General Non-Markovian Quantum Dynamics

A general approach to the construction of non-Markovian quantum theory is proposed. Non-Markovian equations for quantum observables and states are suggested by using general fractional calculus. In the proposed approach, the non-locality in time is represented by operator kernels of the Sonin type. A wide class of the exactly solvable models of non-Markovian quantum dynamics is suggested. These models describe open (non-Hamiltonian) quantum systems with general form of nonlocality in time. To describe these systems, the Lindblad equations for quantum observable and states are generalized by taking into account a general form of nonlocality. The non-Markovian quantum dynamics is described by using integro-differential equations with general fractional derivatives and integrals with respect to time. The exact solutions of these equations are derived by using the operational calculus that is proposed by Yu. Luchko for general fractional differential equations. Properties of bi-positivity, complete positivity, dissipativity, and generalized dissipativity in general non-Markovian quantum dynamics are discussed. Examples of a quantum oscillator and two-level quantum system with a general form of nonlocality in time are suggested.


Introduction
The dynamics of open quantum systems (OQS) is usually described by Markovian equation for quantum observables and quantum states [1][2][3][4][5]. The most general explicit form of such equations were first proposed by Gorini, Kossakowski, Sudarshan, and Lindblad in 1976 [6][7][8]. Quantum mechanics of OQS has been actively developing (see [1][2][3][4][5][9][10][11][12]). The Markovian dynamics of OQS is characterized by non-standard properties that lead to a change in the usual relations and approaches, for example, such as the uncertainty relation for open quantum systems [12,13], path integrals [5] (pp. 475-485) and [14], pure stationary states [5] (pp. 453-462) and [15,16], and quantum computer with mixed states [5] (pp. 487-520) and [17]. We can say that the theory of open quantum systems is the most general type of modern quantum mechanics such as fundamental theory [5]. Note that OQS are non-Hamiltonian quantum systems, that is, it is not enough to specify the Hamilton operator to describe the dynamics of such systems [5]. This theory has great practical importance for the creation of quantum computers and quantum informatics due to the fact that the influence of the environment on the process of quantum computation, which is realized by the dynamics of the quantum systems of qubits.
Currently, modern quantum mechanics is faced with the question of the most general form of the equations that describe the non-Markovian dynamics of quantum systems. A general form of non-Markovian character of quantum processes can be interpreted as kernels). In this paper, we consider general fractional operators with kernels that belong to the Sonin set. The general form of non-Markovian dynamics of open quantum systems is described by the equations with general fractional derivatives and integrals [58,59]. The exact solutions of these equations were derived by using the general operational calculus proposed in [60].

Markovian Dynamics of Quantum States and Observables
In this section, we will briefly describe the basics of Markovian quantum dynamics for fixing concepts and notations.
Quantum states can be described by density operators ρ that are normalized (Tr[ρ] = 1), self-adjoint (ρ † = ρ), positive (ρ ≥ 0) operators. All these properties must be conserved in the time evolution S t : ρ → ρ t . Therefore to describe Markovian quantum dynamics in general form, we should find such a time evolution for the density operators ρ t = S t ρ, which satisfy the following conditions for all t ∈ (0, ∞).
(1) The self-adjoint condition (2) The positivity condition (3) The normalization condition If we consider the Markovian dynamics, then the semigroup condition is used S t S τ = S t+τ (4) for all t, τ ≥ 0, and the condition S 0 ρ = ρ is represented as S t (ρ) → ρ at t → 0 in the trace norm.
The Markovian quantum dynamics of quantum observables A t = Φ t (A) is described by the dual dynamical maps Φ t , where the duality is represented by the condition Tr[Aρ t ] = Tr[A t ρ], (forallt > 0).
Let A be an algebra of bounded operators (for example, C * -algebra, or algebra of B(H) of bounded operators on the Hilbert space H).
Normalization condition (3) means that Φ t (I) = I, where I is the identity operator. Positivity condition (2) means that Φ t A † A ≥ 0. The positivity condition is usually replaced by the complete positivity condition for the dynamical maps Φ t : for all A j , B j ∈ A, j = 1, . . . n and all n ∈ N.
To describe the Markovian dynamics of quantum observables, the semigroup condition is used for all t, τ ≥ 0, and the condition Φ 0 (A) = A is represented as Φ t (A) → A at t → 0 in ultraweak operator topology. The ultraweak operator topology can be defined as the topology induced on A by the set of all seminorms of the form For the complete positive semigroup {Φ t : t ∈ (0, ∞)}, for which Φ t (I) = I and Φ t (A) → A ultraweakly at t → 0 , there exists a superoperator L for which d dt Φ t (A) = LΦ t (A) (9) holds for all A ∈ D(L), where D(L) is a ultraweakly dense domain of A. The superoperator L is called the generator of the semigroup Φ t . The dual generator Λ of the semigroup S t is connected with L through the equation For superoperator Λ, we have The bounded superoperators are defined on the normed operator space A: The superoperator L is called bounded, if for some constant c and all A ∈ A. Inequality (12) means that L transforms norm bounded sets of A into the norm bounded sets. The least value of c equal to is called the norm of the superoperator L. If A is a normed space, and L is a bounded superoperator, then Λ = L .
Let us define the class of real superoperators: A real superoperator is a superoperator L on A, such that [L(A)] † = L A † (15) for all A ∈ D(L) ⊂ A and A † ∈ D(L), where A † be adjoint of A ∈ A. If L is a real superoperator, then Λ is real. The Lindblad proved theorem defines the structure of the superoperators L and Λ. This theorem is proved for the completely positive dynamical semigroup Φ t , which is norm continuous lim Equation (16) is a more restrictive condition than ultraweak continuity. For such semigroup. the superoperator L is bounded.
For completely positive maps Φ t , the bi-positivity condition holds for all t > 0 with equality at t = 0. The differentiation of inequality (17) gives at t = 0 the property of the superoperator for all A ∈ A. The bounded superoperators L, which satisfy conditions L(I) = 0, L A † = (LA) † and inequality (18), are called dissipative. The real superoperator L is called completely dissipative, if where k, l = 1, . . . , n, for all n ∈ N and all A † k , A l , A † k A l ∈ D(L) ⊂ A, where A † k , A † k A l ∈ D(L). The Lindblad theorem gives the most general form of completely dissipative superoperators L.
The Lindblad theorem states [7] that the superoperator L is completely dissipative and ultraweakly continuous if and only if it has the form where Equations (20) and (21) give explicit forms of most general Markovian dynamics of quantum observables and states.
The quantum Markovian equation, which are also called the Lindblad equations, are written in the form where A(t) is a quantum observable; ρ(t) is a quantum state; H is the Hamiltonian operator; and V k are the Lindblad operators [5]. Equations (22) and (23) are the standard time-local (memoryless) Markovian equations for quantum observables and states [6][7][8].
If V k = 0 for all k ∈ N, then Equations (22) and (23) give the standard Heisenberg equation and von Neumann equation, respectively. In this case, Equations (22) and (23) describe Markovian dynamics of Hamiltonian quantum systems without nonlocality in time.

General Non-Markovian Dynamics of Quantum Observables and States
For the description of non-Markovian quantum processes, we can take into account nonlocality in time, which means that the behavior of the quantum observable A(t) (or state ρ(t)) and its derivatives may depend on the history of previous changes of this operator. To describe this type of behavior, we cannot use differential equations of integer orders.
To take into account nonlocality in time (non-Markovality) in open quantum systems in work [37] (Chapter 20) and [38,39], it has been proposed to use the derivatives of non-integer orders instead of integer-order derivatives with respect to time.
In this section, we proposed using general fractional calculus and general fractional derivatives (GFD) as mathematical tools to allow us to take into account the general nonlocality in time for non-Markovian quantum processes.

Generalization of Lindblad Equation for Quantum Observables
The Lindblad equation for quantum observables is described by the operator differential equation of the first order where the superoperator L is defined by expression (20). Equation (24) can be written in the integral form The nonlocality in time can be taken into account by using an integral kernel in generalization of Equation (25) in the form where the function M(t − τ) describes nonlocality in time. If M(t) = 1 for all t ∈ (0, ∞), then Equation (26) gives standard Equation (24), which describes the Markovian quantum dynamics.
Obviously, not all kernels M(t − τ) can describe nonlocality in time. The nonlocality requirement can be formulated as follows. If integral Equation (26) can be written as a differential equation of an integer order (or a finite system of such equations), then the process, which is described by Equation (26), is local in time. The first obvious example of such "local" kernel is with positive integer values of α = n ∈ N. Examples of "local" kernels are also the probability density functions of the exponential distribution and the gamma distribution with integer shape parameters (the Erlang distribution) [73,74]. Note that kernel (27) with non-integer values of α > 0 define "non-local" kernel of the Riemann-Liouville and Caputo fractional operators [29].
Let us assume that the functions M(t) belongs to the space C −1,0 (0, ∞), and suppose that there exists a function K(t) ∈ C −1,0 (0, ∞), such that the Laplace convolution of these functions is equal to one for all t ∈ (0, ∞). The function X(t) belongs to the space C −1,0 (0, ∞), if this function can be represented in the form X(t) = t p Y(t), where −1 < p < 0, and Y(t) ∈ c [0, ∞). Definition 1. The functions M(t), K(t) are a Sonin pair of kernels, if the following conditions are satisfied (1) The Sonin condition for the kernels M(t) and K(t) requires that the relations holds for all t ∈ (0, ∞).
(2) The functions M(t), K(t) belong to the space The set of such Sonin kernels is denoted by S −1 .
If the kernel pair (M(t), K(t)) belongs to Sonin set S −1 [59], the kernel K(t) is called the associated kernel to M(t). Note that if K(t) is the associated kernel to M(t), then M(t) is the associated kernel to K(t). Therefore, if (M(t), K(t)) belongs to set S −1 , then I t (M) , D t (K) , and I t (K) , D t (M) can be used as the general fractional integrals (GFI) and general fractional derivatives (GFD).
To define GFI and GFD, we used Luchko's approach to general fractional calculus, which is proposed in [58,59].
that is defined by the equation If the functions M(t) and K(t) belong to the Sonin set, then we can define general fractional derivatives D t (K) and D t, * (K) that are associated with GFI I t (M) .
As proven in [58,59], operators (32) and (33) are connected (see Equation (47) in Definition 4 of [58] (p. 8)) by the equation The proposed GFI and GFD can be used to formulate non-Markovian dynamics in the general form, where the nonlocality in time is described by the kernel pairs that belong to the Sonin set S −1 .
If (M(t), K(t)) ∈ S −1 , then Equation (26) can be written through the GFI with kernel M(t) as The action of the GFD D s (K) [t] with kernel K(t), which is associated to M(t), in Equation (35), gives For the right-hand side of Equation (36), we use the first fundamental theorem of GFC (see Theorem 3 of [58] (p. 9)). This theorem states that the equation holds for f (t) ∈ C −1 (0, ∞).
Using Equation (37) and the equality Equation (36) is written in the form The left-hand side of Equation (39) can be expressed through GFD D s, * (K) by using Equation (34) in the form for A(t) ∈ C 1 −1 (0, ∞) (see also Equations (47) and (49) in [58] (p. 8)). As a result, Equation (36) can be written as where L is the Lindblad superoperator (the Lindbladian, quantum Liouvillian). Equation (41) describes non-Markovian dynamics of quantum observables in the general form, where the kernel belongs to the Sonin set S −1 .
As a result, we proved the following theorem.

Generalization of Lindblad Equation for Quantum States
Equation (23) is the standard memoryless Markovian quantum master equation [6][7][8]. Equation (23) can be written in the integral form The nonlocality in time can be taken into account by using an integral kernel in Equation (44) in the form If M(t) = 1 for t ∈ (0, ∞), then Equation (45) gives standard Equation (44). In general, the kernel M(t − τ) can be used to describe non-Markovian quantum dynamics.
As a result, the non-Markovian master equation for quantum states takes the form if (M(t), K(t)) ∈ S −1 . Equation (46) describes non-Markovian dynamics of quantum states in the general form, where the kernel K(t) belongs to the Sonin set S −1 .
The solutions of Equations (41) and (46) can be derived by using the Luchko operational calculus [60]. To describe the solution, we give the following theorem and define the Luchko function. Theorem 2. Let M(t) be a kernel from the Sonin set S −1 and the power series has non-zero convergence radius r. Then the convolution series is convergent for all t ∈ (0, ∞) and the function F(x, λ, t) belongs to the ring R −1 .

Definition 4.
Let M(t) be a kernel from the Sonin set S −1 , and M * ,j (t) is the convolution j-power: where M k (t) = M(t) for all k = 1, . . . , j, and t ∈ (0, ∞). Then, the function will be called the first Luchko function. (1) For the Sonin kernel the first Luchko function has the form Here, E α,β (z) is the two-parameters Mittag-Leffler function [76] that is defined as where α > 0, β ∈ R (or β ∈ C).
Then, the function will be called the second Luchko function.
Note that Equation (60) contains the GFI with kernel K(t) rather than kernel M(t).
The second Luchko function (60) is used [60] in solution of equations with GFD that is defined by the kernel K(t) associated with the kernel M(t) of the GFI.
Therefore, these Luchko functions belong to the ring R −1 . Thess statements are based on the fact that GFI I t (K) [τ] is the operator on C −1 (0, ∞) (see Equation (30) and reference [60]). Note that the second Luchko function (60) can be considered as independent of the kernel K(t) since the Sonin condition (K * M)(t) = {1} is satisfied for all t ∈ (0, ∞).
Using the superoperator form of the first Luchko function F(M, λ, t) and the second Luchko function L(M, λ, t), we can propose solutions of equations for non-Markovian open quantum systems with nonlocality in time.

General Form of Solutions for Non-Markovian Equations
Let L be a bounded superoperator on the normed operator algebra A, in other words, and L 0 = L I is the unit superoperator (L I A = A for all A ∈ A). The superoperator power series converges in norm and the radius of convergence is equal to if L is the bounded superoperator on the normed operator algebra A.
Using Theorem 2, we can state that the series The solution of Equation (41) can be expressed through the superoperator (65).
To obtain the solution of the general non-Markovian equation for quantum observable A ∈ A, we will use Theorem 5.1 of [60] (p. 366).
is Sonin pair from S −1 , and L is a bounded superoperator on A. Then the initial value problem has the unique solution The proof of this theorem is based on Theorem 5.1, which was proven in [60] (pp. 366).
Using condition (69), we get the convolution of kernel K = K(t) and L = F(M, L, t) in the form where L 0 = L I is the unit superoperator (L I A = A for all A ∈ A). Therefore the map Φ t (M) is written as In the next section, we give some examples of the general non-Markovian Equation (66) and solutions (67).

Example of General Non-Markovian Dynamics
Let us consider some special cases of the proposed general non-Markovian equations and its solutions that are derived by the Luchko operational calculus [60].
(1) In the first example, we consider the Sonin pairs of the kernels where t > 0, and 0 < α < 1. In this case, the GFD D t, * where the Mittag-Leffler function E α (z) = E α,1 (z) [76]. The fractional differential equation has the solution This type of non-Markovian quantum dynamics was first described in [37] (Chapter 20) and [38,39].
(2) In the second example, we consider the Sonin pairs of the kernels where The GFD has the form The superoperator Φ t (M) is written as For nonlocality (76), the equation of non-Markovian dynamics has the solution (3) In the third example, we consider the Sonin pairs of the kernels where 0 < α < β < 1. In this case, we use the GFD that has the Mittag-Leffler function in the kernel Then, the non-Markovian dynamics is described by the superoperator where the function E (1−β,1−β+α),1 is a special case (binomial) of the multinomial Mittag-Leffler function [77]. The equation with GFD (83) and initial condition has the solution These examples of equations and their solutions describe the non-Markovian dynamics of quantum observables.

Violation of Semigroup Property for Non-Markovian Maps
The non-Markovian maps Φ  The set Φ (M) t t > 0 , is called a quantum dynamical groupoid [5,38]. Note that the following properties are realized Φ where L I is an identity superoperator (L I A = A). As a result, the non-Markovian map Φ (M) t , t > 0, are real and unit preserving maps on the operator algebra of quantum observables. For the GFD is the Caputo fractional derivative of the order α, and the non-Markovian map has the form that is described in [37,38], where E α [z] is the Mittag-Leffler function [76].
The superoperators Φ t form a semigroup such that This property holds since For α ∈ N, we have the violation of the semigroup property [78][79][80]: Therefore, the semigroup property is not satisfied for non-Markovian dynamics As a result, the non-Markovian maps Φ

Some Properties of Markovian Maps
In this section, we will briefly describe the properties of Markovian maps Φ t and the superoperator L, for the convenience of generalizations to non-Markovian dynamics.

Bi-Positivity and Dissipativity in Markovian Theory
The Markovian quantum dynamics is described by the map where h n+1 (t) = t n Γ(n + 1) = t n n! .
We will assume that D(L) = A to simplify the description. To this purpose, we will also consider the bi-positivity condition instead of complete positivity condition.
The bi-positivity condition for the Markovian map Φ t can be considered in the form which should be satisfied for all t > 0 and A ∈ A. The importance of condition (88) is due to the fact that it leads to the positive condition As a result, we obtain If the real superoperator L is completely dissipative, for which the inequality is satisfied for all A k , A l ∈ A, then the quantum Markovian map map Φ t is completely positive, if Φ t A † = (Φ t (A)) † for all A ∈ A. This statement can be proved similarly by using the following transformations Let us consider two approaches to find a condition that the real superoperator L must satisfy in order for the bi-positivity condition to be satisfied for all t > 0 in the Markovian quantum dynamics, and a possibility to generalize these approaches to the general non-Markovian maps.

Markovian Case: First Approach
The condition on L to have the bi-positivity of maps Φ t is usually obtained by differentiating inequality (88) with respect to time and using the standard Leibniz rule The Markovian equations for quantum observables and Equation (91) allow us to get inequality (90) in the form In the limit t → 0 , we get the condition Unfortunately, this approach cannot be used for equations with fractional derivatives and GFD because the standard Leibniz rule (the product rule) is violated For example, in the non-Markovian quantum theory, which was proposed in [37] (pp. 477-482) and [38,39], the Sonin pair of kernels is used in the form with 0 < α < 1. In this case, the GFI and GFD are the Riemann-Liouville fractional integral and the Caputo fractional derivative. The generalized Leibniz rule (see Theorem 3.17 in [30] (p. 59)) has the form The violation of the standard Leibniz rule is a characteristic property of fractional derivatives of non-integer order [81].
For GFD, there is no rule for differentiating the product in the general case. Therefore, we should use another approach to derive the conditions on the superoperator L. For Markovian dynamics, another method of obtaining the condition on L can be used, and this method can be generalized to the case of non-Markovian quantum theory.

Markovian Case: Second Approach
Let us consider the bi-positivity condition in the form which should be satisfied for all t > 0 and all A ∈ A, where the average value of the quantum observable

Substitution of expression (87) into inequality (95) gives
where we use the linearity of the average value. Using n = m + k, and Since the bi-positivity condition must be satisfied for all t > 0, we obtain which should be satisfied for all n ∈ N. For n = 1, inequality (98) has the form which should be satisfied for all A ∈ A. The bounded superoperators L, which satisfy conditions L(I) = 0, L A † = (LA) † and inequality (99), are called dissipative.
Inequality (99) for real superoperator L is a necessary and sufficient condition in order for the Markovian quantum map Φ t to have a bi-positivity property.
Theorem 4. Let L be real superoperator, which satisfies the dissipativity condition for all A ∈ A. Then, the Markovian map Φ t satisfies the bi-positive condition for all t > 0 and all A ∈ A.
Proof. Using the series representation of Φ t A A † and inequality we get

General Non-Markovian Maps: Bi-Positivity and Complete Positivity
In this section, we will consider the bi-positivity condition instead of the complete positivity condition to simplify the description and proofs. Condition of complete positivity proved similar to the proofs for bi-positivity, and conditions for complete dissipativity of the superoperator L are written analogously to conditions for general dissipativity. We will also assume that D(L) = A to simplify the description.
Let us give the definition for the non-Markovian quantum maps that are described in the paper.
will be called the general non-Markovian quantum map.
which holds for all t > 0 and all A ∈ A. The average value of the quantum observable for all t > 0 and all A ∈ A. Then, the positivity condition holds for all t > 0 and all A ∈ A.
Proof. The following equalities hold for all t > 0 and for all A ∈ A. Therefore, bi-positivity condition (104) leads to the positivity which holds for all t > 0, and for all A j , B j ∈ A, j = 1, . . . n and all n ∈ N.
holds for all t > 0.
Proof. Using A k = A and B k = I, inequality (105) takes form (106).

From Bi-Positivity to General Dissipativity
Let us find a condition that the real superoperator L must satisfy in order to bipositivity condition (104) to be satisfied for all t > 0 for the general non-Markovian quantum maps.
This required condition is given by the following theorem.

Theorem 5. Let the bi-positivity condition
be satisfied for all t > 0 for the general non-Markovian map where M n (t) > 0 (109) for all n ∈ N and all t > 0.
Then real superoperator L satisfies the inequality for all n ∈ N and all t > 0, where Proof. Substitution of expression (102) in the right side of inequality (104) gives where we use the linearity of the average value. Using m = n − k, Equation (112) takes the form Using (102) and (113), bi-positivity condition (104) is represented by the inequality Since inequality (114) must hold for all t > 0, we will look for conditions on L by using the inequalities Using the assumption that the inequality holds for all n ∈ N and all t > 0, we obtain the condition where and k, n ∈ N 0 and 0 ≤ k ≤ n.
Definition 9. Let a pair of kernels (M(t), K(t)) belong to the Sonin set S −1 , and for all n ∈ N and all t > 0. Then the function where k, n ∈ N 0 and 0 ≤ k ≤ n, will be called the general binomial coefficients.

Remark 2. If the kernel M(t) is positive
for all t > 0, then condition (119) holds.

Definition 10.
Let the real operator L satisfy the inequalities for all n ∈ N 0 and all A ∈ A. Then L will be called the general dissipative superoperator. The general complete dissipativity condition is defined in the form which holds for all t > 0, and for all A i , A j ∈ A, j = 1, . . . m and all n, m ∈ N.
We will consider the bi-positivity condition instead of the complete positivity condition to simplify the description and proof in the next sections. Conditions for complete positivity were proved similarly to the proofs for bi-positivity. The condition for general complete dissipativity of the real superoperator L will be used analogously as the condition for general dissipativity.
In connection with general dissipativity condition (121), which must be satisfied for any value of all n ∈ N, two questions arise: (1) What condition must the superoperator L 1 = L satisfy in order for the general dissipativity condition (121) to hold for all n ∈ N? (2) What is the connection between the general dissipativity condition, and the condition of the dissipativity? Answers to these questions will be offered in the following sections.

General Dissipativity for n = 1
Let us consider the general dissipativity condition for the superoperator L n with n = 1. Proof. Using the definition of the general binomial coefficients, we have Using we get Theorem 7. Let a pair of kernels (M(t), K(t)) belongs to the Sonin set S −1 , and M n (t) > 0 for all n ∈ N and all t > 0. Then the general dissipative superoperator L n with n = 1 satisfies the condition for all A ∈ A.
Proof. For n = 1, inequality (110) takes the form where Using Theorem 7, we have Therefore, inequality (127) has the form be general non-Markovian map with the positive kernel M(t) > 0 for all t > 0 such that the bi-positivity condition is satisfied for all t > 0, and all A ∈ A. Then, the real superoperator L satisfies the dissipativity condition holds for all A ∈ A.
Proof. Using Theorem 5, we get that the bi-positivity condition (132) for general non- leads to the general dissipativity condition for the superoperator L n .
Then, using Theorem 6, the general dissipativity condition leads to the dissipativity condition (133).

From General Dissipativity to Bi-Positivity
Let us prove the theorem that is converse to Theorem 5.
is satisfied for all t > 0 and all A ∈ A for the general non-Markovian map Proof. Let us use condition (134) in the form Using assumption that M n (t) > 0 holds for all n ∈ N and all t > 0, inequality (138) can be written as Since, by condition of the theorem, inequality (138) holds for any n ∈ N 0 . Then, summing from 0 to ∞ inequality (139), we obtain Using m = n − k, condition (140) takes the form As a result, Theorems 5 and 9 allow us to formulate the following statement.
that holds for all t > 0, is the general dissipativity of the real superoperator L in the form that satisfies for all n ∈ N.

Remark 3. Let the inequality
be satisfied for all n ∈ N and all A i , A j ∈ A, where i, j = 1, . . . , m. Then, the completely positive condition holds in the form for all n ∈ N and all A i , B i ∈ A, where i, j = 1, . . . , m. The proof of this statement is realized similarly to Theorem 9.

Corollary 2.
The condition that a superoperator is dissipative is a necessary condition for non-Markovian maps to be bi-positive, but it is not a sufficient condition.

From General Dissipativity to Dissipativity
Let us consider the relationship between the concepts of dissipativity and general dissipativity. Then, the superoperator L satisfies the dissipativity condition for all n ∈ N.
Proof. Using Theorem 7, we get that general dissipativity condition (148) with n = 1 has the form Using inequality (150) two times for the expression L 2 A A † and the linearity property of the superoperator L, we obtain Using inequality (150) n-times for the expression L n A A † and the linearity property of the superoperator L in a similar way, we obtain inequality (149).
which holds for all t > 0 and all A ∈ A. Then, the superoperator L satisfies the dissipativity condition for all n ∈ N, and all A ∈ A.

Remark 4. The dissipativity condition
which holds for all n ∈ N does not lead to the fact that the general non-Markov map Φ If we assume in non-Markovian quantum theory that L is dissipative superoperators only, then the standard bi-positive condition that should be satisfied for all t ≥ 0. Let us prove this statement.
Theorem 11. Let real superoperator L satisfy the inequality Then, the inequality is general non-Markovian map.
Proof. Using the proof of Theorem 10, we get that the repeated action of the inequality (154) gives the condition Using the series representation of Φ t A A † and inequality (18), we get Using m + k k ≥ 1 for k, n ∈ N 0 , 0 ≤ k ≤ n, and M n+1 (t) > 0 for all n ∈ N 0 , we can write the inequality The convolution with {1} means the integration in the form Therefore, multiplying the left and right sides of the inequality by {1} and using Sonin's condition

Remark 6.
Similarly, we can prove the inequality

From Dissipativity to Bi-Positivity
Let us consider the relationship between the dissipativity and bi-positivity in general non-Markovian dynamics.

Theorem 12.
Let real superoperator L satisfy the dissipativity condition for all n ∈ N and all A ∈ A, and the general binomial coefficients satisfy the condition for all t > 0 and all k, n ∈ N 0 , 0 ≤ k ≤ n. Then, the superoperator L satisfies the general dissipativity condition for all n ∈ N and all A ∈ A. Then, the general non-Markovian map Φ (M) t satisfies the bi-positivity condition which holds for all t > 0.
which holds for all t > 0 and all A ∈ A, and the general binomial coefficients satisfy the condition for all t > 0 and all k, n ∈ N 0 , 0 ≤ k ≤ n. Then, the superoperator L satisfies the dissipativity condition for all n ∈ N and all A ∈ A.
Proof. Using Theorem 5, bi-positivity condition (161) gives the general dissipativity condition for all n ∈ N and all A ∈ A. Then using the condition (162) for the general binomial coefficients, we obtain for all n ∈ N and all A ∈ A. Then, using conditions (164) and (165), we derive the dissipativity condition (163).

Examples of General Binomial Coefficients
Let us consider the general binomial coefficients for the non-Markovian quantum maps with the pair of kernels that belongs to the Sonin set for α ∈ (0, 1). We see that the condition M(t) > 0 holds for all t > 0. Let us give a definition of the generalized binomial coefficients (see [26] (p. 14-15) and [29] (p. [26][27]). These well-known "generalized" coefficients should not be confused with the "general" coefficients suggested in this article. Definition 11. The generalized binomial coefficients are defined by the equation where α, β ∈ C, α = −1, −2, . . .
Let us prove that the general binomial coefficients for kernel pair (166) are expressed through the generalized binomial coefficients.
Then, we have the equation which satisfies the condition M n (t) > 0 for all n ∈ N and all t > 0.

Proof. Using the property
that holds for α, β > 0, we obtain Then, We see that the inequality M n+1 (t) > 0 holds for all n ∈ N and for all t > 0. Then, using (169), we get where the generalized binomial coefficients αn αk are defined by (167).
Let us prove the following statement about the properties of the general binomial coefficients (171) for the kernels (166).

Theorem 15. Let us consider the function
where α > 0, 0 ≤ k ≤ n, k, n ∈ N 0 . This function is increasing with respect to α > 0 with fixed parameters k, n ∈ N 0 , and the following inequality holds Proof. Let us prove that d dα f n,k (α) ≥ 0.
Using the digamma function ψ(α) of real argument α > 0, increases function with respect to α > 0: for k ≥ 0, and that the generalized binomial coefficients are positive functions for α > 0, 0 ≤ k ≤ n, k, n ∈ N 0 , we derive that the generalized binomial coefficients f n,k (α) as a function of the variable α > 0 has a non-negative derivative d f n,k (α)/dα ≥ 0 with respect to α.

Remark 7. For general binomial coefficients
we have the inequalities

Examples of Inequalities for General Binomial Coefficients
Let us give some examples of inequalities for general binomial coefficients.

Example 1.
Let us consider the pair of the kernels This pair belongs to the Sonin set S −1 , if α ∈ (0, 1). For these kernels, the general binomial coefficients are given by the equations .
If the parameter α satisfies the condition α ∈ (0, 1), then the inequalities hold for all k, n ∈ N 0 , when 0 ≤ k ≤ n and all t > 0. We can assume that within the framework of the GFC of arbitrary order [59,71], the expression of the general non-Markovian map Φ (M) t and the general binomial coefficients E n k (K, M) can also be derived. For the kernels with α ∈ (1, 2), the non-Markovian equation for quantum observables has the form where D t h 2−α [τ] is the Caputo fractional derivative of the order α ∈ (1, 2) that is defined as To solve Equation (175) and derive general binomial coefficients, we can use Theorem 4.3 and Example 4.10 of [29] (p. 231-231). The solution is described by the equation For A(0) = A and A (1) (0) = 0, we have the non-Markovian map

Corollary 4. Let the general dissipativity condition
with the kernels (174), be satisfied for all n ∈ N and A ∈ A. Then, the general binomial coefficients have the form where α ∈ (1, 2). For the non-Markovian map (176), the bi-positivity condition is satisfied for all t > 0 and all A ∈ A.

Non-Markovian Quantum Oscillator with Nonlocality in Time
Let us consider a non-Markovian quantum oscillator with nonlocality in time.
As is usually assumed for oscillators that are open quantum system [10][11][12]37], the general form of a bounded completely dissipative superoperator holds for an unbounded superoperator L. Then, the general non-Markovian dynamics of coordinate Q and momentum P is described by the equations where L is defined by Equation (20). For the linear quantum oscillator, the operators V k = V k (Q, P) and H = H(Q, P) are the functions of the coordinate and momentum operators in the form where a k , and b k , k = 1, 2, are complex numbers. The term with parameter µ can be interpreted as friction, for which force is proportional to the velocity m −1 P.

Remark 8.
In general, the operators coordinate Q and momentum P are unbounded operators. Due to this, instead of the Hilbert space, one can use the so-called rigged Hilbert space (the Gelfand triplet). A rigged Hilbert space is the ordered triplet where H is a Hilbert space, B is a Banach space, and B * is dual of B. The term "rigged Hilbert space" is also used to describe the dual pairs (B, B * ) generated from a Hilbert space H. The term "Gelfand triplet" is sometimes used instead of the term "rigged Hilbert space". Example of a rigged Hilbert space is the triple of spaces that consists of the Banach space J(R n ) of test functions, the Hilbert space L 2 (R n ) of square integrable functions, and the Banach space J * (R n ) of the linear functionals on J(R n ). For details, see Chapter 2 in [5].
Remark 9. The Lindblad result has been extended by E.B. Davies [9] to a class of quantum dynamical semi-group with unbounded generating superoperators.

Remark 10.
In this linear model, the parameters a k , b k ∈ C cannot be arbitrary. Let us consider the real parameters d aa := 2 |a 1 | 2 + |a 2 | 2 , d bb := 2 |b 1 | 2 + |b 2 | 2 , There is a fundamental constraint [12] on the parameters in the form which follows from the Schwartz inequality Using the canonical commutation relations for operators Q and P, we obtain Equations (180) and (181) for operators Q(t) and P(t) in the form where where D t, * (K) is the general fractional derivative with the kernel K(t) ∈ C 1 (0, ∞), for which M(t) is the associated kernel so that the pair (M(t), K(t)) belongs to the Sonin set S −1 .
Let us represent Equations (184) and (185) in the matrix form. Using the matrices Equations (184) and (185) are written as where we used LA(t) = N A(t).
Theorem 16. Let the function K(t), M(t) ∈ C 1 (0, ∞) belong to the Sonin set S −1 . Then, the initial value problem for the equation and the condition A(0) = A 0 , where A(t) and N are defined by (187), has the solution in the form with the quantum dynamical map where L(M, N, t) is the second Luchko function with the matrix argument N.
The statement of Theorem 16 follows directly from Theorem 3.
the GFD is the Caputo fractional derivative of the order α ∈ (0, 1), and solution (191) has the form For α = 1, solution (192) gives that describes the Markovian quantum dynamics of open system without nonlocality in time.
Let us prove the following theorem.

Proof.
To get exact expression of the solution for coordinate and momentum operators, we represent the matrix N in the form where and Using (198), the non-Markovian quantum dynamical map Φ (M) t is given as As a result, we have Φ Substituting expression (199) and (200) into Equation (203), we get the dynamical map where we use the functions (196) and (197).
For α = 1, map (205) is given by the standard expression   (185), which describe non-Markovian dynamics of quantum system with power-law memory, were first derived in [37][38][39], where For the case with α = 1, equations describe the Markovian dynamics of open quantum systems without nonlocality in time (α = 1), since E 1 [z] = exp(z), and where sinh and cosh are hyperbolic sine and cosine.

Non-Markovian Quantum Dynamics of Two-Level System
In the case of an N-level open quantum system, the problem was investigated by V. Gorini, A. Kossakowski, and E.C.G. Sudarshan [6]. The general form of the generating superoperator of a completely positive dynamical semi-group of this system has been established [6]. In the case of the N-level quantum system, the Hilbert space H has the dimension dimH = N. Each N-dimensional separable Hilbert space over C is isomorphic to C n .
Let us consider a general non-Markovian dynamics of quantum states. The non-Markovian dynamics of the density operator ρ(t) can be described by the equations with GFD in the form where {A, B} = A B + B A. Let us consider the non-Markovian two-level quantum systems with general nonlocality in time, which is described by the Sonin kernel K(t). The Hamiltonian will be considered in the form where ω 0 > 0 is the transition frequency. Then, the Hamiltonian of the two-level quantum system is diagonal in the basis |0 , |1 . The operators V k = 0 with k = 1, 2 will be considered in the form where η 1 , η 2 ∈ R, and ρ 10 (t) = L(M, λ 2 , t) ρ 10 (0).
The second pair is the equations To get solutions of Equations (235) and (236), we considered these equations in the matrix form where and γ 1 > 0, and γ 2 > 0. The solution of Equation (237) has the form where The matrix G can be diagonalized as where Exact expression for solution (239) of Equation (237) is derived by the transformations Therefore, we get S t (α) = L(M, G, t) + S(γ 1 , γ 2 ) = where γ = γ 1 + γ 2 , and the solution for ρ 00 (t) and ρ 11 (t) has in the form As a result, we obtain the solution for components of the density matrix ρ kl (t) in the form and where γ = γ 1 + γ 2 .

Entropy for General Non-Markovian Quantum Dynamics
In quantum mechanics and quantum statistics, the concept of entropy S is defined through the density operator ρ, which is a positive, normalized self-adjoint linear operator. John von Neumann defines [82] the entropy as an extension of the Gibbs entropy concepts from classical mechanics to the quantum mechanics.
For a quantum-mechanical system described by a density operator, the von Neumann entropy is defined (see Sections V.2 and V.3 in [82]) by the equation where Tr is the trace and ln denotes the (natural) matrix logarithm.
For non-Markovian and non-Hamiltonian systems, the von Neumann entropy changes in the general case.
Let us consider the von Neumann entropy for general non-Markovian dynamics of two-level quantum systems.
Using representation (258) of density operator (257), we can derive an explicit form of the von Neumann entropy for non-Markovian two-level quantum system.

Conclusions
In this paper, we proposed the formulation of non-Markovian quantum theory in the general form. The non-locality in time is represented by kernels of integral and integrodifferential operators. These kernels are described by functions that belong to the Sonin set of kernel pairs. The results can be derived in the general form without using special realization of these kernels. Therefore, these results are valid for any operator kernels from the Sonin set. This approach to non-Markovian quantum theory is directly connected with the concept of general fractional dynamics suggested in [72].
Non-Markovian equations for quantum observables and states are suggested by using general fractional calculus. The exact solutions of these equations are derived by using the operational calculus, which is proposed by Luchko in [60] for equations with general fractional derivatives. A wide class of the exactly solvable models of non-Markovian quantum dynamics is suggested. These models describe non-Markovian open quantum systems with the general form of nonlocality in time. The non-Markovian models of quantum oscillator and two-level quantum system with general form of nonlocality in time are described. The exact solutions of equations for these models are proposed.
This paper proposes a general approach to describing non-Markov quantum dynamics. Many important issues are not covered in this article. This work does not offer solutions to all the problems of constructing general non-Markov dynamics of open quantum systems. Let us note some unresolved questions that await their solution in future research.
(1) A quantum system can be embedded in some environments and therefore the system is not isolated. The environment of a quantum system is in principle unobservable or is unknown. This would render the non-Markovian theory of open quantum systems a fundamental generalization of quantum mechanics. However, for practical applications, it is useful to have models of open quantum systems that can be derived from some closed systems including the system under study and some environments. In this regard, the problem arises of constructing models of such closed systems and obtaining general non-Markov dynamics, for example, within the framework of the Caldeira-Leggett approach [83]. At the moment, this problem has not been solved, and the question remains open. We think that the construction of such models is possible. This opinion is based on the following: in the framework of the simplest models, the kernels of fractional derivatives, which describe nonlocality in time, were obtained in [84]. (2) For open quantum systems, its "reduced" dynamics not to violate thermodynamics must not decrease entropy of the evolving state [85]. In this regard, the problem arises of a detailed study of the behavior of entropy for general non-Markov dynamics. At the moment, this problem has not been solved. This question is interesting for further research and computer simulation of the behavior of entropy. (3) The form of the superoperator L was determined by the Lindblad theorem, which describes the relationship between a completely positive semigroup and a completely dissipative superoperator. The condition for the dissipativity of the superoperator is in fact the standard Leibniz rule, in which equality is replaced by inequality. In non-Markovian dynamics, the semigroup property is violated, and the fractional derivative violates the standard Leibniz rule. In this regard, the question arises about the existence of a generalization of the Lindblad superoperator, in the framework of the proposed general non-Markovian dynamics. In our opinion, such a possibility exists and may be associated with the fractional powers of Lindblad superoperators and the models proposed in the works [5] (pp. 433-444) and [37] (pp. 458-464, 468-477), and [41,42].
Funding: This research received no external funding.

Conflicts of Interest:
The author declares no conflict of interest.