---
abstract: 'Investigating the influence of quantum information scrambling on quantum correlations in a physical system is an interesting problem. In this article we establish the mathematical connections among the quantifiers known as quantum information scrambling, Uhlmann fidelity, Bures metric and bipartite concurrence. We study these relations via four point out-of-time-order correlation function (OTOC) used for quantum information scrambling. It is investigated that, the quantum information scrambling is a squared root function with Uhlmann fidelity, squared function with Bures metric and liner function with bipartite concurrence. Further we study the dynamics of all the quantifiers and investigate the influence of quantum information scrambling on entanglement in two qubits prepared in Bell states. We also figure out the quantum information scrambling and entanglement balancing point in bipartite system.'
author:
- |
    Kapil K. Sharma$^\ast$\
    **\
    E-mail: $^\ast$iitbkapil@gmail.com\
    **\
title: 'Quantum Information Scrambling and Entanglement in Bipartite Quantum States.'
---

Introduction
============

Quantum information scrambling (QI scrambling)[@qi1; @qi2; @qi3; @vqi] is known as the spreading of the quantum information over the physical system. The phenomenon of quantum information scrambling may takes place in any physical system because of the chaotic situations. From classical mechanics perspective, the chaos is observed by studying the dynamics of classical trajectories in phase space. If the initial condition is very sensitive than the trajectories diverge in the space and follow the Lyapunov exponent as $e^{\lambda t}$[@lay1; @lay2]. This diversion of trajectories in the phase space with the sensitivity of initial conditions is known as butterfly effect[@bf1]. It is difficult to study butterfly effect in quantum mechanics because of the missing notion of trajectories. The chaos in quantum domain does not have a single definition; there are many ways to look into quantum chaos from quantum mechanics point of view[@qc1]. A popular approach to study the chaos in the physical system is to measure the degree of Irreversibility by using the mismatch between forward-backward evolution of the system. To quantify the chaos by using forward-backward evolution approach; the famous quantifiers are known as Loschdimt Echo and irreversible entropy production[@los1; @los2; @ep1]. These quantifiers also have experimental manifestations in varieties of physical systems. In the connection of forward-backward evolution approach, recently out of time order correlators (OTOC) attracted much attention to measure the quantum information scrambling in thermal density matrices[@ot1; @ot2]. OTOC originally discovered by Larkin and Ovchinnikov, while studying the quasi classical method in theory of superconductivity in 1968[@ot1]. He has been studied the behavior of classical correlator function ie. $C_{c}(t)=\langle [p(t)p(0)]^{\dagger}.[p(t)p(0)]\rangle=e^{2\lambda t}$ in Fermi gas with the Laypunov exponent $\lambda$. The Laypunov exponent quantifies the strength of the chaos which is unbounded for classical physical systems while it has bound for quantum systems with the limit $\lambda\leq 2\pi KT/\hbar$. Recent trends in quantum chaos deal with the quantum mechanical version of $C_{c}(t)$, represented in terms of quantum operators. The quantum version of OTOC has close connection with black hole information problem, holographic theories and quantum chaos in many body physics. It has been shown by Hayden and Preskill by considering a simple model of random unitary evolution that black holes rapidly process the quantum information and exhibit the fastest information scrambling[@qi1]. Temperature is natural recourse of energy for black holes, which is a major factor for information scrambling; hence it is customary to study the quantum information scrambling in thermal density matrices. In continuation of OTOC discussion; here we mention that, the investigation of different versions of OTOC is also an active area. The impact of OTOC can disturb the quantum correlations in a physical system but simultaneously the deep distinction between quantum information scrambling and decoherence is not clear[@qid1]. A lot of work have been carried out on OTOC in different physical systems dealing with varieties of domains like conformal field theories, quantum phase transition, Luttinger liquids, quantum Ising chain, symmetric Kitaev chain, quadratic fermions, hardcore boson model, XX spin chain with random filed [@cf1; @lt1; @lt2; @qch1; @fot1; @xot1]. Often, the OTOC in spin chains have been studied as a function of the distance between two arbitrary spins which are imposed by actions of the local non-commutative operators[@scv1]. Further the laypunov exponent as a function of the velocity ie. $\lambda(v)$, has been studied in classical and semi classical regime and early time behavior of quantum information scrambling has been investigated with Baker-Campbell-Hausdorff (BCH) formula[@bch1; @bch2]. Here we recall, at present the quantum chaos community is bound to study OTOC in thermal density matrices by following the analogy of the involvement of the temperature with black holes. However dynamical studies in varieties of non thermal quantum states are missing. To the best of our knowledge, the present study is totally new in this direction. In the present work, we establish the direct mathematical connection between quantum information scrambling and concurrence in pure bipartite quantum states[@con1; @con2; @con3]. With this mathematical connection, it is easy to study the direct influence of quantum information scrambling on the entanglement. However establishing such direct connections is a hard problem in larger Hilbert spaces. We also establish mathematical connections between quantum information scrambling, Uhlamann fidelity and Bures metric[@uh1; @bm1]. These quantifiers are helpful to study the degree of mismatch between forward and backward evolution and the influence of this mismatch on QI scrambling. Further we study the dynamical behavior of above mentioned quantifiers in two qubits Bell states[@bl1], which prove the strength of of mathematical relations. To start with, the quantum mechanical version of OTOC is given below, $$\langle C(t) \rangle=\langle [W(t),V]^{\dagger}].[W(t),V]\rangle \label{e1}$$ The expression of $\langle C(t) \rangle$ is represented in Heisenberg picture. Here we consider that the operators $W(t)$ and $V$ are Hermitian as well as Unitary. For initial stage at $(t=0)$, the operators $\{W(0),V\}$ commute and no QI scrambling takes place; this is expressed with the following condition, $$[W(0),V]=0.\label{c1}$$ As time advances, the commutativity between $W(t)$ and $V$ may break; which produce QI scrambling. So the condition for the existence of QI scrambling can be considered as, $$[W(t),V]\neq 0.$$ The unitary time evolution of the operator $W(t)$ under certain Hamiltonian govern the degree of commutativity and hence information scrambling. The unitary time evolution of the operator $W(t)$ is give by the series $$\begin{aligned}
W(t)=e^{iHt}W(0)e^{-iHt}
=W(0)+it[H,W(0)]+\nonumber
\\ \frac{t^{2}}{2!}[H,[H,W(0)]]+\frac{it^{3}}{3!}[H,[H,[H,W(0)]]]+......... \label{se1}\end{aligned}$$ At $(t=0)$, the series terminates to $W(0)$ and no scrambling takes place. For the existence of QI scrambling, the following condition should also satisfy, $$[H,W(0)]\neq 0.$$ Here we mention that if $\{H,W(0)\}$ are bounded operators with $||H||\leq \epsilon$ and $||W(0)||\leq \epsilon$, then the series is a convergent series.

Linking QI scrambing, Uhlamann Fedility and Bures metric {#a1}
========================================================

In this section we explore the derivation of the OTOC and establish the mathematical connections among the QI scrambling, Uhlmann fidelity and Beures metric for pure quantum states. Here we recall the definition of OTOC given in Eq.\[e1\] as below, $$\langle C(t)\rangle=\langle [W(t),V]^{\dagger}.[W(t),V]\rangle=\langle P.Q\rangle$$ Where, $$\begin{aligned}
P=[W(t),V]^{\dagger}=V^{\dagger}.W(t)^{\dagger}-W(t)^{\dagger}.V^{\dagger}\end{aligned}$$ and $$Q=[W(t),V]=W(t).V-V.W(t)$$ In brief, we can obtain the factor $P.Q$ as, $$\begin{aligned}
P.Q=V^{\dagger}.W(t)^{\dagger}.W(t).V-V^{\dagger}.W(t)^{\dagger}.V.W(t)\\-W(t)^{\dagger}.V^{\dagger}.W(t).V+W(t)^{\dagger}.V^{\dagger}.V.W(t)\label{eq:7}\end{aligned}$$ Here $W(0)$and $V$ are Hermitian Operators, so these must satisfy the following conditions, $$W(0)^{\dagger}=W(0); \quad W(t)^{\dagger}=W(t) \label{o1}$$ and $$V^{\dagger}=V\label{o2}$$ Applying these conditions to Eq.$\text{\ref{eq:7}}$, we can obtain, $$P.Q=2-V.W(t).V.W(t)-W(t).V.W(t).V\label{eq:12}$$ Taking the average on both the sides we obtain, $$\langle P.Q\rangle=2-\langle\psi|V.W(t).V.W(t)|\psi\rangle-\langle\psi|W(t).V.W(t).V|\psi\rangle$$ $$=2-Tr[V.W(t).V.W(t).\rho]-Tr[W(t).V.W(t).V.\rho.]$$ Assuming $(\rho=I)$ and applying the cyclic property of the trace operation, our simplification leads, $$\langle P.Q\rangle=2.\{1-Tr[W(t).V.W(t).V]\}$$ $$=2.[1-F(t)]\label{ot1}$$ Where, $$F(t)=Tr[W(t).V.W(t).V] \label{ex1}$$ $F(t)$ is called as OTOC with special time ordering of the operators. The expectation value of an operator $O$ in quantum mechanics is defined as, $$<.>=Tr[O.\rho]. \label{ex2}$$ where $O$ is the operator and $(\rho)$ represents the density matrix of the quantum state. Comparing Eq.\[ex1\] with Eq.\[ex2\], we conclude the Eq.\[ex1\] represents the expectation value over the Identity matrix $(I)$. In general this Eq.\[ex1\] can be written over the density matrix $(\rho)$, as below, $$F(t)=Tr[W(t).V.W(t).V.\rho]\label{d1}$$ As per the definition of OTOC; more accurately we can write, $$F(t)=Re[F(t)] \label{r1}$$ Finally the expectation value of $\langle P.Q\rangle$ in Eq.\[ot1\] can be written over a density matrix $(\rho)$ as below, $$\langle P. Q\rangle=2[1-Re.F(t)]\label{is1}$$ Or, $$\langle C(t)\rangle=2[1-Re.F(t)]\label{is1}$$ We emphasize that, the above equations involve $F(t)$ as expressed in Eq.\[r1\] over a density matrix $(\rho)$. To establish the OTOC connection with the Uhlamann fidelity of pure quantum states, we re-look into the Eq.\[d1\] and rewrite the equation by using the cyclic property of trace operation, $$F(t)=\langle\psi|W(t).V.W(t).V|\psi\rangle
=\langle y|x\rangle$$ With, $$|x\rangle=W(t).V|\psi\rangle \label{ef1}$$ and $$|y\rangle=V.W(t)|\psi\rangle \label{eb1}$$ Here the state $|x\rangle$ represents forward evolution of the state $|\psi\rangle$ and $\langle y|$ represent the backward evolution of the state $|\psi\rangle$ under the actions of the operators $\{W(t),V\}$. The operators $W(t)$ and $V$ are unitary, so the state $|\psi\rangle$ does not lost its purity during complete evolution. Here the complete evolution we mean, the total evolution involving the forward and backward evolution. The Uhlamann fidelity of two quantum states gives the information about the overlapping of quantum states and measure the similarity or probability of transition between two states. The Uhlmann fidelity between two pure quantum states $\{|x \rangle,|y\rangle\}$ is defined as, $$f=|\langle y|x\rangle|^{2}$$ The above equation gives the degree of mismatch between the forward and backward evolution of the quantum state $|\psi\rangle$. Exploring the above equation we obtain, $$f=\langle y|x\rangle\langle y|x\rangle^{\dagger}=\langle y|x\rangle.\langle x|y\rangle \label{fu1}$$ We plug-in the values from the Eqs.\[ef1\] and \[eb1\] in Eq.\[fu1\] and applying the cyclic property of the trace operation, we obtain the following, $$f=Tr[W(t).V.W(t).V.\rho].Tr[V.W(t).V.W(t).\rho]$$ Assuming $(\rho=I)$, we obtain, $$f=Tr[W(t).V.W(t).V].Tr[V.W(t).V.W(t)]$$ Applying the cyclic property of trace operation we can get, $$\begin{aligned}
f=Tr[W(t).V.W(t).V].Tr[W(t).V.W(t).V]\\=\{Tr[W(t).V.W(t).V]\}^{2}\end{aligned}$$ By using the definition in Eq.\[ex2\], in general the above equation can be written over the density matrix $(\rho)$, as below, $$f=\{Tr[W(t).V.W(t).V.\rho]\}^{2} \label{fe1}$$ Which leads, $$\sqrt{f}=Tr[W(t).V.W(t).V.\rho]=Tr[M] \label{ft1}$$ Here $M$ represent the density matrix after the complete evolution mentioned below, $$M=W(t).V.W(t).V.\rho \label{ce1}$$ The dimension of the matrix $M$ depends on the size of the Hilbert space, in which the density matrix $(\rho)$ exists. By utilizing the Eqs.\[d1\] and \[r1\], the Eq.\[ft1\] may be rewritten as, $$\sqrt{f}=Re[F(t)]\label{f1}$$ Putting the value of the above factor in Eq.\[is1\] we obtain the connection between QI scrambling and Uhlamann fidelity as below, $$\langle C(t) \rangle=2[1-\sqrt{f}] \label{es1}$$ The above equation shows that, the QI scrambling is the square root function of Ulhmann fidelity for pure states. Referring the Ref\[x\], we know the relation between Bures metric $(D)$ and Uhlamann fidelity as below, $$D=\sqrt{2(1-\sqrt{f})} \label{bm1}$$ Hence the Eq.\[es1\] can be re written as, $$\langle C(t) \rangle=D^{2} \label{bm2}$$ The above equation establish the connection between QI scrambling and Bures metric. We mention that, Bures metric is another measure of closeness of two quantum states and connected with Uhlmann fidelity. Here we find the QI scrambling is a square function with Bures metric. In the present work we only focus on Uhlamann fidelity as this has lucid properties and widely used in literature.

![image](qef)

Important properties of QI scrambling
-------------------------------------

Bases on the Eq.\[es1\] we find the following properties of QI scrambling,

-   Positivity; $\langle C(t) \rangle\geq 0$

-   Bounded limits; $0\leq\langle C(t) \rangle\leq 2.$

-   Unitary invariant; $U\langle C(t) \rangle U^{\dagger}=\langle C(t) \rangle$

-   Exchange symmetry over forward and backward evolution; The Uhlmann fidelity follow the following symmetry, $$f=|\langle y|x\rangle|^{2}=|\langle x|y\rangle^{2}$$ Hence we conclude that under the exchange of forward and backward evolution the QI scrambling $(\langle C(t)\rangle)$ will remain same.

Linking QI scrambling and two qubits concurrence
------------------------------------------------

In this section we establish the mathematical connection between QI scrambling and two qubits concurrence. Here we mention that, concurrence is a good measure of entanglement for a bipartite system which also has its experimental manifestations, it is defined as below, $$C(|\psi\rangle)=|\langle\psi|\sigma_{y}\otimes\sigma_{y}|\psi^{\star}\rangle| \label{conc1}$$ Or, $$C(|\psi\rangle)=|Tr(\sigma_{y}\otimes \sigma_{y}).(|\psi^{\star}\rangle\langle\psi|)| \label{c11}$$ Here $(\sigma_{y} \otimes \sigma_{y})$ is the spin flip matrix, which flip both the spins under the action of Pauli Y operators and $|\psi^{\star}\rangle$ is the complex conjugate of the state $|\psi\rangle$. If we consider the case $(|\psi^{\star}\rangle=|\psi\rangle)$; which means the density matrix $(\rho)$ will have only real elements and the condition $(\rho^{\star}=\rho)$ is satisfied. By using this condition, we may write the Eq.\[c11\] as, $$C(\rho)=|Tr[(\sigma_{y}\otimes \sigma_{y}).(\rho)]|$$ Further we may straightforward write the concurrence in the state $(M)$ after complete evolution given in Eq.\[ce1\], as below, $$C(M)=|Tr[(\sigma_{y}\otimes \sigma_{y}).M|$$ Or, $$C(M)=|Tr[(\sigma_{y}\otimes \sigma_{y}).W(t).V.W(t).V.\rho]| \label{cc1}$$ We know the product of two Hermitian matrices $A$ and $B$ is Hermitian matrix if and only if they commute ie.$[A,B]=0$. The factor $W(t).V$ may not be a hermitian matrix in general since the condition $[W(t).V]\neq 0$ is satisfied for QIS. But this factor may convert into a hermitian matrix, it depends on the parameter time $(t)$ involved in the equation. Hence overall the matrix $M$ may not be a Hermitian matrix in general ie. $(M^{\dagger}\neq M)$. For two qubits the structure of the operators $\{W(t).V\}$ can be expanded in composite Hilbert space $(H_{1}\otimes H_{2})$ with the dimension $(2^{2}\times 2^{2})$. We analytically find the structure of the density matrix $M$, which trace is invariant under the action of the operator $(\sigma_{y} \otimes \sigma_{y})$ in the composite Hilbert space $(H_{1}\otimes H_{2})$. This form of the matrix $M$ with its transpose is given as below, $$M=\left(
\begin{array}{cccc}
 a & b & c & -a \\
 d & e & e & f \\
g & h & h & i \\
 j & k & l & -j \\
\end{array}
\right);\quad M^{T}=\left(
\begin{array}{cccc}
 a & d & g & j \\
 b & e & h & k \\
c & e & h & l \\
 -a & f & i & -j \\
\end{array}
\right) \label{xt}$$ with, $$Tr[M]=Tr[M^{T}]$$ Where $(a,b,c,d,e,f,g,h,i,j,k,l)$ are complex numbers. The transpose of the matrix $M$ ie. $M^{T}$ is also trace invariant under the action of the operator $(\sigma_{y}\otimes \sigma_{y})$. Investigation of the matrix $M$ depends on many factors such as the Hamiltonian of the physical system $(H)$, the nature of the scrambling operators $\{W(0),V\}$ and the density matrix $(\rho)$ of the state. Finding out the exact form of the constitutes $\{H,W(0),V.\rho\}$, which produces the structure of $M$ is mathematically difficult problem, even in larger Hilbert spaces. We recall that the trace of $\{M,M^{T}\}$ is invariant under the action of $(\sigma_{y}\otimes \sigma_{y})$, so we can easily write the following relation, $$Tr[M]=|Tr[(\sigma_{y} \otimes \sigma_{y}).M]| \label{te1}$$ Or, $$Tr[M^{T}]=|Tr[(\sigma_{y} \otimes \sigma_{y}).M^{T}]|$$ By substituting the value of $M$ from Eq.\[ce1\], we can write the Eq.\[te1\] as below , $$Tr[W(t).V.W(t).V.\rho]=|Tr[(\sigma_{y} \otimes \sigma_{y}).W(t).V.W(t).V.\rho]|$$

![image](abc)

Using the above relation, we can easily obtain the new form of the Eq.\[cc1\] as, $$C(M)=Tr[W(t).V.W(t).V.\rho] \label{con1}$$ Putting the value from Eq.\[ft1\] in Eq.\[con1\], we can establish the connection between concurrence and Uhlmann fidelity as, $$C(M)=\sqrt{f} \label{cf}$$ Obtaining the value of the factor $\sqrt{f}$ from the Eq.\[bm1\] and plug-in the Eq.\[cf\], we can establish the direct connection between concurrence and Bures metric as, $$C(M)=1-\frac{D^{2}}{2}$$ Putting the value of the factor $\sqrt{f}$ from Eq.\[cf\] in Eq.\[es1\], we can obtain the direct connection between QI scrambling and concurrence as, $$\langle C(t)\rangle=2[1-C(M)] \label{cm1}$$ Here we find that the QI scrambling is the liner function with the concurrence during complete evolution. Rewriting the above equation we obtain, $$C(M)=1-\frac{\langle C(t)\rangle}{2}\label{qie}$$ This mathematical relation is helpful to study the direct influence of QI scrambling on two qubits concurrence during complete evolution with the following conditions,

-   The density matrix $(\rho)$ deals with pure bipartite quantum states having real elements.

-   The density matrix after the complete evolution has the form given in Eq.\[xt\]

In case if the exact structure of $M$ given in Eq.\[xt\] is not obtained, then one is forced to study the concurrence by using the Eq.\[conc1\]. Uhlamann Fidelity and Bures metric, both are measures of the degree of mismatch of quantum state during forward and backward evolution. We recall that, in the present work we only focus on Uhlamann Fidelity. We plot QI scrambling and concurrence vs. Uhlamann fidelity $(f)$ by using the Eqs.\[es1\] and \[cf\] in the left part of Fig.\[f1\]. We find the QI scrambling is monotonically decreasing and entanglement is monotonically increasing function. The decreasing nature of QI scrambling also comes from the series given in Eq.\[se1\], as this series is a convergent series. Both the graphs intersect at $(f=0.44)$, hence QI scrambling and entanglement both have equal amplitude at this point; we call this point as QI scrambling and concurrence balancing point. At $(f=1)$, the QI scrambling becomes zero; which means the operators $\{W(0),V\}$ commute and contributes in no scrambling but concurrence sustains to $C(\rho)=1$. The decreasing nature of QI scrambling decay the entanglement and may not be a favorable candidate in quantum information. This may also be observed in the right figure incorporated in Fig.\[f1\], which is a plot of entanglement vs. QI scrambling by using the Eq.\[qie\]. We find, when QI scrambling is minimum as $(\langle C(t)\rangle=0)$, the entanglement is high but increasing QI scrambling, decreases the entanglement linearly.

Dynamics of QI scrambling, Ulhmann fidelity and concurrence in Bells states
===========================================================================

In this section we study the dynamics of QI scrambling, Uhlmann fidelity and concurrence by considering two qubits prepared in Bell states, written as below, $$\begin{aligned}
|\psi_{\pm}\rangle_{1}=\frac{1}{\sqrt{2}}[|00\rangle\pm|11\rangle]\label{b1} \\
|\psi_{\pm}\rangle_{2}=\frac{1}{\sqrt{2}}[|01\rangle\pm|10\rangle]\label{b2}\end{aligned}$$ The corresponding density matrices of these states are expressed as $(\rho_{\pm})_{1}$ and $(\rho_{\pm})_{2}$. We assume two qubits are prepared in Bell states and carry the Ising interaction with external imposed magnetic filed in z direction. This Ising Hamiltonian is expressed as, $$H=-j_{z}\sum_{i=1}^{2}\sigma^{z}_{1}\sigma^{z}_{2}-b\sum_{i=1}^{2}\sigma_{z}.$$ The operator $W(0)$ evolve under the Hamilton $(H)$ and follow the series given in Eq.\[se1\]. This series can be easily derived by using the famous Baker-Campbell-Hausdorff formula[@bch1; @bch2]. The rate of change this series play an important role in QI scrambling. To proceed the dynamical study of QI scrambling and concurrence, here we consider the scrambling operators as Pauli operators such that, $$\{W(0),V \}=\{\sigma_{i},\sigma_{j}\}$$ With, $$\{\sigma_{i},\sigma_{j}\} \in \{\sigma_{x},\sigma_{y},\sigma_{z}\};\quad (i,j)\in \{x,y,z\}$$ All the Pauli operators satisfy the property of Hermition as well as Unitary matrix given in the Eqs\[o1\],\[o2\] and fulfill’ the need of choosing $\{W(0),V\}$. We consider that any one of the two qubits goes under the action of Pauli operator; here we consider the action of Pauli operators on first qubit. In this direction we can develop the structure of the operators $\{W(0),V\}$ for two qubits in composite Hilbert space $(H_{1}\otimes H_{2})$ as below, $$\{W(0),V\}_{(i,j)}=\{\sigma_{i}\otimes I,\sigma_{j}\otimes I\}\label{w1}$$ Here we mention that the factor $Tr[W(t).V.W(t).V.\rho]$ involved in Eq.\[f1\] is responsible for QI scrambling and concurrence obtained in Eqs.\[es1\] and \[con1\]. This factor is invariant under the cyclic permutations of the operators; if we consider the following option, $$\{W(0),V\}_{(j,i)}=\{\sigma_{j}\otimes I,\sigma_{i}\otimes I\}\label{w2}$$ Then, because of the cyclic permutation property both the Eqs.\[w1\] and \[w2\] will produce the same results of QI scrambling and entanglement. So we are intended to choose the following combinations of the operators $\{W(t),V\}$ for our study as below, $$\begin{aligned}
\{\sigma_{x}\otimes I,\sigma_{x}\otimes I \};\quad \label{op1}
\{\sigma_{x}\otimes I,\sigma_{y}\otimes I \} \label{op2}\nonumber \\
\{\sigma_{x}\otimes I,\sigma_{z}\otimes I \}\label{op3};\quad
\{\sigma_{y}\otimes I,\sigma_{y}\otimes I \}\label{op4} \nonumber \\
\{\sigma_{y}\otimes I,\sigma_{z}\otimes I \};\quad \label{op5} 
\{\sigma_{z}\otimes I,\sigma_{z}\otimes I \}\nonumber \label{op6} \end{aligned}$$ We study all the Bell states given in Eqs.\[b1\] and \[b2\] under the above mentioned combinations of operators. We find, all the bell states $(\rho_{\pm})_{1}$ and $(\rho_{\pm})_{2}$ under the actions of the combinations $\{\sigma_{x}\otimes I,\sigma_{x}\otimes I \},\{\sigma_{x}\otimes I,\sigma_{y}\otimes I\},\{\sigma_{y}\otimes I,\sigma_{y}\otimes I\}$ produce the same results for all the quantifiers such as QI scrambling, Uhlmann fidelity, Bures metric and concurrence. The functions of these quantifiers are obtained as below, $$\begin{aligned}
\text{QI scrambling:}\quad \langle C(t)\rangle=
2 \left(1-\cos[4 t (b+j_{z})]\right)\label{p1} \\
\text{Uhlmann Fidelity:}\quad f=\cos^2[(4 t (b+j_{z}))]\label{p2} \\
\text{Bures Metric:}\quad D=\sqrt{2}\sqrt{1-\cos[4 t (b+j_{z})]}\label{p3} \\
\text{Concurrence:}\quad C(M)=\cos[4 t (b+j_{z})]\label{p4}\end{aligned}$$ On the other hand the actions of the operators $\{\sigma_{x}\otimes I,\sigma_{z}\otimes I\},\{\sigma_{y}\otimes I,\sigma_{z}\otimes I\},\{\sigma_{z}\otimes I,\sigma_{z}\otimes I\}$ do not produce any scrambling for all the states $(\rho_{\pm})_{1}$ and $(\rho_{\pm})_{2}$. Corresponding to these action operators, the values of all the quantifiers {QI scrambling, Uhlmann Fidelity, Bures Metric and Concurrence} are obtained as $\{0,1,0,1\}$. The functions in Eqs.\[p1\] to \[p4\] of all the quantifiers are oscillating functions with the parameters $(b,j_{z},t)$. Here we plot the dynamical behavior of the quantifiers QI scrambling, Uhlmann fidelity and concurrence with the varying parameters $(b,j_{z})$ vs. the parameter time $(t)$ in Fig.\[f2\]. We find the peak value of the QI scrambling vanish the entanglement in the system and makes it zero. The QI scrambling and entanglement balancing points are periodic in Bell states. On the other hand we also do analysis of QI scrambling vs. Uhlmann fidelity. As the Uhlmann fidelity $(f)$ is zero; it means the degree of mismatch between forward and backward evolution of the quantum state $(|\psi\rangle)$ is very high and hence the QI scrambling is also very high, which consequently kills the entanglement in the system and makes it zero. If Ulhmann fidelity $(f)$ approaches to the amplitude as unity, then the forward and backward evolution of the state $(|\psi\rangle)$ is same and QI scrambling is zero in the states, which helps to keep the high amount of the entanglement in Bell states.

Conclusion
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In this article we established the mathematical relations among the quantifiers QI scrambling, Uhlamann fidelity, Bures metric and bipartite concurrence. Most importantly the mathematical relation is developed between QI scrambling and concurrence, which is useful to study the direct influence of QI scrambling on bipartite entanglement in real density matrices. Further we have studied the dynamical behavior of quantum information scrabling, Uhlamann fidelity and entanglement with Ising Hamiltonian in two qubits prepared in Bell states. We have used the scrambling operators as Pauli operators and find out the combinations of scrambling operators under which no scrambling takes place in Bell states. The influence of QI scrambling has been studied on entanglement and it has been found, the increasing QI scrambling decreases the entanglement in the system and may not be a good candidate in quantum information. Further we also found the dynamics of all the quantifiers in Bell states is periodic. The present work may be explored in larger perspectives and may be helpful for quantum information community.

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