Nonlinear extension of the quantum dynamical semigroup

In this paper we consider deterministic nonlinear time evolutions satisfying so called convex quasi-linearity condition. Such evolutions preserve the equivalence of ensembles and therefore are free from problems with signaling. We show that if family of linear non-trace-preserving maps satisfies the semigroup property then the generated family of convex quasi-linear operations also possesses the semigroup property. Next we generalize the Gorini-Kosakowski-Sudarshan-Lindblad type equation for the considered evolution. As examples we discuss the qubit evolution in our model as well as an appropriate extension of the Jaynes-Cummings model.

Deterministic nonlinear evolutions are believed to allow for signaling [6], it was for the first time explicitly shown by Gisin in [7], compare also [8,9]. Such evolutions are usually defined for pure states: and consequently the evolution of ensembles is assumed to have the following form: If f t (|ψ 1 ) = |ψ 1 (t) , f t (|ψ 2 ) = |ψ 2 (t) then f t : λ|ψ 1 ψ 1 | + (1 − λ)|ψ 2 ψ 2 | → λ|ψ 1 (t) ψ 1 (t)| + (1 − λ)|ψ 2 (t) ψ 2 (t)|, (2) i.e., coefficients of the ensemble do not change under the evolution. This assumption easily implies that deterministic nonlinear evolution breaks the equivalence of ensembles corresponding to the same mixed state and results in the possibility of arbitrary fast signaling [10]. In our recent paper [11] it was shown that if we replace the assumption (2) by the so called convex quasi-linearity, i.e., we allow the appropriate change of the coefficients [see Eq.
(3)], then evolution satisfying such a condition preserves equivalence of ensembles and consequently such evolutions cannot be ruled out by the standard Gisin's argument.
Let us stress that in our approach we do not change anything else in the quantum formalism but only extend admissible set of quantum evolutions by including nonlinear deterministic evolutions that do not admit superluminal signaling. This is in contrast with such nonlinear extensions of quantum mechanics that does not * jaremb@uni.lodz.pl † P.Caban@merlin.phys.uni.lodz.pl allow signaling but demand modifications of other quantum mechanical rules (see, e.g., [12,13]). We also do not consider here stochastic nonlinear evolutions some of which are free from the problems with signaling and have many important applications (like in the collapse models [14,15]).
In this paper we demonstrate that there exits a large class of convex quasi-linear evolutions. These evolutions are generated by linear non-trace-preserving quantum operations. What is interesting, recently considered evolutions generated by non-Hermitian Hamiltonians [16,17] also belong to this class. It shows that convex quasi-linearity might be used as a principle joining nonlinear quantum mechanics and non-Hermitian quantum mechanics.
In Sec. II we remind the definition of convex quasilinearity and demonstrate that each linear non-tracepreserving quantum operation generates convex quasilinear operation. In Sec. III we show that if family of linear non-trace-preserving maps satisfies the semigroup property then the generated family of convex quasilinear operations also possesses the semigroup property. Next we consider Gorini-Kosakowski-Sudarshan-Lindblad type equation for the considered evolution. Sec. IV is devoted to the discussion of qubit evolution in our model. We finish with some conclusions and open problems.

II. ADMISSIBLE NONLINEAR QUANTUM OPERATIONS
We start with recalling the definition of convex quasilinearity [11]. Let us denote by S the convex set of density matrices. We call a map Φ : S → S convex quasi-linear if for all ρ i ∈ S and p i ∈ 0, 1 , i p i = 1 (i = 1, . . . , N ) there existp i such that andp i ∈ 0, 1 , ip i = 1. Below we show that there exists a class of convex quasi-linear operations.

arXiv:2003.09170v1 [quant-ph] 20 Mar 2020
An example is discussed by Kraus [18] as a generalized measurement. For simplicity we restrict here our example to a standard selective, projective measurement. Let Π be a projector. After the measurement of Π = 1 · Π + 0 · (I − Π) we obtain provided that the outcome of the measurement is 1 (which takes place with the probability Tr(Πρ)). Now, if ρ is given as the following ensemble whereλ = λ Tr(Πρa) Tr(Πρ) . We see that after action of the operation Φ we obtain again ensemble Φ(ρ) as a convex combination of images of the initial density operators with a change of coefficients λ →λ. Thus the map Φ(ρ) satisfies the convex quasi-linearity property (compare [11] for more detailed discussion of Φ in this case). Moreover, it is easy to see that the convex quasi-linearity of the considered nonlinear mappings implies preserving of the equivalence classes of ensembles of density matrices. This means that under Φ equivalence class of ensembles is mapped on the equivalence class.
Let us consider a most general linear, completely positive quantum operation ϕ having the Kraus form where Tr(ρ in ) = 1, K α are Kraus operators, α max < N 2 , N < ∞ is the dimension of the Hilbert space H of the considered system. Furthermore, F = α K † α K α ≤ I. To obtain a map into the convex set of density matrices we must normalize the ϕ(ρ in ) in the case F < I. Consequently, the complete quantum operation has the form i.e., Of course Tr[ϕ(ρ in )] = 1 implies linearity of Φ. It is easy to see that Φ is in general convex quasi-linear, i.e., where 0 ≤ λ ≤ 1 and thus so 0 ≤λ ≤ 1. Of course, if Tr[ϕ(ρ in )] = 1 thenλ = λ (compare [19]). From the above definition it follows that the frequently used argument that two equivalent ensembles after nonlinear map lost this equivalence does not apply to the map Φ. This means that Φ can be treated as the generalized form of the acceptable quantum operation.

III. ADMISSIBLE NONLINEAR EVOLUTIONS
Now, the notion of convex quasi-linearity can be extended for time evolution. Namely, the transformation is a convex quasi-linear time evolution if Φ t forms a semigroup, i.e., Φ t1 •Φ t2 = Φ t1+t2 and the condition (3) holds for all times t, i.e., for all ρ 0i ∈ S, p i ∈ 0, 1 , and p i (t) ∈ 0, 1 , i p i (t) = 1. In our paper [11] we have found a toy model of a convex quasi-linear time evolution of a qubit. Here we show that there exists a large class of natural evolutions fulfilling the above conditions.

A. Quasi-linear time evolution generated by linear transformations
According to the above discussion there are no formal objections to identify time evolution of density operators with a family of time dependent extended quantum operations. Consequently we postulate the evolution of the density operator in the form of the nonlinear map with Here Notice that if F (t) = I then the evolution is linear. The initial condition ρ 0 = ρ(0) is related standardly with K 0 (0) = I and K α (0) = 0 for α = 1, 2, . . . , α max .
Let us assume that the family of linear positive maps ϕ t satisfy ϕ τ (ϕ t (M )) = ϕ τ +t (M ) for each M , i.e., {ϕ t } forms a one parameter semigroup. Then using linearity of ϕ t and the definition of Φ t we have for each ρ 0 i.e., Φ τ (Φ t (ρ 0 )) = Φ τ +t (ρ 0 ). We conclude that under our assumptions the family of quantum operations Φ t (ρ 0 ) forms a nonlinear realization of the one-parameter semigroup realized in the convex set of the density matrices.

B. Nonlinear extension of the Gorini-Kosakowski-Sudarshan-Lindblad equation
The linear dynamics of an open quantum system is described by Gorini-Kosakowski-Sudarshan-Lindblad (GKSL) dynamical semigroup which is generated by the GKSL generator.
Using the infinitesimal form of the global time evolution as well as the form of the effect operator F (t) and defining K 0 (δt) ≈ I +δt(G−iH), where G and H are Hermitian while for α = 1, 2, . . . , α max , K α (δt) ≈ √ δtL α , we obtain the generalization of the action of this generator to the forṁ (the detailed derivation of the above equation is given in Appendix). Notice, that the nonlinearity of the dynamics of the density operator is rather weak: It relies on the nonlinear coupling between ρ (operator) and the ρ-dependent trace Tr ρ 2G + α=1 L † α L α . From the above dynamical equation it follows that infinitesimally the operator F (t) is generated by We observe that for F (t) = I, i.e., for 2G+ α=1 L † α L α = 0, we recover the standard form of the GKSL generator: Notice, that vanishing of the Lindblad generators (only K 0 is nonzero) implies that the nonlinear Kraus evolution reduces to with where G † = G, H † = H and we can restrict ourselves to the case of traceless generators G and H. Thus, the family of K(t) operators forms an one-parameter subgroup of the SL(N, C) linear group. The corresponding GKSL equation reduces to the nonlinear generalization of the von Neumann equation, i.e., (compare [16]) with the initial condition ρ(0) = ρ 0 . Let us observe that the pure states form for this evolution an invariant subset in the convex set of density operators. Indeed, we see that in this case ρ 0 = |ψ ψ| with ψ|ψ = 1, so the evolution equation takes the form Therefore i.e., ρ(t) is a pure state. Therefore, we can find a counterpart of Eq. (27) for state vectors. However, the corresponding equation describing the evolution of a state vector is not uniquely determined by Eq. (27). Indeed, the whole family of equations of the form where κ is an arbitrary real function of |ψ , leads to the evolution equation (27) for density matrices. It is worth to mention here that in general the evolution equation for a state vector also does not determine uniquely the evolution equation for density matrices. Evolution of the form (30) for pure states with κ = 0 and G = kH was discussed by Gisin in [20] and subsequently with general G and κ = 0 in [21]. Evolution equation similar to (27) was used in a specific experimental context in [22], where an interaction of a two-level atom with the electromagnetic field was considered.

IV. NONLINEAR QUBIT EVOLUTION
In this section we discuss qubit evolution in our model. Under the special choice of the von Neumann nonlinear equation it was also investigated recently in [16,17,23,24]. In this case the initial density matrix can be taken in the following form: where ξ is a real vector satisfying the condition ξ 2 ≤ 1 and σ is the triple of the Pauli matrices.
A. The case of vanishing Lindblad generators (Lα = 0) The SL(2, C) nonlinear evolution of ρ in the case of vanishing of the Lindblad generators has the form where the evolution matrix reads with G = 1 2 g · σ, H = 1 2 ω · σ, and g, ω are fixed real vectors. Moreover we assume n(0) = ξ.
This evolution can be represented on the Bloch ball as a nonlinear realization of one-parameter subgroup of the orthochronous Lorentz group homomorphic to SL(2, C). Notice that the quantities C 1 = g · ω and C 2 = g 2 − ω 2 are invariant under inner automorphisms of the SL(2, C). Therefore, values of C 1 and C 2 determine different types of evolution.
Let us calculate the time evolution of the probability of finding the evolved state ρ(t) in eigenvectors of the Hamiltonian H. For ω = (0, 0, ω) the normalized eigenvectors of the Hamiltonian H = 1 2 σ 3 have the form: thus the corresponding projectors correspond to the ξ ± = (0, 0, ±1). We obtain: (44) In the asymptotic limit p ± (∞) = 1 2 .
In the asymptotic limit p ± (∞) = 1 2 . g < ω p + (t) = 1 + n 3 (t) 2 = 1 2 The case g < ω (when Ω = ω 2 − g 2 ) is the most interesting one (compare with [17]). In this case the probability of finding the evolved state in one of the eigenstates of the Hamiltonian H, a − = (0, 1), is given by (48). For comparison, let us remind that the counterpart of the probability (48) obtained for the standard Rabi oscillations (i.e. in the case of evolution governed by the Hamiltonian H = ω 2 σ 3 + g 2 σ 1 ) has the form [25] p Rabi It is interesting to notice that both probabilities, p − (t) (48) and p Rabi − (49), for fixed g and ω attain the same maximal value equal to (50) Below we illustrate the behavior of the probabilities (48) and compare it with the probabilities obtained for the standard Rabi oscillations (49). In Figs. 1 and 2 we present the probability (48) for g < ω. In Fig. 3 we compare the probability (48) with the probability obtained it the case of standard Rabi oscillations (49) (for the same values of parameters ω and g).
Since the initial state is pure |ξ| = 1, the vector n(t) has length 1 for all t. It means that the curves in Fig. 4 are situated on the Bloch sphere. For comparison, in Fig. 5 we present similar trajectories for the initial state ρ(0) = 1 2 I (ξ = (0, 0, 0) and again ω = (0, 0, ω) and g = (g, 0, 0), ω, g > 0) [for calculations we used formulas (39,40,41)]. In this case the curves are inside the Bloch sphere on the plane ξ 3 = 0. Therefore, we present the section of the Bloch sphere with this plane. Blue (solid) line corresponds to the case ω > g (for the plot we set ω = 6, g = 4). Red (dotted) line corresponds to the case ω = g (for the plot we set ω = 6). Green (dashed) line corresponds to the case ω < g (for the plot we set ω = 6, g = 8).

B. The case with nonzero Lindblad operators
Finally, we will discuss an example with nonzero Lindblad operators. For simplicity let us assume that only one of them is nonzero, i.e., L 1 = L, L i = 0 for i > 1. In this case the nonlinear GKSL equation takes the forṁ Now, we consider an example of the Kraus form of the qubit evolution with the following choice of generators: with the initial condition given in Eq. (31). The nonlinear GKSL equation implies in this case thaṫ n 3 (t) = − g − l 2 2 (n 3 (t)) 2 − l 2 n 3 (t) + g + l 2 2 , (53) where n + = n 1 + in 2 . Taking into account the initial condition and integrating the system we finally get wherel = 2g+l 2 2g−l 2 . Notice that g ≤ 0 implies −1 ≤l ≤ 1. Moreover, for 0 < g: n 3 (∞) = 1, n + (∞) = 0 so we obtain the pure state while for 0 > g: n 3 (∞) = −l, n + (∞) = 0 so we get in general the mixed state. In Fig. 6 we present an exemplary evolution of a Bloch vector in this case. In Fig. 7 we present the behavior of the von Neumann entropy S(ρ(t)) under the evolution (55), we take the same initial condition and parameters as in Fig. 6.
We can also find an explicit form of Kraus operators in this case. We have and Furthermore so we arrive at the same form of n(t) as the result of the solution of the nonlinear GKSL equation.

C. The Jaynes-Cummings model
In this section we show that the standard Jaynes-Cummings model [26,27] describing the interaction of a two-level atom with a single mode of the electromagnetic (EM) field can be also generalized along the lines discussed in the paper. The standard Jaynes-Cummings Hamiltonian describing the system atom + EM field has the form The creation and annihilation operatorsâ † ,â fulfill the standard bosonic commutation relation.
In order to obtain a nonlinear generalization we replace H JC with Let us notice that subspaces  are invariant under the action of both H and iG. Therefore, H + iG can be written as the following direct sum: where Furthermore, assuming that the initial full density matrix can be written in a similar way where the conditions Tr[ρ(0)] = 1, Tr[ρ n (0)] = 1 imply that n λ n = 1. Under the evolution governed by (62) the density matrix (64) evolves to Notice that choosing λ 0 = 1, λ i = 0 for i > 0 we obtain the model considered in the subsection IV A. It is worth to mention that in [22] the master equation resembling the nonlinear von Neumann equation (27) was derived. The nonlinear equation obtained in [22] governs the evolution of a state of the photon field in the periods between detector clicks while in our examples the nonlinear equation describes the evolution of a state of the atom. In [22] the nonlinearity arises from the necessity of distinguishing between the notions of observation and detection in the statistical analysis of the process. In particular, it explicitly depends on the difference of the efficiencies of detectors.

V. CONCLUSIONS
In this paper we have discussed a generalization of the notion of the quantum operation and the quantum time evolution. The generalization relies on extending the linearity of quantum operations to the quasi-linearity condition. This condition is motivated by the appearance of such operations in a "hidden" form (e.g. selective measurement [18]) in quantum formalism. On the other hand, convex quasi-linearity guarantees absence of superluminal communication. We have identified a natural class of operations satisfying this condition. Moreover, we have generalized the GKSL master equation for quasilinear evolutions. As an example we have considered nonlinear qubit evolution. It is interesting that some of these qubit evolutions were discussed independently in the context of non-Hermitian quantum mechanics [16,17]. In general, the nonlinear time development of qubit is related to an interaction of the quantum system with an environment. Depending on the interrelation with environment the von Neumann entropy of qubit can demonstrate different time dependence. We also discussed modification of the Jaynes-Cummings model [26,27] describing the interaction of a two-level atom with a single mode of the electromagnetic field.
Let us stress here that our goal was to introduce nonlinear evolution with minimal changes in the rest of quantum formalism. In our approach we do not change Born rule or projection postulate as it takes place in other nonlinear extensions of quantum mechanics ( [12,13]). But of course nonlinearity of evolution equations has implications-the superposition principle is broken during the nonlinear evolution.
The following important question remains open: Does the class of maps generated by linear non-tracepreserving quantum operation exhaust the set of convex quasi-linear operations?