Entanglement dynamics of spins using a few complex trajectories

For simplicity, we will begin the present discussion restricted to only one part of the bipartite system. Later on, results will be straightforwardly extended to the whole state. We also need to comment that there are several references dealing with this subject radcliffe ; klauderb ; perelomov ; gilmore ; gazeau , from which this subsection was written.

Essentially, a spin coherent state $|s\rangle$ is interpreted as the most classical spin state, in the sense that it saturates the uncertainty relation ur for the angular momentum operator $\hat{\mathbf{J}}=(\hat{J}_{x},\hat{J}_{y},\hat{J}_{z})$,

where $a$ and $b$ can assume $x$, $y$, or $z$. For the $(2j+1)$-dimensional case, $|s\rangle$ is explicitly given by

where $\hat{J}_{+}=\hat{J}_{x}+i\hat{J}_{y}$ is the raising spin operator and the ket $|-j\rangle$ is the extremal eigenstate of $\hat{J}_{z}$ with eigenvalue $-j$. Notice that $\hat{\mathbf{J}}$ and the complex number $s$, used to label the state, are dimensionless quantities.

An interesting and useful way to represent a coherent state can be reached from its mean value

where $\mathbf{n}$ is a unitary vector conveniently written in spherical coordinates as $\mathbf{n}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$. The connection between the label $s$ and $\mathbf{n}$ becomes particularly simple when a stereographical projection involving the complex plane $s$ and the unitary sphere is considered. Making the south pole correspond to $s=0$, while the north pole corresponds to $s\to\infty$, we find that

Since $|s\rangle$ are minimum-uncertainty states, it becomes natural to connect $\langle\hat{\mathbf{J}}\rangle$ with a classical angular momentum $j\mathbf{n}$, which justifies the appeal to use them in semiclassical approaches.

In practice, we can say that the wide use of coherent states in many research fields is somehow due to their capacity of generating a basis for the states in Hilbert’s space. In particular, it allows for defining an over-complete unity resolution

where $s^{{(R)}}$ and $s^{{(I)}}$ are, respectively, the real and the imaginary part of $s$, and the integral runs from $-\infty$ to $+\infty$. In addition, it is important to mention that spin coherent states are non-orthogonal,

where $s^{*}$ ($\tilde{s}^{*}$) is the complex conjugate of $s$ ($\tilde{s}$).

We can now return to the original problem of entanglement in bipartite states, constraining the initial state as a product of two coherent states, that is,

where both $|s_{0A}\rangle$ and $|s_{0B}\rangle$ are given by Eq. (3). Moreover, according to Eq. (5), we can rewrite Eq. (2) in the spin-coherent-state representation, finding

where $\mbox{d}\boldsymbol{\mu}\equiv\mbox{d}\mu(s_{A})\,\mbox{d}\mu(\tilde{s}_{A})\,\mbox{d}\mu(s_{B})\,\mbox{d}\mu(\tilde{s}_{B})$. In the integrand, we identify four quantum propagators, defined by

where $\hat{H}$ is the Hamiltonian, $\hat{U}_{T}=e^{-i\hat{H}T/\hbar}$, $\mathbf{s}_{\eta}=(s_{\eta A},s_{\eta B})$, and $\mathbf{s}_{\mu}=(s_{\mu A},s_{\mu B})$. The label $\xi$ refers to forward ($\xi=+$) or backward ($\xi=-$) propagators.

Equation (7) is proper to apply our semiclassical approximation, replacing each quantum propagator $\mathrm{K}_{\xi}$ with their respective semiclassical formulas $\mathcal{K}_{\xi}$. For this reason, in the next subsection, we briefly discuss such approximated expressions.

There is a vast literature concerning the application of semiclassical approximations to the forward quantum propagator $\mathrm{K}_{+}$, for both canonical and spin degrees of freedom scsp1 ; scsp2 ; scsp3 ; aguiar01 ; ribeiro04 ; garg1 ; sscsp1 ; sscsp2 ; sscsp3 ; sscsp4 ; sscsp5 ; ribeiro06 ; garg2 ; thiago . On the other hand, concerning the backward propagator $\mathrm{K}_{-}$, we point out that we have worked with its semiclassical version in the last decade pra2010 ; pra2012 ; foggiatto ; arlans . Very shortly, to deduce a semiclassical expression for $\mathrm{K}_{\pm}$, one starts from its path integral formulation, identifying, under proper assumptions ($j\to\infty$ and/or $\hbar\to 0$), certain classical trajectories as the critical paths of integration. Thus, to conclude the approximation, the integrand is expanded up to the second order around them, and the resulting Gaussian integral is computed.

The classical trajectory involved in this kind of calculation has initial and final boundary conditions mandatorily related to the labels of the ket and the bra appearing in Eq. (8). Such constraints are simplified when a new set of classical variables $\mathbf{u}=(u_{A},u_{B})$ and $\mathbf{v}=(v_{A},v_{B})$ is introduced, according to

Using it, the mentioned boundary conditions become

where (here and in the rest of the paper) we use the notation that a single (double) prime stands for initial (final) time. Classical equations of motion in the new variables are

and the equivalent for the variables assigned to part $B$.

The peculiarity of the semiclassical propagators in coherent states resides in the fact that the boundary conditions (10) are generally overdetermined, provided that one keeps the natural assumption that $\mathbf{u}$ is the complex conjugate of $\mathbf{v}$, and vice-versa. In other words, Eq. (10) implies the complete knowledge of both initial and final phase-space points. As a trajectory is determined by only one of them, the possibility of finding one that satisfies (10) is reduced to the very singular case where $\mathbf{s}_{\eta}$ and $\mathbf{s}_{\mu}$ are fortuitously connected by the classical dynamics.

This problem is surpassed by realizing that the theory allows for the analytical continuation of the (real) classical phase space onto the complex domain. In this case, $\mathbf{u}$ and $\mathbf{v}$ are seen as independent variables and the searching for trajectories, which are now complex, becomes possible, in general. From this scenario, we emphasize the importance of distinguishing real and complex trajectories. While the former lives in the ordinary classical phase space, having the property $\mathbf{u}^{*}=\mathbf{v}$, for all instants of time, the last one inhabits an extended (complex) phase space and $\mathbf{u}^{*}\neq\mathbf{v}$. We should also comment that more than one trajectory satisfying Eqs. (10) and (11) may exist, and all of them in principle should be used in the calculation of $\mathcal{K}_{\xi}$.

The stability of the complex trajectories presented above is explicitly involved in the semiclassical calculations here studied. For this reason, it is important to define the stability matrix $\mathbf{M}$ through the expression

Essentially, $\mathbf{M}$ is responsible for the evolution of sufficiently small initial displacements $\delta\mathbf{u}^{\prime}$ and $\delta\mathbf{v}^{\prime}$ until the final time $T$. As we will show [see Eq.(15) below], the elements of $\mathbf{M}$ can be written in terms of the second derivatives of the complex action $\mathcal{S}_{\xi}=\mathcal{S}_{\xi}(\mathbf{s}_{\eta}^{*},\mathbf{s}_{\mu},T)$, which is given by

where $\chi\equiv\chi_{A}+\chi_{B}$ and $\tilde{\Lambda}\equiv\tilde{\Lambda}_{A}+\tilde{\Lambda}_{B}$, with

and the same for part $B$. By differentiating $\mathcal{S}_{\xi}$, we get the relations

and the analog for part $B$, which will be also very useful later. Now, if we differentiate Eq. (14) and properly rearrange the terms (for details, see Appendix A of Ref. pra2012 ), we have

where

In Eq. (15), we also define the auxiliary matrices $\mathbf{C}^{\prime\prime}\equiv{u^{\prime\prime}_{A}}^{2}\mathbf{A}^{\prime\prime}+{u^{\prime\prime}_{B}}^{2}\mathbf{B}^{\prime\prime}$ and $\mathbf{D}^{\prime\prime}\equiv{v^{\prime\prime}_{A}}^{2}\mathbf{A}^{\prime\prime}+{v^{\prime\prime}_{B}}^{2}\mathbf{B}^{\prime\prime}$, with

where

To write the semiclassical formula of $\mathrm{K}_{\xi}$, we still need to define $\mathcal{G}_{\xi}=\mathcal{G}_{\xi}(\mathbf{s}_{\eta}^{*},\mathbf{s}_{\mu},T)$ and $\mathcal{D}_{\xi}=\mathcal{D}_{\xi}(\mathbf{s}_{\eta}^{*},\mathbf{s}_{\mu},T)$, such that

Given all these functions, we finally write

where the term $\Lambda\equiv\Lambda_{A}+\Lambda_{B}$ is originated from the normalization of the states $|\mathbf{s}_{\eta}\rangle$ and $|\mathbf{s}_{\mu}\rangle$, with $\Lambda_{A}\equiv j\ln\left[(1+|s_{\eta A}|^{2})(1+|s_{\mu A}|^{2})\right]$ and the equivalent for $\Lambda_{B}$. The sum runs over all complex trajectories as defined earlier.

As already announced, we will now replace each quantum propagator seen in Eq. (7) with its semiclassical version (17). For clearness, we list below the replacements that we want to do,

Notice that $\mathcal{K}_{1}$ and $\mathcal{K}_{3}$ refer to semiclassical forward propagators, while $\mathcal{K}_{2}$ and $\mathcal{K}_{4}$ are the backward ones. From this point on, we will refer to the trajectory involved in the calculation of $\mathcal{K}_{k}$ simply as $[\mathbf{u}_{k}(t),\mathbf{v}_{k}(t)]$, for $k=1,\ldots,4$. In accordance with the (extended) complex phase space discussed in the last subsection, we will consider the pair $(s_{A},s_{A}^{*})$ as independent variables, and the same for $(\tilde{s}_{A},\tilde{s}_{A}^{*})$, $(s_{B},s_{B}^{*})$, and $(\tilde{s}_{B},\tilde{s}_{B}^{*})$. Thus, taking into account the boundary conditions (10), we rewrite the integration variables of Eq. (7) as

Following this approach, it is also important to revisit Eq. (5), defined as an integration over the whole complex plane $s$. By considering $s$ and $s^{*}$ as independent variables, this integral is reinterpreted as two path integrals, one along the $s$-plane and the other along the $s^{*}$-plane. In this case, we have

a result that should be joined to our calculation. When we accomplish all these tasks, Eq. (7) becomes

where

and

Functions $\mathcal{D}_{k}$, $\mathcal{S}_{k}$, and $\mathcal{G}_{k}$ clearly refer to their respective semiclassical propagator $\mathcal{K}_{k}$. Notice that the sum over complex trajectories was omitted in Eq. (20), for simplicity. This point will be resumed later.

We are now ready to deduce a semiclassical expression for the linear entropy, attacking Eq. (20) through the saddle point method (or steepest descent method) bleistein . To start the computation, we first recognize that the line integral (20) is defined in a space of eight complex variables, which we will rewrite as

where $\mathbf{r}^{\mathrm{T}}$ indicates the transpose of the column vector $\mathbf{r}$. For each point of the integration path, the input parameters of all $\mathcal{K}_{k}$ are automatically determined, which define the four trajectories needed to evaluate the integrand. Then, the direct prescription to compute integral (20) is following the path of integration and summing the contribution of each point.

However, it happens that the integrand of Eq. (20) fastly oscillates around zero along any generic path. It occurs because of the semiclassical regime assumed here: as $j\to\infty$ (with $\hbar\sim 1/j$), a simple inspection in the function $\mathcal{S}_{k}$, present in $\Phi$, assures this behavior. Therefore, one can say that the integral vanishes along generic paths, which is actually the reason why we have so far neglected any information about paths of integration. The purpose of the steepest descent method consists of finding the saddle points of the integrand and, supported by Cauchy’s integral theorem, performing the integral along its steepest descents. By doing so, the rapid oscillations are dropped out because the imaginary part of $\Phi$ is constant along this particular path bleistein . Moreover, in the regime considered, in general, it is enough to replace $\Phi$ with its second order expansion around the saddle point, so that solving Eq. (20) simply becomes computing a Gaussian integral.

The saddle point $\bar{\mathbf{r}}$ is given by the solution of $\nabla\Phi=0$, where the derivatives are taken with respect to the components of $\mathbf{r}$. In the semiclassical limit, the derivatives of $\mathcal{G}_{k}$ can be disregarded in comparison to other terms of $\Phi$. Therefore, with the help of Eq. (14), we find that the saddle point should satisfy

Notice that these relations imply that the final points of the four critical trajectories are mutually connected. For instance, the final point $(\bar{u}^{\prime\prime}_{1A},\bar{u}^{\prime\prime}_{1B},\bar{v}^{\prime\prime}_{1A},\bar{v}^{\prime\prime}_{1B})$ of the trajectory entering in $\mathcal{K}_{1}$ must be equal to $(\bar{u}^{\prime\prime}_{4A},\bar{u}^{\prime\prime}_{2B},\bar{v}^{\prime\prime}_{4A},\bar{v}^{\prime\prime}_{2B})$, which represents a joint constraint with the final points of the trajectories 2 and 4 (analogous relations can be found for other trajectories). Because of this property, we also call these four trajectories, used to evaluate the semiclassical linear entropy, as entangled-boundary-condition trajectories, in accordance with the nomenclature adopted in Ref. arlans .

The eight equalities of Eq. (23), in addition to the initial conditions

give all prescriptions needed to find contributing sets of four complex trajectories. In particular, when $\bar{\mathbf{v}}_{1}^{\prime}=\bar{\mathbf{v}}_{3}^{\prime}=\mathbf{s}_{0}^{*}$ and $\bar{\mathbf{u}}_{2}^{\prime}=\bar{\mathbf{u}}_{4}^{\prime}=\mathbf{s}_{0}$, all trajectories are the same and real, as dicussed in Sect. II.2. As studied in Refs. pra2010 ; pra2012 , these trajectories also satisfy Eq. (23), leading to a semiclassical approximation for the linear entropy which agrees with the quantum result only for short evolution time. As previously mentioned, here we will get a better accuracy by including the complex trajectories.

Once the saddle point of the integrand is understood, we proceed with the calculation, performing the expansion of the integrand around it. As usual, functions $\mathcal{G}_{k}$ and prefactors $\mathcal{D}_{k}$ are just calculated at the saddle point, so that

where $\bar{\mathcal{J}}_{k}=(1+\bar{u}^{\prime\prime}_{kA}\bar{v}^{\prime\prime}_{kA})(1+\bar{u}^{\prime\prime}_{kB}\bar{v}^{\prime\prime}_{kB})$ and $I_{\mathrm{G}}$ is the Gaussian integral

The bar over the functions indicates that they should be evaluated with the saddle point $\bar{\mathbf{r}}$, and $\bar{\mathbf{Q}}$ is the $8\times 8$ matrix

where

The definition of $\bar{\mathbf{A}}^{(k)}$, $\bar{\mathbf{B}}^{(k)}$, $\bar{\mathbf{C}}^{(k)}$, $\bar{\mathbf{D}}^{(k)}$, and $\bar{\mathbf{S}}^{(k)}_{\boldsymbol{\alpha\beta}}$ are presented right below Eq. (15), provided with information about the trajectory number $k$.

Equation (25) can be substantially simplified if we replace the second derivatives of the complex action with the elements of the stability matrix $\mathbf{M}$, according to Eq. (15). Using this strategy, we finally get the semiclassical linear entropy of the reduced state derived from the pure state $|\psi_{0}\rangle$, given by Eq. (6), as a function of the time evolution $T$

Here, for simplicity, we remove the bar over the symbols and define the matrix

and the functions

and $\mathcal{A}\equiv\mathcal{A}_{A}\mathcal{A}_{B}$, with

and the equivalent for $\mathcal{A}_{B}$.

Equation (27) is the main result of the present paper and an example of its application will be presented in Sect. IV. The sum in $S_{\mathrm{sc}}(T)$ indicates that all sets of classical trajectories respecting the boundary conditions (23) and (24) are, in principle, important to approach the quantum linear entropy (1). Notice that this consideration recovers the arbitrary exclusion of the summation in Eq. (20). However, we need to comment that numerical evidence shows that some sets of trajectories furnish unphysical results which give origin to unexpected divergent behaviors, for example. These contributions will be arbitrarily excluded from the calculation. Although we do not mathematically prove this argument, we associate this issue with those saddle points of Eq. (20) whose steepest descents cannot be deformed from the original path of integration. We remind that this kind of problem is very common in applications of coherent state propagators ribeiro04 ; ribeiro05 .

The derivation of $S_{\mathrm{sc}}(T)$ assumes the semiclassical regime, as we have considered large values of $j$. Working with the extremal case $j\to\infty$ and based on the correspondence principle, we can say that real trajectories are enough to reproduce the quantum behavior. By relaxing this condition, we expect that complex trajectories become important to the approximation. Then, we can think that the complex contributions closer to the real one should be among the most important to evaluate $S_{\mathrm{sc}}(T)$. For this reason, in the next section, we investigate general conditions needed for finding sets of quasi-real trajectories, as an attempt to understanding their origin.