# Scale-invariant freezing of entanglement

###### Abstract

We show that bipartite entanglement can be frozen over time with a proper choice of the many-body substrate, which is in contact with one, or more than one environments via a repetitive interaction. Choosing the one-dimensional anisotropic XY model in transverse uniform or alternating fields as the system, we show, in systems of moderate size, that the numerically obtained freezing, or near-freezing of entanglement occurs in all three phases, namely, the dimer, the antiferromagnetic, and the paramagnetic phases of the model. We also show that in the dimer and the paramagnetic phases, the length of the freezing interval, for a chosen pair of nearest-neighbor spins, is independent of the length of the spin-chain, indicating a scale-invariance. This allows us to propose a relation between the length of the freezing interval and the distance of the chosen pair of spins with the environment. Since the phenomenon also occurs in the one-dimensional XXZ model in an external field as well as in the spin chain under the same decoherence scheme, we argue that this feature is potentially generic to the low-dimensional quantum spin models.

## I Introduction

Rapid development of quantum information technology has been possible due to the path-breaking inventions of communication and computational schemes, including classical information transmission via quantum states with or without security densecode (); communication (); crypto (); diqc (), quantum state transfer asdus_physrep (); teleport (), quantum metrology metrology (), and one-way quantum computation onewayQC (). An almost universal feature in all these quantum tasks is the use of quantum correlations in the form of entanglement ent_horodecki () between the constituents of composite quantum systems as resource. Over last few years, highly entangled bipartite and multipartite states have been created in the laboratory using different substrates like photons photons (), trapped ions ions (), superconducting materials sc-qu (), nuclear magnetic resonances (NMR) nmr_multi (), and optical lattices optical-lattice (), making the implementation of quantum information processing tasks using few qubits possible. However, a quantum mechanical advantage in quantum computation, or error correction error-correct () requires realization of entangled states shared between qubits having number of the order of qadvantage (), which remains to be a challenging task even with the cutting edge technology.

A main obstacle in this enterprise is the fragility of entanglement to decoherence, which is exhibited by the rapid decay of entanglement with time in multiparty quantum systems exposed to environments ent_decay (). It has been shown, both theoretically and experimentally, that entanglement in a multiparty system can suddenly disappear when subjected to local environments – a phenomena known as “sudden death” of entanglement sd_group (); sd_group2 (), which restrains the success of realizing quantum information schemes like transmission of information through quantum channels and implementation of quantum gates. On the other hand, under carefully specified initial conditions, quantum correlations that are independent of entanglement, are shown to be robust against similar environmental effects disc_dyn_group (), and can even be preserved for some time disc_freeze_group (); titas_freeze (). However, despite a few attempts ent-freeze (), a realizable situation for preserving entanglement over time, as yet, remains elusive.

In this paper, we present scenarios involving physical systems that are realizable in the laboratory, and a specific model of environment, in which entanglement of the system, even when exposed to the environment, remains constant for a finite interval of time at the beginning of the dynamics. We call this phenomena as freezing of entanglement. More specifically, we consider two paradigmatic spin models as systems, namely, the anisotropic XY model in external uniform as well as alternating transverse fields alt-field-rev (); alt-field-book (); alt-field-pap (), and the XXZ model in an external magnetic field xxz_phase (); xxz_group_old (); xxz_group_qinfo (), both defined over a chain of spin- particles. In both the cases, the environment interacts with one, or more than one spins in the system via repetitive quantum interaction rqim_attal (); rqim () (cf. others ()). The motivation for such choice of the system is due to their potential of realization in different substrates in the laboratory, including ion traps and optical lattices opt_latice (); ion-spinsimulation (). Hence, the implementation of the freezing phenomena reported here can, in principal, be possible in the laboratory by currently available technology.

We show that in the anisotropic XY model in a uniform or an alternating transverse field, bipartite entanglement, as quantified by logarithmic negativity neg_group (); neg_part_group (); logneg_def (), freezes or near-freezes, up to the accuracy of the numerical algorithm, except for the nearest-neighbor spin-pair(s) that is (are) adjacent to the environment(s), for systems of moderate size. In the dimer and paramagnetic phases of the model, the length of the freezing interval, corresponding to a chosen pair of nearest-neighbor spins, increases monotonically as the distance of the spin pair from the environment(s) increases, and is invariant to a change in the system size. This allows one to estimate the length of the freezing interval as a function of the distance of the chosen pair from the environment(s), and the system-size. Such scale-invariance is not present in the case of the antiferromagnetic phase of the anisotropic XY model in a uniform or an alternating transverse field, and in any of the phases of the XXZ model in an external field. Moreover, we observe that in the XXZ model, the duration of freezing or near-freezing of the pairwise entanglement does not show any monotonicity with the increase of the distance between spin-pair and environment. We also point out that the freezing phenomenon is robust against a change in the temperature of the environment(s).

## Ii Methodology & Models

In this section, we provide a brief description of the repeated quantum interaction (RQI) model rqim () corresponding to a quantum spin system in contact with a reservoir. We also discuss the important features of the relevant quantum spin models used in this paper.

### ii.1 Repeated quantum interaction

Let us consider a physical system, , characterized by the state and described by the Hamiltonian , in the Hilbert space . The system is in contact with a collection of identical copies of an environment, , given by , and defined in the Hilbert spaces , where is a large number. Each copy of the environment, described by the Hamiltonian , is in a state , where are identical to each other, and so are . The total Hamiltonian, , describing the combination of the system, , any one copy of the environment, , and the interaction between them is defined in the Hilbert space . We consider the system-environment interaction to be such that interacts with only one chosen copy of , say, , at a given time instant, and the interaction lasts for a very short time-interval, . During this interval, all the other copies of environment, , remain isolated from as well as . Without any loss of generality, we assume that during the first interval , interacts with , and the duo of and , denoted by , are isolated from . At , the state of is denoted by , where is the state of at .

The unitary evolution generated by in the interval is given by , where . In the next interval , the system, having an initial state given by , interacts with only, and the initial state of the system-environment composition, , is given by . In this interval, the dynamics is governed by the Hamiltonian , which is defined in a way similar to . Note here that and are identical to each other. Continuation of this procedure in all subsequent intervals leads to a repetitive interaction between the system, , and the environment, . At the beginning of every time interval, the initial state of the system-environment duo, , is reset to the product of the state of the environment, (which is the Markovian approximation) and the evolved state of , obtained by tracing out the environment from the evolved state of at the end of the previous interval.

Let us consider the time interval, given by , during which interacts with only, . The evolution of the complete state, , of the system, , and copies of the environment, , in this interval is achieved by , where and are defined in the Hilbert space given by . The operation is equivalent to operating in the space , and identity over the other copies of , except , i.e.,

(1) |

where is the total Hamiltonian of the system, the environment and their interactions in the interval. Here, is the identity operator defined in the environment Hilbert space. A collective evolution of the system-environment combination, up to a time is given by , where the sequence of unitaries, , satisfies the relations

(2) |

with being the identity operator in . We will consider the unitary evolution given in Eq. (2) up to a time , in the limit and , such that remains finite.

### ii.2 Lindblad master equation

Under the setting of RQI, we focus on a system, , consisting of interacting spin- particles, described by the Hamiltonian . The system is connected to one, or more than one reservoir(s), in the form of large collection(s) of decoupled, yet identical spins. The Hamiltonian describing each such spin in the collection is given by , with being the energy of one spin, and , are Pauli matrices. An individual spin in the collection represents an individual copy of a thermal reservoir, whose temperature is given by , where is the absolute temperature of the bath , and is the Boltzmann constant. Typically, the access of the environment to the system is mediated by the coupling between the environment and one, or several spins in the system, which we cast as the “doors” to the system. In this paper, we restrict ourselves to “single-door” and “double-door” systems. However, in principle, a system can have any number of doors, and the result obtained here can be helpful to infer properties in multiple “door” systems.

Single-door access: Let us first assume that the environment has access to the system via any one of the spins constituting the system (see Fig. 1). The Hamiltonian , describing the system , any one of the decoupled spins as environment, and their interactions can be written as

(3) |

where represents the system-environment interaction, and and are the identity operators in the Hilbert spaces of the system and the environment, respectively. Note here that we have discarded the subscript “” of the Hamiltonian for brevity. We consider the interaction between the door and the environment to be of the form

(4) |

where the superscript “d” indicates the door spin, and is a constant. In all our calculations, we set . Under RQI with a single-door system, the Lindblad master equation is given by rqim ()

(5) | |||||

where , , and , with , and .

Note. It is natural to ask what happens if the system is accessed by not one, but more than one collections of decoupled spins through the door – a situation where the system is subjected to RQI with more than one environments via the same door. In this case, following the same route for determining Eq. (5), one can show that the Lindblad master equation takes the form

(6) | |||||

where is the number of baths accessing the system via the same door.

Double-door access: Let us now increase the number of doors to two, at positions and . We consider two independent environments, in the form of two individual spins each of which is described by the Hamiltonian and at temperature (see Fig. 1). Each of the environments is interacting with the system via any one of the two doors, and only one environment has access to the system through a single door. The interaction of the environments via their respective doors is simultaneous. The total Hamiltonian, , in the present scenario, would be

(7) | |||||

where, and describe the Hamiltonians of the environments and respectively, and

(8) |

Following a similar procedure as before, the Lindblad master equation corresponding to the Hamiltonian (Eq. (7)) can be obtained as rqim (),

(9) | |||||

### ii.3 Choice of systems: Quantum spin models

Let us now discuss the choice of the physical system . In recent times, a wide spectrum of substrates is probed in the laboratories all over the world, thereby providing a large set of physical systems to search for the frozen entanglement. A priori, it is not at all clear which of these systems are more preferable for exhibiting such phenomena in comparison to the others. In this paper, we consider generic quantum many-body Hamiltonians as , which can be realized in the laboratory using several such substrates and currently available technology. In this respect, low-dimensional quantum spin Hamiltonians stand out as excellent candidates.

We choose a generic quantum spin chain of sites, in the presence of a transverse site-dependent magnetic field that changes its direction from to depending on whether the lattice site is even, or odd. The Hamiltonian of the model is given by

(10) | |||||

where is the strength of the exchange interaction between nearest-neighbor spins, is the anisotropy, is the anisotropy in the direction. Here, and are respectively the strengths of the uniform and site-dependent magnetic fields. Note here that we consider the open boundary condition (OBC) for the spin chain, the implication of which will be clear in subsequent discussions.

From the Hamiltonian in Eq. (10), two different quantum spin models belonging to two different paradigms emerge. For , the Hamiltonian describes an one-dimensional (1d) anisotropic model in the presence of a transverse uniform and an alternating magnetic field alt-field-rev (); alt-field-pap (); alt-field-book (). On the other hand, the Hamiltonian corresponding to an 1d anisotropic XXZ model in an external uniform magnetic field (TXXZ) is given by with and xxz_phase (); xxz_group_old (); xxz_group_qinfo (). The main motivation of choosing the anisotropic model with a transverse alternating field (ATXY) over the widely studied model in a transverse uniform field is the richer phase diagram of the former, where along with the antiferromagnetic (AFM) and quantum paramagnetic (PM) phases, a dimer (DM) phase appears, which is not present in the latter model. In the thermodynamic limit and also with the periodic boundary condition (PBC), the phase boundaries of the ATXY model are given by the lines (PM AFM) and (AFM DM) on the plane, where we choose and as the system parameters alt-field-book (); alt-field-pap (). In the case of OBC, these phase boundaries change only slightly, even with a moderately small system size, and the AFM region shrinks on the phase plane.

The TXXZ model also shows three phases, namely, an AFM, a ferromagnetic (FM), and an XY (spin flopping) phase, among which the first two are gaped, while the third has a gapless spectrum. Specifically, without the external magnetic field, the FM XY transition occurs at , while at , the XY AFM transition takes place. With increasing the strength of the external field, the quantum phase transition (QPT) points, , shifts to the left (see xxz_phase () for the phase diagram of the model). Here, we point out that in the FM phase (), the bipartite entanglement vanishes for all values of Ferro_ent_vanish ().

## Iii Freezing of Entanglement

In this section, we discuss the main results, the freezing of entanglement in quantum spin models, using logarithmic negativity (LN) logneg_def (); neg_group (); neg_part_group (), denoted by , as the entanglement measure. Although entanglement is considered to be a fragile quantity against environmental noise, we point out that with suitable tuning of the parameters governing the system, the environment, and the system-environment interaction, a constant, or near-constant behavior of LN against time can be achieved. If LN is found to be unchanged for a finite time interval, we say that a freezing of LN, in that interval, has occurred.

Let us denote the LN corresponding to any two nearest-neighbor spins at and , , by . Evidently, during the evolution of the system in contact with the environment, is a function of time as well as the relevant parameters defining the system, the environment, and the system-environment interactions. In this paper, we shall focus only on the time-variation of , keeping all the other parameters fixed. The variation of as a function of can be determined by computing LN for . Here, is the time-dependent reduced density matrix of the nearest-neighbor spin pair , which can be obtained by tracing out all spins, except the spins at and , from . Our main aim is to find out the dynamics of up to a large time, , by solving Eq. (5) in different phases of the chosen quantum spin model. Note here that the typical value of has to be chosen by a careful inspection of the physical quantity, which is LN in the present case. A time-span, , is considered to be large if LN saturates to a fixed value for , due to, for LN of the system, some accidental cancellations within the expressions representing the physical quantities of interest, which is not necessarily equivalent to the equilibriation of the entire system. In the case of the spin models considered in this paper, we find that . Let us denote the initial value of at by . We consider to be frozen over a certain time interval , if for every time point in the interval,

(11) |

where is a pre-decided small number. For , the freezing of entanglement breaks, and we call the quantity, , to be the freezing terminal. Evidently, the value of is a characteristic of the chosen nearest-neighbour spin pair as well as the values of the parameters that define the system, the environment, and the system-environment interaction. In our calculations throughout this paper, we choose . We point out here that the saturation of LN is different from the concept of freezing of LN. This is in the sense that the former occurs only at “large” time, while the latter takes place right after, or close to the time when the system starts interacting with the environment.

### iii.1 Freezing in single-door dynamics

We first consider the single-door dynamics, as discussed in Sec. II.2. We assume that the first spin in the ATXY model, labeled by “”, is the door, via which the model is subjected to RQI. Unless otherwise stated, here and in the rest of this paper, we consider a chain of spins as the “system” for the purpose of demonstration . We shall show that in specific phases of the ATXY model, the findings remain qualitatively unchanged even when the system size is varied, exhibiting a scale-invariance. In the cases where the scale-invariance does not hold, the results are tested and found to be true for larger system-sizes , which can be handled by available numerical resources. Using the paradigmatic quantum spin models discussed in Sec. II.3, one can determine the time-evolved state, , of the system by using Eq. (5). The initial state of the time-evolution of the system is chosen to be a thermal state, given by , where is the partition function of the system. Unless otherwise stated, we set , , and for all our calculations. We shall discuss the effects of varying the system and the environment temperatures on the freezing phenomena in subsequent sections.

Freezing. For the purpose of demonstration, we choose specific values of and from the DM phase of the ATXY model (Eq. (10) with ) with , and study the dynamics of LN. For all our computations throughout this paper, we fix . However, the results remain qualitatively unchanged if one chooses different values of . Since the spin “” in the chain with OBC is used as the door, LN corresponding to the pair of spins and is expected to be maximally exposed to the noise due to RQI between the system and the environment. In agreement to this intuition, is found to be a highly oscillatory function of throughout the investigated interval of time. Interestingly, , , are found to remain constant over a finite time interval at the beginning of their dynamics, and then decay rapidly to zero with increasing time, thereby exhibiting a freezing of entanglement. A typical freezing profile for is shown in Fig. 2(a), where the values of the system parameters, and , are in the DM phase.

Scale invariance. It is important to investigate the effect of varying system-size on the freezing of entanglement. Towards this aim, we fix a specific nearest-neighbor spin pair , and determine the effect of varying over the freezing terminal by computing in the DM phase of the ATXY model. Remarkably, in the DM phase, the value of , for a specific choice of , is found to be unaffected with a change in the size of the system, indicating a scale invariance, i.e., for fixed ,

(12) |

where . For the purpose of demonstration, we chose a point in the DM phase. As shown in Fig. 2(b) (solid line), the curves representing the variations of against , corresponding to different values of , coincide. This is a result of the coincidence of the points representing the values of for a fixed choice of , corresponding to different system sizes upto our numerical accuracy (). This indicates an invariance of for a fixed , as well as the variation of with , against varying . The scale-invariance motivates us to determine an analytical form for the variation of as a function of , which remain independent of . We find that for , where the values of are considerably high, the variation of with is a parabolic one, given by

(13) |

where and are the fitting parameters, the values of which depend on the chosen values of the system parameters. The importance of Eq. (13) lies in providing us an estimation of corresponding to with increasing distance from the door, which remains invariant against a change in the value of . Note also that for a desired value of of the freezing terminal, Eq. (13) provides an estimate of the minimum size of the system, given by , required to attain this value, where is obtained as a solution of Eq. (13), by using .

Note that the values of , for all values of , are considerably small when , i.e., when the spin-pairs are comparatively closer to the door. For , achieves higher values. We observe from Fig. 2(b) that in the DM phase, the value of the freezing terminal increases monotonically as one moves away from the spin which is directly exposed to the environment, i.e., , thereby imposing a hierarchy among the different nearest-neighbor pairs in terms of the values of . For example, in the case of with the spin “” attached to the environment, is found to be the highest among all the dynamics terminals . Although at this point, such hierarchy may seem obvious due to the increasing distance of the chosen spin pair from the door, we shall demonstrate in succeeding discussions that such a simplified argument of propagation of noise is not valid. Note also that the existence of the scale-invariance is independent of whether the trend of with is monotonic, or non-monotonic. We shall elaborate on this point in the succeeding sections.

Freezing of entanglement is present also in the AFM and the PM phases of the ATXY model. However, the nearest-neighbour entanglement in the DM phase is much higher compared to the AFM or the PM phase, and therefore the freezing in this phase is more useful for several quantum information processing tasks that use entanglement as resources. The qualitative features like the hierarchy and the scale invariance are observed in the PM phase also, but not in the AFM phase. An example of scale-invariance of in the PM phase is depicted by the dashed line-point curve in Fig. 2(b), which shows the monotonic increase of with increasing , where the system parameters are chosen to be . Similar to the DM phase, here also the variation of with can be represented by Eq. (13), where the fitting parameters are governed by the values of the system parameters in the PM phase. On the other hand, Fig. 2(c) exhibits the non-monotonic variation of with in the AFM phase of the ATXY model, which can be clearly seen from the continuous line joining the data points corresponding to , where the system parameters are chosen as . This implies a possibility for for , which indicates that the mechanism of freezing of entanglement for nearest-neighbour spin pairs away from the door is an intricate one, and can not be explained simply by considering the increasing distance of the chosen spin pair from the door. Although it is clear from Fig. 2(c) that , as well as are independent of in the present case, showing that the non-monotonic nature of with may not be connected to size-independence. Interestingly, irrespective of the phases, we find no trade-off between the frozen value of LN, i.e., , and the freezing terminal, , for a chosen pair of nearest-neighbor spins of the ATXY model. This is in contrast to the previous findings of the freezing phenomena of other quantum correlation measures titas_freeze ().

Note here that the choice of the OBC allows one to increase the distance of the chosen nearest-neighbor spin pair from the door, thereby increasing the value of in the DM and the PM phases. In contrast, use of PBC imposes an extra symmetry in the system, which incorporates an upper bound on the maximum distance of a chosen nearest-neighbor spin pair from the door, given by when is even, and when is odd. Therefore, OBC is advantageous over PBC in context of achieving longer freezing of entanglement in the DM and the PM phases of this model. Moreover a major difference between the dynamics of nearest-neighbour LN in the DM phase, and the same in the AFM or the PM phase is the occurrence of non-monotonic variation of LN with for in the latter ones, which is absent in the DM phase.

Invariance against thermal noise. Another remarkable feature of the freezing phenomena of entanglement in the ATXY model under RQI is its robustness against the choice of the environment-temperature. Although entanglement is known to be extremely fragile against thermal noise, in the present case, the quantitative results regarding the frozen bipartite entanglement and the freezing terminal, along with its scale-invariance, remain unchanged with a change in , although the entanglement decays more rapidly for when the environment temperature is high. However, note that to obtain the freezing phenomena, the temperature, , of the initial state of the system at has to be such that the value of the bipartite entanglement is non-zero at the beginning of the dynamics.

Freezing in other quantum spin models. As a second example, we consider the TXXZ model (Eq. (10) with ) with OBC, and find that the freezing phenomena is present in all the phases of the model, except the FM phase, where bipartite entanglement vanishes at due to the alignment of the spins, and does not revive when interaction with the environment is turned on. This indicates the possibility of the freezing phenomena to be generic to the phases of the low-dimensional quantum spin models. A typical freezing profile of is depicted in Fig. 2(d), where the system parameters are chosen from the AFM phase of the TXXZ model (). However, unlike the DM and the PM phase of the ATXY model, the scale-invariance of and the monotonic increase of with are absent in the AFM and the XY phases of the TXXZ model. For example, in the TXXZ model with and , we find that , although , which demonstrates non-monotonicity of with . The non-monotonic variation of with , and its dependence on the system-size, , are prominent from the dashed line-point curve in Fig. 2(c), which corresponds to a system-size . Interestingly, in the TXXZ model, starts increasing (decreasing) with when when is even (odd), indicating an alternating dynamics at for alternate nearest-neighbor spin pairs, which is not present in the ATXY model. However, similar to the ATXY model, the invariance of the freezing of entanglement with a change in the temperature of the environment remains unchanged in the TXXZ model also.

Keeping the model for system-environment interaction unchanged, freezing of entanglement is also found when XY anisotropy is introduced in the TXXZ model (), in the isotropic Heisenberg model (), and in different phases of the 1d model, represented by the Hamiltonian j1j2 (). Note that the model differs from the former models by the addition of the next-nearest-neighbour interaction term. This emphasizes the possibility of the freezing phenomenon to be generic to the phases of the 1d quantum spin models. However, scale-invariance of the variation of with , as well as the monotonic increase of with are absent in all the cases except the dimer and the paramagnetic phases of the 1d anisotropic XY model with uniform or alternating field, which counters the intuition that the environmental effects propagate, and decrease with increasing distance from the environment(s).

Collection of Environments: The above analysis can also be carried out by considering a collection of environments interacting with the system via a single door at spin “”, where the dynamics of the system is governed by Eq. (6). However, qualitatively similar results regarding freezing of entanglement, as that in the case of , are obtained for both the ATXY and TXXZ model. Interestingly, in the AFM phase of the ATXY model, the value of , for a fixed pair of nearest-neighbour spins, decreases monotonically with increasing approximately as . However, in the DM and the PM phases, and for fixed , both monotonic and non-monotonic variation of with increasing are found. The non-monotonic variation of with is abundant when one moves away from the phase boundaries. Also, counter-intuitively, with , with is found to remain non-zero for a longer time after , compared to the same in the case of , thereby indicating a robustness of entanglement against the increase of the number of environments.

### iii.2 Freezing in double-door dynamics

We now discuss the case of a double-door access of the environments to the system, via two chosen spins in the spin-chain, as discussed in Sec. II.2. We assume that the spins “” and “” in the chain are attached to one bath each, and are interacting with the baths via RQI. We do an analysis similar to the single-door dynamics discussed in the previous section, by obtaining the time evolved state as a solution of Eq. (9), where we keep all the values of the relevant parameters fixed at the same values as in the case of single-door dynamics. We first look into the dynamics of LN in different phases of the ATXY model. In all the three phases of the model, namely, the DM, PM, and AFM phases, LN corresponding to a chosen pair of nearest-neighbor spins, , with , is found to freeze for a finite time interval, . We demonstrate this with the example of , where for a particular point in the DM phase, the freezing profile of , is depicted in Fig. 3(a). Once again, the value of depends on the choice of the values of the relevant parameters characterizing the system, the environment, and the system-environment interaction. Similar to the single-door dynamics, LN corresponding to the spin pairs adjacent to the doors, i.e., and , are found to exhibit a highly oscillatory dynamics with several revivals and collapses, with no freezing phenomena.

In comparison to the results in the case of a single door, the value of the freezing terminal corresponding to a fixed pair of nearest-neighbor spins is found to decrease with the introduction of the second door. For example, we find that for , freezes upto in the single door case, while the freezing terminal in the double door scenario reduces to . However, the increasing trend of the values of , as the position of the chosen nearest-neighbor spin pair moves away from the doors, remains unchanged in the DM and the PM phases. Also, similar to the earlier case, the dynamics of LN in the DM phase can be distinguished from the same in the AFM or the PM phase by the non-monotonic behavior of for .

The scale invariance of the variation of with , in the DM and the PM phase of the ATXY model, i.e., , holds true even for the double-door dynamics, and a similar relation between the freezing terminal and the distance from the doors, as given in Eq. (13), exists, where the values of the fitting parameters are governed by the choice of the system parameters. However, similar to the single-door dynamics, no such scale invariance is observed in the AFM phase. Therefore, the DM and PM phases of the ATXY model serve as the generator of the scale-invariant freezing phenomena for both single-, and double-door dynamics with RQI. Also, as in the previous case, freezing of entanglement is found in the AFM and the XY phases of the TXXZ model also (see Fig. 3(b) for a typical freezing profile for different nearest-neighbour spin pairs), which strengthens the possibility of the phenomena being generic to the phases of low-dimensional quantum spin models. Note here that the robustness of the freezing phenomena of entanglement against environment-temperature remains unaltered even in the case of the double-door dynamics.

## Iv Concluding remarks

Entanglement is known to be an important resource in a large class of quantum information protocols. Therefore, finding robustness of entanglement under different decoherence models has attracted a lot of attention. In this paper, we show that under a specific type of environment that interacts with the system via repeated quantum interaction, bipartite entanglement of a quantum many-body system is constant, or neare-constant, within numerical accuracy, over a finite interval of time, called the freezing terminal. We call this feature as freezing of entanglement, and demonstrate this in the case of two paradigmatic quantum spin models, namely, the anisotropic XY model with uniform or alternating magnetic fields, and the XXZ model in an external magnetic field. In all three phases of the anisotropic XY model in external transverse uniform or alternating fields, namely, the dimer, paramagnetic, and antiferromagnetic phases, and in the AFM and the XY phases of the XXZ model in an external field, freezing or near-freezing of bipartite entanglement for a chosen pair of nearest-neighbor spins take place. Interestingly, in the dimer and the paramagnetic phases of the alternating-field XY model, the freezing terminal for a chosen spin-pair is found to be independent of the system size, allowing one to estimate the freezing duration with the distance of the chosen spin-pair from the environment. Such “scale-invariance” of the freezing phenomenon is absent in the antiferromagnetic phase of the alternating-field XY model, and in all the phases of the XXZ model in an external field. In the dimer and the paramagnetic phases of the alternating-field XY model, the duration of freezing increases monotonically with increasing the distance of the chosen spin-pair from the spin(s) that interact(s) directly with the environment, while this intuitive feature does not hold in the antiferromagnetic phase of the model as well as all the phases in the XXZ model in an external field. Interestingly, the freezing of entanglement in both the models is found to be unaffected with a change in the temperature of the environment, thereby exhibiting a robustness against thermal noise. Our results are expected to be important in developing schemes for protecting entanglement against decoherence.

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