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Hierarchicalenvironmentassisted nonMarkovian and its effect on thermodynamic properties
EPJ Quantum Technology volume 8, Article number: 9 (2021)
Abstract
We consider a microscopic collision model, i.e., a quantum system interacts with a hierarchical environment consisting of an auxiliary system and a reservoir. We show how the nonMarkovian character of the system is influenced by the coupling strength of systemauxiliary system and auxiliary systemreservoir, coherence of environment and initial systemenvironment correlations. And we study the nonMarkovianity induced by coherence of environment from the perspective of energy, further the relationship between information backflow and energy flux is obtained. Then we study the effect of nonMarkovianity on thermodynamic properties. By studying the entropy change of system especially that from heat exchanges with the environment, we reveal the essence of entropy change between positive and negative values during nonMarkovian evolution is due to the contribution of heat flux induced by coherence. And compared with the case of Markovian dynamics, we observe that the entropy production decreases in some specific time intervals under nonMarkovian dynamics induced by the coupling strength. And this is different to the case of nonMarkovianity caused by initial systemenvironment correlation, that we show the possibility of positive entropy production during the whole dynamics.
Introduction
The study of open quantum systems is of great importance in quantum information and computation recently. Because the dynamics of open quantum systems is greatly affected by its environment, and the environments are often very complex, solving the dynamics of open quantum systems has always been a challenge. The Markovian approximation is important to describe the dynamics of open quantum system either in terms of maps and Kraus operators or in terms of master equations [1]. One advantage of this approximation is that the dynamics of the system will be a Markovian process and can be described by a standard Markovian master equation.
However, it has been shown that the Markovian approximation fails in many situations [2–5], and the nonMarkovian dynamics have been received considerable attention and have been extensively studied recently [6–13]. Based on this, several measures of nonMarkovianity(NM) have been proposed [14–16]. With the help of these measures, one can claim that an evolution is nonMarkovian if a nonzero degree of NM is detected. These measures have been applied to many models to investigate their nonMarkovian characteristics [17–24]. Furthermore, the transition from Markovian to nonMarkovian dynamics has also been theoretically and experimentally implemented based on these measures [25–31]. For example, Brito et al. have implemented the transitions from Markovianity to NM by preparing different system initial states or dynamically manipulating the subsystem coupling [26]. Ma et al. have showed how the nonMarkovian character of the system is influenced by the coupling strength between the qubit and cavity and the correlation time of the reservoir, and they have found a phenomenon whereby the qubit Markovian and nonMarkovian transition exhibits a anomalous pattern in a parameter space depicted by the coupling strength and the correlation time of the reservoir [27]. In Ref. [31], initial systemenvironment correlations have been showed to substantially increase the distance between two qubit states evolving to longtimelimit states according to exact nonMarkovian dynamics. And in Ref. [32], it have showed that the trace distance between two states of the open system can increase above its initial value when the system and its environment are initially correlated. In particular, Smirne et al. [33] have provided experimental evidence of the behavior showed in Ref. [32]. All of these factors together make it difficult to understand their independent role in the nonMarkovian dynamics of open quantum system. However we have not seen any reports about the effect of coherence of environment on the nonMarkovian dynamics. Thus an interesting question concerns how the independent role of these factors to influence the system dynamics, specifically systemenvironment coupling, initial systemenvironment correlations and the intraenvironment coherence.
As one of the representative models for studying open quantum systems, collision model, also called repeated interaction framework, has been extensively studied during recent decades [34–40]. A quantum collision model is a microscopic framework to describe the open dynamics of a system interacting with a reservoir assumed to consist of a large collection of smaller constituents (ancillas), and the system is assumed to interact (collide) sequentially with an ancilla at each time step [34, 41, 42]. It offers a bottomtop description of an environment, where one has precise theoretical control of the microscopic aspects that give rise to macroscopic characteristics of the reservoir. The collision model has been applied in nonMarkovian dynamics widely [43–57]. For example, Ciccarello et al. have endowed the reservoir with memory by introducing interancillary collisions between next systemancilla interactions [44]. Bernardes et al. have investigated the Markovian to nonMarkovian transitions in collision models by introducing correlations in the state of the environment [46]. In Ref. [47], the use of collision model with interenvironment swaps has displayed a signature of strongly nonMarkovian dynamics that is highly dependent on the establishment of systemenvironment correlations. Campbell et al. have also identified the relevant systemenvironment correlations that lead to a nonMarkovian evolution in a collision model [52]. Kretschmer et al. have studied the applicability of collisional models for nonMarkovian dynamics of open quantum systems, and they have discussed the possibility to embed nonMarkovian collision model dynamics into Markovian collision model dynamics in an extended state space [48]. Lorenzo et al. have shown that the composite quantum collision models they studied can accommodate some known relevant instances of nonMarkovian dynamics [53]. In Ref. [54], a nonMarkovian dynamics is established under a structured environment based on collision model. In Ref. [56], it has studied the effects of different strategies of systemenvironment interactions and states of the blocks on the nonMarkovianities by introducing a block (a number of environment particles) as the unit of the environment instead of a single particle. Ref. [57] has found that the information is scrambled if the memory and environmental particles are alternatively squeezed along two directions which are perpendicular to each other.
Recently, the relation between NM and thermodynamics in open quantum system has attracted considerable attention. In Ref. [58], the heat flux has exhibited a nonexponential time behavior in the case of nonMarkovian dynamics of the subsystem. In Ref. [59], the heat flux changes between positive to negative values for a nonMarkovian evolution of the subsystem, which leads to a violation of opensystem formulation of Landauer’s principle for the heat and entropy fluxes. A similar result has also been obtained in Refs. [60, 61] that the Landauer’s principle is violated in nonMarkovian dynamics. Raja et al. have investigated how memory effects influence the ability to perform work on the driven qubit, and they have showed that the average work performed on the qubit can be used as a diagnostic tool to detect the presence or absence of memory effects [29]. Abiuso et al. have found that the nonMarkovian effects can fasten the control and improve the power output of a quantum thermal engine [62]. Pezzutto et al. have addressed the effects that NM of the opensystem dynamics of the work medium can have on the efficiency of the thermal machine [63]. Katz et al. have studied the performance characteristics of a heat rectifier and a heat pump in a nonMarkovian framework [64]. Ref. [65] has studied the effects of environmental temperature on the NM of an open quantum system by virtue of collision models.
As one of the important thermodynamic quantities, the entropy production and the associated entropy production rates are crucial in the thermodynamic characterization of a given process. And the exploration of the relation between NM and entropy production has provoked great interest recently [66–73]. Refs. [68, 69] have shown that the entropy production can become transiently negative in the nonMarkovian dynamics compared with the Markovian case, and the transient negativity of the entropy production rate is a sufficient sign of NM [69]. Further, Strasberg et al. have explored the link between a negative entropy production rate and NM precisely by showing under which conditions a negative entropy production rate implies NM and when it does not [70]. And Ref. [71] has shown that the possibility of positive entropy production rate with the initial correlation between the system and its heat reservoir.
In this paper, we consider a twolevel system coupled to a structured environment consisting of a auxiliary system and a reservoir, and the reservoir is of a large collection of initially uncorrelated systems which we call ancillas (see Fig. 1). Based on this structured environment model, there can be different factors to influence the nonMarkovian character of the system, and we mainly consider the effects of coherence of environment and initial systemenvironment correlations on system dynamics. And we study the relationship between NM and thermodynamic properties. For example, information backflow and energy flux, nonMarkovian dynamics and entropy change of system, including entropy flux (entropy change of system induced by heat exchanges with environment) and entropy production.
Methods
Model and solution
We consider a qubit (system S) couples to a hierarchical environment, which contains a auxiliary qubit \(A_{Q}\) and a collection of N identical noninteracting ancillas (qubits) \(\{\mathcal{R}_{1},\mathcal{R}_{2},\ldots,\mathcal{R}_{N}\}\) that consists a reservoir \(\mathcal{R}\), and this reservoir is in the product state \(\eta _{\mathrm{tot}}=\otimes ^{N}_{j=1}\eta _{j}\). In this way, the auxiliary qubit \(A_{Q}\) and the reservoir hierarchically constitute the whole big reservoir E, which is called the environment of system S. And the general scheme is illustrated in Fig. 1. The Hamiltonians of system and a generic environment particle \(E_{j}\) including the auxiliary qubit and ancillas are
where \(\hat{\sigma }_{z}\) is the Pauli matrices and we set \(\hbar =1\) throughout this paper.
The evolution of system S and its interaction with the environment are proceeded as follows. S interacts with the environment first: Specifically S and \(A_{Q}\) interact and then \(A_{Q}\) collides with the individual ancilla of the reservoir. As the assumption of a big reservoir \(\mathcal{R}\) that \(A_{Q}\) never interacts twice with the same ancilla, i.e., at each collision the state of the ancilla is refreshed. And this process is implemented through the unitary operator
where \(\hat{U}_{S,A_{Q}}=e^{i\hat{H}^{\mathrm{int}}_{S,A_{Q}}\tau }\), \(\hat{V}_{A_{Q},\mathcal{R}_{j}}=e^{i\hat{H}^{\mathrm{int}}_{A_{Q}, \mathcal{R}_{j}}\tau }\). Here \(\hat{H}^{\mathrm{int}}_{S,A_{Q}}\) and \(\hat{H}^{\mathrm{int}}_{A_{Q},\mathcal{R}_{j}}\) are the interaction between ‘\(SA_{Q}\)’, ‘\(A_{Q}\mathcal{R}_{j}\)’ respectively, and τ is the interaction time.
In our model, we consider a coherent interaction between the bipartite systems including ‘\(S{A}_{Q}\)’ and ‘\(A_{Q}\mathcal{R}_{j}\)’, i.e., a mechanism that can be described by a Hamiltonian model of some form, specifically in this paper we suppose that the interaction Hamiltonian is
where \(\hat{\sigma }^{S}_{i}\), \(\hat{\sigma }^{A_{Q}}_{i}\) and \(\hat{\sigma }^{\mathcal{R}_{j}}_{i}\) (\(i=x,y,z\)) are the Pauli matrices, and \(g_{1(2)}\) is a coupling constant. And we use the result [34]
where \(\hat{\mathbb{I}}\) is the identity operator, and \(\hat{S}^{\mathrm{sw}}\) is the twoparticle swap operator, i.e., it is the unitary operation whose action is \(\psi _{1}\rangle \otimes \psi _{2}\rangle \rightarrow \psi _{2} \rangle \otimes \psi _{1}\rangle \) for all \(\psi _{1}\rangle ,\psi _{2}\rangle \in \mathbb{C}^{2}\). We can now write the unitary timeevolution operator \(\hat{U}_{S,A_{Q}}\) in Eq. (2) as
where \(\gamma =2g_{1}\tau \) is a dimensionless interaction strength. And when \(\gamma =0\) Eq. (5) is reduced into an identity operator and indicates that there is no interaction between S and \(A_{Q}\); and when \(\gamma =\pi /2\) Eq. (5) is reduced into a fully swap operator and represents a complete exchange of quantum state information between S and \(A_{Q}\). Thus in the range of \(\gamma \in [0,\pi /2]\), the larger the γ, the stronger the coupling. And in the ordered basis \(\{00\rangle , 01\rangle , 10\rangle , 11\rangle \}\), \(\hat{S}_{S,A_{Q}}^{\mathrm{sw}}\) in Eq. (5) reads [74]
Similarly \(\hat{V}_{A_{Q},\mathcal{R}_{j}}\) in Eq. (2) can be written as
with \(\delta \neq \gamma \), in general, and the analog of the operations introduced above applies to \(\hat{\mathbb{I}}_{A_{Q},\mathcal{R}_{j}}\) and \(\hat{S}^{\mathrm{sw}}_{A_{Q},\mathcal{R}_{j}}\) (swap gate between \(A_{Q}\) and \(\mathcal{R}_{j}\)). As mentioned above the dynamics of system S consists of sequential systemenvironment interaction and each step is treated in the following process: First S and \(A_{Q}\) interact and then subsequently \(A_{Q}\) collides with \(\mathcal{R}_{j}\) (one of the ancillas in \(\mathcal{R}\)). Thus the system is brought from step n to step \(n+1\) through the process
where \(\rho ^{S,A_{Q}}_{n}\) is the state of ‘\(SA_{Q}\)’ after the nth interaction. Hence after the \((n+1)\)th interaction, we can obtain the reduced system state, \(\rho ^{S,A_{Q}}_{n+1}=\operatorname{Tr}_{\mathcal{R}}[\rho ^{\mathrm{SE}}_{n+1}]\) (the state of ‘\(SA_{Q}\)’), \(\rho ^{S}_{n+1}=\operatorname{Tr}_{A_{Q}}[\rho ^{S,A_{Q}}_{n+1}]\) (the state of S) and \(\rho ^{A_{Q}}_{n+1}=\operatorname{Tr}_{S}[\rho ^{S,A_{Q}}_{n+1}]\) (the state of \(A_{Q}\)), where \(\operatorname{Tr}_{x}[\cdots ]\) means the trace of x degree of freedom.
NM
The trace distance between two quantum states is one of the most important measures of distinguishability of quantum states [74], which is given by
where \(A=\sqrt{A^{\dagger }A}\) for any operator A. It is obvious that for any pair of states \(\rho _{1}\) and \(\rho _{2}\) the trace distance satisfies the inequality \(0\leqslant \mathcal{D}(\rho _{1},\rho _{2})\leqslant 1\). For the time evolution of a quantum state described by a tracepreserving completely positive map, the trace distance is always less than or equal to the initial value [75]; that is,
In particular, for a Markovian evolution it can always be represented by a dynamical semigroup of completely positive and tracepreserving maps [76], and we obtain the inequality
for any positive τ, which indicates that the trace distance decreases monotonically with time. The decrease of trace distance corresponds to the reduction of distinguish ability between the two states, and this could be interpreted as an outflow of information from the system to the environment. In contrast to this, if the time derivative of the trace distance becomes positive in some time intervals, the time evolution is nonMarkovian [14, 17]. Furthermore, if the trace distance exceeds the initial value, the time evolution cannot be described by a tracepreserving completely positive map. Based on this, a measure of NM can be defined by [14]
where \(\sigma (t,\rho _{1}(0),\rho _{2}(0))=\frac{\mathrm{d}}{\mathrm{d}t} \mathcal{D}(\rho _{1}(t),\rho _{2}(t))\). Conceptually, \(\mathcal{N}\) accounts for all regions where the distance between two arbitrary input states increases, thus witnessing a backflow of information from the environment to system. And in this case, an evolution is nonMarkovian if and only if \(\mathcal{N}>0\).
As the evolution in our model proceeds in discrete steps, we will employ the discretized version of Eq. (12), which is obtained as [17, 77]
with \(\sigma ^{+}=\bigcup_{n}(n,n+1)\) is the union of all the interaction steps \((n,n+1)\) within which \(\mathcal{D}(\rho _{1,n+1},\rho _{2,n+1})\mathcal{D}(\rho _{1,n}, \rho _{2,n})>0\), and \(\{\rho _{1,n+1},\rho _{2,n+1}\}\) a pair of state of system obtained starting from the corresponding pair of orthogonal state \(\{\psi _{+}\rangle , \psi _{}\rangle \}\) after \(n+1\) steps of our protocol,
where \(\theta \in [0,\frac{\pi }{2}]\) and \(\varphi \in [0,2\pi ]\). The maximization in Eq. (13) performed over all possible values of θ and φ, i.e., all possible orthogonal pairs of initial system states.
Results and discussion
NonMarkovian dynamics of the system
In this section we study how the system dynamics can be affected by different ways, including the coupling strength between the bipartite systems (‘\(S{A}_{Q}\)’ and ‘\(A_{Q}\mathcal{R}\)’), coherence of the environment and initial systemenvironment correlation. We consider the initial state of each ancilla of reservoir \(\mathcal{R}\) as
where \(p\in [0,1]\), \(\psi \rangle =\frac{1}{\sqrt{Z}}(e^{\frac{1}{4}\omega _{E}\beta }0 \rangle +e^{i\phi _{1}+\frac{1}{4}\omega _{E}\beta }1\rangle )\) with a relative phase \(\phi _{1}\), and \(\rho _{\beta }\) is the thermal state assumed to be of canonical equilibrium form, i.e., \(\rho _{\beta }=\frac{1}{Z}e^{\beta \hat{H}_{E}}\). Here \(\beta =1/T\) and are the inverse temperature and the partition function respectively. Note that the diagonal elements of states \(\rho _{E}\) and \(\rho _{\beta }\) are identical, and compared with the thermal state, the offdiagonal elements of state \(\rho _{E}\) are nonzero if \(p\neq 0\). Therefore, Eq. (15) can also be written as
where \(\rho _{\mathrm{coh}}\) is the nondiagonal part of state \(\psi \rangle \langle \psi \), i.e., the offdiagonal elements of \(\rho _{\mathrm{coh}}\) are the same as that of state \(\psi \rangle \langle \psi \) and the diagonal elements are zero.
Effect of the coupling strength on NM
We suppose that the environment is in thermal state, i.e., all environment particles including \(A_{Q}\) and each ancilla are in the state \(\rho _{\beta }\) with \(T=\omega _{E}=1\). We numerically calculate the degree of NM for different γ and δ which is presented in Fig. 2. We can see that the whole diagram is divided into two regions, where the green stars represent the degree of NM being equal to zero (Markovian region) and the red dots represent the degree of NM being larger than zero (nonMarkovian region). It shows that the nonMarkovian dynamics of the system is determined by a delicate balance between the two parameters γ and δ. Specifically the system dynamics is Markovian for small γ and larger δ, and the nonMarkovian region increases with the increase of γ. Physically this can be understood as following. When the interaction between S and \(A_{Q}\) is small (small γ) and with a relatively large interaction between \(A_{Q}\) and \(\mathcal{R}_{j}\) (larger δ), the information obtained by \(A_{Q}\) from S is less and all of which flow into the reservoir \(\mathcal{R}\), which forms Markovian dynamics of the system. In other words, the system is losing information at a slower rate than that of the evolution of environment, thus the backflow of information cannot happen now. However with the increase of γ, more and more information flows from S into \(A_{Q}\) which leads to only part of the information flows into the reservoir and the rest is reserved and flows back to S, and in this case the nonMarkovian dynamics of system is formed.
Effects of coherence of environment on NM and energy flux
We consider the case of environment with coherence, i.e., \(A_{Q}\) and each ancilla are in state (16) with a relative phase \(\phi _{2}\) and \(\phi _{1}\) respectively. Thus the phase difference between reservoir \(\mathcal{R}\) and \(A_{Q}\) is \(\phi =\phi _{1}\phi _{2}\). In Fig. 3, we plot the variation of the NM with respect to p for fixed ϕ (\(\phi =0\)) (Fig. 3(a)), and ϕ for fixed p (\(p=0.4\)) (Fig. 3(b)), and the coupling strength γ and δ are of the Markovian region of coupling presented in Fig. 2. From numerical calculations we find that the system dynamics is Markovian for \(p\in [0,0.4]\), and in the region \(p\in (0.4,1]\) the increase of p leads to an increase of NM. An interesting feature here is that a transition from Markovian to nonMarkovian dynamics is observed. Besides parameter p phase difference ϕ is also one of the influence factor of coherence of environment. The system dynamics is Markovian for \(\phi \in [0,\pi /4]\), and in the region \(\phi \in (\pi /4,\pi ]\) the increase of ϕ leads to an increase of NM, in the region \(\phi \in [0,2\pi ]\) the change of NM is symmetrical about \(\phi =\pi \). Also a transition from Markovian to nonMarkovian dynamics is observed by means of ϕ. Physically this can be understood as following. As we consider energyconserving interactions, i.e., \([\hat{H}^{\mathrm{int}}_{S,A_{Q}},(\hat{H}_{S}+\hat{H}_{A_{Q}})]=0\), \([\hat{H}^{\mathrm{int}}_{A_{Q},\mathcal{R}_{j}},(\hat{H}_{A_{Q}}+\hat{H}_{ \mathcal{R}_{j}})]=0\), leading to
where \(\Delta E_{n}=\operatorname{Tr}\{\hat{H}_{S}[\rho _{S}(n+1)\rho _{S}(n)]\}\), is the change in energy of the system in each interaction, and \(Q_{A_{Q},\mathcal{R}_{n}}\) is the change in energy of the environment. From Eq. (16), compared to initial thermal state of each element of the environment, coherence, i.e., the second term in Eq. (16), is added, and the amount of coherence increases with the increase of parameter p. Therefore, \(\Delta E_{n}\) in Eq. (17) can be divided into two parts:
where the first term, \(\Delta Q^{\beta }_{n}\), is the contribution of the first term in Eq. (16), i.e., the thermal state of environment, and the second term \(\Delta Q^{\mathrm{coh}}_{n}\) is the contribution of coherence of environment (the second term in Eq. (16)). In Fig. 4, we plot \(\Delta E_{n}\), \(\Delta Q^{\beta }_{n}\) and \(\Delta Q^{\mathrm{coh}}_{n}\) with respect to n in two cases: \(p=0.4\) of the Markovian regime (Fig. 4(a)) and \(p=0.9\) of nonMarkovian regime (Fig. 4(b)). It shows that in the Markovian regime \(\Delta Q^{\beta }_{n}\) plays a major role in \(\Delta E_{n}\), which suppress energy backflow from the environment to system. In contrast to this, in the nonMarkovian regime \(\Delta Q^{\mathrm{coh}}_{n}\) plays a major role of the contribution to \(\Delta E_{n}\) and the energy backflow appears.
In order to study the relationship between the energy backflow of interest and the NM of the system dynamics definitely, we consider the effect of ϕ on NM presented in Fig. 3(b). In Fig. 5, we plot \(\Delta E_{n}\), \(\Delta Q^{\beta }_{n}\) and \(\Delta Q^{\mathrm{coh}}_{n}\) with respect to n for different ϕ, \(\phi =\pi /2\) (a) and \(\phi =\pi \) (b) which are in the nonMarkovian regime. We find that for the nonMarkovian regime caused by ϕ, \(\Delta Q^{\beta }_{n}\) plays a major role in Eq. (18), and the energy flows from S to its environment unidirectionally, i.e., energy backflow is suppressed. Above all, in the Markovian regime energy backflow is suppressed. However the opposite is in general not true; namely, the absence of energy backflow does not imply absence of information backflow. Note that similar results have been obtained that NM allows for the observation of energy backflow [78], and the information backflow from the reservoir to the system does not necessarily correlate with the backflow of heat [79].
Effect of initial systemenvironment correlations on NM
Compared to the cases discussed above without any initial correlation between system and its environment (i.e., product state), we consider a group of two initial states
where \(\psi \rangle =\xi 01\rangle +\sqrt{1\xi ^{2}}10\rangle \). It is worth noting that the reduced density matrixs of these two initial states are identical, and \(\rho _{S,A_{Q}}^{1}(0)\) has quantum correlations (quantum entanglement and quantum discord) between S and \(A_{Q}\) initially, \(\rho _{S,A_{Q}}^{2}(0)\) has classical correlation. We adopt Wootter’s concurrence [80] as the entanglement measure. For the Xstructure density matrix \(\rho _{AB}\) of bipartite system the concurrence is given by
where \(\rho _{ii}\) (\(i=1,2,3,4\)) are the matrix elements of \(\rho _{AB}\). For the X state described by the density matrix \(\rho _{AB}\), the analytic expression of quantum discord has been reported [81] and expressed by
where \(Q_{j}=H(\rho _{11}+\rho _{33})+\sum^{4}_{i=1}\lambda _{i}\log _{2} \lambda _{i}+D_{j}\) with \(\lambda _{i}\) being the four eigenvalues of \(\rho _{AB}\), \(D_{1}(\tau )=H(\tau )\), \(D_{2}(\tau )=\sum^{4}_{i=1}\rho _{ii}\log _{2}\rho _{ii}H(\rho _{11}+ \rho _{33})\) with \(\tau =(1+ \sqrt{[12(\rho _{33}+\rho _{44})]^{2}+4(\rho _{14}+ \rho _{23})^{2}})/2\) and \(H(\tau )=\tau \log _{2}\tau (1\tau )\log _{2}(1\tau )\). And the classical correlation can be expressed as
where \(I(\rho _{AB})=S(\rho _{A})+S(\rho _{B})S(\rho _{AB})\) is the mutual information, \(S(\rho )=\operatorname{Tr}\rho \log _{2}\rho \) is the von Neumann entropy. And in the case of initial correlation between S and \(A_{Q}\), the nonMarkovian dynamics of system can be witnessed by the nonmonotonicity in evolution of trace distance between two states of system
Here \(\rho _{S}(n+1)\) and \(\tilde{\rho }_{S}(n+1)\) are a pair of states of system after the \((n+1)\)th step, which corresponds to the pair of initial states of the composite system ‘\(SA_{Q}\)’, \(\{\rho _{S,A_{Q}}(0), \tilde{\rho }_{S,A_{Q}}(0)\}\), and \(\tilde{\rho }_{S,A_{Q}}(0)=\operatorname{Tr}_{A_{Q}}(\rho _{S,A_{Q}}(0)) \otimes \operatorname{Tr}_{S}(\rho _{S,A_{Q}}(0))\). And in this section we suppose that each ancilla in \(\mathcal{R}\) are initially in the thermal state \(\rho _{\beta }\) with \(T=\omega _{E}=1\).
From Eqs. (21) and (22), the amount of entanglement and quantum discord may change with ξ and this change is showed in Fig. 6(a). They (entanglement and quantum discord) have the similar behaviors, and in the region \(\xi \in [0,0.7]\) the amount of entanglement and quantum discord increase with the increase of ξ. In Fig. 6(b) we plot the trace distance Eq. (24) against the number of collisions n for initial state (19) and thermal state of each ancilla of the reservoir for different ξ (\(\xi =0.3,0.5,0.7\)), \(\gamma =\frac{\pi }{14}\) and \(\delta =\frac{\pi }{6}\) which correspond to a Markovian region of coupling presented in Fig. 2. It shows that the trace distance increases from zero to a maximum and then decreases until to zero, which implies that a nonMarkovian dynamics of system, and the amount of information backflow increases with the increase of initial quantum correlation (entanglement and quantum discord) between system and environment. It is noted that the trace distance here exceeds the initial value. Laine et al. have pointed out that the trace distance between two states of the open system can increase above its initial value when system and its environment are initially correlated [32]. And in our case it can be written as
where \(\rho _{S}(n)\) and \(\tilde{\rho }_{S}(n)\) are the reduced state of system after the nth interaction corresponding to the initial state \(\rho _{S,A_{Q}}^{1}(0)\) and \(\operatorname{Tr}_{A_{Q}}(\rho _{S,A_{Q}}^{1}(0))\otimes \operatorname{Tr}_{S}( \rho _{S,A_{Q}}^{1}(0))\), respectively. This inequality shows how far from each other two initially indistinguishable reduced states can evolve when only one of the two initial states is correlated. And physically this can be understood as following: The maximal amount of information the open system can gain from the environment is the amount of information flowed out earlier from the system since the initial time, plus the information which is initially outside the open system. Thus the increase of the trace distance is bounded from above by the correlations in the initial state. We calculate the bound of Eq. (25) in different cases of ξ which is showed in Fig. 6(b), and the inequality Eq. (25) is well satisfied. We notice that the maximum value of the trace distance at a certain n in Fig. 6(b) is much smaller than the bound of Eq. (25) for a fixed ξ, i.e., the bound of Eq. (25) is actually loose. This means that only less of the information in the composite system ‘\(S+A_{Q}\)’ initially transfers to the reduced system S during the evolution, and this is due to the Markovian reservoir \(\mathcal{R}\). Moreover, Smirne et al. have provided experimental evidence that if the environmental state is fixed, the trace distance between two states of an open quantum system can increase over its initial value only in the presence of initial correlations [33].
From the discussion above, it is always able to induce a transition from Markovian to nonMarkovian dynamics for initial quantum correlation between system and its environment. For initial classical correlation state (20) and a thermal state of the reservoir, from numerical calculation we find that the trace distance Eq. (24) is always zero with the number of collisions n within the Markovian region of coupling presented in Fig. 2. In order to study the effect of initial classical correlation on NM more comprehensively, we use the measure of the degree of NM in the Appendix (Eq. (32)), and we find the similar result that \(\mathcal{N}\) in Eq. (32) is also zero. However it is worth noting that Eq. (32) can only be used to witness the occurrence of nonMarkovian dynamics rather than to confirm a Markovian dynamics. Therefore, from now on it cannot guarantee that the dynamics of system must be Markovian for initial state (20). And thus for initial classical correlation we do only claim that the \(\mathcal{N}\) in Eq. (32) is zero in the case of thermal reservoir comparing to the case of reservoir with coherence (see below). In Fig. 7, we plot the trace distance Eq. (24) against the number of collisions n for initial classical correlation state (20); \(\gamma =\frac{\pi }{14}\), \(\delta =\frac{\pi }{6}\), and a state with coherence (Eq. (16) with \(p\neq 0\)) of each ancilla of the reservoir, \(p=0.4\) in Fig. 7(a) and \(p=\{0.1,0.2,0.4\}\) in Fig. 7(b) which corresponds to a Markovian regime presented in Fig. 3(a). Obviously the change of trace distance is similar to the case of initial quantum correlation presented in Fig. 6(b), the trace distance increases from zero to a maximum and then decreases until to zero. This also indicates a nonMarkovian dynamics of system and the trace distance here exceeds the initial value. And the amount of information backflow can be increased by two ways, on the one hand with the increase of initial classical correlation between system and environment, on the other hand with the increase of coherence of reservoir. Here from numerical calculation we find that the change of amount of classical correlation with ξ in Eq. (20) is the same to quantum correlation, i.e., \(C_{\mathrm{cla}}(\rho ^{2}_{S,A_{Q}}(0))\) increases with the increase of ξ in the region \(\xi \in [0,0.7]\). Note that in Ref. [32] it has pointed out that the effects of initial classical correlation are related to the form of interaction, and they have verified that the existence of the initial classical correlation will not make the trace distance of the system exceed the initial value if two qubits are under the action of controlledNOT gate only; and if first apply the controlledNOT gate and then a swap operation, it can obtain a growth of the trace distance. In our case, a growth of the trace distance and a nonMarkovian dynamics are emerged by means of coherence of reservoir in the case of initial classical correlation. And Eq. (25) is also satisfied now.
In summary, we study the effect of initial systemenvironment correlations on system dynamics, including quantum correlation and classical correlation. We realize a growth of the trace distance and a nonMarkovian dynamics with the help of initial quantum correlation, however for initial classical correlation this can only be confirmed to occur when there is coherence of the reservoir simultaneously.
Effect of NM on thermodynamic properties
In this section we consider the system in contact with a thermal environment, i.e., the initial state of each ancilla of the reservoir is thermal state \(\rho _{\beta }\), and the reduced state \(\rho ^{A_{Q}}_{n}\) maintain the form of thermal state, \(\rho ^{A_{Q}}_{n}=\rho ^{A_{Q}}_{\beta }=\frac{1}{Z}e^{\beta _{A_{Q}} \hat{H}_{A_{Q}}}\), with the ndependent inverse temperatures \(\beta _{A_{Q}}=\frac{1}{T_{A_{Q}}}\).
Entropy change and heat flux
It is known that the total von Neumann entropy of the composite system ‘\(S{A}_{Q}\)’ under the unitary evolution \(U_{S,A_{Q}}\) is invariant during each step, i.e., \(S(\rho ^{S,A_{Q}}_{n})=S(\tilde{\rho }^{S,A_{Q}}_{n+1})\), here \(\tilde{\rho }^{S,A_{Q}}_{n+1}=U_{S,A_{Q}}(\gamma )(\rho ^{S,{A}_{Q}}_{n})U^{ \dagger }_{S,A_{Q}}(\gamma )\). Based on this, the change in entropy of system during the \((n+1)\)th interaction can be expressed as [61]
where \(\tilde{\rho }^{S}_{n+1}=\operatorname{Tr}_{A_{Q}}[\tilde{\rho }^{S,A_{Q}}_{n+1}]\), \(\tilde{\rho }^{A_{Q}}_{n+1}=\operatorname{Tr}_{S}[\tilde{\rho }^{S,A_{Q}}_{n+1}]\), \(D(\rho _{1}\\rho _{2})\equiv \operatorname{Tr}(\rho _{1}\ln \rho _{1}) \operatorname{Tr}(\rho _{1}\ln \rho _{2})\) is the quantum relative entropy between two density matrices \(\rho _{1}\) and \(\rho _{2}\), and the mutual information \(I(\rho ^{S,A_{Q}}_{n})=\mathrm{S}(\rho ^{S}_{n})+\mathrm{S}(\rho ^{A_{Q}}_{n}) \mathrm{S}(\rho ^{S,A_{Q}}_{n})\), measures the correlation between S and \(A_{Q}\), and this correlation has been established after their collision in the first step. According to the definition of \(\rho ^{A_{Q}}_{\beta }\) above, we can obtain \(\textrm{Tr}_{{A}_{Q}}[(\tilde{\rho }^{A_{Q}}_{n+1}\rho ^{A_{Q}}_{ \beta })\ln \rho ^{A_{Q}}_{\beta }]=\beta _{{A}_{Q}}\Delta Q_{n+1}\), here
representing the heat flowing from auxiliary qubit \({A}_{Q}\) to system S. Therefore, Eq. (26) can also be written as
Notice that we choose energypreserving interactions between the bipartite systems, ‘\(S{A}_{Q}\)’, ‘\(A_{Q}\mathcal{R}_{j}\)’. Mathematically, this translates as \([\hat{U}_{S,A_{Q}},\hat{H}^{\mathrm{fre}}_{S,A_{Q}}]=0\), \([\hat{V}_{A_{Q},\mathcal{R}_{j}},\hat{H}^{\mathrm{fre}}_{A_{Q},\mathcal{R}_{j}}]=0\), that is, \([\hat{H}^{\mathrm{int}}_{S,{A}_{Q}},\hat{H}^{\mathrm{fre}}_{S,{A}_{Q}}]=0\), \([\hat{H}^{\mathrm{int}}_{A_{Q},\mathcal{R}_{j}},\hat{H}^{\mathrm{fre}}_{A_{Q}, \mathcal{R}_{j}}]=0\), here \(\hat{H}^{\mathrm{fre}}_{S,{A}_{Q}}=\hat{H}_{S}+\hat{H}_{A_{Q}}\), \(\hat{H}^{\mathrm{fre}}_{A_{Q},\mathcal{R}_{j}}=\hat{H}_{A_{Q}}+\hat{H}_{ \mathcal{R}_{j}}\). So that the heat given by the system is completely transferred to the environment, and vice versa. In other words, no heat is given or taken in the form of thermodynamic work while performing the unitary operations. Thus the canonical definition of heat flow \(\Delta Q_{n+1}\) in Eq. (27) is valid and compatible with thermodynamics, and the term \(\beta _{{A}_{Q}}\Delta Q_{n+1}\) in Eq. (28) is associated with the system entropy change due to heat exchanges, i.e., entropy flux.
In order to study the system entropy change especially that results from heat exchanges with the environment, in Fig. 8(a)–(b) we plot \(\Delta S_{n+1}\) and \(\beta _{A_{Q}}\Delta Q_{n+1}\) against the number of collisions n of a Markovian region of coupling (\(\gamma = \frac{\pi }{14}\), \(\delta =\frac{\pi }{6}\)) in Fig. 8(a) and a nonMarkovian region of coupling (\(\gamma =\frac{\pi }{14}\), \(\delta =\frac{\pi }{9}\)) in Fig. 8(b). The initial state of system is ground state \(1\rangle \), and the initial states of auxiliary qubit and reservoir qubits are in the same thermal state \(\rho _{\beta }\) (\(p=0\) in Eq. (16)) with \(T=1\). It shows that the changes of \(\Delta S_{n+1}\) and \(\beta _{A_{Q}}\Delta Q_{n+1}\) are almost consistent with the increase of n, increasing first and then oscillating decay. However \(\Delta S_{n+1}\) and \(\beta _{A_{Q}}\Delta Q_{n+1}\) are always larger than zero for Markovian environment (Fig. 8(a)), and which can be less than zero during some time intervals for nonMarkovian environment (Fig. 8(b)). Physically this can be understood as following. We define \(\rho _{ij}\) (\(i,j=1,2,3,4\)) are the matrix elements of state \(\rho ^{S,{A}_{Q}}_{n}\) of ‘\(S{A}_{Q}\)’ before their \((n+1)\)th collisions. Due to the correlations between S and \({A}_{Q}\), \(\Delta Q_{n+1}\) in Eq. (27) can be divided into two different contributions:
where
are the heats determined, respectively, by the diagonal and coherent (offdiagonal) elements of state \(\rho ^{S,{A}_{Q}}_{n}\), and \(\omega =\omega _{S}=\omega _{E}\) is the resonance frequency of S, \(A_{Q}\) and \(\mathcal{R}_{j}\). The nonzero coherent term \(\rho _{23}\) of \(\rho ^{S,A_{Q}}_{n}\) is a direct witness of correlation between S and \(\rho ^{A_{Q}}_{n}\), which in turn gives the correlationdependent heat \(\Delta Q^{\mathrm{coh}}_{n+1}\). For fixed parameter γ, the relatively large values of δ lead to Markovian dynamics, and the established systemenvironment correlations are weak, so the contribution \(\Delta Q^{\mathrm{dia}}_{n+1}\) plays a major role in determining the behavior of total heat \(\Delta Q_{n+1}\). This can be verified by Fig. 8(c): the correlations \(I(\rho ^{S,A_{Q}}_{n})\) established within the dynamical process decrease with the increase of δ for fixed γ. Differently, when δ is sufficiently small (nonMarkovian dynamics) the behavior of \(\Delta Q_{n+1}\), especially its transition from positive to negative values, is mainly determined by the contribution \(\Delta Q^{\mathrm{coh}}_{n+1}\), as showed in Fig. 8(d).
Irreversible entropy production
The entropy production can be defined as the difference between the change in entropy of the reduced system state and the mean exchanged heat with a reservoir at fixed temperature, T, divided by T [66, 69, 71]. From Eq. (28), the irreversible contribution to the entropy production during the \((n+1)\)th interaction can be written as
it provides the contribution in entropy change of system which cannot be traced back to a reversible heat flow. In Fig. 9, we plot entropy production of system Σ, \(D(\tilde{\rho }^{S,A_{Q}}_{n+1}\parallel \tilde{\rho }^{S}_{n+1}\rho ^{A_{Q}}_{ \beta })\) (the first term in Eq. (31)) and the established systemenvironment correlation \(I(\rho ^{S,A_{Q}}_{n})\) (the second term in Eq. (31)), with respect to n for different dynamics of system. As expected, we find the entropy production of system can become transiently negative for the nonMarkovian dynamics compared with the corresponding Markovian case. In other words, in some specific time intervals the entropy production can decrease, provided that the quantum dynamics fails to be positive divisible, i.e. it is essentially nonMarkovian. And the multipleinteraction entropy production, Σ is zero regardless of whether the underlying dynamics is Markovian or nonMarkovian, which is due to thermalization of system with the environment, i.e., S is in a thermal equilibrium state \(\rho _{\beta }\) in the longtime limit.
In Fig. 10, we study the entropy production of system Σ against the number of collisions n for initial quantum correlation between the system and its environment Eq. (19). We find that it can undergoes a negative Σ during the dynamics in some cases, for example, \(\xi =\{0.3,0.5,0.7\}\) in Eq. (19). However Σ can always positive during the whole dynamics for \(\xi =0.9\). In other words, we see the possibility of positive Σ induced by the kind of NM of the initial quantum correlation. And this is different from the kind of NM induced by the coupling strength of systemauxiliary system and auxiliary systemreservoir mentioned above, that there is a corresponding relationship between nonMarkovian dynamics and a negative Σ. Consequently, the NM originated from the coupling strength induces a negative Σ definitely, whereas from the initial quantum correlation may be positive or negative. And a similar result has been obtained that the nonMarkovian effect regarding the initial correlation may yield positive entropy production rate [71].
Conclusion
In this paper, we have investigated the nonMarkovian character of the system and its effect on thermodynamic properties by means of a collision model, that a system is coupled to a structured environment consisting of a auxiliary system and a reservoir. We have studied how the systemauxiliary system and auxiliary systemreservoir coupling strength, coherence of environment and initial systemenvironment correlation affect the nonMarkovian character of the system. Especially we have studied the nonMarkovian dynamics induced by coherence of environment from the perspective of energy, and the relationship between information backflow and energy flux. And we have shown the growth of trace distance regarding initial classical correlation between the system and its environment by means of the coherence of reservoir, and this is different from the result showed in Ref. [32] that the effect of initial classical correlation on the growth of trace distance is related to the form of interaction.
Then we have studied the effects of NM on the entropy change of the system. We have shown that the essence of entropy flux (the system entropy change induced by heat exchange with the environment) between positive and negative values under nonMarkovian evolution is due to the contribution of heat flux induced by coherence. And we have observed a onetoone correspondence between a transient negative values of the entropy production and nonMarkovian dynamics induced by the coupling strength. On the contrast, we have shown the possibility of positive entropy production during the whole nonMarkovian dynamics induced by initial systemenvironment correlation.
Note that in this paper we have used the collision model to investigate the influences of nonMarkovian dynamics, and the relation of NM and thermodynamics. The reason to consider this simple model is that exact solutions can be obtained for a general class of initial systemenvironment correlations and the initial states of reservoirs with coherence. We expect that some features of the NM and thermodynamics in this simple model might be similar to those in more involved but less tractable structuredenvironment models, so we can gain some insight into the general feature of the effects of initial systemenvironment correlations and reservoirs with coherence on NM, and hence the relation between NM and thermodynamic properties.
Availability of data and materials
The data sets used or analysed during the current study are available from the corresponding author on reasonable request.
Abbreviations
 NM:

NonMarkovianity
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Acknowledgements
The authors are grateful to Prof. Jian Zou, Prof. Zhongxiao Man and Dr. Chaoquan Wang for fruitful discussions.
Funding
The National Natural Science Foundation of China (Grant No. 11947047).
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YWL performed calculations, analysed the results and wrote the manuscript. LL originated the idea, directed the project, analysed the results and helped in writing. All authors discussed the results and commented on the manuscript. All authors read and approved the final manuscript.
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Appendix: NM witness with initial classical correlation
Appendix: NM witness with initial classical correlation
We introduce the degree of NM which makes use of the nonmonotonicity of the trace distance between two states of system to witness the effect of initial classical correlation on the nonMarkovian dynamics of the system [54],
where \(\Delta \mathcal{D}(n+1)=\mathcal{D}[\rho _{s}(n+1),\tilde{\rho }_{s}(n+1)] \mathcal{D}[\rho _{s}(n),\tilde{\rho }_{s}(n)]\), and \(\rho _{s}(n+1)\) is the same as that in Eq. (19), \(\tilde{\rho }_{s}(n+1)\) is the reduced state of system after the \((n+1)\)th interaction corresponding to the initial state \(\rho _{S,A_{Q}}(0)=\tilde{\rho }_{s}(0)\otimes \operatorname{Tr}_{S}(\rho _{S,A_{Q}}^{2}(0))\) with the initial system state \(\tilde{\rho }_{s}(0)=\cos \frac{\theta }{2}0\rangle +e^{i\varphi } \sin \frac{\theta }{2}1\rangle , \theta \in [0,\pi ],\varphi \in [0,2 \pi ]\). The maximization is performed by taking all possible system states \(\tilde{\rho }_{s}(0)\) over the Bloch sphere. And here the definition of \(\sigma ^{+}\) is the same as that in Eq. (13), in which \(\Delta \mathcal{D}(n)>0\).
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Li, Y., Li, L. Hierarchicalenvironmentassisted nonMarkovian and its effect on thermodynamic properties. EPJ Quantum Technol. 8, 9 (2021). https://doi.org/10.1140/epjqt/s40507021000988
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Keywords
 Open quantum systems
 Collisionbased models
 Quantum nonMarkovianity
 Quantum coherence
 Initial correlation
 Quantum thermodynamics