0d−1⟨j|⟨j|)]\displaystyle\frac{1}{dN_{d}}\bigg{[}qTr\bigg{[}(|k\rangle_{A_{a}}\langle k|\otimes|l\rangle_{B_{b}}\langle l|)\bigg{(}\sum^{d-1}_{j=0}|j\rangle|j\rangle\bigg{)}\bigg{(}\sum^{d-1}_{j=0}\langle j|\langle j|\bigg{)}\bigg{]}
+q​(ξ2−q)​T​r​[(|k⟩Aa​⟨k|⊗|l⟩Bb​⟨l|)​((∑j=0d−1|j⟩​|j⟩)​⟨00|+|00⟩​(∑j=0d−1⟨j|​⟨j|))]\displaystyle+\sqrt{q}(\xi^{2}-\sqrt{q})Tr\bigg{[}(|k\rangle_{A_{a}}\langle k|\otimes|l\rangle_{B_{b}}\langle l|)\bigg{(}\bigg{(}\sum^{d-1}_{j=0}|j\rangle|j\rangle\bigg{)}\langle 00|+|00\rangle\bigg{(}\sum^{d-1}_{j=0}\langle j|\langle j|\bigg{)}\bigg{)}\bigg{]}
+(1−q)​ξ2​T​r​[(|k⟩Aa​⟨k|⊗|l⟩Bb​⟨l|)​|0⟩​⟨0|⊗∑j=1d−1|j⟩​⟨j|]\displaystyle+(1-q)\xi^{2}Tr\bigg{[}(|k\rangle_{A_{a}}\langle k|\otimes|l\rangle_{B_{b}}\langle l|)|0\rangle\langle 0|\otimes\sum^{d-1}_{j=1}|j\rangle\langle j|\bigg{]}
+((1−q)​ξ4q+(ξ2−q)2)Tr[(|k⟩Aa⟨k|⊗|l⟩Bb⟨l|)|00⟩⟨00|]]\displaystyle+\bigg{(}\frac{(1-q)\xi^{4}}{q}+(\xi^{2}-\sqrt{q})^{2}\bigg{)}Tr\bigg{[}(|k\rangle_{A_{a}}\langle k|\otimes|l\rangle_{B_{b}}\langle l|)|00\rangle\langle 00|\bigg{]}\bigg{]}

Except qq factor, first term of PQ​MF​(Aa=k,Bb=l)P^{F}_{QM}(A_{a}=k,B_{b}=l) is same as the joint probability PQ​M​(Aa=k,Bb=l)P_{QM}(A_{a}=k,B_{b}=l) of unfiltered state. Using the quantum Fourier transformation Nielsen2010QuantumCA for |k⟩Aa|k\rangle_{A_{a}} and |l⟩Bb|l\rangle_{B_{b}} for the second term, the joint probability PQ​MF​(Aa=k,Bb=l)P^{F}_{QM}(A_{a}=k,B_{b}=l) can be re-written as

PQ​MF​(Aa=k,Bb=l)\displaystyle P^{F}_{QM}(A_{a}=k,B_{b}=l) =\displaystyle= 1d3​Nd[q|∑j=0d−1exp[i2​π​jd(k−l+αa+βb)]|2\displaystyle\frac{1}{d^{3}N_{d}}\bigg{[}q\left|\sum^{d-1}_{j=0}\exp{\bigg{[}i\frac{2\pi j}{d}(k-l+\alpha_{a}+\beta_{b})\bigg{]}}\right|^{2}
+q​(ξ2−q)​(∑j=0d−1exp⁡[−i​2​π​jd​(k−l+αa+βb)]+∑j=0d−1exp⁡[i​2​π​jd​(k−l+αa+βb)])\displaystyle+\sqrt{q}(\xi^{2}-\sqrt{q})\bigg{(}\sum^{d-1}_{j=0}\exp{\bigg{[}-i\frac{2\pi j}{d}(k-l+\alpha_{a}+\beta_{b})\bigg{]}}+\sum^{d-1}_{j=0}\exp{\bigg{[}i\frac{2\pi j}{d}(k-l+\alpha_{a}+\beta_{b})\bigg{]}}\bigg{)}
+(1−q)(d−1)ξ2+((1−q)​ξ4q+(ξ2−q)2)]\displaystyle+(1-q)(d-1)\xi^{2}+\bigg{(}\frac{(1-q)\xi^{4}}{q}+(\xi^{2}-\sqrt{q})^{2}\bigg{)}\bigg{]}
=\displaystyle= 1d3​Nd[q​sin2⁡[π​(k−l+αa+βb)]sin2⁡[π​(k−l+αa+βb)/d]+q​(ξ2−q)​sin⁡[2​π​(k−l+αa+βb)]sin⁡[π​(k−l+αa+βb)/d]\displaystyle\frac{1}{d^{3}N_{d}}\bigg{[}\frac{q\sin^{2}{[\pi(k-l+\alpha_{a}+\beta_{b})]}}{\sin^{2}{[\pi(k-l+\alpha_{a}+\beta_{b})/d]}}+\frac{\sqrt{q}(\xi^{2}-\sqrt{q})\sin{[2\pi(k-l+\alpha_{a}+\beta_{b})]}}{\sin{[\pi(k-l+\alpha_{a}+\beta_{b})/d]}}
+(1−q)(d−1)ξ2+(1−q)​ξ4q+(ξ2−q)2]\displaystyle+(1-q)(d-1)\xi^{2}+\frac{(1-q)\xi^{4}}{q}+(\xi^{2}-\sqrt{q})^{2}\bigg{]}

Last line is obtained using ∑m=0M−1exp⁡(i​m​x)=sin⁡(M​x/2)sin⁡(x/2)​exp⁡(i​x​(M−1)/2)\sum^{M-1}_{m=0}\exp{(imx)}=\frac{\sin{(Mx/2)}}{\sin{(x/2)}}\exp{(ix(M-1)/2)}. Substituting the values of αa\alpha_{a} and βb\beta_{b} in Eq(9), we have

PQ​MF​(Aa=k,Bb=l)\displaystyle P^{F}_{QM}(A_{a}=k,B_{b}=l) =\displaystyle= 1d3​Nd[q2​sin2⁡[π​(k−l+αa+βb)/d]+q​(ξ2−q)sin⁡[π​(k−l+αa+βb)/d]\displaystyle\frac{1}{d^{3}N_{d}}\bigg{[}\frac{q}{2\sin^{2}{[\pi(k-l+\alpha_{a}+\beta_{b})/d]}}+\frac{\sqrt{q}(\xi^{2}-\sqrt{q})}{\sin{[\pi(k-l+\alpha_{a}+\beta_{b})/d]}}
+(1−q)(d−1)ξ2+(1−q)​ξ4q+(ξ2−q)2]\displaystyle+(1-q)(d-1)\xi^{2}+\frac{(1-q)\xi^{4}}{q}+(\xi^{2}-\sqrt{q})^{2}\bigg{]}

Using this joint probability the optimal quantum value of CGLMP functional after local filtering operation can be obtained.

Appendix B When |ψd⟩|\psi_{d}\rangle is maximally violating state

B.1 Detailed calculation for d=4d=4

The CGLMP inequality for d=4d=4 is given by

(I4)L\displaystyle(I_{4})_{L} =\displaystyle= P​(A1=B1)+P​(B1=A2+1)+P​(A2=B2)+P​(B2=A1)\displaystyle P(A_{1}=B_{1})+P(B_{1}=A_{2}+1)+P(A_{2}=B_{2})+P(B_{2}=A_{1})
−\displaystyle- (P​(A1=B1−1)+P​(B1=A2)+P​(A2=B2−1)+P​(B2=A1−1))\displaystyle(P(A_{1}=B_{1}-1)+P(B_{1}=A_{2})+P(A_{2}=B_{2}-1)+P(B_{2}=A_{1}-1))
+\displaystyle+ 13(P(A1=B1+1)+P(B1=A2+2)+P(A2=B2+1)+P(B2=A1+1)\displaystyle\frac{1}{3}\bigg{(}P(A_{1}=B_{1}+1)+P(B_{1}=A_{2}+2)+P(A_{2}=B_{2}+1)+P(B_{2}=A_{1}+1)
−\displaystyle- (P(A1=B1−2)P(B1=A2−1)+P(A2=B2−2)+P(B2=A1−2)))≤2.\displaystyle(P(A_{1}=B_{1}-2)P(B_{1}=A_{2}-1)+P(A_{2}=B_{2}-2)+P(B_{2}=A_{1}-2))\bigg{)}\leq 2.

We take shared entangled states of the form

ρ4=q​|ψ4⟩​⟨ψ4|+(1−q)|0⟩​⟨0|⊗𝕀44\displaystyle\rho_{4}=q|\psi_{4}\rangle\langle\psi_{4}|+(1-q)|0\rangle\langle 0|\otimes\frac{\mathbb{I}_{4}}{4} (35)

where

|ψ4⟩=γ1​|00⟩+γ2​|11⟩+γ3​|22⟩+γ4​|33⟩\displaystyle|\psi_{4}\rangle=\gamma_{1}|00\rangle+\gamma_{2}|11\rangle+\gamma_{3}|22\rangle+\gamma_{4}|33\rangle (36)

is the maximally violating state for suitable value of γ1,γ2,γ3\gamma_{1},\gamma_{2},\gamma_{3} and γ4=1−(γ12+γ22+γ32)\gamma_{4}=\sqrt{1-(\gamma_{1}^{2}+\gamma_{2}^{2}+\gamma_{3}^{2})}. For this shared state and operators AaA_{a} (a=1,2a=1,2) and BbB_{b} (b=1,2b=1,2) with eigenvectors given in Eq.(7) and Eq.(8), we derive the maximum quantum value of CGLMP inequality (I4)L(I_{4})_{L} as

I4,Q=2.972​q\displaystyle I_{4,Q}=2.972q (37)

at γ1=0.5686,γ2=0.4204,γ3=0.4204\gamma_{1}=0.5686,\gamma_{2}=0.4204,\gamma_{3}=0.4204, and γ4=0.5686\gamma_{4}=0.5686 fonseca18 ; roy20 . The state |ψ4⟩|\psi_{4}\rangle is a maximally violating state with these values of coefficient. We get (I4)L>2(I_{4})_{L}>2 when 0.672<q≤10.672<q\leq 1 and hence the state ρ4\rho_{4} is local in the range of 0<q≤0.6720<q\leq 0.672.

In order to reveal hidden nonlocality in 0<q≤0.6720<q\leq 0.672, let us apply a local filtering operation defined by

FA=ξ​|0⟩​⟨0|+|1⟩​⟨1|+|2⟩​⟨2|+|3⟩​⟨3|\displaystyle F_{A}=\xi|0\rangle\langle 0|+|1\rangle\langle 1|+|2\rangle\langle 2|+|3\rangle\langle 3| (38)
FB=δ​|0⟩​⟨0|+|1⟩​⟨1|+|2⟩​⟨2|+|3⟩​⟨3|\displaystyle F_{B}=\delta|0\rangle\langle 0|+|1\rangle\langle 1|+|2\rangle\langle 2|+|3\rangle\langle 3|

on their respective part of the shared state. The quantum value of (I4)L(I_{4})_{L} for the filtered state

ρ4F=(FA⊗FB)​ρ4​(FA⊗FB)†T​r​[(FA⊗FB)​ρ4​(FA⊗FB)†]\displaystyle\rho^{F}_{4}=\frac{(F_{A}\otimes F_{B})\rho_{4}(F_{A}\otimes F_{B})^{\dagger}}{Tr[(F_{A}\otimes F_{B})\rho_{4}(F_{A}\otimes F_{B})^{\dagger}]} (39)

and the measurement settings in Eq.(7) and Eq.(8) is derived as

(I4F)Q=−2.562​ξ2​q3/2−1.401​q2−0.902​q2+ξ4​(−0.097​q−0.333)−ξ2​q​(1−q)\displaystyle(I^{F}_{4})_{Q}=\frac{-2.562\xi^{2}q^{3/2}-1.401q^{2}}{-0.902q^{2}+\xi^{4}(-0.097q-0.333)-\xi^{2}q(1-q)} (40)

In this case we obtain quantum violation of CGLMP inequality for the range of 0.585<q≤10.585<q\leq 1 at ξ=0.64\xi=0.64, which is wider in comparison to 0.648<q≤10.648<q\leq 1 obtained for ρ4\rho_{4} with |ψ4⟩|\psi_{4}\rangle as a maximally entangled state.

B.2 Detailed calculation for d=5d=5

Substituting d=5d=5 in Eq.(II.2), the CGLMP inequality reduces to

(I5)L\displaystyle(I_{5})_{L} =\displaystyle= P​(A1=B1)+P​(B1=A2+1)+P​(A2=B2)+P​(B2=A1)\displaystyle P(A_{1}=B_{1})+P(B_{1}=A_{2}+1)+P(A_{2}=B_{2})+P(B_{2}=A_{1})
−\displaystyle- [P​(A1=B1−1)+P​(B1=A2)+P​(A2=B2−1)+P​(B2=A1−1)]\displaystyle[P(A_{1}=B_{1}-1)+P(B_{1}=A_{2})+P(A_{2}=B_{2}-1)+P(B_{2}=A_{1}-1)]
+\displaystyle+ 12(P(A1=B1+1)+P(B1=A2+2)+P(A2=B2+1)+P(B2=A1+1)\displaystyle\frac{1}{2}\bigg{(}P(A_{1}=B_{1}+1)+P(B_{1}=A_{2}+2)+P(A_{2}=B_{2}+1)+P(B_{2}=A_{1}+1)
−\displaystyle- [P(A1=B1−2)P(B1=A2−1)+P(A2=B2−2)+P(B2=A1−2)])≤2\displaystyle[P(A_{1}=B_{1}-2)P(B_{1}=A_{2}-1)+P(A_{2}=B_{2}-2)+P(B_{2}=A_{1}-2)]\bigg{)}\leq 2

We take the mixed entangled state

ρ5=q​|ψ5⟩​⟨ψ5|+(1−q)|0⟩​⟨0|​𝕀55\displaystyle\rho_{5}=q|\psi_{5}\rangle\langle\psi_{5}|+(1-q)|0\rangle\langle 0|\frac{\mathbb{I}_{5}}{5} (42)

where,

|ψ5⟩=γ1​|00⟩+γ2​|11⟩+γ3​|22⟩+γ4​|33⟩+γ5​|44⟩\displaystyle|\psi_{5}\rangle=\gamma_{1}|00\rangle+\gamma_{2}|11\rangle+\gamma_{3}|22\rangle+\gamma_{4}|33\rangle+\gamma_{5}|44\rangle (43)

is maximally violating state for suitable value of γ1,γ2,γ3,γ4\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4} and γ5=1−(γ12+γ22+γ32+γ42)\gamma_{5}=\sqrt{1-(\gamma_{1}^{2}+\gamma_{2}^{2}+\gamma_{3}^{2}+\gamma_{4}^{2})}. For the shared state |ψ5⟩|\psi_{5}\rangle and operators AaA_{a} (a=1,2a=1,2) and BbB_{b} (b=1,2b=1,2) with eigenvectors given in Eq.(7) and Eq.(8), the maximum quantum value of CGLMP inequality (I5)L(I_{5})_{L} is

I5,Q=3.0158​q\displaystyle I_{5,Q}=3.0158q (44)

obtained at γ1=0.5368,γ2=0.3859,γ3=0.3548\gamma_{1}=0.5368,\gamma_{2}=0.3859,\gamma_{3}=0.3548, and γ4=0.3859\gamma_{4}=0.3859 and γ5=0.5368\gamma_{5}=0.5368 fonseca18 ; roy20 . In this case, nonlocality is observed in the range of 0.663<q≤10.663<q\leq 1, and hence ρ5\rho_{5} is local in the range of 0<q≤0.6630<q\leq 0.663.

In order to reveal hidden nonlocality in 0<q≤0.6630<q\leq 0.663, we consider the local filtering operations defined by

FA=ξ​|0⟩​⟨0|+|1⟩​⟨1|+|2⟩​⟨2|+|3⟩​⟨3|+|4⟩​⟨4|\displaystyle F_{A}=\xi|0\rangle\langle 0|+|1\rangle\langle 1|+|2\rangle\langle 2|+|3\rangle\langle 3|+|4\rangle\langle 4| (45)
FB=δ​|0⟩​⟨0|+|1⟩​⟨1|+|2⟩​⟨2|+|3⟩​⟨3|+|4⟩​⟨4|\displaystyle F_{B}=\delta|0\rangle\langle 0|+|1\rangle\langle 1|+|2\rangle\langle 2|+|3\rangle\langle 3|+|4\rangle\langle 4|

on their respective part of the shared state. The quantum value of (I5)L(I_{5})_{L} for the filtered state

ρ5F=(FA⊗FB)​ρ5​(FA⊗FB)†T​r​[(FA⊗FB)​ρ5​(FA⊗FB)†]\displaystyle\rho^{F}_{5}=\frac{(F_{A}\otimes F_{B})\rho_{5}(F_{A}\otimes F_{B})^{\dagger}}{Tr[(F_{A}\otimes F_{B})\rho_{5}(F_{A}\otimes F_{B})^{\dagger}]} (46)

and the measuring operators AaA_{a} (a=1,2a=1,2) and BbB_{b} (b=1,2b=1,2) with eigenvectors given in Eq.(7) and Eq.(8) is given by

(I5F)Q=−2.172​ξ2​q3/2−1.597​q2−0.889​q2+ξ4​(−0.110​q−0.25)−ξ2​q​(1−q)\displaystyle(I^{F}_{5})_{Q}=\frac{-2.172\xi^{2}q^{3/2}-1.597q^{2}}{-0.889q^{2}+\xi^{4}(-0.110q-0.25)-\xi^{2}q(1-q)} (47)

We obtain the quantum violation of CGLMP inequality ((I5F)Q>2(I^{F}_{5})_{Q}>2) for the range of 0.539<q≤10.539<q\leq 1 at ξ=0.54\xi=0.54. This range is wider than the range 0.627<q≤10.627<q\leq 1 obtained for ρd\rho_{d} with |ψd⟩|\psi_{d}\rangle as a maximally entangled state. Hence, the range of the mixing parameter showing hidden nonlocality using a maximally violating state is 0.539<q<0.6270.539<q<0.627.