Dynamics of Pair of Entangled Spin-1/2 Particles and Quantification of the Dynamics in terms of Correlations

. Te dynamics of an identical pair of entangled spin-1/2 particles, both subjected to the same random magnetic feld, are studied. Te dynamics of the pure joint state of the pair are derived using stochastic calculus. An ensemble of such pure states is combined using the modifed spin joint density matrix, and the joint relaxation time for the pair of spin-1/2 particles is obtained. Te dynamics can be interpreted as a special kind of correlation involving the spatial components of the Bloch polarization vectors of the constituent entangled spin-1/2 particles.


Introduction
Entanglement is an important feature of quantum mechanics that is useful in the area of "quantum computing and information." Two-qubit maximally entangled states are of particular interest in the implementation of quantum communication protocols like quantum teleportation and superdense coding [1,2]. Tey are also useful to generate a secured key distribution between sender and receiver for communicating information [3]. Tis is because entangled qubits are strongly correlated such that the behaviour of one of the qubits can decide the behaviour of the other, irrespective of the separation, provided the qubits are undisturbed. For example, consider an entangled state in which a pair of spin-1/2 particles' component of the angular momentum along a preferred direction (usually the z-axis) is zero, i.e., (1/ � 2 √ )(|↑〉 ⊗ |↓〉 + |↓〉 ⊗ |↑〉). If one of them is measured along the z-axis and the component of the angular momentum is found to be (1/2)Z, then the result of the other will be forced to be − (1/2)Z. It is important to understand the properties of the entangled states and their behaviour in the environment in which they exist. Tis paper considers one such idea, in which the entangled state of a pair of spin-1/2 particles is considered and subjected to a random magnetic feld. Te random feld arises due to various couplings and molecular motions [4]. Te dynamics of a single spin-1/2 particle in its pure state under a random magnetic feld has already been studied [5]. An ensemble of spins is combined using modifed spin density to notice the fuctuations, and the relaxation times were obtained in the context of nuclear magnetic resonance (a phenomenon in which nuclei respond to the surrounding magnetic felds [5,6]). Relaxation theory and spin entanglement fnd relevance in the context of nuclear magnetic resonance (NMR) (for example, [7,8]). In the current article, we restrict our attention to the "extreme narrowed" case in which the autocorrelation time of the random magnetic felds driving relaxation is negligibly small and thereby the transverse/longitudinal relaxation times are equal (this same assumption applies to our considerations for both a single spin-1/2 particle and a pair of entangled spin-1/2 particles.) Additionally, here, we do not include interaction operators in the Hamiltonian explicitly; cf. Te ideas pertaining to longlived spin states described in [7,8]. Te ideas in [5] are used and extended for an entangled state of a pair of spin-1/2 particles, and the dynamics are derived. Te defnition of modifed spin density is also extended to a pair of entangled spins (modifed spin joint density), and the joint relaxation time is obtained.
Tis paper is organized as follows: in Section 2, we recall important defnitions related to spin-1/2 and pairs of spin-1/ 2 systems and the tensor representation of their density matrices to familiarize the reader with the notations. In Section 3, some important ideas in [5] are explained. Section 3 is based on the dynamics of a single spin-1/2 particle under a random magnetic feld, in which the dynamics of the density matrix in the form of a stochastic diferential equation (SDE) gives relaxation time and steady state. It demonstrates that in the extreme narrowed approximation, the autocorrelation frequency of the dynamics of the spin corresponds to the relaxation time of the spin, which is a familiar parameter in the context of standard approaches to NMR. Section 4 is an extended study on the dynamics of an entangled pair of spin-1/2 particles, motivated by the ideas in Section 3. Te SDE pertaining to the entangled pair gives the steady state and the timescale associated for it to be reached by the joint state of the pair of spins. Te correlation matrix concept is invoked to elucidate the entanglement notion and is motivated by some other entanglement-based measures in the literature. Te SDE and the associated volatility are discussed for the correlation matrix components to illustrate that a maximally entangled state has stronger fuctuations than its unentangled counterpart. Tese ideas were developed in the present article to convey some extra intuition concerning the properties of entangled spins. Section 4 thus contains the central results of the paper, beginning with the idea of the correlation matrix, whose components are the correlations involving the Bloch polarization vector components of the constituent spins. We defne the Hamiltonian for the pair of spins and derive the dynamics for a single pair of entangled spins. An ensemble of these pure states is combined using the modifed spin joint density, and fuctuations in various components of the density matrix are noted. Tese dynamics are interpreted as correlations from the tensor representation of the density matrix. Using the idea of a correlation matrix, it is shown that entangled states have stronger scales of fuctuations than those of the unentangled states. We obtain the joint relaxation time for the pair of spins and conclude in Section 5 with a discussion of the results, including the steady state density matrix and persistence of entanglement when the constituent spins are subjected to the same random magnetic feld and a timescale associated with the joint relaxation time.

Spin States and Teir
Representations. Te spin-1/2 is the fundamental unit of spin system. Te spin-j angular momentum observed along a preferred direction (usually zaxis) takes the eigenvalues m � − j to j. Te spin-1/2 eigenstates in |j, m〉 notation are (1) In the presence of a strong and constant magnetic feld applied along the z-axis (say B 0 z), the spin-1/2 particle tends to align along or opposite to the direction of the magnetic feld, following the Boltzmann fraction [5,6], where the minimum energy dominates (elaborated in Section 3). Tese states are denoted by |↑〉 and |↓〉, respectively. Any other direction parameterized by θ, ϕ is a superposition of these two eigenstates. For example, Tis direction gives the polarization of the spin-1/2 particle. We say that the particle is polarized along p → � (sin θ cos ϕ, sin θ sin ϕ, cos θ).

Te Spin Density Matrix and Tensor Representation.
Te quantum mechanical density matrix of a system containing the states {|ϕ m 〉} occurring with the probability {p m } is given by where m p m � 1. If the decomposition above has only one state, then it is called a pure state. Otherwise, it is called a mixed state. Geometrically, spin-1/2 pure states correspond to points on the Bloch sphere, whereas mixed states correspond to points inside the sphere. Tis can be elaborated from the tensor representation, as we are about to demonstrate. Te density matrix of a spin-j system can be represented using some special tensors with "2j" indices [9]. Te spin-1/2 density matrix of a pure state is given by the projection operator |↗〉〈↗|(� ρ).
Tis can be expressed in terms of Pauli matrices and the identity matrix as (cf. [9]) where X i � Tr(ρσ i ), X 0 � 1, σ 0 � I, and the Pauli matrices are (X 1 , X 2 , X 3 ) is the Bloch polarization vector whose modulus is unity. Hence, pure states correspond to points on the unit sphere. Now, consider a mixture of two pure states ρ 1 and ρ 2 with probabilities p, 1 − p. Let the Bloch unit vectors be (X 1 , X 2 , X 3 ) and (Y 1 , Y 2 , Y 3 ), respectively. Te density matrix of the mixed state is From the tensor representation of ρ, the polarization vector of the mixed state ρ is Concepts in Magnetic Resonance Part A, Bridging Education and Research Te magnitude of P → is less than unity because it lies on the line segment joining the points (X 1 , X 2 , X 3 ) and (Y 1 , Y 2 , Y 3 ). Hence, mixed states correspond to points inside the unit sphere (the appendix).
Te density matrix for a pair of spin-1/2 particles can be expressed in terms of tensors with two indices. Te density matrix of a state |↗〉 1 ⊗ |↗〉 2 whose constituent spin-1/2 particles are polarized along diferent directions is is the polarization of the frst spin and (Y 1 , Y 2 , Y 3 ) is the polarization of the second spin. Te coefcients X a Y b can be interpreted as correlations between the constituent qubits [10]. It should be noted that the indices 1, 2 in the above equation indicate that the constituents are polarized along diferent directions.
Te density matrix of |↗〉 ⊗ |↗〉 can be expressed as where X ab � X a X b . In this case, the constituents are polarized along same direction. From now on, we consider the states in which the constituents are polarized along same or opposite directions. Te density matrix of a pair of entangled spins can also be expressed in a similar way.
Te various terms arising above can be expressed as Writing in terms of (σ a ⊗ σ b ), we get with X ab (� X ba ) defned as In this way, we could proceed for higher spins and also extend for mixed states. We restrict ourselves to the pair of entangled spin-1/2 particles. It is interesting to see that the coefcients in the entangled state are diferent from those in unentangled states. Later, we also prove a well-known fact that the entangled states have stronger correlations than the unentangled ones by defning the correlation matrix.
In the presence of a random magnetic feld, the polarization of the spin-1/2 pure state is afected. It changes randomly as the magnetic feld changes. Te polarization is a vector random process whose dynamics were derived in [5]. In the case of an entangled pair of spin-1/2 particles, the correlation matrix is a random process whose dynamics can be derived from the ideas in [5]. For more discussion on spin states, see [11,12].

Dynamics of a Spin-1/2 Particle
In this section, we briefy explain the idea of [5], which considers a single spin-1/2 particle in its pure state. Te dynamics in the presence of random magnetic feld modelled as Gaussian white noise process can be derived using the stochastic calculus. An ensemble of spins is combined in a special way using the modifed spin density as opposed to the conventionally used ensemble density matrix, which conceals the information about the fuctuations. Te ensemble density matrix and associated relaxation times can be obtained from the modifed spin density via ensemble averaging (law of large numbers), thereby making contact with conventional nuclear magnetic resonance (NMR).
Here, we extract and present some important ideas from [5] and use them to derive the dynamics of the entangled state. Te random magnetic feld experienced by the spins may be the result, inter alia, of molecular motions and dipolar interactions; this constitutes the environment [4]. Such microfeld dynamics (here treated classically, i.e., frst quantized) leads to relaxation in the case of (an ensemble of) unentangled spin-1/2 particles, wherein, in the present context, the standard relaxation time is obtained from the autocorrelation properties of the spin noise process. In this way, the amplitude of the magnetic feld fuctuations determines the corresponding relaxation time. In a similar way, by extending these considerations for a single spin-1/2 particle to a pair of entangled spin-1/2 particles, the random environment is principally due to these kinds of dipole-dipole interactions. Te total magnetic feld experienced by a spin-1/2 particle is the sum of main feld B 0 z and the random feld B Te vector random process is zero mean Gaussian whose autocorrelation function is . Te Hamiltonian associated with the main feld B 0 is Te spin-1/2 particle precesses about z-axis under constant feld. Te random Hamiltonian H 1 (t) is Te Hamiltonian above, which is fundamentally here an operator-valued stochastic process, is represented (in appropriate basis) as a random matrix process. In terms of Wiener diferentials, we can write [13] Following (23), we emphasize that the Hamiltonian (to be precise) is not expressed in terms of Wiener diferentials. Te Hamiltonian is modelled using a Gaussian white noise process Γ t (as in a Langevin approach [14]); thereby the Hamiltonian H 1 (t) does not follow the Wiener process. It is actually the operator H 1 (t)dt that corresponds to Wiener In other words, the Wiener process (which corresponds directly to the spin phase) is obtained from the Gaussian white noise process via the time integral; thus Γ t � dW t /dt, in the sense of Langevin. In such an extreme narrowed treatment the autocorrelation time of the random magnetic feld is negligibly small (corresponding mathematically to the Dirac delta-function autocorrelation for the white noise process); in a more general description, beyond extreme narrowing, and of relevance to slow molecular motions for instance, the magnetic feld noise could instead be modelled by a stochastic process with fnite (i.e., positive) autocorrelation time.
Te total Hamiltonian afecting the spin-1/2 is H 0 + H 1 (t). Te dynamics can be derived assuming the spin is initially polarized along an arbitrary direction with H 0 � 0 (switching of the main feld). Te solution to the equation of motion of the density matrix integrated up to second order gives the dynamics of the spin.
Te equation of motion integrated up to second-order gives [5,15] Te Wiener terms appearing in the expression of H 1 (t)dt in (23) are such that dW 2 t � dt and all other higher powers dW n t � 0 for n > 2. Terefore, the second-order solution is exact [4,5]. Assuming the initial state ρ(0) as the pure state projection operator (10).
Te components of dρ(t) on the left side of the equation of motion (25) (up to second-order following the property of Wiener diferential) are Concepts in Magnetic Resonance Part A, Bridging Education and Research Comparing the stochastic diferential equations (SDEs) on both sides of equation (25), we get where Te term with the double commutator in the evolution equation (25) contains the following integral which is evaluated using the properties of Wiener diferentials (cf. [5]): where a, b � x, y, z. Te product "∘" in the integral should be understood in the Stratonovich sense (cf. [5]). From the above SDEs equations (28) and (29), we can obtain the dynamics of the density matrix as where f � − sin θ t dW θ t and g � e iϕ t (cos θ t dW θ t + idW ϕ t ). Te dynamics can also be derived using the concept of rotational difusion on a unit sphere by considering Laplacian in spherical coordinates (cf. [5]). Te joint probability distribution of the difusion process is θ, ϕ are statistically independent. Terefore, Te dynamics of a spin-1/2 particle implies the dynamics of the spin-1/2 Bloch vector. Using the tensor representation, the dynamics of the components of the vector (X 1 , X 2 , X 3 ) are obtained as For an ensemble of spin-1/2 population, the modifed spin density is defned [5] as as opposed to the mean density matrix Σ t which can be expressed in terms of the modifed spin density as Since the density matrix can be associated with the Bloch vector, the advantage of modifed spin density is that it gives information about the stochastic volatility/variances due to the random felds afecting the spin Bloch vector. Pertaining to each pure state in the ensemble, we defne dχ For some Wiener process W t , such that dW 2 t � dt. Also consider dζ (j) for some complex Wiener process ξ t , such that dξ 2 t � 0, dξ t dξ * t � dt. Te coefcients 2/3 and 4/3 we obtained above correspond to the variances in the longitudinal and transverse spin components (i.e., X 3 � cos θ and X 1 + iX 2 � e iϕ sin θ), respectively. When the efect of random feld is prevalent, the statistical information associated with it cannot be ignored. Te SDE of the modifed spin density is Te mean density matrix can be obtained from (38) as Te modifed spin density matrix Σ t can be expressed in tensors to reveal the variances in each of the spatial components of the spin Bloch vector. Σ t can also be expressed using tensors to see how the mean Bloch vector components X 1 , X 2 , X 3 decay exponentially to zero (geometrically the centre of the sphere). From the expression of the ensemble density matrix (43), we see that the steady state as t ⟶ ∞ is the maximally mixed state (1/2)I, which is geometrically the centre of the Bloch sphere. In the presence of the main feld, the steady state density matrix is given by the Boltzmann fraction [5]. Te spin longitudinal and transverse relaxation times are each equal to k − 1 .

Te Boltzmann Density Matrix.
In the presence of main feld B 0 z, the density matrix ρ 0 under steady state assumes Boltzmann distribution [5,15].
where k B is the Boltzmann constant, T is the temperature, and H 0 is the Hamiltonian associated with the main feld B 0 z (see equation (21)).
Terefore, the matrix exponential in the expression of ρ 0 is For practical purposes, |Zω 0 /2k B T| ≪ 1 (cf. [6]). So, we can approximate the exponential terms in the above matrix expression as Te density matrix ρ 0 can be written as Since ω 0 < 0, the quantity − (Zω 0 /4k B T) > 0. We denote it as p. So, ρ 0 becomes which can be expressed as Tis means that spins in the lower energy state |↑〉 slightly outnumber the spins in higher energy state |↓〉 (as p is a small positive number). Te dynamics were obtained assuming B 0 � 0. Te steady state density matrix in this case is ρ 0 � (1/2)I, which complies with equation (43) as t ⟶ ∞.

The Dynamics of Entangled State of Pair of Spin-1/2 Particles
Te entangled pair of spin-1/2 particles is studied in the same way as we did for the case of spin-1/2. We begin with a single pair of entangled spin-1/2 particles in their pure state and combine using the modifed spin joint density to capture the fuctuations; the dynamics are then interpreted in terms Concepts in Magnetic Resonance Part A, Bridging Education and Research of correlations. Tis section is organized as follows: we defne the correlation matrix and prove that the entangled states have stronger correlations than the unentangled states based on the positive semi-defniteness of the correlation matrix. We consider the entangled state |s〉 and derive the dynamics using the equation of motion. We interpret the dynamics of the components of the density matrix |s〉〈s| using tensor representation.

Te Correlation Matrix.
Te correlation matrix we defne here quantifes quantum correlations between the constituent spins in the language of classical probability.
Recall the density matrix of a pure state consisting of a pair of spins polarized in diferent directions, |↗〉 1 ⊗ |↗〉 2 is As explained in Section 2, (X 1 , X 2 , X 3 ) is the polarization of the frst spin and (Y 1 , Y 2 , Y 3 ) is the polarization of the second spin. We defne a tensor C ij as Te components of the correlation matrix are defned as In case of two spins with diferent polarization vectors, Similarly, the correlation matrices can be defned for states like |↗〉 ⊗ |↗〉, |↗〉 ⊗ |↙〉, |s〉 ≜ (1/ � 2 √ )(|↗〉 ⊗ |↙〉 + |↙〉 ⊗ |↗〉). Te state |s〉 is called maximally entangled state as we demonstrate using the correlation matrix. A general entangled state can be represented as (55) For α � 0, β � 0 and α � π, β � 0, we obtain the unentangled states |↗〉 ⊗ |↙〉 and |↙〉 ⊗ |↗〉 as special cases, respectively. Now the Bloch vectors of |↗〉 and |↙〉 are X → and − X → , respectively. Consider the correlation matrix defned from the tensor representation of density matrix |χ〉〈χ|. Te components of the matrix C are (56) If we represent X → by the column vector X, the correlation matrix C is For α � 0, π and β � 0, the correlation matrix corresponds to the unentangled states |↗〉 ⊗ |↙〉 and |↙〉 ⊗ |↗〉. Te correlation coefcient is attained for α � (π/2), β � 0 and α � − (π/2), β � π. Te quantum states corresponding to these pairs are |s〉 and − i|s〉. Since the overall phase does not distinguish quantum states as both correspond to the same point in their state space (the complex projective space CP n for an n + 1 dimensional quantum system). Both the pairs of (α, β) correspond to |s〉. Te correlation matrix of the maximally entangled state C (s) is Te correlation matrix of the general entangled state: Also, (1 − sin α cos β) ≥ 0 and the matrix (I − XX T ) has the positive semi-defnite property as its eigenvalues are 0, 1, 1. Terefore, the matrix C (s) − C≽0. So, we can write Tus, C (s) is the maximal correlation matrix corresponding to the maximum value of sin α cos β.Consistently, |s〉 is referred to as maximally entangled state with strong correlations between the constituent spins. Magnetic Field. In this section, we derive the dynamics of a pair of entangled spin-1/2 particles experiencing the same random magnetic feld. Te joint random Hamiltonian affecting the entangled pair is H (tot) � H (1) ⊗ I + I ⊗ H (2) .

Dynamics of Entangled Spin-1/2 Pair under Random
(63) Since both the spins are under the same feld H (1) � H (2) � H, In terms of Wiener diferentials, we can write [13] (according to Langevin) Te total joint Hamiltonian can be expressed as where dW (w) t . Let the initial joint density matrix of the entangled pair be ρ(0) � |s〉〈s|. Te equation of motion of the joint density matrix integrated up to second-order [5,15] is then Comparing the expressions on both sides of the equation, we get the SDEs of θ, ϕ as where dW θ (71) Te term with the double commutator in the equation of motion (67) contains the following integral [5]: where a, b � x, y, z as in case of spin-1/2. Te product "∘" in the integral should be understood in the Stratonovich sense (cf. [5]). Te dynamics of the density matrix can be obtained as where the fuctuating terms in the matrix can be read of as Te density matrix ρ M in (73) is given by which can be expressed as By the simulation of noise associated with the random magnetic feld, ρ M can be computed via ensemble averaging of SDE (73). For instance, let Q t be a stochastic process being characterized by the SDE.
where b t is the drift and S t is the volatility. Following [16], we can obtain b t and S t from the process Q t that is being observed. b t � E(dQ t )/dt where E(.) is the expectation (mean/ average) operator. Tis means that we need an ensemble of Concepts in Magnetic Resonance Part A, Bridging Education and Research sample paths Q t to compute the drift b t . In other words, generate a large number of Q(t) and Q(t + Δt) and compute the ensemble mean of the diference Q(t + Δt) − Q(t) so that via the law of large numbers the drift can be computed as b t � lim N⟶∞ (1/NΔt) N i�1 (Q i (t + Δt) − Q i (t)). Using this idea, the steady state density matrix ρ M can be verifed from the SDE (73). Te steady state density matrix denoted by ρ M,num computed by simulating noise pertaining to random magnetic feld looks as shown below: where a ≈ (1/3), b ≈ (1/6) and |δ 1 |, |δ 2 | ≈ 0 (cf. ρ M in (75)).
Using the modifed spin density, we can combine an ensemble of pairs to reveal the fuctuations that are the origin of joint relaxation. Later, we give the interpretation for various components of the matrix in the SDE (73). Te modifed spin pair density Σ t is defned [5] as Now we calculate the following terms to write the SDE of Σ t : for some Wiener process W t , such that dW 2 t � dt. Tis is possible from the stability property of the Gaussian distribution. Te sum of Gaussian distributed random variables is Gaussian distributed [5,14]. From the independence of the random variables g (j) 1 , the squared sum, [G 2 1 ], corresponds to the sample mean of the independent and identically distributed (iid) random variables sin 2 θ (j) t cos 2 θ (j) t dt via the law of large numbers, the sample mean tends to the expected value as N ⟶ ∞. Similarly, Since θ, ϕ are statistically independent, for some complex Wiener process ξ t , such that dξ t dξ * t � dt and dξ 2 t � 0.
3 . (83) Since θ, ϕ are statistically independent, Tus, we can write for some complex Wiener process ζ t , such that dζ 2 t � 0 and dζ t dζ * t � dt. It is to be noticed that G i ′ s and g i ′ s are stochastic diferential forms. All the expected values are calculated based on the SDEs obtained from θ, ϕ whose joint probability density is Now we can write the SDE of the modifed spin pair density.
In the above SDE, N 1/2 in the numerator shows that the components of Σ t become large as N ⟶ ∞. Physically, it means that the overall signal obtained from an ensemble of particles also becomes large as N becomes large. Following [5], we can now defne the mean density matrix Σ t � E(ρ). Te SDE (73) of ρ t is mean reverting. Te entangled pair of spins asymptotically attain the state ρ M of (75) at the rate 3k. Tis can be understood better by calculating E(ρ) via the law of large numbers, Te (deterministic) diferential equation in Σ t akin to conventional NMR can be obtained from the SDE (86) of Σ t as dΣ t � ρ M − Σ t 3k dt.

Interpretation of Spin Dynamics in terms of Correlations.
Te tensor representations of the density matrix yield various correlations involving the spatial components of the Bloch vectors of the constituent spins. Te dynamics of the density matrix determine the dynamics of these correlation components. As we already mentioned, in the case of spin-1/ 2, the Bloch vector is a random process, whereas in the case of an entangled pair of spin-1/2 particles, it is the correlation matrix that is the random process of primary relevance. Te SDEs can be derived from the correlation components of a single pair of entangled spins. Let C (s) denote the correlation matrix of the maximally entangled state |s〉. Te SDEs of various components of the (symmetric) matrix C (s) (see equation (59)) are Figures 1-6 show the sample paths for the six correlation components of the correlation matrix C (s) pertaining to α � (π/2), β � 0 (maximally entangled state). Tese are obtained numerically from the solution of the SDEs (90)-(95). Tese SDEs can also be derived from the SDEs of X 1 , X 2 , X 3 that we obtained in case of a single spin-1/2. We can also derive the SDEs for the correlation components of a general entangled state. Let C denote the correlation matrix of a general entangled state: Tese plots are obtained via the SDEs of θ t , ϕ t , assuming the Wiener fuctuating terms of the random magnetic feld components are zero mean where N is the normal random variable (zero mean and variance unity).
Te dynamics of the general correlation matrix components can be derived from the defnition, using the SDEs of X 1 , X 2 , X 3 , as dC 11 � k dt 2 sin α cos β − 1 − 3C 11 − 2k 1/2 (1 + sin α cos β) sin θ t cos θ t cos 2 ϕ t dW θ Te stochastic volatility S(R t , t) in a general SDE, is a measure of how the magnitude of fuctuations in a stochastic process R t vary randomly. In the above SDE, the drift term b(R t , t) is in general a function of R t , t, in which case, the SDE corresponds to an Ito process R t . Else, R t is a difusion. Finally, we consider the mean squared volatility in the SDEs of the correlation components (97) from which it is easy to see that Tese results can be numerically verifed by considering an ensemble of sample paths and calculating the mean squared volatility. To illustrate, consider equation (103) to generate a large number of sample paths to obtain their squared diferences (R i (t + Δt) − R i (t)) 2 . Now, fnd the sample mean of the squared diferences, which tends to the expected value of the squared stochastic volatility via the law of large numbers. In other words, E(S 2 t ) is equal to lim 2 , which is obtained numerically to verify the above results for an ensemble of sample paths of the correlation components to obtain the matrix elements of (105). To elaborate, fuctuating random magnetic feld components are generated following the Wiener property dW t � N �� dt √ , where N is the standard normal random variable has a zero mean and unit variance.
Following equations (68)-(71), the quantities θ(t), θ(t + dt) and ϕ(t), ϕ(t + dt) can be obtained, which could be used to generate the components C (s) ab (t), C (s) ab (t + dt) via the SDEs (90)-(95). By considering an ensemble of sample paths for the correlation components, the expected value of the squared stochastic volatility can be calculated using the law of large numbers, as mentioned above. As an illustration, we shall simulate as demonstrated above to calculate [E(dC (s) 2 ab )/k dt] numerically to obtain the matrix [E(dC (s) 2 ab )/k dt] num components in equation (105) (notice k in the denominator to avoid the dependence of k for convenience). Te (expected) squared volatility matrix evaluated numerically for two runs is given below (for comparing with equation (105) Te correlation matrix structure we have described is therefore an efective measure to quantify the entanglement between constituent spins. Indeed, there are other measures to quantify entanglement, such as concurrence, which leads to the idea of entanglement of formation (cf. [17] for the defnitions of concurrence and entanglement of formation). In this case, we assumed that the initial state is maximally entangled and the constituents are subjected to the same random magnetic feld, meaning the maximally entangled state evolves into another maximally entangled pure state. As a result of the defnition [17], the concurrence (and thus the entanglement of formation) remains constant for the maximally entangled state. In other words, the plot of concurrence versus time is a constant function. Te correlation matrix we defned in this article for a maximally entangled state evolves as a stochastic matrix process as the random magnetic feld fuctuates and thus is an Ito process in its own right via Ito's formula, the trajectories of which are plotted (Figures 1-6). Te compact matrix SDE (of the maximally entangled state) for the process C (s) t can be obtained from the component SDEs as where F t is a matrix containing the fuctuation terms appearing from (90)-(95). It is necessary to mention the general Ito formula for any kind of entanglement quantifer that could be defned depending on the context. Let G t � g(X t ) be a function/functional (an entanglement quantifer in this context), dependent on a stochastic process X t , where