Introduction

One of the most intriguing and fascinating aspects of quantum mechanics is quantum nonlocality. Nonlocality is expressed by correlations of entangled quantum states of quantum objects. As was proven theoretically by John Stuart Bell1 nonlocality is an immanent aspect of quantum reality which was confirmed for elementary systems in series of correlation experiments in different scales of energy-momentum as well as different conditions for space-time localization2,3,4,5,6,7,8,9. Recently, the possibility of testing Bell-type inequality violation under extremal conditions arising in a decay of the Higgs boson has been put forward10,11,12,13,14,15,16,17. Indeed, in the most interesting case of Higgs decay into two gauge bosons (ZZ or WW) we have possibility to measure quantum spin states of unstable vector bosons. This is possible by means of registration of their decay products (leptons) which are detected in polarization states determined by kinematics. The decay process of weak bosons takes place in the extremely short time \(10^{-25}\) s and at last one of decaying bosons is in a virtual state, out of the mass shell. Therefore, investigation of possible correlations of entangled vector bosons can provide a test of quantum mechanics under completely new conditions.

It should be stressed here that analysis of spin correlations in the relativistic regime is hindered by problems with the definition of a proper spin observable for a relativistic particle. Various operators have been proposed in the literature, see, e.g.18,19,20,21,22,23 and references therein. In our paper we used the so called Newton-Wigner spin operator corresponding to a spin of a particle in its rest frame, we justified this choice in, e.g.11. We will also shortly comment this point in Conclusions.

Einstein–Podolsky–Rosen (EPR) type correlation experiment with relativistic vector bosons in the simplest scalar state was considered in Ref.24 for the first time. The possibility of observing the violation of Clauser–Horn–Shimony–Holt (CHSH) and Collins–Gisin–Linden–Massar–Popescu (CGLMP) inequalities by WW bosons arising in the actual decay \(H \rightarrow WW\) for the first time was analyzed in Ref.10 (notice, that the first proper discussion of spin correlations in this decay was given in Ref.25). In Ref.11 the possibility of violation of CHSH, Mermin and CGLMP inequalities by a boson-antiboson system in the most general scalar state (introduced in this context in Ref.26) was discussed. In Ref.12 the entanglement and violation of CGLMP inequality by a pair of Z bosons arising in the decay \(H\rightarrow ZZ\) was considered. The authors of Ref.12 assumed the Standard Model interaction of H with the daughter Z bosons and showed that ZZ state produced in such a process is highly entanglement. In our paper16 we also discussed entanglement and the violation of CGLMP inequality by a ZZ pair arising in the decay \(H\rightarrow ZZ\) but assuming anomalous (beyond the Standard Model) structure of the vertex describing interaction of a Higgs particle with two daughter bosons. It can be shown (compare e.g.27,28) that the amplitude corresponding to the most general Lorentz-invariant, CPT conserving coupling of the pseudoscalar/scalar particle X with two vector bosons \(V_1\), \(V_2\) in the decay

$$\begin{aligned} X\rightarrow V_1 V_2, \end{aligned}$$
(1)

depends on three parameters, denoted in Eq. (2) by \(v_1\), \(v_2\) and \(v_3\). The Standard Model Higgs decay \(H\rightarrow ZZ\) corresponds to \(v_1=1\), \(v_2=v_3=0\) while \(v_3\not =0\) implies the possibility of CP violation and a pseudoscalar component of H. In16 we considered anomalous coupling but limited ourselves to the case of a scalar Higgs (\(v_1\not =0\), \(v_2\not =0\), \(v_3=0\)).

In this paper we extend our analysis and discuss entanglement and CGLMP inequality violation in the state of two vector bosons arising in the decay of a pseudoscalar/scalar particle X given in Eq. (1) assuming that both bosons decay into leptons. As an example of such a process we use the decay of the Higgs particle into a pair of Z bosons: \(H\rightarrow ZZ\). That is, we assume that both anomalous couplings \(v_2\) and \(v_3\) are nonzero, we also assume \(v_1\not =0\) to have the possibility to use the actual decay \(H\rightarrow ZZ\) as an example. Notice that even for \(v_1=0\) we can apply the same methods, we shortly comment this point after Eq. (8).

Anomalous coupling parameters for the decay \(H\rightarrow ZZ\) are constrained by measurements of Higgs properties performed at the LHC29, they are also constrained from the theoretical point of view by perturbative unitarity. We discuss this point in “Experimental and theoretical bounds on c, \(\tilde{c}\) for the process \(H\rightarrow ZZ\)” section. In the present paper we do not limit values of anomalous couplings to these bounds since we treat the Higgs decay as an exemplary process only, our considerations are more general.

It is worth noticing that the Higgs decay is not the only process which was proposed as a test bed for exploring fundamental quantum properties like entanglement or Bell-type inequality violation in high energy physics. In fact, the first propositions of EPR-like experiments in particle physics were put forward more then forty years ago30,31. In recent years, the first system considered in this context was a system of top quarks produced in colliders32,33,34,35,36,37,38,39. Moreover, the only experimental observation of entanglement at such high energy scale has been recently reported by the ATLAS collaboration at the LHC in a tt system40 and subsequently confirmed by the CMS collaboration41.

Other high energy processes have also been proposed in this context, including, among other: various scattering processes42,43, \(B^0\bar{B}^0\) mesons44, tW systems45, WW pairs produced in electron-positron colliders46, \(\tau \tau\) pairs47 and tripartite systems48,49,50.

Furthermore, the possibility of detecting (or bounding) new physics effects with the help of quantum information techniques has been discussed51,52,53. For a recent review of the subject of quantum entanglement and Bell inequality violation at colliders see54.

We use the standard units (\(\hbar =c=1\), here c denotes the velocity of light), the Minkowski metric tensor \(\eta =\textrm{diag}(1,-1,-1,-1)\) and assume \(\varepsilon _{0123}=1\).

Decay of a pseudoscalar/scalar particle into two vector bosons

We consider here the decay (1), where, in general, V bosons can be off-shell. We will treat off-shell particles like on-shell ones with reduced invariant masses, similarly as it was done in previous papers12,15,27,28. Let us denote by M the mass of the pseudoscalar/scalar particle X and by \(k,m_1\) and \(p,m_2\) the four-momenta and invariant masses of the daughter particles. The amplitude corresponding to the most general Lorentz-invariant, CPT conserving coupling of the (pseudo)scalar particle with two vector bosons can be written as (see e.g.27,28)

$$\begin{aligned} \mathcal {A}_{\lambda \sigma }(k,p) \propto \big [v_1 \eta _{\mu \nu } + v_2 (k+p)_\mu (k+p)_\nu +v_3 \varepsilon _{\alpha \beta \mu \nu } (k+p)^\alpha (k-p)^\beta \big ] e_{\lambda }^{\mu }(k) e_{\sigma }^{\nu }(p), \end{aligned}$$
(2)

where \(\lambda ,\sigma\) are spin projections of the final states, \(v_1\), \(v_2\), \(v_3\) are three real coupling constants, and \(\varepsilon _{\alpha \beta \mu \nu }\) is a completely antisymmetric Levi-Civita tensor. Moreover, amplitude \(e_{\lambda }^\mu (q)\) for the four-momentum \(q=(q^0,\vec {q})\) with \({q^0}^2-{\vec {q}}^2=m^2\) reads24

$$\begin{aligned} e(q) = [e^{\mu }_{\sigma }(q)] = \begin{pmatrix} \tfrac{\vec {q}^T}{m}\\ {\mathbb {I}}+ \tfrac{\vec {q}\otimes \vec {q}^T}{m(m+q^0)} \end{pmatrix} V^T, \end{aligned}$$
(3)

with

$$\begin{aligned} V=\frac{1}{\sqrt{2}} \begin{pmatrix} -1 & i & 0 \\ 0 & 0 & \sqrt{2} \\ 1 & i & 0 \\ \end{pmatrix}. \end{aligned}$$
(4)

These amplitudes fulfill standard transversality condition

$$\begin{aligned} e^{\mu }_{\sigma }(q) q_\mu = 0. \end{aligned}$$
(5)

We do not provide a Lagrangian of the interaction, since the anomalous vertices are obtained from operators carrying a dimension larger than four. Thus, it would be necessary to collect all dimension-six operators in the SMEFT

(Standard Model Effective Field Theory) associated with this interaction which is out of the scope of the present work.

For our exemplary decay \(H\rightarrow ZZ\) the Standard Model interaction corresponds to \(v_1=1\), \(v_2=v_3=0\). Therefore, since we want to use actual experimental values of masses of the Higgs particle and Z bosons in our numerical examples, from now on we will assume that \(v_1\not =0\). Moreover, we admit nonzero \(v_2\) and \(v_3\). Note that experimental data regarding Higgs decay admit nonzero \(v_2\) and \(v_3\) but give strong bounds on their values29, we will discuss these bounds later on.

With these assumptions, the most general pure state of two vector bosons arising in the decay (1) can be parametrized with the help of two parameters, c, \(\tilde{c}\), as

$$\begin{aligned} |\psi _{VV}(k,p)\rangle = \big [ \eta _{\mu \nu } + \tfrac{c}{(kp)}(k_\mu p_\nu + p_\mu k_\nu ) + \tfrac{\tilde{c}}{(kp)} \varepsilon _{\alpha \beta \mu \nu } (k+ p)^\alpha ( k - p)^\beta \big ] e_{\lambda }^{\mu }(k) e_{\sigma }^{\nu }(p) |(k,\lambda );(p,\sigma )\rangle , \end{aligned}$$
(6)

where

$$\begin{aligned} c = (kp) \tfrac{v_2}{v_1},\quad \tilde{c} = (kp) \tfrac{v_3}{v_1}, \end{aligned}$$
(7)

and \(|(k,\lambda );(p,\sigma )\rangle\) is the two-boson state, one boson with the four-momentum k and spin projection along z axis \(\lambda\), second one with the four-momentum p and spin projection \(\sigma\). For \(k\not =p\) states \(|(k,\lambda );(p,\sigma )\rangle\) are orthonormal:

$$\begin{aligned} \langle (k,\lambda );(p,\sigma )|(k,\lambda ^\prime );(p,\sigma ^\prime )\rangle = \delta _{\lambda \lambda ^\prime } \delta _{\sigma \sigma ^\prime }. \end{aligned}$$
(8)

In this paper we consider the case \(v_1\not =0\) but even for \(v_1=0\) we can apply the same methods. For instance, if \(v_3=0\) as well then the state is always separable, while if \(v_3\not =0\) one can define a single parameter, e.g. \((kp)v_2/v_3\), and perform very similar analysis.

The state (6) is not normalized, with the help of Eq. (8) we find

$$\begin{aligned} \langle \psi _{VV}(k,p)|\psi _{VV}(k,p)\rangle = 2 + \Big [ (1+c) \tfrac{(kp)}{m_1 m_2} - c \tfrac{m_1 m_2}{(kp)} \Big ]^2 + 8 \tilde{c}^2 \Big [ 1 - \Big ( \tfrac{m_1 m_2}{(kp)} \Big )^2 \Big ]. \end{aligned}$$
(9)

We will use center of mass (CM) frame for our further computations. The kinematics of the decay (1) in the CM frame is briefly summarized in “Kinematics of the decay \(X\rightarrow VV\) in the center of mass frame” section. Using formulas from this Appendix we find that in the CM frame normalization of the state \(|\psi _{VV}(k,p)\rangle\) depends only on masses M, \(m_1\), \(m_2\) and the parameters c, \(\tilde{c}\):

$$\begin{aligned} \langle \psi _{VV}(k,p)|\psi _{VV}(k,p)\rangle |_{CM} = 2 (1+\tilde{\kappa }^2) + \kappa ^2, \end{aligned}$$
(10)

where, in analogy with our previous paper16, we have introduced the following notation

$$\begin{aligned} \kappa = \beta + c (\beta - 1/\beta ), \quad \tilde{\kappa } = 2 \tilde{c} \sqrt{1- 1/\beta ^2}, \end{aligned}$$
(11)

and \(\beta\) is given in Eq. (58).

The ranges of possible values of \(\kappa\), \(\tilde{\kappa }\) depend on the values of c, \(\tilde{c}\), respectively:

$$\begin{aligned}&\kappa \in (-\infty ,1] & \text {for}\quad & c\in (-\infty ,-1), & \end{aligned}$$
(12)
$$\begin{aligned}&\kappa \in [0,1] & \text {for}\quad & c=-1, & \end{aligned}$$
(13)
$$\begin{aligned}&\kappa \in [2\sqrt{-c(1+c)},\infty ) & \text {for}\quad & c\in (-1,-\tfrac{1}{2}), & \end{aligned}$$
(14)
$$\begin{aligned}&\kappa \in [1,\infty ] & \text {for}\quad & c\in [-\tfrac{1}{2},\infty ), & \end{aligned}$$
(15)

and

$$\begin{aligned}&\tilde{\kappa } \in (2\tilde{c},0] & \text {for}\quad & \tilde{c}\in (-\infty ,0), & \end{aligned}$$
(16)
$$\begin{aligned}&\tilde{\kappa } \in [0,2\tilde{c}] & \text {for}\quad & \tilde{c}\in [0,\infty ). & \end{aligned}$$
(17)

We have given the admissible ranges of \(\kappa\), \(\tilde{\kappa }\) for all values of c, \(\tilde{c}\). However, for a real decay, like our exemplary process \(H\rightarrow ZZ\), there exist further experimental and theoretical bounds on possible values of c, \(\tilde{c}\); for the mentioned process \(H\rightarrow ZZ\) we discuss these bounds in “Experimental and theoretical bounds on c, \(\tilde{c}\) for the process \(H\rightarrow ZZ\)” section.

Next, without loss of generality we can assume that bosons arising in the decay (1) move along z-axis, i.e. we can take \(k^\mu =(\omega _1,\vec {k})\), \(p^\mu =(\omega _1,-\vec {k})\), where \(\vec {k}=(0,0,|\vec {k}|)\) and energies \(\omega _1\), \(\omega _2\) are given explicitly in (55, 56). We also simplify the notation of basis two-boson states in this case

$$\begin{aligned} |\lambda ,\sigma \rangle \equiv |(\omega _1,0,0,|\vec {k}|);(\omega _2,0,0,-|\vec {k}|)\rangle . \end{aligned}$$
(18)

In this notation, with the help of Eqs. (3, 6, 10), the normalized state of two bosons reads

$$\begin{aligned} |\psi _{VV}^{\textsf{norm}}(m_1, m_2, c, \tilde{c})\rangle = \frac{1}{\sqrt{2 (1+\tilde{\kappa }^2) + \kappa ^2}} \big [ (1-i\tilde{\kappa }) |+,-\rangle - \kappa |0,0\rangle +(1+i\tilde{\kappa }) |-,+\rangle \big ]. \end{aligned}$$
(19)

It should be noted that when \(\tilde{\kappa }=0\) the above state coincides with the state discussed in our previous paper16.

Bosons arising in a single decay (1) have definite masses \(m_1\) and \(m_2\); thus two-boson state is pure and has the following form

$$\begin{aligned} \rho (m_1,m_2,c,\tilde{c}) = |\psi _{VV}^{\textsf{norm}}(m_1, m_2, c, \tilde{c})\rangle \langle \psi _{VV}^{\textsf{norm}}(m_1, m_2, c, \tilde{c})|. \end{aligned}$$
(20)

However, when one determines two-boson state from experimental data then averaging over various kinematical configurations is necessary and the state becomes mixed

$$\begin{aligned} \rho _{VV}(c,\tilde{c}) = \int dm_1\, dm_2\, \mathcal {P}_{c,\tilde{c}}(m_1,m_2) \rho (m_1,m_2,c,\tilde{c}), \end{aligned}$$
(21)

where \(\mathcal {P}_{c,\tilde{c}}(m_1,m_2)\) is a normalized probability distribution. The explicit form of this probability distribution can be determined for different channels of the subsequent decay of daughter vector bosons VV arising in (1). In our exemplary case of the Higgs decay into Z bosons, the decay chain \(H\rightarrow Z Z^* \rightarrow (f^+ f^-)(f^+ f^-)\) constitutes one of the most promising channels to certify entanglement at colliders in a qutrit-qutrit system. Thus we stick to this channel in our general analysis. In the case when the daughter bosons decay into massless fermions

$$\begin{aligned} X \rightarrow VV \rightarrow f_{1}^{+} f_{1}^{-} f_{2}^{+} f_{2}^{-}, \end{aligned}$$
(22)

we can use the results from Ref.16,27 and following exactly the same line of reasoning as in our previous paper (compare Eqs. (33–38) from Ref.16) we find

$$\begin{aligned} \mathcal {P}_{c,\tilde{c}}(m_1,m_2) = N \frac{\lambda ^{\frac{1}{2}}(M^2,m_1^2,m_2^2) m_1^3 m_2^3}{D(m_1) D(m_2)} \big [2(1+\tilde{\kappa }^2)+\kappa ^2\big ], \end{aligned}$$
(23)

with

$$\begin{aligned} D(m) = \big (m^2-m_V^2\big )^2 + (m_V \Gamma _V)^2, \end{aligned}$$
(24)

where \(m_V, \Gamma _V\) denote the mass and decay width of the on-shell V boson and the normalization factor N can be determined numerically for given values c and \(\tilde{c}\).

Therefore, introducing the notation

$$\begin{aligned} \textsf{B}(n)&= \int \limits _{S} \! dm_1 dm_2 \frac{\lambda ^{1/2}(M^2,m_1^2,m_2^2) m_1^3 m_2^3}{D(m_1) D(m_2)} \beta ^n, \end{aligned}$$
(25)
$$\begin{aligned} \tilde{\textsf{B}}(n)&= \int \limits _{S} \! dm_1 dm_2 \frac{\lambda ^{1/2}(M^2,m_1^2,m_2^2) m_1^3 m_2^3}{D(m_1) D(m_2)} \beta ^n (\beta ^2 -1)^{1/2}, \end{aligned}$$
(26)

for \(n=-2,-1,0,1,2\), where \(S=\{ (m_1,m_2): m_1\ge 0, m_2\ge 0, m_1+m_2 \le M \}\), the state averaged over kinematical configurations (21) can be written as

$$\begin{aligned} \rho _{VV}(c,\tilde{c}) = \frac{1}{\textsf{b}+2\textsf{e}} \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & \boxed {\textsf{e}} & 0 & \boxed {\textsf{f}} & 0 & \boxed {\textsf{h}} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & \boxed {\textsf{f}^*} & 0 & \boxed {\textsf{b}} & 0 & \boxed {\textsf{f}} & 0 & 0\\ 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & \boxed {\textsf{h}^*} & 0 & \boxed {\textsf{f}^*} & 0 & \boxed {\textsf{e}} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}, \end{aligned}$$
(27)

where for better visibility we have framed the non-zero matrix elements:

$$\begin{aligned} \textsf{b}&= -2c(1+c)\textsf{B}(0)+ (1+c)^2\textsf{B}(2) + c^2 \textsf{B}(-2), \end{aligned}$$
(28)
$$\begin{aligned} \textsf{e}&= (1+4\tilde{c}^2)\textsf{B}(0) - 4\tilde{c}^2 \textsf{B}(-2), \end{aligned}$$
(29)
$$\begin{aligned} \textsf{f}&= -(c+1)\textsf{B}(1) + c \textsf{B}(-1) +2i\tilde{c}\big [ (1+c) \tilde{\textsf{B}}(0) -c \tilde{\textsf{B}}(-2) \big ], \end{aligned}$$
(30)
$$\begin{aligned} \textsf{h}&= (1-4\tilde{c}^2) \textsf{B}(0) +4\tilde{c}^2 \textsf{B}(-2) -4i\tilde{c} \tilde{\textsf{B}}(-1), \end{aligned}$$
(31)

and star denotes complex conjugation.

When we want to obtain two-boson density matrix for our exemplary decay \(H\rightarrow ZZ\), we have to insert into Eqs. (23), (24), (25), (26) the measured values for the Higgs mass, Z mass and Z decay width, i.e., \(M=M_H=125.25\; \textsf{GeV}\), \(m_V=m_Z=91.19\; \textsf{GeV}\), \(\Gamma _V=\Gamma _Z=2.50\; \textsf{GeV}\)55. With these values, from (28,29,30,31) we receive

$$\begin{aligned} \textsf{b}_Z&= 9431.55 + 12883.6 c + 4983.07 c^2, \end{aligned}$$
(32)
$$\begin{aligned} \textsf{e}_Z&= 2989.76 + 5834.84 \tilde{c}^2, \end{aligned}$$
(33)
$$\begin{aligned} \textsf{f}_Z&= -4819.07 - 2752.19 c + 7052.85 i \tilde{c} + 4477.64 i c \tilde{c}, \end{aligned}$$
(34)
$$\begin{aligned} \textsf{h}_Z&= 2989.76 - 8031.86 i \tilde{c} - 5834.84 \tilde{c}^2. \end{aligned}$$
(35)

Entanglement

To check whether the state (27) is entangled and to estimate how much it is entangled we can use one of entanglement measures. In our previous paper16 we used the logarithmic negativity56,57 which is a computable entanglement measure and is defined as

$$\begin{aligned} E_N(\rho _{AB}) = \log _3(||\rho ^{T_B}||_1), \end{aligned}$$
(36)

where \(T_B\) denotes partial transposition with respect to the subsystem B and \(||A||_1={{\,\textrm{Tr}\,}}(\sqrt{A^\dagger A})\) is the trace norm of a matrix A. \(||A||_1\) is equal to the sum of all the singular values of A; when A is Hermitian then it is equal to the sum of absolute values of all eigenvalues of A. \(E_N(\rho )>0\) implies that the state \(\rho\) is entangled.

It is worth noticing that the general structure (the number and positions of non-zero entries) of the density matrix (27) is the same as the structure of the density matrix describing a ZZ pair produced in the decay of the Standard Model Higgs particle analyzed in12. In this paper it was shown that for a density matrix with such a structure the Peres–Horodecki criterion is not only sufficient but also necessary for the state to be entangled. And this implies that the state (27) is entangled iff at least one off-diagonal matrix entry is non-zero.

In Fig. 1 we have plotted the logarithmic negativity of the state (27) for the decay \(H\rightarrow ZZ\), i.e. with matrix elements \(\textsf{b}_Z\), \(\textsf{e}_Z\), \(\textsf{f}_Z\), \(\textsf{h}_Z\) given in Eqs. (32), (33), (34), (35). In this case numerically obtained maximal value of the logarithmic negativity is equal to 0.99638. This value is attained for \(c=-0.73719\), \(\tilde{c}=0.00005\). Moreover, \(E_N>0\) for all values of c, \(\tilde{c}\) and in the limit \(c\rightarrow \infty\) the logarithmic negativity tends to zero.

Fig. 1
figure 1

In this figure we present logarithmic negativity of the state (27), \(E_N(\rho _{VV}(c,\tilde{c}))\) as a function of c, \(\tilde{c}\). To obtain this plot we have inserted the measured values for the Higgs mass, Z mass and Z decay width, i.e., we put \(\textsf{b}_Z\), \(\textsf{e}_Z\), \(\textsf{f}_Z\), \(\textsf{h}_Z\) given in Eqs. (32), (33), (34), (35).

Full size image

Violation of Bell inequalities

Now, let us consider the violation of Bell inequalities in the state (27). The optimal Bell inequality for a two-qudit system was formulated in58 and is known as the Collins–Gisin–Linden–Massar–Popescu (CGLMP) inequality. For two qubits (\(d=2\)) the CGLMP inequality reduces to the well known Clauser–Horn–Shimony–Holt (CHSH) inequality59. Here we are interested in the CGLMP inequality for a two-qutrit system (for spin-1 particle there are three possible outcomes of a spin projection measurements). In this case the CGLMP inequality has the following form

$$\begin{aligned} \mathcal {I}_3 \le 2, \end{aligned}$$
(37)

where

$$\begin{aligned} \mathcal {I}_3 = \big [ P(A_1=B_1) + P(B_1=A_2+1) + P(A_2=B_2) +P(B_2=A_1) \big ] \nonumber \\ -\big [ P(A_1=B_1-1) + P(B_1=A_2) +P(A_2=B_2-1) + P(B_2=A_1-1) \big ], \end{aligned}$$
(38)

and \(A_1\), \(A_2\) (\(B_1\), \(B_2\)) are possible measurements that can be performed by Alice (Bob). Each of these measurements can have three outcomes: 0,1,2. Moreover, \(P(A_i=B_j+k)\) denotes the probability that the outcomes \(A_i\) and \(B_j\) differ by k modulo 3, i.e., \(P(A_i=B_j+k) = \sum _{l=0}^{l=2} P(A_i=l,B_j=l+k \mod 3)\). As usual, we assume that Alice can perform measurements on one of the bosons, Bob on the second one, i.e., we take Alice (Bob) observables as \(A\otimes I\) (\(I\otimes B\)).

To answer whether and how much a given quantum state \(\rho\) violates the CGLMP inequality we have to find such observables \(A_1\), \(A_2\), \(B_1\), \(B_2\) for which the value of \({\mathcal {I}}_3\) is maximal in the state \(\rho\) (so called optimal observables). But, in general, there does not exist a procedure of finding such optimal observables.

The CGLMP inequality (37) can be written as

$$\begin{aligned} {{\,\textrm{Tr}\,}}\big ( \rho \mathcal {O}_{\textsf{Bell}} \big ) \le 2, \end{aligned}$$
(39)

where \(\mathcal {O}_{\textsf{Bell}}\) is a certain operator depending on the observables \(A_1\), \(A_2\), \(B_1\), and \(B_2\).

Each Hermitian \(3\times 3\) matrix A can be represented with the help of the \(3\times 3\) unitary matrix \(U_A\), columns of \(U_A\) are normalized eigenvectors of A in a given basis. Using this notation in Ref.12 it was shown that

$$\begin{aligned} \mathcal {O}_{\textsf{Bell}}(U_{A_1},U_{A_2},U_{B_1},U_{B_2})= -[U_{A_1}\otimes U_{B_1}] P_1 [I\otimes S^3] P_{1}^{\dagger } [U_{A_1}\otimes U_{B_1}]^\dagger + [U_{A_1}\otimes U_{B_2}] P_0 [I\otimes S^3] P_{0}^{\dagger } [U_{A_1}\otimes U_{B_2}]^\dagger \nonumber \\ + [U_{A_2}\otimes U_{B_1}] P_1 [I\otimes S^3] P_{1}^{\dagger } [U_{A_2}\otimes U_{B_1}]^\dagger - [U_{A_2}\otimes U_{B_2}] P_1 [I\otimes S^3] P_{1}^{\dagger } [U_{A_2}\otimes U_{B_2}]^\dagger , \end{aligned}$$
(40)

where \(S^3\) is the standard spin z component matrix, \(S^3=\textrm{diag}(1,0,-1)\), and \(P_0\), \(P_1\) are \(3^2\times 3^2\) block-diagonal permutation matrices:

$$\begin{aligned} P_n = \begin{pmatrix} C^n & \mathcal {O} & \mathcal {O} \\ \mathcal {O} & C^{n+1} & \mathcal {O} \\ \mathcal {O} & \mathcal {O} & C^{n+2} \end{pmatrix}, \quad n=0,1, \end{aligned}$$
(41)

where \(\mathcal {O}\) is the \(3\times 3\) null matrix and C is the \(3\times 3\) cyclic permutation matrix

$$\begin{aligned} C = \begin{pmatrix} 0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0 \end{pmatrix}. \end{aligned}$$
(42)

Each U from (40) can be taken as an element of SU(3) group which has 8 parameters. Therefore, to perform the full optimization of \(\mathcal {O}_{\textsf{Bell}}\) for a given state one should optimize over the 32 dimensional parameter space which is computationally challenging. Thus, usually, one applies a certain optimization procedure in order to find optimal observables.

In Appendix B of our previous paper16 we described in detail two such procedures we used in the case \(\tilde{c}=0\) (for the state of Z bosons arising in the Higgs decay). The first of these procedures, originally introduced in Ref.12, worked very well for c close to 0. The second one, inspired by the proof of Theorem 2 in Ref.60, allowed us to show that the CGLMP inequality is violated for all c. We will apply here this second procedure to show explicitly that CGLMP inequality is violated for all c, \(\tilde{c}\) for all states (27) for which at least one off-diagonal element is non-zero (\({\textsf{f}}\not =0\) or \({\textsf{h}}\not =0\)). To this end, let us notice that the density matrix (27) can be written as

$$\begin{aligned} \rho _{VV}(c,\tilde{c}) = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & \boxed {a_{11}} & 0 & \boxed {a_{12}} & 0 & \boxed {a_{13}} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & \boxed {a_{12}^*} & 0 & \boxed {a_{22}} & 0 & \boxed {a_{23}} & 0 & 0\\ 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & \boxed {a_{13}^*} & 0 & \boxed {a_{23}^*} & 0 & \boxed {a_{33}} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}, \end{aligned}$$
(43)

where \(a_{11}+a_{22}+a_{33}=1\), \(a_{11},a_{22}, a_{33}\in {\mathbb {R}}_{+}\). Moreover, we define unitary matrices

$$\begin{aligned} U_V(t,\theta ) = \begin{pmatrix} \cos \tfrac{t}{2} & 0 & e^{i\theta }\sin \tfrac{t}{2}\\ 0 & 1 & 0\\ -e^{-i\theta } \sin \tfrac{t}{2} & 0 & \cos \tfrac{t}{2} \end{pmatrix} \qquad \text {and} \qquad O_A = \begin{pmatrix} 0 & 0 & 1\\ 0 & -1 & 0\\ 1 & 0 & 0 \end{pmatrix}. \end{aligned}$$
(44)

Now, we calculate the mean value of the operator

$$\begin{aligned} (O_A \otimes I) \mathcal {O}_{\textsf{Bell}}(U_V(0,0),U_V(\tfrac{\pi }{2},0),U_V(t,\theta ),U_V(-t,\theta )) (O_A \otimes I), \end{aligned}$$
(45)

in the state (43). The result can be written as

$$\begin{aligned} {\mathcal {I}}_3 = 2 + \tfrac{3}{2}\big [ a(\cos {t} -1) - 2 |a_{13}| \cos (\alpha +\theta ) \sin t \big ], \end{aligned}$$
(46)

where we used the following notation:

$$\begin{aligned} a_{11}+a_{33}=a, \qquad a_{13} = |a_{13}| e^{i\alpha }, \end{aligned}$$
(47)

and \(a\ge 0\) as a sum of diagonal elements of a positive semidefinite matrix. The maximal value of (46) is attained for

$$\begin{aligned} \sin (\alpha +\theta ) = 0, \qquad \cos (\alpha +\theta ) = \pm 1,\qquad \cos t = \tfrac{a}{\sqrt{a^2+4|a_{13}|^2}},\qquad \sin t = \mp \tfrac{2|a_{13}|}{\sqrt{a^2+4|a_{13}|^2}}, \end{aligned}$$
(48)

and is equal to

$$\begin{aligned} ({\mathcal {I}}_3)_{\textrm{max}} = 2 + \tfrac{3}{2} \big [ \sqrt{a^2 + 4 |a_{13}|^2} -a \big ]. \end{aligned}$$
(49)

Thus, we see that \(({\mathcal {I}}_3)_{\textrm{max}}>2\) for \(|a_{13}|\not =0\).

To show that \(({\mathcal {I}}_3)_{\textrm{max}}>2\) for \(a_{12}\not =0\) or \(a_{23}\not =0\) we need to change the block structure of the unitary matrices \(U_V\) (44) putting a nontrivial \(2\times 2\) block in the upper-left corner for \(a_{12}\not =0\) or in the lower-right corner for \(a_{23}\not =0\) (notice that in our case \(a_{12}=a_{23}\)). We do not present the detailed calculations here since they are similar to those given above.

Summarizing, we have shown that CGLMP inequality can be violated for all values of c, \(\tilde{c}\) if at least one of the elements \(a_{12}\), \(a_{13}\), \(a_{23}\) is non-zero.

It is very interesting that, as we have noticed in the “Entanglement” section, the same condition holds for the state (27) to be entangled. In other words, we have shown that the state \(\rho _{VV}(c,\tilde{c})\) (27) violates the CGLMP inequality iff it is entangled. It is a non-trivial observation since for an arbitrary \(3\times 3\) quantum state \(\rho\) such a statement is true only if \(\rho\) is pure.

This also leads to strong phenomenological implications, since we have proven that in a pair of vector bosons one can indirectly test the violation of the Bell inequality by checking that the pair is entangled, which is a much easier experimental task. And this regardless of the interaction among the spin-0 particle and the gauge bosons (provided that they are CPT conserving and Lorentz invariant), so no future new physics can change the statement of entanglement and Bell violation in this system.

In our exemplary decay \(H\rightarrow ZZ\) from Eqs. (32)–(35) we see that \(a_{13}\not =0\). Therefore, in this decay the CGLMP inequality can be violated for all values of c, \(\tilde{c}\). In Fig. 2 we have plotted the value of \(({\mathcal {I}}_3)_{\textrm{max}}\) obtained with the help of the above optimization procedure for that decay. In Fig. 3 we have plotted \(({\mathcal {I}}_3)_{\textrm{max}}\) as a function of c for three chosen values of \(\tilde{c}=0,0.5,1.5\). The value \(\tilde{c}=0\) corresponds to the scalar Higgs, \(\tilde{c}=0.5\) corresponds to the boundary value given in Eq. (63) while \(\tilde{c}=1.5\) is given for comparison.

Fig. 2
figure 2

In this figure we present the maximal value of \(\mathcal {I}_3\) in the state (27) as a function of c, \(\tilde{c}\). We have inserted the measured values for the Higgs mass, Z mass and Z decay width, i.e., we put \(\textsf{b}_Z\), \(\textsf{e}_Z\), \(\textsf{f}_Z\) and \(\textsf{h}_Z\) given in Eqs. (32)–(35).

Full size image
Fig. 3
figure 3

In this figure we present the maximal value of \(\mathcal {I}_3\) in the state (27) as a function of c, for three chosen values of \(\tilde{c}\). We have inserted the measured values for the Higgs mass, Z mass and Z decay width, i.e., we put \(\textsf{b}_Z\), \(\textsf{e}_Z\), \(\textsf{f}_Z\) and \(\textsf{h}_Z\) given in Eqs. (32)–(35). Notice that the choice \(\tilde{c}=0\) corresponds to the Standard Model value, \(\tilde{c}=0.5\) to the experimental bound (63). The value \(\tilde{c}=1.5\) is presented for comparison.

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Comparing plots presented in Figs. 1 and 2 we see that state with the highest entanglement do not correspond to the state with the highest violation of the CGLMP inequality. This observation is consistent with the general property of CGLMP inequality61.

Experimentally, the state \(\rho _{ZZ}(c,\tilde{c})\) is reconstructed in collider experiments via quantum tomography methods13,62. In such a case the presence of errors and background in the process \(H\rightarrow ZZ \rightarrow f_1^+ f_1^- f_2^+ f_2^-\) modifies the state (27). To estimate how this modification influences the violation of the CGLMP inequality we consider the resistance of this violation with respect to the white noise. It is worth noticing that the addition of white noise can also model, in the first approximation, decoherence of the final state. The noise resistance we define as a minimal value of \(\lambda\), \(\lambda _{\textsf{min}}\), for which the state

$$\begin{aligned} \lambda \rho _{ZZ}(c,\tilde{c}) + (1-\lambda ) \tfrac{1}{9} I_9,\qquad \lambda \in (0, 1], \end{aligned}$$
(50)

violates the CGLMP inequality. Inserting the state (50) into the CGLMP inequality (39) we obtain

$$\begin{aligned} \lambda _{\textsf{min}} = \frac{2}{\max \{{{\,\textrm{Tr}\,}}(\rho _{ZZ}(c,\tilde{c})\mathcal {O}_{\textsf{Bell}}) \}}. \end{aligned}$$
(51)
Fig. 4
figure 4

In this figure we present \(\lambda _{\textsf{min}}\) (51), as a function of c and \(\tilde{c}\). We have inserted the measured values for the Higgs mass, Z mass and Z decay width, i.e., we put \(\textsf{b}_Z\), \(\textsf{e}_Z\), \(\textsf{f}_Z\) and \(\textsf{h}_Z\) given in Eqs. (32)–(35).

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We have plotted this value in Fig. 4 assuming that \(\max \{{{\,\textrm{Tr}\,}}(\rho _{ZZ}(c,\tilde{c})\mathcal {O}_{\textsf{Bell}}) \}\) is calculated with the help of the optimization procedure below Eq. (45). From this plot we can see that for values c, \(\tilde{c}\) close to 0 we can tolerate up to almost a 20% of noise and still attain a violation of the CGLMP inequality and hence an entangled state.

As we mentioned before, there exist other optimization procedures then the one used in this paper. For example, one can focus on specific region of the \((c,\tilde{c})\) parameter space (e.g. close to (0, 0)). Using such procedures one can attain larger violation of the CGLMP inequality in the considered region, which in consequence leads to higher noise tolerance.

Notice that values of c, \(\tilde{c}\) close to 0 are expected for the decay \(H\rightarrow ZZ\) due to experimental bounds on anomalous couplings for the HZZ vertex63. We discuss these bounds in more details in “Experimental and theoretical bounds on c, \(\tilde{c}\) for the process \(H\rightarrow ZZ\)” section.

Conclusions

We have discussed the CGLMP inequality violation and entanglement in a system of two vector bosons \(V_1 V_2\) produced in the decay of a pseudoscalar/scalar particle X. As an example of such a process we use the decay of the Higgs particle into two Z bosons. We have assumed the most general CPT conserving, Lorentz-invariant coupling of the particle X with the daughter bosons \(V_1 V_2\) (compare Eq. (2)). The amplitude of such a coupling depends on three parameters \(v_1\), \(v_2\), \(v_3\). In the case of \(H\rightarrow ZZ\), the Standard Model interaction corresponds to \(v_1=1\), \(v_2=v_3=0\). On the other hand \(v_3\not =0\) implies the possibility of CP violation and a pseudoscalar component of H. Thus, we have assumed that \(v_1\not =0\). In such a case, the state of produced bosons, beyond four-momenta and spins, can be characterized by two parameters c, \(\tilde{c}\) which, up to normalization are equal to \(v_2/v_1\) and \(v_3/v_1\), respectively (cf. Eq. (7)). Next, in the center-of-mass frame, we have determined the most general pure state of \(V_1 V_2\) boson pair for a particular event \(X\rightarrow V_1 V_2\) and the \(V_1 V_2\) density matrix \(\rho _{VV}(c,\tilde{c})\) obtained by averaging over kinematical configurations with an appropriate probability distribution (which can be obtained when both bosons subsequently decay into leptons). Finally, we have shown that this matrix is entangled and violates the CGLMP inequality for all values of c and \(\tilde{c}\) if at least one of off-diagonal elements of the density matrix is non-zero.

In Introduction we noticed that different spin operators have been proposed in the literature. In this context it is interesting to mention the paper22 in which spin correlations of an electron-positron pair arising in a decay of a pseudo-scalar particle were discussed. The authors of22 use different spin operator from the operator used in the present work. As a result they obtain that the considered \(e^+ e^-\) pair becames unentangled in the high energy limit.

It is also worth noticing here that spin correlation functions in colliders are determined from cross-sections (or angular distribution of momenta). This allows for construction of peculiar local hidden variable models duplicating experimental statistics64,65. One can hope that this loophole will be closed with future technical developments as it was the case in the standard low-energy Bell tests7,66,67. The problem of loophole-free tests of Bell nonlocality at colliders has been recently addressed in68. The authors of this paper take a different point of view. They point out that by performing the quantum tomography of the system we are able to reconstruct the full two-partite density matrix. Then, any realization of the observables \(A_1\), \(A_2\), \(B_1\), \(B_2\) over the density matrix can be theoretically performed.

In this context it is very interesting that, as we have shown, the state \(\rho _{VV}(c,\tilde{c})\) (27) violates the CGLMP inequality iff it is entangled. It is a non-trivial observation since for an arbitrary \(3\times 3\) quantum state \(\rho\) such a statement is true only if \(\rho\) is pure. This also leads to strong phenomenological implications, since we have proven that in a pair of vector bosons one can indirectly test the Bell-type inequality violation by checking that the pair is entangled. And this seems to be a much easier experimental task (compare the recent paper40). Moreover, this observation holds regardless of the interaction among the spin-0 particle and the gauge bosons (provided that they are CPT conserving and Lorentz invariant, a very sensible requirement), so no future new physics can change the statement of entanglement and Bell violation in this system.

It is also important to stress that this relation between entanglement and the violation of the CGLMP inequality is solely based on the texture of the matrix (which is a consequence of the symmetries involved in the decay) and not on the experimental way of getting the density matrix itself.

In the paper we have considered the case \(v_1\not =0\) in order to compare the results with the actual decay \(H\rightarrow ZZ\). However, even for \(v_1=0\) we can use the same methods and obtain similar results. For instance, if \(v_3=0\) as well then the state is always separable, while if \(v_3\not =0\) one can define a single parameter, e.g. \((kp)v_2/v_3\), and perform an analogous analysis.

Finally, In our paper as an exemplary process we considered the decay \(H\rightarrow ZZ\). However, some scalar or pseudo-scalar mesons could also decay in a similar way. For scalar mesons \(\tilde{c}=0\) but one could have \(c\not =0\) while for pseudo-scalar ones \(\tilde{c}\not =0\).