Dissipative generators, divisible dynamical maps and Kadison-Schwarz inequality

TL;DR

This paper introduces Kadison-Schwarz divisible dynamical maps as a generalization of CP-divisibility, characterizing quantum Markovian evolution through time-local dissipative generators, with illustrative qubit examples.

Contribution

It defines and characterizes Kadison-Schwarz divisible maps, extending the framework of quantum Markovian dynamics beyond CP-divisibility.

Findings

Kadison-Schwarz divisible maps are characterized by time-local dissipative generators

The concept generalizes CP-divisibility in quantum Markovian evolution

Illustrative qubit evolution demonstrates the new concept

Abstract

We introduce a concept of Kadison-Schwarz divisible dynamical maps. It turns out that it is a natural generalization of the well known CP-divisibility which characterizes quantum Markovian evolution. It is proved that Kadison-Schwarz divisible maps are fully characterized in terms of time-local dissipative generators. Simple qubit evolution illustrates the concept.

Equations

$${\dot{\Lambda }}_t=\mathcal{L}{\Lambda }_t,$$

$$\mathcal{L}(\rho )=-i[H,\rho ]+\sum _k{\gamma }_k\left(V_k\rho V_k^{†}-\frac{1}{2}\{V_k^{†}{V}_k,\rho \}\right),$$

$${\Lambda }_t={\mathcal{T}}_{\leftarrow }\exp \left({\int }_0^t{\mathcal{L}}_{\tau }d\tau \right),$$

$${\Lambda }_t=V_{t,s}{\Lambda }_s,t>s,$$

$$\frac{d}{dt}\Vert {\Lambda }_t(\rho )-{\Lambda }_t(\sigma ){\Vert }_1\le 0,$$

$$\frac{d}{dt}\Vert {\Lambda }_t(X){\Vert }_1\le 0,$$

$$\mathcal{L}(\rho )=-i[H,\rho ]+\Phi (X)-\frac{1}{2}\{{\Phi }^{♯}(1\mathrm{l}),X\}.$$

$$\Phi (X{X}^{†})\ge \Phi (X)\Phi (X^{†}),$$

$$T(X^{†}X)=|1⟩⟨1|,T(X^{†})T(X)=|2⟩⟨2|,$$

$${\mathrm{id}}_k:M_k(\mathbb{C})\otimes \mathcal{B}(\mathcal{H})\to M_k(\mathbb{C})\otimes \mathcal{B}(\mathcal{H})$$

$$\mathrm{CP}={\mathrm{P}}_d\subset {\mathrm{P}}_{d-1}\subset \dots \subset {\mathrm{P}}_2\subset \mathrm{KS}\subset {\mathrm{P}}_1,$$

$$\Psi (X)=\Phi (1\mathrm{l}{)}^{-\frac{1}{2}}\Phi (X)\Phi (1\mathrm{l}{)}^{-\frac{1}{2}}.$$

$$\Phi (X{X}^{†})\ge \Phi (X)\Phi (1\mathrm{l}{)}^{-1}\Phi (X^{†}).$$

$$\Phi =p_1{\Phi }_1+p_2{\Phi }_2+p_3{\Phi }_3,$$

$${\Phi }_k(X)=\frac{1}{2}({\sigma }_{k}X{\sigma }_k+X),$$

$${\Phi }_k(X)=\sum _{\mu =1}^2{P}_k^{(\mu )}X{P}_k^{(\mu )},$$

$$\Phi ({\sigma }_k)=p_k{\sigma }_k,$$

$$\Phi ={\Phi }_1+{\Phi }_2-{\Phi }_3,$$

$$\Phi (X^{†}X)=|1⟩⟨1|,\Phi (X^{†})\Phi (X)=|2⟩⟨2|.$$

$$p_1^2+p_2^2+p_3^2\le 1+2p_1{p}_2{p}_3,$$

$${\mathcal{L}}^{♯}(X^{†}X)\ge {\mathcal{L}}^{♯}(X^{†})X+X^{†}{\mathcal{L}}^{♯}(X),$$

$${\mathcal{L}}^{♯}(X)=i[H,X]+\Phi (X)-\frac{1}{2}\{\Phi (1\mathrm{l}),X\},$$

$$\Phi (X^{†}X)\ge \Phi (X^{†})X+X^{†}\Phi (X)-X^{†}\Phi (1\mathrm{l})X,$$

$$V_{t,s}^{♯}={\mathcal{T}}_{\to }\exp \left({\int }_s^t{\mathcal{L}}_{\tau }^{♯}d\tau \right),$$

$$\mathrm{CP}={\mathrm{P}}_2\subset \mathrm{KS}\subset {\mathrm{P}}_1,$$

$${\mathcal{L}}_t(\rho )=\gamma (t)({\sigma }_3\rho {\sigma }_3-\rho ),$$

$$\Lambda_{t}(\rho)=\left(\begin{array}[]{cc}\rho_{11}+(1-|G(t)|^{2})\rho_{22}&G(t)\rho_{12}\\ G^{*}(t)\rho_{21}&|G(t)|^{2}\rho_{22}\end{array}\right),$$

$${\mathcal{L}}_t(\rho )=-i[H(t),\rho ]+\gamma (t)({\sigma }_{-}\rho {\sigma }_{+}-\frac{1}{2}\{{\sigma }_{+}{\sigma }_{-},\rho \}),$$

$${\mathcal{L}}_t(\rho )=\frac{1}{2}\sum _{k=1}^3{\gamma }_k(t)({\sigma }_k\rho {\sigma }_k-\rho ),$$

Full text

I Introduction

Evolution of a quantum system is represented by a family of quantum channels $\Lambda_{t}:\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{H})$ ($t\geq 0$) such that $\Lambda_{0}={\rm id}$ (identity map). In what follows $\mathcal{B}(\mathcal{H})$ denotes an algebra of bounded linear operators acting in the Hilbert space $\mathcal{H}$ (actually, in this paper we consider only finite dimensional $\mathcal{H}$). Such family is usually called a dynamical map. For isolated system the dynamical map has a well known structure $\Lambda_{t}(\rho)=U_{t}\rho U_{t}^{\dagger}$, where $U_{t}=e^{-iHt}$ and $H$ denotes the Hamiltonian of the (closed) system (we keep $\hbar=1$). For an open quantum system one often considers a dynamical semigroup governed by the Markovian master equation Breuer ; RivasHuelga

[TABLE]

where the generator $\mathcal{L}:\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{H})$ is given by the celebrated Gorini-Kossakowski-Sudarshan-Lindblad formula GKLS ; Lindblad

[TABLE]

with positive rates $\gamma_{k}>0$. This structure guarantees that the solution $\Lambda_{t}=e^{t\mathcal{L}}$ defines a legitimate dynamical map – completely positive and trace preserving (CPTP). To go beyond dynamical semigroup one considers master equation (1) with time dependent generator $\mathcal{L}_{t}$. It has exactly the same form as in (2) with time dependent $H(t)$, $V_{k}(t)$ and $\gamma_{k}(t)$. The formal solution for $\Lambda_{t}$ reads

[TABLE]

where $\mathcal{T}_{\leftarrow}$ stands for chronological operator. In this case, however, we do not known necessary and sufficient conditions for $\mathcal{L}_{t}$ which guarantee that (3) is CPTP for all $t>0$. Time dependent generators are recently analyzed in connection to quantum non-Markovian evolution NM1 ; NM2 ; NM3 ; NM4 . Recall, that a dynamical map $\Lambda_{t}$ is called divisible if

[TABLE]

with a family of ‘propagators’ $V_{t,s}:\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{H})$. For invertible maps such propagator always exists and it is given by $V_{t,s}=\Lambda_{t}\Lambda_{s}^{-1}$. Maps which are not invertible require special treatment PRL-2018 ; Sagnik (for a recent review of various concepts of divisibility for quantum channels and dynamical maps see Mario ). Now, being divisible one calls $\Lambda_{t}$ P-divisible if $V_{t,s}$ is positive and trace-preserving, and CP-divisible if $V_{t,s}$ is CPTP. Following RHP one calls the evolution represented by $\Lambda_{t}$ Markovian if $\Lambda_{t}$ is CP-divisible. Actually, for invertible maps CP-divisibility is fully controlled by the properties of time-local generator $\mathcal{L}_{t}$, that is, all time-dependent rates satisfy $\gamma_{k}(t)\geq 0$. Authors of BLP proposed another approach based on distinguishability of quantum states: $\Lambda_{t}$ is Markovian if for any pair of initial states $\rho$ and $\sigma$ one has

[TABLE]

where $\|X\|_{1}={\rm Tr}\sqrt{X^{\dagger}X}$. Interestingly, for invertible maps P-divisibility is equivalent to

[TABLE]

for all $X^{\dagger}=X$. Hence, CP-divisibility implies P-divisibility and this implies BLP condition (5).

In this paper we introduce another notion of divisibility based on the Kadison-Schwarz (KS) inequality. Let us recall that a linear map $\Phi:\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{H})$ is trace-preserving iff its dual map $\Phi^{\sharp}$ is unital. $\Phi^{\sharp}$ is defined by ${\rm Tr}(X\Phi(Y))={\rm Tr}(\Phi^{\sharp}(X)Y)$ for all $X,Y\in\mathcal{B}(\mathcal{H})$. Hence, if $\Phi(X)=\sum_{k}\lambda_{k}A_{k}XB_{k}^{\dagger}$, then $\Phi^{\sharp}(X)=\sum_{k}\lambda^{*}_{k}B_{k}XA_{k}^{\dagger}$. In particular, if $\Lambda_{t}(\rho)=U_{t}\rho U^{\dagger}_{t}$ represents Schrödinger evolution of the density operator, then $\Lambda^{\sharp}_{t}(X)=U^{\dagger}_{t}XU_{t}$ represents Heisenberg evolution of the observable $X$. Introducing a completely positive map $\Phi(\rho)=\sum_{k}\gamma_{k}V_{k}\rho V_{k}^{\dagger}$, the GKSL generator (2) can be rewritten in a compact form as follows

[TABLE]

Now, a unital map $\Phi:\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{H})$ satisfies Kadison-Schwarz (KS) inequality Paulsen ; Stormer ; Kadison ; Bhatia if

[TABLE]

for all $X\in\mathcal{B}(\mathcal{H})$. We say that dynamical map $\Lambda_{t}$ is KS-divisible if the propagator $V^{\sharp}_{t,s}$ satisfies (8). In this paper we analyze this concept and relate it to P- and CP-divisibility. Interestingly, for invertible maps KS-divisibility is fully controlled by the property of $\mathcal{L}_{t}$ called dissipativity Lindblad . Finally, we illustrate KS-divisibility by simple example of qubit evolution.

II Kadison-Schwarz maps

Celebrated Cauchy-Schwarz inequality found a lot of applications in mathematics, physics and engineering. It states that for any $x,y\in\mathcal{H}$ one has $|\langle x|y\rangle|^{2}\leq\langle x|x\rangle\langle y|y\rangle$. Kadison found elegant generalization of this inequality for linear maps in operator algebras Kadison ; Paulsen ; Stormer ; Bhatia : a linear map $\Phi:\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{H})$ is positive if for any $A\geq 0$ one has $\Phi(A)\geq 0$. Equivalently, for any $X\in\mathcal{B}(\mathcal{H})$ one has $\Phi(X^{\dagger}X)\geq 0$. A linear map is unital if $\Phi({\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}})={\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}$, with ${\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}$ being an identity operator in $\mathcal{H}$. Now, a unital map $\Phi:\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{H})$ satisfies Kadison-Schwarz (KS) inequality Paulsen ; Kadison ; Bhatia if

[TABLE]

for all $X\in\mathcal{B}(\mathcal{H})$. It immediately follows from (9) that Kadison-Schwarz map is positive. However, the converse needs not be true. An example of a positive unital map which is not KS is provided by the transposition map. Indeed, for $d=2$ taking $X=|1\rangle\langle 2|$ one finds

[TABLE]

and hence (9) is violated. Interestingly, any unital positive map satisfies (9) but for normal operators (i.e. $X^{\dagger}X=XX^{\dagger}$). Denote by $M_{k}(\mathbb{C}$ linear space of $k\times k$ complex matrices. Recall, that a linear map $\Phi$ is $k$-positive if the extended map

[TABLE]

is positive (${\rm id}_{k}$ denotes an identity map). A map which is $k$-positive for $k=1,2,\ldots$ is called completely positive (CP). If the dimension of $\mathcal{H}$ is ‘$d$’, then complete positivity is equivalent to $d$-positivity. Among unital maps one has the following hierarchy

[TABLE]

where ${\rm P}_{k}$ denotes $k$-positive unital maps.

If $\Phi$ and $\Psi$ are KS, then $\Phi\Psi$ is KS as well. Moreover, the convex combination $\lambda\Phi+(1-\lambda)\Psi$ is again KS Farrukh .

Actually, the concept of unital KS maps may be generalized for maps which are not unital. Consider a map $\Phi:\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{H})$ such that $\Phi({\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}})>0$, and define

[TABLE]

Clearly, one has $\Psi({\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}})={\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}$. Now, if $\Psi$ satisfies KS condition (9), then

[TABLE]

Example 1

Consider a qubit map

[TABLE]

where $p_{1}+p_{2}+p_{3}=1$, and

[TABLE]

with $\sigma_{k}$ being Pauli matrices. Note, that

[TABLE]

and $P^{(\mu)}_{k}$ are eigen-projectors of $\sigma_{k}$. It is clear that $\Phi$ is unital. It is CP iff $p_{k}\geq 0$. Note, that

[TABLE]

and hence one easily finds that $\Phi$ is positive iff $|p_{k}|\leq 1$. For example the map

[TABLE]

is positive (but of course not CP). This map is not KS. Indeed, taking $X=|1\rangle\langle 2|$ one gets

[TABLE]

It is shown in the Appendix that if

[TABLE]

then $\Phi$ is KS (cf. Figure 1). Interestingly, the constraint (14) provides good approximation for a set of CP maps (cf. Figure 1). For example all three vertices of the inscribed triangle satisfy (14) with equality.

III KS-divisibility

It was already observed by Lindblad Lindblad that $\Lambda^{\sharp}_{t}=e^{t\mathcal{L}^{\sharp}}$ is KS iff $\mathcal{L}^{\sharp}$ is dissipative, that is,

[TABLE]

for all $X\in\mathcal{B}(\mathcal{H})$. Any dissipative generator has the following structure

[TABLE]

where the map $\Phi:\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{H})$ satisfies the following condition

[TABLE]

that is, it has exactly the same structure as (7) but the CP map $\Phi$ is replaced by the map satisfying (17) (note that (7) is represented in the Schrödinger picture whereas (16) in the Heisenberg picture). Actually condition (17) is weaker than generalized KS condition (12). One has

Proposition 1

Any $\Phi$ satisfying (12) satisfies (17).

For the proof see Appendix.

Consider now a dynamical map $\Lambda_{t}$ satisfying time-local master equation $\dot{\Lambda}_{t}=\mathcal{L}_{t}\Lambda_{t}$, that is, $\Lambda_{t}$ is represented as in (3).

Theorem 1

If $\Lambda_{t}$ is invertible, then

• •

it is KS-divisible if and only if $\mathcal{L}^{\sharp}_{t}$ is dissipative,

• •

it is CP-divisible if and only if $\mathcal{L}^{\sharp}_{t}$ is completely dissipative

for all $t\geq 0$.

Proof: the existence of $V_{t,s}$ follows from invertibility of $\Lambda_{t}$, that is, $V_{t,s}=\Lambda_{t}\Lambda_{s}^{-1}$. One has

[TABLE]

where now $\mathcal{T}_{\rightarrow}$ stands for anti-chronological operator. Now, if $\mathcal{L}^{\sharp}_{t}$ is dissipative, then $V_{t,s}^{\sharp}$ is unital KS. If $V_{t,s}^{\sharp}$ is KS for any $t>s$, then for $\epsilon\to 0+$ one has $V^{\sharp}_{t+\epsilon,t}\to e^{\epsilon\mathcal{L}_{t}^{\sharp}}$ which implies that $\mathcal{L}^{\sharp}_{t}$ is dissipative. $\Box$

IV Examples: KS-divisible qubit dynamical maps

In this section we consider several simple examples of qubit evolution. In this case the hierarchy (10) reduces to

[TABLE]

that is, KS maps interpolate between CP and positive maps.

Example 2** (Qubit dephasing)**

For a qubit dephasing governed by

[TABLE]

P-, KS-, and CP-divisibility coincide and they are equivalent to $\gamma(t)\geq 0$. Note, that in this case one has $\mathcal{L}^{\sharp}_{t}=\mathcal{L}_{t}$.

Example 3** (Amplitude damping channel)**

The evolution of amplitude-damped qubit is governed by a single function $G(t)$

[TABLE]

where the function $G(t)$ depends on the form of the reservoir spectral density $J(\omega)$ Breuer . This evolution is generated by the following time-local generator

[TABLE]

where $\sigma_{\pm}$ are the spin lowering and rising operators, $H(t)=\frac{\omega(t)}{2}\sigma_{+}\sigma_{-}$, together with $\omega(t)=-2{\rm Im}\frac{\dot{G}(t)}{G(t)}$, and $\gamma(t)=-2{\rm Re}\frac{\dot{G}(t)}{G(t)}$. . Again in this case P-, KS-, and CP-divisibility coincide and they are equivalent to $\gamma(t)\geq 0$.

Example 4** (Pauli channel)**

Consider the qubit evolution governed by the following time-local generator

[TABLE]

which leads to the following dynamical map (time-dependent Pauli channel):

[TABLE]

where

[TABLE]

and

[TABLE]

with $\Gamma_{k}(t)=\int_{0}^{t}\gamma_{k}(\tau)d\tau$. It was shown Filip-PRA that P-divisibility is equivalent to the following conditions:

[TABLE]

Now, it is shown in the Appendix that KS-divisibility is equivalent to the following the stronger conditions:

[TABLE]

In Erika authors considered so called eternally non-Markovian evolution corresponding to

[TABLE]

It gives

[TABLE]

Clearly, the map $\Lambda_{t}$ is CPTP and P-divisible. Now, let us consider a simple modification

[TABLE]

It gives

[TABLE]

Again, the map $\Lambda_{t}$ is CPTP due to the fact that $p_{\alpha}(t)\geq 0$. Indeed, one finds

[TABLE]

due to $\cosh t\geq 1$. Hence, it provides an example of KS-divisible qubit evolution since conditions (25) are trivially satisfied. Clearly, the evolution governed by (26) is P-divisible but not KS-divisible.

V Conclusions

In this paper we introduced the concept of KS-divisibility which is based on the Kadison-Schwarz inequality (9). This concept interpolates between CP-divisibility(often assumed as a definition of Markovianity RHP ) and P-divisibility (which is closely related to the well known notion of information flow BLP ). Any CP-divisible map in KS-divisible, and any KS-divisible map is P-divisible and hence does not display information backflow. For dynamical maps satisfying time-local master equation with time-dependent generator $\mathcal{L}_{t}$ KS-divisibility is fully controlled by the property of the generator $\mathcal{L}_{t}^{\sharp}$ (Heisenberg picture), that is, the maps is KS-divisible if and only if $\mathcal{L}_{t}^{\sharp}$ is dissipative. This concept is illustrated by several examples of qubit evolution. Interestingly for the evolution governed by well known generator $\mathcal{L}_{t}(\rho)=\frac{1}{2}\sum_{k}\gamma_{k}(t)(\sigma_{k}\rho\sigma_{k}-\rho)$ we found necessary and sufficient conditions for dissipativity: $\gamma_{i}(t)+2\gamma_{j}(t)\geq 0$ for $i\neq j$. I shows that so called eternally non-Markovian evolution proposed in Erika is P-divisible but not KS-divisible. However, we proposed a simple modification which is again eternally non-Markovian (one of the rate is always negative) but displays KS-divisibility. Actually, condition $\gamma_{i}(t)+2\gamma_{j}(t)\geq 0$ for $i\neq j$ has a clear physical interpretation: note that the initial Bloch vector $\mathbf{x}=(x_{1},x_{2},x_{3})$ evolves according to

[TABLE]

where $\lambda_{k}(t)=\exp(-\int_{0}^{t}1/T_{k}(\tau)d\tau)$, and the local relaxation times read

[TABLE]

for mutually different $i,j,k$. Now, assuming that $\gamma_{3}(t)<0$, one finds the following constraints

[TABLE]

Note, that if the map is only P-divisible one has $T_{3}(t)\leq\frac{1}{2|\gamma_{3}(t)|}$ and no additional constraints for $T_{1}(t)$ and $T_{2}(t)$. This shows that these two concepts of divisibility have different physical flavour.

It would be interesting to investigate KS-divisibility for higher dimensional systems.

Acknowledgments

DC was supported by the National Science Center project No 2018/30/A/ST2/00837.

Appendix A Condition (14) for KS map

We note that every matrix $X\in\textit{M}_{2}(\mathbb{C})$ can be written in this basis as $X=w_{0}{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}+{\mathbf{w}}\cdot\sigma$ with $w_{0}\in\mathbb{C},{\mathbf{w}}=(w_{1},w_{2},w_{3})\in\mathbb{C}^{3},$ here by ${\mathbf{w}}\cdot\sigma$ we mean the following

[TABLE]

One finds

[TABLE]

where the $3\times 3$ matrix $T_{ij}$ reads $T_{ij}=p_{i}\delta_{ij}$. Taking into account a result of Farrukh the KS conditions yields

[TABLE]

where

[TABLE]

Let us assume $\|{\mathbf{w}}\|=1$. Hence, taking into account $w_{1}=r_{1}e^{i\alpha_{1}}$, $w_{2}=r_{2}e^{i\alpha_{2}}$, $w_{3}=r_{3}e^{i\alpha_{3}}$, the inequality (30) reduces to

[TABLE]

where $\theta_{1}+\theta_{2}+\theta_{3}=2\pi$. Clearly, the last inequality is satisfied if one has

[TABLE]

Introducing $x:=r_{1}^{2},y:=r_{2}^{2},z:=r_{3}^{2}$, the last one is equivalent to

[TABLE]

for all $x+y+z=1$, $x,y,z\geq 0$. Let us introduce the following function

[TABLE]

where the arguments $(x,y)$ satisfy $0\leq x+y\leq 1$. One can check that this function reaches its maximum on the boundary of $0\leq x+y\leq 1$. Hence, it is enough to study the following function

[TABLE]

on the interval $[0,1]$. One shows that the maximum of $g(y)$ on the interval $[0,1]$ is less or equal than 0 if and only if one has $A\leq\beta\gamma$. Similarly, one finds the other two conditions $B\leq\alpha\gamma$ and $C\leq\alpha\beta$. Hence, if

[TABLE]

then $\Psi$ is KS-operator. Now, taking into account (31) and $|q_{k}|\leq 1$, the last conditions (36) reduce to

[TABLE]

Appendix B proof of Proposition 1

Denoting $Y:=\Phi({\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}})$ one has

[TABLE]

which implies

[TABLE]

This together with KS-condition yields the assertion.

Appendix C KS Divisibility

In this section, we show that the generator $\mathcal{L}^{\sharp}$ is dissipative which means that the mapping $\Phi$ in (16) satisfies (17).

[TABLE]

where $\Phi_{t}(X)=\frac{1}{2}\sum_{k=1}^{3}\gamma_{k}(t)\sigma_{k}X\sigma_{k}$. This genetaror gives rise to KS-divisible evolution iff

[TABLE]

To simplify notation we skip time-dependence. Note, that $\Phi({\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}})=\gamma{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}$, with

[TABLE]

Let us observe that the condition (17) can be rewritten as follows:

[TABLE]

So, we have

[TABLE]

and hence

[TABLE]

Now, to simply analysis without loosing generality we put $\gamma=1$. We show that if condition (25) is satisfied for any $t\geq 0$, then $\Phi$ satisfies

[TABLE]

One has for $X=w_{0}{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}+{\mathbf{w}}\cdot\sigma$

[TABLE]

where the $3\times 3$ matrix $T_{ij}$ reads $T_{ij}=\lambda_{i}\delta_{ij}$, and the eigenvalues of the map read

[TABLE]

One finds

[TABLE]

where $\mathbf{a}\times\mathbf{b}$ stands for the vector product of 3-dimensional vectors. One obtains

[TABLE]

and

[TABLE]

Hence, we find

[TABLE]

with

[TABLE]

Now, (41) is equivalent to

• •

$a\geq 0$

• •

$a\geq|{\bf b}|$.

Using

[TABLE]

one finds

[TABLE]

and hence taking into account that $\gamma_{1}+\gamma_{2}+\gamma_{3}=2$, one finds

[TABLE]

which reproduces condition (24). Hence, condition $a\geq 0$ is equivalent to P-divisibility. The second condition $a\geq|{\bf b}|$ provides further restriction which clearly shows that KS-divisibility implies P-divisibility. One finds for the 3-vector ${\bf b}$:

[TABLE]

where the $3\times 3$ matrix $S$ reads $S_{ij}=\mu_{i}\delta_{ij}$, with

[TABLE]

where again we took into account $\frac{1}{2}(\gamma_{1}+\gamma_{2}+\gamma_{3})=1$. The condition $\|{\bf b}\|\leq a$ reads

[TABLE]

Introducing the following parametrization: $w_{1}=xe^{i\alpha_{1}}$, $w_{2}=ye^{i\alpha_{2}}$, $w_{3}=ze^{i\alpha_{3}}$, with $x,y,z\geq 0$, it reduces to

[TABLE]

where $\theta_{1}+\theta_{2}+\theta_{3}=2\pi$. The inequality is clearly satisfied if all $\gamma_{k}\geq 0$. Now, suppose that $\gamma_{3}<0$ (note, that only one $\gamma_{k}$ can be negative) and let us consider the worst case scenario maximizing LHS and minimizing RHS of (53). Since

[TABLE]

and

[TABLE]

let us take $z=0$ and $\sin\theta_{3}=1$. It leads to

[TABLE]

and hence

[TABLE]

for all $x,y\geq 0$.

Lemma 1

Let $\alpha,\beta>0$. Condition

[TABLE]

is satisfied for all $x,y\geq 0$ iff $\alpha\geq 1$ and $\beta\geq 1$.

Clearly, $\alpha\geq 1$ and $\beta\geq 1$ is sufficient. To show that it is also necessary take $y=1/x$. It gives

[TABLE]

The RHS is minimal for $x=\sqrt{\beta/\alpha}$ and hence

[TABLE]

which give $\alpha\geq 1$. In the same way we prove that $\beta\geq 1$. Summarising, we showed that

[TABLE]

which can be rewritten as

[TABLE]

Clearly $\gamma_{1}+2\gamma_{2}\geq 0$ and $\gamma_{2}+2\gamma_{1}\geq 0$.