{'fraction_non_alphanumeric': 0.04310319956146678, 'fraction_numerical': 0.023628862410201668, 'mean_word_length': 4.074963396778917, 'pattern_counts': {'":': 0, '<': 66, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Entanglement protection in higher-dimensional systems

The inevitable dissipative interaction of an entangled quantum system with its environment causes degradation in quantum correlations present in the system. This can lead to a finite-time disappearance of entanglement, which is known as entanglement sudden death (ESD). Here, we consider an initially entangled qubit-qutrit system and a dissipative noise which leads to ESD, and propose a set of local unitary operations, which when applied on the qubit, qutrit, or both subsystems during the decoherence process, cause ESD to be hastened, delayed, or avoided altogether, depending on its time of application. The physical implementation of these local unitaries is discussed in the context of an atomic system. The simulation results of such ESD manipulations are presented for two different classes of initially entangled qubit-qutrit systems. A prescription for generalization of this scheme to a qutrit-qutrit system is given. This technique for entanglement protection in the noisy environment is compared with other related techniques such as weak measurement reversal and dynamic decoupling.


Introduction
Quantum decoherence [i] is a ubiquitous and unavoidable phenomenon arising because of entanglement between quantum systems and their environment. Entanglement, on the other hand, is a fundamentally important phenomenon in the studies of quantum foundations and has great significance in quantum technologies. It is now seen as an indispensable resource in Quantum Information Processing (QIP) for various tasks such as quantum computation, teleportation, superdense coding, cryptography, etc., which are either impossible or less efficient using classical correlations [ii-iv].
The inevitable dissipative interaction of an entangled quantum system with its environment leads to an irreversible loss of single particle coherence as well as degradation of entanglement present in the system [v]. For some initial states, in the presence of an Amplitude Damping Channel (ADC), entanglement degrades asymptotically, whereas for others it can disappear in finite time, also known as early stage disentanglement or Entanglement Sudden Death (ESD), in literature [vi-viii]. Soon after its theoretical prediction, ESD was experimentally demonstrated in atomic [ix] and photonic [x] systems. The real world success of several quantum information, communication, and computation tasks depends on the resilience of entanglement from noises present in the environment, and longevity of the entanglement. Thus, ESD poses a practical limitation on QIP tasks. Therefore, strategies which make entanglement robust against the detrimental effects of the noise are of practical interest in QIP.
Owing to the simplicity of two-qubit entangled systems and its usefulness as a resource in QIP, there have been several theoretical proposals to combat decoherence and finite-time disentanglement in these systems [xi-xxiv]. Some of these proposals have been experimentally demonstrated in atomic, photonic, and solid state systems [xxv-xxxiv]. One of the entanglement protection schemes [xxi] considered a class of two-qubit entangled states which undergo ESD in the presence of an ADC. For such systems, a Local Unitary Operation (LUO), in this case the Pauli σ x operator also known as NOT operation, has been proposed. When LUO is applied on one or both the subsystems during the process of decoherence, it can hasten, delay, or completely avoid the ESD, all depending on the time when NOT operation is applied. This proposal was later transformed by our group into an all-optical experimental setup to study the effect of LUOs on the disentanglement dynamics in a photonic system [xxii]. This work proposed an experimentally feasible architecture for the implementation of ADC and the local NOT operation to suitably manipulate the ESD of a two-qubit system in a controlled manner.
Higher-dimensional entangled quantum systems (qudits, d ≥ 3) can offer practical advantages over the canonical two-qubit entangled systems in QIP protocols. These systems are more resilient to errors than their qubit counterparts in quantum cryptography, and they offer practical advantages; for example, increased channel capacity in quantum communication, enhanced security in QIP protocols, efficient quantum gates, and in the tests of the foundations of quantum mechanics [xxxv]. It is therefore important to study the effects of decoherence on these systems. Entanglement evolution has been studied in higher-dimensional systems present in the noisy environment, and despite the resilience of these systems to noise [xxxvi], ESD is established to be ubiquitous in all dimensions of the Hilbert spaces [xxxvii, xxxviii].
As a first step towards generalization to higher dimensional systems, a qubitqutrit system is the natural choice as it is intermediate in complexity to a qubitqubit and qudit-qudit system. Given the availability of full-fledged separability criteria for pure as well as mixed state qubit-qutrit systems, experimental feasibility of state preparation, and the significance of higher-dimensional systems in QIP, they become an important architecture towards understanding and controlling the effects of decoherence in higher-dimensional systems. Some attempts have been made to locally manipulate the disentanglement dynamics in qubit-qutrit [xxxix, xl], and qutrit-qutrit [xli-xliii] systems to retain the entanglement for longer duration. It is found that the quantum interference between two upper levels of the qutrit can also be used to control the disentanglement dynamics in higher-dimensional systems [xxxix, xliv].
Consider a practical scenario where Charlie prepares a bipartite entangled state for some QIP task and he has to send the entangled particles to Alice and Bob through a quantum channel which is noisy and can potentially cause disentanglement before the particles reach the two parties. In this scenario, we ask the following question: given a higher-dimensional bipartite entangled state which would undergo ESD in the presence of an ADC, can we alter the time of disentanglement by some suitable LUOs during the process of decoherence? As a possible answer to this question, we explore the generalization of the proposal in Ref. [xxi] for higher-dimensional systems, say qubitqutrit or qutrit-qutrit system, in the presence of an ADC. If such systems undergo ESD, we propose a set of LUOs, such that when they are applied on the subsystems during the process of decoherence, they manipulate the disentanglement dynamics, in particular, delay the time at which ESD occurs. Such a study was partially done in Ref. [xxxix], and here we generalize their scheme and propose a more general class of LUOs, which can always suitably manipulate ESD for an arbitrary initially entangled state. In the case of qubits, Ref. [xxi] found it to be NOT operation which is operation corresponding to the Pauli σ x operator. In this work, for qutrits, we propose a set of LUOs, which allow flipping the population between different levels of the qutrit. Depending on the combination of LUOs, and the time of their application, this method is shown to be able to hasten, delay or completely avoid the ESD in qubit-qutrit as well as qutrit-qutrit systems.
In the decoherence process, pure states evolve to mixed states. Therefore, a computable measure of entanglement for mixed state is required to quantify the entanglement. Negativity is such a measure, based on the Positive Partial Transpose (PPT) criterion due to Peres and Horodecki [xlv, xlvi]. For 2 ⊗ 2 and 2 ⊗ 3 dimensional systems, all PPT states are separable and Negative Partial Transpose (NPT) states are entangled. Negativity is defined as the sum of the absolute values of all the negative eigenvalues of the partially transposed density matrix with respect to one of the subsystems. While the enigmatic features of entanglement were predicted back in 1935, its witnessing and quantification continues to be at the forefront of current research. For a specific class of higher dimensional states ρ which are supported on d ⊗ D Hilbert space (d ≤ D) and with rank r(ρ) ≤ D, PPT criterion has been found to be necessary and sufficient condition for separability [xlvii]. But for more general higher-dimensional systems (d ⊗ D, ∀ d, D ≥ 3 & r(ρ) > D), PPT criterion is only a necessary but not sufficient condition for separability.
For these systems, we study Negativity Sudden Death (NSD), whose non-occurrence guarantees Asymptotic Decay of Entanglement (ADE). But for 3 ⊗ 3 dimensional PPT states with r(ρ) > 3 which have zero Negativity, we cannot comment on the separability and we need some other measure. For this purpose, we have used matrix realignment method for detecting and quantifying entanglement [xlviii-l] after state is found to have zero negativity using PPT criterion. Realignment criterion states that for any separable state ρ, the trace norm of the realignment matrix m| ⊗ µ|ρ R |n ⊗ |ν = m| ⊗ n|ρ|µ ⊗ |ν never exceeds one. Therefore, if the realigned negativity given by R(ρ) = max(0, ||ρ R ||−1), is non-zero then state ρ is entangled, where ρ R ij,kl = ρ ik,jl . This criterion can detect some of the bound-entangled states which may not be detected by PPT criterion. We use PPT as well as realignment criterion for entanglement detection.
The rest of the paper is organized as follows: In section (2), we briefly discuss the physical model in which qubit-qutrit entangled system can be realized in an atomic system and the natural presence of ADC in such a system. Next, Kraus operators governing the evolution of quantum system in the presence of ADC are given. We then propose a set of LUOs for qubit, and qutrit for manipulating the ESD, and their physical implementation for an atomic system is discussed. In section (3) and (4), we present our calculations implementing the proposed LUOs on two different class of qubit-qutrit system and their results. In section (5), we briefly comment on the generalization of this proposal to higher dimensions, taking an example of a two-qutrit system. In section (6), the results of ESD manipulations are compared and contrasted for a given initial state with respect to the choice of various LUOs. In the end, we conclude with the advantages and limitations of our proposal for ESD manipulation with other existing schemes.

Physical Model
Consider a hybrid qubit-qutrit entangled system in the presence of an ADC. The qubit and qutrit systems can be realized by a two-level, and a three-level atom in Vconfiguration (see Fig. 1), respectively, in an optical trap for instance. The qubit-qutrit entangled state can be prepared by methods discussed in Refs. [li-lv]. Afterwards, the entangled atoms are put into two cavities separated far from each other -by distances larger than the wavelength of the photons emitted from these atoms. The cavities are taken to be at absolute zero temperature and in vacuum state. The spontaneous emission due to atoms interacting locally with their cavities with electromagnetic field in vacuum state, forms identical but independent ADC.
The evolution of a qubit in the presence of an ADC is given by the following quantum map: where |0 S and |1 S are the levels of qubit, and |0 E and |1 E are vacuum state and single-photon Fock state of the environment (cavity), respectively. In the Born-Markov approximation: p = 1 − exp(−Γt), which is the probability of de-excitation of the qubit from the higher level |1 S to the lower level |0 S . The subscripts 'S' and 'E' refer to the system and environment, respectively. The quantum map of a qutrit (in V -configuration ) in the presence of an ADC is Figure 1. Two-level atom (qubit), and a three-level atom (qutrit) in V-configuration. The parameters p, p 1 , and p 2 denote the decay probabilities between the levels |1 S → |0 S of the qubit, and |1 S → |0 S , and |2 S → |0 S levels of the qutrit, respectively. The qutrit is taken to be in V-configuration such that the dipole transitions are allowed only between the levels |1 S ↔ |0 S and |2 S ↔ |0 S .
given by where |0 S , |1 S , and |2 S are three levels of the V-type qutrit. There are two decay probabilities for such a qutrit corresponding to the transitions |1 S → |0 S and |2 S → |0 S , and given by p 1 = 1 − exp(−Γ 1 t) and p 2 = 1 − exp(−Γ 2 t), respectively. Here, Γ 1 and Γ 2 represent the decay rate of levels |1 S , and |2 S , respectively. The Kraus operators governing the evolution of the system in the presence of ADC, for qubit (M i ) and qutrit (M i ), are obtained by tracing over the degrees of freedom of the environment from Eqs. (1) and (2) , respectively, and given by The Kraus operators for qubit-qutrit system (M ij ) are obtained by taking appropriate tensor products of the qubit and qutrit Kraus operators as follows: If initial state of the system is ρ(0) then evolution in the presence of ADC is given by, If such a system undergoes ESD, then manipulation of ESD is achieved by applying LUOs (NOT operations) on qubit and/or qutrit at appropriate time t = t n as shown in Fig. 2 Figure 2. Scheme for prolonging the entanglement in the presence of ADC. Initial state ρ i undergoes ADC decoherence then NOT operations are performed at t = t n and system is left to evolve in the ADC. The clock measuring the time is reset at t = t n and time after the application of NOT operation is measured by t = t − t n . Such a splitting of time is essential for later discussion.
The NOT operation on qubit is Pauli spin operator σ x . For qutrit, we propose here a set of operations which swap the populations between the two, or all the three levels of qutrit, namely trit-flip operations: F 01 , F 02 , F 102 , and F 201 , where In an atomic system, the NOT operation on the qubit (σ x ) can be applied by a π-pulse on the transition |0 S ↔ |1 S , which interchanges the population between the levels |0 S and |1 S of the qubit. The trit-flip operation F 01 ( F 02 ) on the qutrit can be applied by a π-pulse on the transitions |0 S ↔ |1 S ( |0 S ↔ |2 S ) of the qutrit, which interchanges the population between the respective two levels. The tritflip operation F 102 (F 201 ) on the qutrit can be realized by a π-pulse applied on the transition |1 S ↔ |0 S ( |2 S ↔ |0 S ) followed by another π-pulse to interchange the populations between |0 S and |2 S (|0 S and |1 S ). That is, by a series of two π-pulses Mathematically, the application of LUOs at p = p n can be represented by where U 1 = σ x or I 2 , and U 2 = F 01 , F 02 , F 102 , F 201 or I 3 . Let us label another set of Kraus operators (M ij ) with the parameter p replaced by p (t replaced by t , t = t − t n ); p = 1 − exp(−Γt ), and of the form similar to Eq. (5). These Kraus operators are applied to the state (8) to see the evolution of the system when it undergoes ADC after the application of LUOs as follows: When both the LUOs are identity operations, i.e., U 1 = I 2 and U 2 = I 3 , we have the uninterrupted system evolving in the ADC. The state of the system in this case is given by Our aim is to investigate whether the phenomenon of hastening, delay and avoidance of ESD also occurs in higher-dimensional entangled systems as seen in 2 ⊗ 2 systems [xxi] or not. For this purpose, we compare the Negativity of the system manipulated using LUOs (9) with that of the uninterrupted system (10). For the purpose of current analysis, we choose the decay probabilities for the qutrit as p 1 = 0.8p and p 2 = 0.6p, where p is the decay probability of the qubit.
3. X-type qubit-qutrit entangled system: State-I Let us consider the one-parameter qubit-qutrit entangled state given by, where 0 ≤ x < 1/3. Assuming that both the subsystems, qubit as well as qutrit, undergo local and independent ADC, evolution of the system is given by Eq.(6). For state (11), the mathematical form of Negativity in the presence of ADC and in terms of x and p is given as Evolution of the entanglement vs. ADC parameter (p) for 0 ≤ x < 1/3 is shown in Fig. 3. The entangled state (11) undergoes ADE for 0 ≤ x ≤ 0.2, and ESD for 0.2 < x < 1/3. This can be easily verified by Eq. (12).

ESD in the presence of ADC
We choose x = 0.25 such that the initial state (11) undergoes ESD at p = 0.6168. The mathematical form of Negativity in terms of p and p is given as Using Eq. (13), the plot of Negativity vs. ADC probability (p, p ) for the state (11) is shown in Fig. 4. For p = 0, ESD occurs at p = 0.6168 and for arbitrary values for p, ESD occurs along the non-linear curve in pp plane as shown in Fig. 4. For entanglement protection, we apply different combinations of LUOs and their effects are discussed below.
3.2. σ x applied to qubit and F 01 applied to qutrit A NOT operation (σ x ) is applied to the qubit and trit-flip operation F 01 applied to qutrit part of the state (11) at p = p n as in Eq. (9).
For uninterrupted system, end of entanglement p depends on p as follows: When LUOs are applied at p = p n , the end of entanglement p depends on p n as follows: The Fig. 5 shows the non-linear curvature in p vs. p or p n for ESD (red curve) and its manipulation (green curve). The manipulation leads to avoidance of ESD for 0 ≤ p n ≤ 0.0615, delay for 0.0615 < p n < 0.1641, and hastening of ESD for 0.1641 < p n < 0.6168 as the green curve dips below red curve in this range. This can be easily verified through Eqs. (14) and (15).  Figure 5. Plot shows the non-linear curvature in p vs. p or p n for ESD (red curve) and its manipulation (green curve) such that the NOT operation (σ x ) is applied on the qubit and trit-flip operation F 01 applied on the qutrit at p = p n for x = 0.25. The action of LUOs give rise to avoidance for 0 ≤ p n ≤ 0.0615, delay for 0.0615 < p n < 0.1641, and hastening of ESD for 0.1641 < p n < 0.6168 as the green curve lies below red curve in this range.

F 01 applied to qutrit only
The trit-flip operation F 01 is applied to the qutrit part of the state (11) at p = p n as in Eq. (9). The Fig. 6 shows the non-linear curvature in p vs. p or p n in ESD (red curve) and its manipulation (green curve). The manipulation leads to avoidance for 0 ≤ p n ≤ 0.2941, delay for 0.2941 < p n < 0.6168, and hastening of ESD does not occur in this case.  Figure 6. Plot shows the non-linear curvature in p vs. p or p n in ESD (red curve) and its manipulation (green curve) such that trit-flip operation F 01 is applied on the qutrit at p = p n for x = 0.25. The action of LUOs give rise to avoidance for 0 ≤ p n ≤ 0.2941, delay for 0.2941 < p n < 0.6168, and hastening of ESD does not occur is this case.
The LUOs σ x ⊗ F 102 and I 2 ⊗ F 102 applied on the state (11) lead to same effect as σ x ⊗ F 01 and I 2 ⊗ F 01 , respectively. The other combination of LUOs such as σ x ⊗ F 02 , σ x ⊗ F 201 , σ x ⊗ I 2 , I 2 ⊗ F 02 , and I 2 ⊗ F 201 applied on the state (11) give rise to only hastening of ESD in the entire range 0 < p n < 0.6168.
The physical reason behind the action of different LUOs resulting in hastening, delay, or avoidance of ESD for this class of state can be understood as follows: when we apply the trit-flip operation F 01 (for example) on the system (11) after it has evolved in the ADC, it changes the instantaneous population between different levels of the qutrit in such a way that the elements of the density matrix (i) ρ 11 and ρ 22 , (ii) ρ 44 and ρ 55 get swapped, and (iii) the coherence term ρ 34 (ρ 43 ) change their position to ρ 35 (ρ 53 ). When this flipped state evolves in the ADC, new coherence terms ρ , which looks similar to the earlier condition for ESD but it depends on new terms of the density matrix after the LUOs. The instantaneous population of different levels of the qubit-qutrit system depends on the decay rate of the different levels of the qubit-qutrit system, the time when LUO (tritflip operation) is applied, and time elapsed after the application of LUO. Again, due to cumulative evolution of different terms of the density matrix, when N becomes zero, system becomes separable. The basic idea behind this entanglement protection scheme is to choose correct combination of LUOs depending on the decay rate of different levels of the qubit-qutrit system and the time of application of LUOs such that the state after the application of LUOs results in the delay or avoidance of ESD.

X-type qubit-qutrit entangled system: State-II
Let us consider another class of one-parameter qubit-qutrit entangled state given by, where 1/3 < x ≤ 1/2. Assuming that both the subsystems; qubit as well as qurit, undergo local and independent ADC, evolution of the system is given by Eq. (6). For state (16), evolution of the entanglement vs. ADC parameter (p) for 1/3 ≤ x ≤ 1/2 is computed. The initial entangled state (16) undergoes ESD in the entire range 1/3 < x ≤ 1/2. For p = 0, ESD occurs at p = 0.8452 and for arbitrary values for p, ESD occurs along the non-linear curve in pp plane similar to Fig. 4. We choose x = 0.5 such that ESD occurs at p = 0.8452, and study the effect of NOT operation (σ x ) applied to the qubit, and/or trit-flip operations F 01 , F 02 , F 102 , F 201 applied to the qutrit, in manipulating the ESD. The effect of different combinations of LUOs in entanglement protection are discussed below.

σ x applied to qubit and F 01 applied to qutrit
The NOT operation (σ x ) is applied to the qubit and trit-flip operation F 01 is applied to the qutrit part of the state (16) at p = p n as in Eq. (9). The Fig. 7 shows the non-linear curvature in p vs. p or p n in ESD (red curve) and its manipulation (green curve). The manipulation leads to avoidance for 0 ≤ p n ≤ 0.3586, delay for 0.3586 < p n < 0.4177, and hastening of ESD for 0.4177 < p n < 0.8452 as green curve lies below the red curve in this range. . Plot shows the non-linear curvature in p vs. p or p n for ESD (red curve) and its manipulation (green curve) such that the NOT operation (σ x ) is applied on the qubit and trit-flip F 01 applied on the qutrit at p = p n for x = 0.5. The action of LUOs give rise to avoidance for 0 ≤ p n ≤ 0.3586, delay for 0.3586 < p n < 0.4177, and hastening of ESD for 0.4177 < p n < 0.8452 as green curve lies below the red curve in this range.

σ x operation applied to qubit only
The NOT operation (σ x ) is applied to the qubit part of the state (16) at p = p n as in Eq. (9). The Fig. 8 shows the non-linear curvature in p vs. p or p n in ESD (red curve) and its manipulation (green curve). The manipulation leads to avoidance for 0 ≤ p n ≤ 0.2309, delay for 0.2309 < p n < 0.2964, and hastening of ESD for 0.2964 < p n < 0.8452 as green curve lies below the red curve in this range.  Figure 8. Plot shows the non-linear curvature in p vs. p or p n in ESD (red curve) and its manipulation (green curve) such that NOT operation is applied only on the qubit at p = p n for x = 0.5. The action of LUOs give rise to avoidance for 0 ≤ p n ≤ 0.2309, delay for 0.2309 < p n < 0.2964, and hastening of ESD for 0.2964 < p n < 0.8452 as green curve lies below the red curve in this range.

F 01 applied to qutrit only
The trit-flip operation (F 01 ) is applied only to the qutrit part of the state (16) at p = p n as in Eq. (9). The Fig. 9 shows the non-linear curvature in p vs. p or p n in ESD (red curve) and its manipulation (green curve). The manipulation of ESD leads to avoidance for 0 ≤ p n ≤ 0.7143, and delay of ESD for 0.7143 < p n < 0.8452 as green curve lies above the red curve but less than one in this range. The hastening of ESD does not occur in this case. . Plot shows the non-linear curvature in p vs. p or p n in ESD (red curve) and its manipulation (green curve) such that only trit-flip operation F 01 is applied on the qutrit at p = p n for x = 0.5. The action of LUOs give rise to avoidance for 0 ≤ p n ≤ 0.7143, and delay of ESD for 0.7143 < p n < 0.8452 as green curve lies above the red curve but less than one in this range. The hastening of ESD does not occur in this case.

F 02 applied to qutrit only
The trit-flip operation F 02 is applied to the qutrit part of the state (16) at p = p n as in Eq. (9). The Fig. 10 shows the non-linear curvature in p vs. p or p n in ESD (red curve) and its manipulation (green curve). The manipulation leads to avoidance for 0 ≤ p n ≤ 0.2032, delay for 0.2032 < p n < 0.2693, and hastening of ESD for 0.2693 < p n < 0.8452 as green curve lies below the red curve in this range.  Figure 10. Plot shows the non-linear curvature in p vs. p or p n in ESD (red curve) and its manipulation (green curve) such that only trit-flip operation F 02 is applied on the qutrit at p = p n for x = 0.5. The action of LUOs give rise to avoidance for 0 ≤ p n ≤ 0.2032, delay for 0.2032 < p n < 0.2693, and hastening of ESD for 0.2693 < p n < 0.8452 as green curve lies below red curve in this range.

F 201 applied to qutrit only
The trit-flip operation F 201 is applied to the qutrit part of the state (16) at p = p n as in Eq. (9). The Fig. 11 shows the non-linear curvature in p vs. p or p n in ESD (red curve) and its manipulation (green curve). The manipulation leads to avoidance for 0 ≤ p n ≤ 0.2059, delay for 0.2059 < p n < 0.2676, and hastening of ESD for 0.2676 < p n < 0.8452 as green curve lies below the red curve in this range.  Figure 11. Plot shows the non-linear curvature in p vs. p or p n in ESD (red curve) and its manipulation (green curve) such that only trit-flip operation (F 201 ) is applied on the qutrit at p = p n for x = 0.5. The action of LUOs give rise to avoidance for 0 ≤ p n ≤ 0.2059, delay for 0.2059 < p n < 0.2676, and hastening of ESD for 0.2676 < p n < 0.8452 as green curve lies below the red curve in this range.
The LUOs σ x ⊗ F 102 and I 2 ⊗ F 102 applied on the state (16) lead to same effect as σ x ⊗ F 01 and I 2 ⊗ F 01 , respectively. The other combination of LUOs such as σ x ⊗ F 02 , and σ x ⊗ F 201 applied on the state (16) only give rise to hastening of ESD in the entire range 0 < p n < 0.8452.

Generalization to qutrit-qutrit system
In the framework of ADC, the dissipative interaction of system and environment causes the flow of population of the system from excited state to ground state. Therefore, any LUO which reverses this effect will change the disentanglement time. Thus, generalization of the above proposal to higher-dimensions is reasonably straight forward. For example, the form of unitary operations for two-qutrits will be same as in Eq. (7). However, lack of a well defined universal entanglement measure for mixed entangled states of dimension greater than six, makes it difficult to study the disentanglement dynamics in these systems because even an initial pure entangled state becomes mixed during the evolution.
For the purpose of our study, we use Negativity as a witness for entanglement, as in general, qutrit-qutrit entanglement is not known to be characterized fully, and that negativity is a sufficient but not necessary condition for entanglement. Thus, if negativity undergoes asymptotic decay then this implies that ESD does not happen.
However, if negativity undergoes sudden death (NSD), this may be suggestive of (but does not imply) ESD. Here, we take a = 1(p 1 = ap) and b = 0.75 (p 2 = bp).
Let us consider an initially entangled two-qutrit system in the presence of ADC as given below.
where 0 ≤ x < 1/3. For two-qutrit system, Kraus operators are given by Assuming that both the qutrits suffer identical but independent ADC, evolution of the system is given by, Evolution of the entanglement vs. ADC parameter (p) for 0 ≤ x < 1/3 is computed. It is found that the entangled state (17) undergoes NSD in the entire range of xparameter for 0 < x < 1/3. Let us label another set of Kraus operators (M ij ) with the parameter p replaced by p (t replaced by t , t = t − t n ); p = 1 − exp(−Γt ), and of the form similar to (18), and apply it to the state (17) to get the state of the uninterrupted system evolving in the ADC.
We choose x = 0.25 such that the initial state (17) undergoes NSD at p = 0.3636. For p = 0, NSD occurs at p = 0.3636 and for arbitrary values for p, NSD occurs along the non-linear curve in pp plane similar to Fig. 4.
For protecting entanglement, trit-flip operation (F 01 ) is applied to only one of the qutrits at p = p n as follows, Evolution of the system afterwards in ADC is given by, The Fig. 12 shows the non-linear curvature in p vs. p or p n for NSD (red curve) and its manipulation (green curve). The action of LUO (F 01 ) gives rise to avoidance of NSD for 0 ≤ p n ≤ 0.0238, delay for 0.0238 < p n < 0.3636, and hastening of NSD does not occur for this choice of parameter.  Figure 12. Plot shows the non-linear curvature in p vs. p or p n in NSD (red curve) and its manipulation (green curve) such that the trit-flip operation (F 01 ) is applied on only one of the qutrits at p = p n for x = 0.25. The action of LUOs give rise to avoidance for 0 ≤ p n ≤ 0.0238, delay for 0.0238 < p n < 0.3636, and hastening of NSD does not occur in this case.

Summary and discussion
We have proposed a set of Local Unitary Operations (LUOs) for qubit-qutrit system undergoing Entanglement Sudden Death (ESD) in the presence of an amplitude damping channel (ADC), such that when they are applied locally on one or both subsystems, then depending on the initial state, choice of the operation, and its time of application, one can always suitably manipulate the ESD. We have considered two different classes of initially entangled qubit-qutrit systems which undergo ESD, and we find that for a given initial state, one can always find a suitable combination of LUOs, such that when applied at appropriate time it can always delay the time of ESD, and therefore facilitate the tasks which would not have been possible under shorter entanglement lifetime.
The results of different combinations of LUOs applied on the qubit-qutrit system on the manipulation of ESD for two different initially entangled states are summarized in the table (1) below. In some cases, ESD can be hastened, delayed, as well as avoided, whereas in other cases, it can be only delayed and avoided, or ESD can be only hastened. Due to symmetry in the population of initial entangled state, the NOT operations σ x ⊗ F 102 and I 2 ⊗ F 102 applied on the either states lead to same effect as σ x ⊗ F 01 and I 2 ⊗ F 01 , respectively. Based on the results in table (1), the following LUOs are advisable: for example, for state-II, σ x on the first qubit and no action on the qutrit suffices to guarantee avoidance of ESD provide it is applied sufficiently early. Since this is the simplest of all possible combination of operations, this may be called the optimal in terms of gate operations. For state-I, all operations allowing avoidance are two-sided. It is worth noting here that although the noise is acting on both the subsystems, still even LUO applied on only one of the subsystems can suitably delay or avoid the ESD as in the case of a two qubit system.
Such a scheme will find application where two parties, say Alice and Bob, share an entangled pair for some quantum information processing task and they know a- . priori that ADC is present in the environment, and therefore they can decide whether they are faced with the prospect of ESD. Then, they can locally apply suitable LUOs at appropriate time to delay or avoid the ESD. Our proposed scheme for preserving entanglement longer will also find application in entanglement distillation protocols. Furthermore, these LUOs can also be effective in protecting higher-dimensional entangled quantum systems in the presence of generalized amplitude damping channel as in the case of two-qubit system [xxiv]. Applying LUOs in the presence of generalized ADC towards the entanglement protection of higher dimensional systems could be an interesting work for future study. However, unlike ADC or generalized ADC, LUOs are not effective in protecting entanglement in the presence of bit-flip or phase-flip channel or both and the argument is as follows: under the action of ADC, population of a qubit (or qutrit) system flows from excited state to the ground state. The action of NOT operation (or more general LUO) is to reverse this effect by pumping the population from the ground to an excited state. While LUOs don't change the amount of entanglement present in the system, they change the subsequent entanglement dynamics leading to hastening, delay or avoidance of ESD. This is the reason for effectiveness of LUOs in controlling the disentanglement dynamics in the presence of ADC. However, same reasoning doesn't hold true for the bit-flip or phase flip channel as these are nondissipative noises.
Other schemes which aim to protect entanglement in a noisy environment include dynamic decoupling [lvi], weak measurement (partial-collapse measurement) and quantum measurement reversal [lvii], and quantum Zeno effect [lviii]. Dynamic decoupling uses a sequence of π-pulses to protect the quantum states from noise. This scheme can potentially freeze the initial state and thereby preserve the quantum coherence stroboscopically to infinite time. The weak measurement and reversal scheme is based on applying a weak measurement prior to decoherence and then probabilistically reversing this operation on the decohered state. The measurement induced quantum Zeno effect [lviii] for entanglement protection in cavity QED architecture, on the other hand, utilizes a very simple method of monitoring the population of the cavity modes that results in the entanglement protection exceeding its natural life-time. In our proposed method, LUOs alter the state in such a way that the decoherence effect on entanglement is minimized, i.e., for the states which were initially undergoing ESD, either time of ESD is delayed or it is averted altogether. Resource-wise, our scheme is simpler compared to the aforementioned schemes as it requires just one time intervention using LUOs but unlike the weak measurement and reversal protocol it does not restore the initial state after the decoherence.
To conclude, we have presented a scheme based on local unitary operations to protect entanglement from undergoing sudden death in the presence of amplitude damping channel for higher-dimensional systems. We have also compared and contrasted our entanglement protection scheme with other existing schemes. An interesting future line of theoretical study could be the entanglement protection in the presence of generalized amplitude damping channel in higher-dimensional systems. Our scheme will also attain more practical significance through commensurate future experiments.