Reachability in Infinite Dimensional Unital Open Quantum Systems with Switchable GKS-Lindblad Generators

TL;DR

This paper extends the understanding of reachability in infinite dimensional open quantum systems, demonstrating approximate controllability under certain conditions with switchable noise and bounded control Hamiltonians.

Contribution

It generalizes finite-dimensional majorization results to infinite dimensions for quantum control systems with switchable noise and bounded Hamiltonians.

Findings

Systems can approximately reach any state majorized by the initial state.

The results apply to systems with switchable noise and bounded control Hamiltonians.

Majorization-based controllability extends to infinite-dimensional quantum systems.

Abstract

In quantum systems theory one of the fundamental problems boils down to: given an initial state, which final states can be reached by the dynamic system in question. Here we consider infinite dimensional open quantum dynamical systems following a unital Kossakowski-Lindblad master equation extended by controls. More precisely, their time evolution shall be governed by an inevitable potentially unbounded Hamiltonian drift term $H_0$, finitely many bounded control Hamiltonians $H_j$ allowing for (at least) piecewise constant control amplitudes $u_j(t)\in{\mathbb R}$ plus a bang-bang (i.e. on-off) switchable noise term $\mathbf{\Gamma}_V$ in Kossakowski-Lindblad form. Generalizing standard majorization results from finite to infinite dimensions, we show that such bilinear quantum control systems allow to approximately reach any target state majorized by the initial one, as up to now only has been known in finite dimensional analogues.---The proof of the result is currently limited to the control Hamiltonians $ H_j$ being bounded and noise terms $\mathbf{\Gamma}_V$ with compact normal $V$.

Equations

$$\dot{F}(t)=-\big{(}i\,\mathbf{H}(t)+\mathbf{\Gamma}(t)\big{)}F(t),\quad F(0)=\mathbf{1}\;,$$

$$\displaystyle\dot{\rho}(t)=-\big{(}i\,\mathbf{H}(t)+\mathbf{\Gamma}(t)\big{)}(\rho(t))\,,$$

$$\mathbf{\Gamma }(t):=\sum _k{\gamma }_k(t){\mathbf{\Gamma }}_{V_k}\text{with}{\mathbf{\Gamma }}_{V_k}(\rho ):=\frac{1}{2}(V_k^{†}{V}_k\rho +\rho V_k^{†}{V}_k)-V_k\rho V_k^{†}.$$

$$\mathbf{\Gamma}(t)\big{(}\rho\big{)}:=\gamma(t)\mathbf{\Gamma}_{V}$$

$$\dot{X}(t)=-(A+\sum _{j=0}^m{u}_j(t)B_j)X(t)\text{with}X(0)=X_0,$$

$${\mathfrak{reach}}_{{\Sigma }_V}({\rho }_0):={\mathcal{S}}_{{\Sigma }_V}\cdot {\rho }_0:=\{F({\rho }_0)|F\in {\mathcal{S}}_{{\Sigma }_V}\},$$

$$\big{(}\mathrm{e}^{-t(i\,\mathrm{ad}_{H_{0}}+i\sum\limits_{j=1}^{m}u_{j}\mathrm{ad}_{H_{j}}+u_{0}\mathbf{\Gamma}_{V})}\big{)}_{t\in\mathbb{R}_{+}}\quad\text{with}\;u_{1},\dots,u_{m}\in{\mathbb{R}}\,,\quad u_{0}\in\{0,1\}\,.$$

$$⟨i{H}_0,i{H}_j|j=1,2,\dots ,m{⟩}_{\mathrm{Lie}}=\mathfrak{su}(\mathcal{H})(\text{or}=\mathfrak{u}(\mathcal{H})).$$

$${\mathfrak{reach}}_{{\Sigma }_0}({\rho }_0)=\{U{\rho }_0{U}^{†}|U\in \mathcal{U}(\mathcal{H})\}.$$

$$\overline{{\mathfrak{reach}}_{\Sigma }({\rho }_0)}\subseteq \{\rho \in \mathbb{D}(\mathcal{H})|\rho \prec {\rho }_0\}$$

$$\dot{U}(t)=-iH(t)U(t)\text{with}U(0)=1,$$

$${\overline{⟨i{H}_0,i{H}_j,i1|j=1,2,\dots ,m{⟩}_{\mathrm{Lie}}}}^s=\mathfrak{u}(\mathcal{H}).$$

$$\overline{\langle iH_{0},iH_{j}\,|\,j=1,2,\dots,m\rangle_{\rm Lie}}^{u}{}=\mathfrak{u}\bigl{(}\mathcal{K}(\mathcal{H})\bigr{)}\,,$$

$$\dot{\rho}(t)=-i\Big{[}H_{0}+\sum_{j=1}^{m}u_{j}(t)H_{j},\rho\Big{]}-\gamma(t)\big{(}\tfrac{1}{2}(V^{\dagger}V\rho+\rho V^{\dagger}V)-V\rho V^{\dagger}\big{)}\,,\;\text{where}$$

$${\overline{{\mathfrak{reach}}_{{\Sigma }_V}({\rho }_0)}}^1=\{\rho \in \mathbb{D}(\mathcal{H})|\rho \prec {\rho }_0\}.$$

$$W_C(T):=\{\mathrm{tr}(C{U}^{†}TU)|U\in \mathcal{U}(\mathcal{H})\}$$

$$\displaystyle P_{C}(T):=\Big{\{}\sum\nolimits_{j=1}^{\infty}\lambda_{j}(C)\lambda_{\sigma(j)}(T)\,\Big{|}\,\sigma:\mathbb{N}\to\mathbb{N}\text{ is a permutation}\Big{\}}$$

$$\underset{U\in \mathcal{U}(\mathcal{H})}{\sup }\mathrm{tr}(C{U}^{†}TU)={\sum }_{j=1}^{\infty }{\lambda }_j^{\downarrow }(C^{+}){\lambda }_j^{\downarrow }(T^{+})+{\sum }_{j=1}^{\infty }{\lambda }_j^{\downarrow }(C^{-}){\lambda }_j^{\downarrow }(T^{-}),$$

$$K_{\rho }(T)=\max \overline{W_{\rho }(T)}={\sum }_{j=1}^{\infty }{\lambda }_j^{\downarrow }(\rho ){\lambda }_j^{\downarrow }(T^{+})$$

$${\sum }_{j=1}^n{\lambda }_j^{\downarrow }(\rho ){\lambda }_j^{\downarrow }(T^{+})\le {\sum }_{j=1}^n{\lambda }_j^{\downarrow }(\omega ){\lambda }_j^{\downarrow }(T^{+})$$

$${\sum }_{j=1}^k{\lambda }_j^{\downarrow }(\rho )=K_{\rho }({\Pi }_k)\le K_{\omega }({\Pi }_k)={\sum }_{j=1}^k{\lambda }_j^{\downarrow }(\omega )$$

$$\dot{\rho}(t)=-i\Big{[}H_{0}+\sum_{j=1}^{m}u_{j}(t)H_{j},\rho\Big{]}-\gamma(t)\big{(}\tfrac{1}{2}(V^{\dagger}V\rho+\rho V^{\dagger}V)-V\rho V^{\dagger}\big{)}\,,\;\text{where}$$

$${\overline{{\mathfrak{reach}}_{{\Sigma }_V}({\rho }_0)}}^1=\{\rho \in \mathbb{D}(\mathcal{H})|\rho \prec {\rho }_0\}.$$

$$\begin{split}\langle f_{j},\mathbf{\Gamma}_{V}(X)f_{k}\rangle&=\big{(}\tfrac{1}{2}{|v_{j}-v_{k}|^{2}}-i\operatorname{Im}(v_{j}\overline{v_{k}})\big{)}\langle f_{j},Xf_{k}\rangle\end{split}$$

$$\exp(-t\mathbf{\Gamma}_{V})(\langle f_{k},\cdot\rangle f_{j})=\exp\big{(}-\tfrac{t}{2}|v_{j}-v_{k}|^{2}\big{)}\exp(it\operatorname{Im}(v_{j}\overline{v_{k}}))\langle f_{k},\cdot\rangle f_{j}$$

$$\lim_{n\to\infty}\Big{\|}\Big{(}\exp\Big{(}\frac{it\mathbf{H}_{0}}{n}\Big{)}\exp\Big{(}\frac{-it\mathbf{H}_{0}-t\mathbf{\Gamma}_{V}}{n}\Big{)}\Big{)}^{n}-\exp(-t\mathbf{\Gamma}_{V})\Big{\|}_{\text{op}}=0\,.$$

$${\overline{{\mathfrak{reach}}_{{\Sigma }_V}({\rho }_0)}}^1\subseteq {\overline{\{\rho \in \mathbb{D}(\mathcal{H})|\rho \prec {\rho }_0\}}}^1=\{\rho \in \mathbb{D}(\mathcal{H})|\rho \prec {\rho }_0\}.$$

$$\begin{array}{rl}{\rho }_0=W_2\mathrm{diag}(y)W_2^{†}\stackrel{\text{Step }1}{⟶}U\mathrm{diag}(y)U^{†}& \stackrel{\text{Step }2}{⟶}\mathrm{diag}(x)\\ & \stackrel{\text{Step }3}{⟶}{W}_1\mathrm{diag}(x)W_1^{†}=\rho .\end{array}$$

$$T={\sum }_{j=1}^{\infty }{\tau }_j⟨e_j,\cdot ⟩e_j$$

$${\mathcal{B}}^1(\mathcal{H}):=\{C\in \mathcal{B}(\mathcal{H})|\Vert C{\Vert }_1:=\mathrm{tr}(\sqrt{C^{†}C})<\infty \}\subseteq \mathcal{K}(\mathcal{H}),$$

Full text

1. Introduction and Overview

1.1. Markovian Bilinear Quantum Control

The Kossakowski-Lindblad equation [25, 26, 30, 21] plays a central role in quantum dynamics since it characterizes the infinitesimal generators of the semigroup of all (invertible1a1a1aHere invertibility only means invertible as linear map, not necessarily as quantum map.) Markovian quantum maps.

As in [46] a quantum map (cptp map1b1b1b cptp maps are linear completely positive and trace-preserving.) is called (time-dependent) Markovian, if it is the solution of a (time-dependent) Markovian master equation

[TABLE]

where Markovianity and cptp property are guaranteed by the Kossakowski-Lindblad form of $\mathbf{\Gamma}(t)$ in Eq. (3). Then for finite dimensional Hamiltonians and noise terms one can show that those Markovian quantum maps (including both, time-dependent and time-independent ones) are infinitesimal divisible into products of exponentials of Kossakowski-Lindblad generators [46] hence leading to Lie semigroup structure [14]. In contrast, non-Markovian quantum maps (existing even arbitrarily close to the identity map) are Kraus maps [27] that are not solutions of a Kossakowski-Lindblad master equation and hence the set of all invertible quantum maps (including Markovian and non-Markovian ones) has no Lie-semigroup structure (details in [46, 14, 39]).

For most of the work, we focus on the corresponding induced system $\Sigma$ acting on the state space $\mathbb{D}(\mathcal{H})$ of all density operators $\rho$ and following the time-dependent Kossakowski-Lindblad master equation of the form

[TABLE]

where $\mathbf{H}(t)$ denotes the adjoint action of some time-dependent Hamiltonian $H(t)$ on $\mathbb{D}(\mathcal{H})$, i.e. $\mathbf{H}(t)(\rho):=\mathrm{ad}_{H(t)}(\rho)=[H(t),\rho]\,.$ For Markovianity take the noise term $\mathbf{\Gamma}(t)$ in the usual Kossakowski-Lindblad form

[TABLE]

The time dependence of $H(t)$ is brought about by adding to the (usually inevitable, possibly unbounded) system Hamiltonian $H_{0}$, bounded control Hamiltonians of the type $u_{j}(t)H_{j}$ to give $H(t):=H_{0}+\sum_{j=1}^{m}u_{j}(t)H_{j}$, where the control amplitudes $u_{j}(t)\in\mathbb{R}$ are typically modulated in a manner at least allowing for piecewise constant controls.

In the finite-dimensional case, an unambiguous separation of the dissipative part and the coherent part results by choosing the $V_{k}$ traceless—as described by Kossakowski, Gorini and Sudarshan in the celebrated work of [21]. In infinite dimensions, this separation is a bit delicate yet not crucial for the sequel. More important is the restriction to compact noise terms $V_{k}$. In our case of interest, the noise terms can be individually switched on and off (as ‘bang-bang controls’), so it suffices to study a single noise term1c1c1cClearly, collectively switched noise in the sense of $\gamma(t)=\gamma_{k}(t)$ for all $k$ is more subtle.

[TABLE]

with $\gamma(t)\in\{0,\gamma_{*}\}$ and $\gamma_{*}>0$—w.l.o.g. we always assume $\gamma_{*}=1$. With these stipulations, we refer to the master equation (2) as gksl-equation henceforth. In the limiting case of $\gamma(t)=0$ for all times, the control system of Eq. (2) turns into a closed Hamiltonian system referred to as $\Sigma_{0}$, while for switchable noise with a single $V$-term the system will be labelled $\Sigma_{V}$.

Note that with the identifications $A:=i\,\mathbf{H}_{0}$, $B_{j}:=i\,\mathbf{H}_{j}$, $B_{0}:=\mathbf{\Gamma}_{V}$ and $X(t)=\rho(t)$ one formally gets a standard bilinear control system [41, 18]

[TABLE]

also identifying $u_{0}(t)=\gamma(t)$. This covers a broad class of quantum control problems including coherent and incoherent feedback [33, 17, 38, 22]. Accessibility of such bilinear Markovian quantum systems (in finite dimensions) was analysed i.a. in terms of symmetries in previous work [39].

In the following, we are interested in characterising the reachable sets of $\Sigma_{V}$ which take the form of a semigroup orbit

[TABLE]

where $\rho_{0}\in\mathbb{D}(\mathcal{H})$ denotes an arbitrary initial density operator and $\mathcal{S}_{\Sigma_{V}}$ is the semigroup generated by the one-parameter semigroups

[TABLE]

If all involved operators are bounded, $\mathrm{e}^{-t(i\,\mathrm{ad}_{H_{0}}+i\sum_{j=1}^{m}u_{j}\mathrm{ad}_{H_{j}}+u_{0}\mathbf{\Gamma}_{V})}$ is given by the exponential series and $\mathfrak{reach}_{\Sigma_{V}}(\rho_{0})$ can alternatively be defined as the collection of all endpoints $\rho(T)$, $T\geq 0$ of trajectories of (2) for piecewise constant controls and initial value $\rho(0)=\rho_{0}$. For the general case ($H_{0}$ unbounded), defining $\mathfrak{reach}_{\Sigma_{V}}(\rho_{0})$ via trajectories is problematic since Eq. (2) allows classical solutions only on a dense domain of initial states. Yet, Eq. (5) also works for unbounded $H_{0}$, as $-(i\,\mathrm{ad}_{H_{0}}+i\sum_{j=1}^{m}u_{j}\mathrm{ad}_{H_{j}}+u_{0}\mathbf{\Gamma}_{V})$ does generate a unique strongly continuous semigroup (details in Appendix Appendix D: Theorem 1 for Unbounded Drift $H_{0}$).

We start the discussion by $\Sigma_{0}$, assuming for the moment that the noise is switched off, i.e. $\gamma(t)=0$. In finite dimensions such a system is fully unitarily controllable if it satisfies the Lie-algebra rank condition [42, 23, 6, 7, 13]

[TABLE]

Then reachable sets are unitary group orbits of the respective initial states

[TABLE]

If the Lie closure $\mathfrak{k}:=\langle iH_{0},iH_{j}\,|\,j=1,2,\dots,m\rangle_{\rm Lie}$ in Eq. (6) is but a proper compact subalgebra $\mathfrak{k}\subsetneq\mathfrak{su}(\mathcal{H})$, one likewise gets a subgroup orbit now by limiting $U$ to elements of $\mathbf{K}:=\exp\mathfrak{k}\subsetneq\mathcal{U}(\mathcal{H})$, see, e.g., [13, 39].

Yet already in open finite dimensional quantum systems $\Sigma_{V}$, it is more intricate to characterise reachable sets: In the unital case, i.e. for $\mathbf{\Gamma}(t)(\mathbf{1})=0$, one finds by the seminal work of [43, 1] and [2] on majorization the inclusion

[TABLE]

as used in [47]. Henceforth, $\overline{\mathfrak{reach}_{\Sigma}(\rho_{0})}$ denotes the closure1d1d1dIn both, finite and infinite dimensions, there is a canonical choice for the topology on $\mathbb{D}(\mathcal{H})$—we will come back to this point later. of $\mathfrak{reach}_{\Sigma}(\rho_{0})$. In the special case $\mathbf{\Gamma}(t)=\gamma(t)\mathbf{\Gamma}_{V}\neq 0$ (where unitality of $\mathbf{\Gamma}(t)$ boils down to normality of $V$) one can obtain equality even for unswitchable noise if there are no bounds on the coherent controls $u_{k}(t)$ and already the control Hamiltonians (without the drift $iH_{0}$) satisfy $\langle iH_{j}\,|\,j=1,2,\dots,m\rangle_{\rm Lie}=\mathfrak{su}(\mathcal{H}),$ a scenario we called Hamiltonian controllable (fully $H$-controllable) [14, 35].

For many experiments this is hopelessly idealising unless one can switch off the noise—a scenario studied below—because then one is allowed to “use” also the drift Hamiltonian $H_{0}$ for controlling the system in the course of noise-free evolution. However, for all physical scenarios (requiring the drift Hamiltonian $H_{0}$ for full controllability of its Hamiltonian part) with sizeable constant noise, the above inclusion is far from being tight and—even worse—the overestimation of the reachable set increases with system size. In these cases, Lie-semigroup techniques help to estimate the reachable set [14, 35].

Yet there are indeed instances of unitarily controllable systems of the type $\Sigma_{V}$ in which the noise can be switched as bang-bang control. An important experimental incarnation are superconducting qubits coupled to an open transmission line [10]. Then, for normal $V$, one can saturate the above inclusion to get $\overline{\mathfrak{reach}(\rho_{0})}=\{\rho\in\mathbb{D}(\mathcal{H})\,|\,\rho\,{\prec}\,\rho_{0}\}$ as shown in [4, 39]. In another extreme, $\mathbf{\Gamma}_{V}$ models coupling the system to a bath of temperature zero (entailing $V$ is the nilpotent matrix $\sigma_{-}$). In this case $\overline{\mathfrak{reach}(\rho_{0})}=\mathbb{D}(\mathbb{C}^{n})$ [16].

Here the goal is to transfer the former result (with normal $V$) from finite to infinite-dimensional systems on separable complex Hilbert spaces $\mathcal{H}$.

1.2. Main Result

In infinite dimensions, establishing unitary controllability for $\Sigma_{0}$ is more intricate. One of the most general results currently known is the following [24]:

Let $H_{0},...,H_{m}$ be selfadjoint operators on a separable Hilbert space $\mathcal{H}$. Further assume that

1. (1)

$H_{0}$ is bounded or unbounded, but has only pure point spectrum. The eigenvalues $x_{k},k\in\mathbb{N}$ are non-degenerate and rationally independent. 2. (2)

The operators $H_{1},...,H_{m}$ are bounded and the set $\{H_{1},...,H_{m}\}$ is connected1e1e1eThis means that the associated graph (which roughly speaking indicates where a transition from energy level $k$ to $l$ is possible) has to be connected, cf. [24]. with respect to the complete set of eigenvectors $\phi_{k}\in\mathcal{H}$, $k\in\mathbb{N}$ of $H_{0}$.

Then the unitary system

[TABLE]

is strongly approximately operator controllable in the following sense:

Definition 1**.**

The unitary control system (7) is called strongly approximately operator controllable, if the strong closure (in $\mathcal{U}(\mathcal{H})$) of the reachable set $\mathfrak{reach}(\mathbf{1})$ coïncides with $\mathcal{U}(\mathcal{H})$.

The result can be generalized to eigenvalues $x_{k}$, $k\in\mathbb{N}$ with finite multiplicities, but this requires more technical conditions on the control Hamiltonians: We have to ensure that trace-free finite rank operators commuting with all eigenprojections of $H_{0}$ are contained in the strong closure of the Lie algebra generated by the $H_{j}$, $j=1,\dots,m$. More challenging are drift Hamiltonians with rationally dependent eigenvalues. However, they can be studied in terms of certain non-Abelian von Neumann algebras; cf. [24] for details. Similar results were derived earlier in terms of Galerkin approximations in [5] and were refined more recently in [8].

If all Hamiltonians (including $H_{0}$) are bounded, we use approximate versions of the Lie algebra rank condition, the most straightforward one being

[TABLE]

Using the continuity of the exponential map in the strong topology [24] it is easy to see that this condition is sufficient for strong operator controllability of (7). Our conjecture is that it is not necessary, but counter examples are not known (their construction is subject of current research). Stronger types of convergence can be achieved if all the Hamiltonians $H_{j}$, $j=0,\dots,m$ are even compact. Since the strong closure of the algebra $\mathcal{K}(\mathcal{H})$ of compact operators is $\mathcal{B}(\mathcal{H})$, it is clear that Eq. (8) is implied by

[TABLE]

where the closure is taken in the uniform (operator norm) topology. If the other implication also holds is still unclear, but unlikely. Note that a compact operator can be the strong limit of a sequence in $\mathcal{K}(\mathcal{H})$ without being the uniform limit.

Let us fix some final notations with regard to the gksl-equation: $\mathcal{B}(\mathcal{H})$ and $\mathcal{B}^{1}(\mathcal{H})$ denote the spaces of all bounded and trace-class operators on $\mathcal{H}$, respectively. Thus $\mathbb{D}(\mathcal{H})\subset\mathcal{B}^{1}(\mathcal{H})$ is precisely the set of all positive semi-definite (selfadjoint) trace-class operators with trace $1$. Moreover, $\|\cdot\|_{1}$ stands for the trace norm on $\mathcal{B}^{1}(\mathcal{H})$ (see Appendix Appendix A: Notation and Basics for more detail on the trace class). To begin with, all Hamiltonians $H_{0},H_{j}$ are assumed to be taken from $\mathcal{B}(\mathcal{H})$, while later $H_{0}$ may be any unbounded selfadjoint operator.

In this setting, the operator solutions of (1) are globally well-defined (with respect to $t\in\mathbb{R}$) for arbitrary piecewise continuous controls (even more irregular controls are admissible) and for each fixed $t\in\mathbb{R}^{+}$ the corresponding map is ultraweakly continuous (cf. footnote 4a) and cptp. In particular for constant controls they form uniformly continuous semigroups of ultraweakly continuous cptp-maps, [30, Thm. 1 & 2].

With these notions and notations and taking majorization from finite to infinite dimensions by way of sequence spaces as introduced by Gohberg and Markus [20] (see Sec. 2.1.), our main result for $\mathbf{\Gamma}(t)=\gamma(t)\mathbf{\Gamma}_{V}(t)$ reads:

Theorem 1**.**

Given the Markovian control system $\Sigma_{V}$

[TABLE]

• (1)

the drift $H_{0}$ is selfadjoint and the controls $H_{1},\ldots,H_{m}$ are selfadjoint and bounded,

• (2)

the Hamiltonian part $\Sigma_{0}$ is strongly (approximately) operator controllable in the sense of Def. 1,

• (3)

the noise term $V\in\mathcal{K}(\mathcal{H})\setminus\{0\}$ is compact, normal and switchable by $\gamma(t)\in\{0,1\}$.

Then the $||\cdot||_{1}$-closure of the reachable set of any initial state $\rho(0)=\rho_{0}\in\mathbb{D}(\mathcal{H})$ under the system $\Sigma_{V}$ exhausts all states majorized by the initial state $\rho_{0}$

[TABLE]

In order to arrive at this result, the paper is organised as follows: Section 2. first takes majorization from finite to infinite dimensions in 2.1. before combining ideas of von Neumann with $C$-numerical ranges for majorization in infinite dimensions 2.2. Section 3. then presents the idea of the main theorem, the proof details themselves being relegated to the Appendix. Appendix Appendix A: Notation and Basics contains technical basics, while Appendix Appendix B: Proof of von Neumann Type of Trace Inequality gives the proofs to Section 2.2. Finally Appendix Appendix C: Proof of the Main Theorem for Bounded $H_{0}$ provides the proof of the main theorem for bounded $H_{0}$, while Appendix Appendix D: Theorem 1 for Unbounded Drift $H_{0}$ relaxes it to unbounded $H_{0}$.

2. From Majorization via $C$-Numerical Range to Reachability

2.1. Majorization in Finite and Infinite Dimensions

Generalizing majorization to infinite dimensions is somewhat delicate. Following [20], one may define majorization first on the space of all real null sequences $c_{0}(\mathbb{N})$ and then on the space of all absolutely summable sequences $\ell^{1}(\mathbb{N})$. As we need a concept of majorization on density operators, for our purposes it suffices to introduce majorization solely on the summable sequences of non-negative numbers $\ell^{1}_{+}(\mathbb{N})$, which is rather intuitive.

In the notation of [2, 31], take a real vector $x=(x_{1},\ldots,x_{n})^{T}\in\mathbb{R}^{n}$ and let $x^{\downarrow}=(x_{1}^{\downarrow},\ldots,x_{n}^{\downarrow})^{T}$ denote its decreasing re-arrangement $x_{1}^{\downarrow}\geq x_{2}^{\downarrow}\geq\ldots\geq x_{n}^{\downarrow}\,.$ For two vectors $x,y\in\mathbb{R}^{n}$, we say $x$ is majorized by $y$ (written $x\prec y$) if $\sum_{j=1}^{k}x_{j}^{\downarrow}\leq\sum_{j=1}^{k}y_{j}^{\downarrow}$ for all $k=1,\ldots,n-1$ and $\sum_{j=1}^{n}x_{j}=\sum_{j=1}^{n}y_{j}$. By definition $x\prec y$ depends only on the entries of $x$ and $y$ but not on their initial arrangement, so $\prec$ is permutation invariant.

Now for sequences $x\in\ell^{1}_{+}(\mathbb{N})$, this re-arrangement procedure works just the same way, and all the non-zero entries of $x$ are again contained within the rearranged sequence $x^{\downarrow}$. However, be aware that $x$ and $x^{\downarrow}$ may differ in the number of their zero entries.

Definition 2**.**

Consider $x,y\in\ell^{1}_{+}(\mathbb{N})$ and $\rho,\omega\in\mathbb{D}(\mathcal{H})$.

• (a)

We say that $x$ is majorized by $y$, denoted by $x\prec y$, if the sum inequalities $\sum_{j=1}^{k}x_{j}^{\downarrow}\leq\sum_{j=1}^{k}y_{j}^{\downarrow}$ hold for all $k\in\mathbb{N}$, and $\sum_{j=1}^{\infty}x_{j}=\sum_{j=1}^{\infty}y_{j}$.

• (b)

$\omega$ majorizes $\rho$, denoted by $\rho\prec\omega$, if $\lambda^{\downarrow}(\rho)\prec\lambda^{\downarrow}(\omega)$ where $\lambda^{\downarrow}(\cdot)\in\ell^{1}_{+}(\mathbb{N})$ denotes the (non-modified) eigenvalue sequence2a2a2aUsually, the eigenvalue sequence $\lambda^{\downarrow}(T)$ of a compact operator $T$ on $\mathcal{H}$ is obtained by arranging its non-zero eigenvalues in the decreasing order of their magnitudes and each eigenvalue is repeated as many times as its (necessarily finite) algebraic multiplicity. If the spectrum of $T$ is finite itself, then the sequence is filled with zeros, cf. [32, Ch. 15]. However, in order to get the result of Lemma 8 with respect to an orthonormal basis (and not just an orthonormal system), and also to properly define the $C$-spectrum of $T$ later on, a modified eigenvalue sequence has to be introduced, as in [15, Ch. 3.2]. If the range of $T$ is infinite-dimensional and the kernel of $T$ finite-dimensional then put $\operatorname{dim}(\operatorname{ker}T)$ zeros at the beginning of the eigenvalue sequence of $T$. If the range and the kernel of $T$ are infinite-dimensional, mix infinitely many zeros into the eigenvalue sequence of $T$ (since for the $C$-spectrum arbitrary permutations will be applied to the modified eigenvalue sequence, we need not specify this mixing procedure further). If the range of $T$ is finite-dimensional, leave the eigenvalue sequence of $T$ unchanged. of the respective state.

Remark 1*.*

In Definition 2 (b) it does not matter whether one considers the usual (non-modified) or the modified eigenvalue sequence (for the purpose of this remark denoted by $\lambda^{\downarrow}$ and $\lambda_{m}$, respectively). More precisely, these sequences by construction share the same non-zero entries so $\lambda^{\downarrow}={\lambda^{\downarrow}_{m}},$.

As in finite dimensions, majorization in infinite dimensions has a number of different characterizations, the following two being particularly advantageous for our purposes.

Lemma 1** ([29], Thm. 3.3).**

For $\rho,\omega\in\mathbb{D}(\mathcal{H})$ the following are equivalent:

• (a)

$\rho\prec\omega$.

• (b)

There exists a bi-stochastic quantum map $T\in\mathbb{S}(\mathcal{H})$ (cf. Def. 4 in Appendix Appendix A: Notation and Basics) such that $T(\omega)=\rho$.

Proposition 1**.**

Let $x,y\in\ell^{1}_{+}(\mathbb{N})$ be non-increasing and let $(e_{n})_{n\in\mathbb{N}}$ be some orthonormal basis of $\mathcal{H}$. Then the following statements are equivalent:

• (a)

$x\prec y$**

• (b)

There exists a selfadjoint $A\in\mathcal{B}^{1}(\mathcal{H})$ with diagonal entries $(x_{n})_{n\in\mathbb{N}}$ and eigenvalues $(y_{n})_{n\in\mathbb{N}}$.

• (c)

There exists a unitary $U\in\mathcal{B}(\mathcal{H})$ such that $U\operatorname{diag}(y)U^{\dagger}$ has diagonal entries $(x_{n})_{n\in\mathbb{N}}$.

Here, “diagonal” always refers to the orthonormal basis $(e_{n})_{n\in\mathbb{N}}$, i.e. the map $\operatorname{diag}:\ell^{1}_{+}(\mathbb{N})\to\mathcal{B}^{1}(\mathcal{H})$ is given by $x\mapsto\sum_{n=1}^{\infty}x_{n}\langle e_{n},\cdot\rangle e_{n}$.

Proof.

“(a) $\Rightarrow$ (b)”: Assume $x\prec y$. By [20, Prop. IV] there exist orthonormal bases $(\phi_{n})_{n\in\mathbb{N}}$ and $(\psi_{n})_{n\in\mathbb{N}}$ of $\mathcal{H}$ such that $H=\sum_{n=1}^{\infty}y_{n}\langle\psi_{n},\cdot\rangle\psi_{n}$ satisfies $\langle\phi_{n},H\phi_{n}\rangle=x_{n}$ for all $n\in\mathbb{N}$. Consider the unitary operator $U\in\mathcal{B}(\mathcal{H})$ which transforms $(\phi_{n})_{n\in\mathbb{N}}$ into $(e_{n})_{n\in\mathbb{N}}$, then $A:=UHU^{\dagger}\in\mathcal{B}^{1}(\mathcal{H})$ does the job. “(b) $\Rightarrow$ (a)”: follows from [19]. “(b) $\Leftrightarrow$ (c)”: The statement is obvious. ∎

We conclude with a classical result on sub-majorization (without proof) which will be needed in the following subsection.

Lemma 2** ([31], 3.H.3.b).**

Let $x,y\in\mathbb{R}^{n}$ such that $\sum_{j=1}^{k}x_{j}^{\downarrow}\leq\sum_{j=1}^{k}y_{j}^{\downarrow}$ for all $k=1,\ldots,n$. Then for arbitrary $c_{1}\geq c_{2}\geq\ldots\geq c_{n}\geq 0$ one has $\sum\nolimits_{j=1}^{n}c_{j}x_{j}^{\downarrow}\leq\sum\nolimits_{j=1}^{n}c_{j}y_{j}^{\downarrow}\,.$

2.2. Combining a von Neumann Idea with $C$-Numerical Ranges

In finite dimensions, Ando [2, Thm. 7.4] has shown that majorization can be characterized in an elegant way via the $C$-numerical range [28, 11]

[TABLE]

with $C\in\mathcal{B}^{1}(\mathcal{H})$, $T\in\mathcal{B}(\mathcal{H})$ and $\mathcal{U}(\mathcal{H})\subset\mathcal{B}(\mathcal{H})$ being the unitary group on $\mathcal{H}$. Here, we generalize his approach to infinite dimensions (Prop. 2 below) using a recent result in [15]. Later on, this characterization will greatly simplify handling continuity properties of majorization (cf. Lemma 5).

For our purpose, we need a relation connecting the $C$-numerical range and $C$-spectrum of a compact operator $T$ given by

[TABLE]

on one hand-side with $(\lambda_{j}(C))_{j\in\mathbb{N}}$ and $(\lambda_{j}(T))_{j\in\mathbb{N}}$ being the modified eigenvalue sequences (cf. footnote 2a) of $C$ and $T$ on the other. Note that each element in $W_{C}(T)$ and $P_{C}(T)$ is bounded by $\|C\|_{1}\|T\|_{\text{op}}$—thus the closures of $W_{C}(T)$ and $P_{C}(T)$ constitute compact subsets of $\mathbb{C}$.

If $C,T$ are normal, one has the inclusion $\overline{W_{C}(T)}\subseteq\operatorname{conv}(\overline{P_{C}(T)})$. Yet under further assumptions on the operators one can even achieve equality.

Lemma 3** ([15], Coro. 3.1).**

Let $C\in\mathcal{B}^{1}(\mathcal{H})$ and $T\in\mathcal{K}(\mathcal{H})$ both be normal, such that the eigenvalues of $C$ are collinear, i.e. the eigenvalues all lie on a common line. Then $\overline{W_{C}(T)}=\operatorname{conv}(\overline{P_{C}(T)})\,.$

In fact, Lemma 3 induces a von Neumann-type of trace (in-)equality [45] for compact, selfadjoint operators. Its proof is in Appendix Appendix B: Proof of von Neumann Type of Trace Inequality.

Corollary 1**.**

Let $C\in\mathcal{B}^{1}(\mathcal{H})$ and $T\in\mathcal{K}(\mathcal{H})$ both be selfadjoint. Then

[TABLE]

where $\lambda_{j}^{\downarrow}(C^{+})$, $\lambda_{j}^{\downarrow}(T^{+})$ and $\lambda_{j}^{\downarrow}(C^{-})$, $\lambda_{j}^{\downarrow}(T^{-})$ denote the decreasing eigenvalue sequences of the positive semi-definite operators $C^{+}$, $T^{+}$ and $C^{-}$, $T^{-}$, respectively, where $C=C^{+}-C^{-}$ and $T=T^{+}-T^{-}$ as usual.

To simplify notation, we use the following abbreviation.

Definition 3**.**

Let $C\in\mathcal{B}^{1}(\mathcal{H})$, $T\in\mathcal{B}(\mathcal{H})$ both be selfadjoint. We define $K_{C}(T):=\sup_{U\in\mathcal{U}(\mathcal{H})}\operatorname{tr}(CU^{\dagger}TU)\in\mathbb{R}$ or, equivalently, $K_{C}(T):=\sup W_{C}(T)=\max\overline{W_{C}(T)}$.

Note that if $C$ and $T$ are positive semi-definite, then $K_{C}(T)$ turns into the $C$-numerical radius $r_{C}(T)$ of $T$. Now this definition gives rise to the following result, whose finite-dimensional analogue can be found, e.g., in [2, Thm. 7.4].

Proposition 2**.**

For $\rho,\omega\in\mathbb{D}(\mathcal{H})$ the following statements are equivalent.

• (a)

$\rho\prec\omega$**

• (b)

$K_{\rho}(T)\leq K_{\omega}(T)$* for all selfadjoint $T\in\mathcal{K}(\mathcal{H})$.*

Proof.

“(a) $\Rightarrow$ (b)”: Keeping in mind that $\rho,\omega\geq 0$, Coro. 1 yields

[TABLE]

and similarly for $K_{\omega}(T)$. Moreover, Lemma 2 implies

[TABLE]

for all $n\in\mathbb{N}$ and thus it follows $K_{\rho}(T)\leq K_{\omega}(T)$ for all selfadjoint $T\in\mathcal{K}(\mathcal{H})$.

“(b) $\Rightarrow$ (a)”: Let $k\in\mathbb{N}$ and let $(e_{n})_{n\in\mathbb{N}}$ be any orthonormal basis of $\mathcal{H}$. Consider the (finite-rank) projection $\Pi_{k}=\sum\nolimits_{j=1}^{k}\langle e_{j},\cdot\rangle e_{j}$. As $\Pi_{k}$ is compact and selfadjoint with eigenvalues $1$ (of multiplicity $k$) and [math] (of infinite multiplicity), Coro. 1 yields $K_{\rho}(\Pi_{k})=\sum\nolimits_{j=1}^{k}\lambda_{j}^{\downarrow}(\rho)$ and $K_{\omega}(\Pi_{k})=\sum\nolimits_{j=1}^{k}\lambda_{j}^{\downarrow}(\omega)\,.$ Now by assumption, one has

[TABLE]

for all $k\in\mathbb{N}$ which shows $\rho\prec\omega$ and thus concludes this proof. ∎

3. Idea behind the Main Result

Below we sketch the proof of our main result Thm. 1. A full proof will be given in Appendix Appendix C: Proof of the Main Theorem for Bounded $H_{0}$. Here, we sketch central ideas and key lemmas, the proofs of which are either straightforward or postponed to Appendices Appendix C: Proof of the Main Theorem for Bounded $H_{0}$ and Appendix D: Theorem 1 for Unbounded Drift $H_{0}$. For convenience, let us first recall the precise statement of Thm. 1.

Theorem 1. Given the Markovian control system $\Sigma_{V}$

[TABLE]

• (1)

the drift $H_{0}$ is selfadjoint and the controls $H_{1},\ldots,H_{m}$ are selfadjoint and bounded,

• (2)

the Hamiltonian part $\Sigma_{0}$ is strongly (approximately) operator controllable in the sense of Def. 1,

• (3)

the noise term $V\in\mathcal{K}(\mathcal{H})\setminus\{0\}$ is compact, normal and switchable by $\gamma(t)\in\{0,1\}$.

Then the $||\cdot||_{1}$-closure of the reachable set of any initial state $\rho(0)=\rho_{0}\in\mathbb{D}(\mathcal{H})$ under the system $\Sigma_{V}$ exhausts all states majorized by the initial state $\rho_{0}$

[TABLE]

The following lemmas play a crucial role in the proof of Theorem 1. The first one reveals a beautiful eigenspace structure of the noise generators $\mathbf{\Gamma}_{V}$ whenever $V$ is normal and compact, and it follows by direct computation.

Lemma 4**.**

Let $V\in\mathcal{K}(\mathcal{H})$ be normal, $(f_{j})_{j\in\mathbb{N}}$ its orthonormal eigenbasis and $(v_{j})_{j\in\mathbb{N}}$ its modified eigenvalue sequence, hence $V=\sum_{j=1}^{\infty}v_{j}\langle f_{j},\cdot\rangle f_{j}$ (cf. Appendix Appendix A: Notation and Basics). Then for all $X\in\mathcal{B}(\mathcal{H})$, the noise operator $\mathbf{\Gamma}_{V}$ given by Eq. (4) acts like

[TABLE]

for all $j,k\in\mathbb{N}$. In particular, each rank-$1$ operator of the form $\langle f_{k},\cdot\rangle f_{j}$ is an eigenvector of $\mathbf{\Gamma}_{V}$ to the eigenvalue $\tfrac{1}{2}{|v_{j}-v_{k}|^{2}}-i\operatorname{Im}(v_{j}\overline{v_{k}})$ and the kernel of $\mathbf{\Gamma}_{V}$ contains $\operatorname{span}\{\langle f_{j},\cdot\rangle f_{j}\,|\,j\in\mathbb{N}\}$. Moreover, it follows

[TABLE]

for all $t\in\mathbb{R}$ and $j,k\in\mathbb{N}$.

The following lemmas provide two crucial approximation results.

Lemma 5**.**

For all $\rho_{0}\in\mathbb{D}(\mathcal{H})$ the set $\{\rho\in\mathbb{D}(\mathcal{H})\,|\,\rho\prec\rho_{0}\}$ is closed w.r.t. the trace norm $\|\cdot\|_{1}$.

Lemma 6** (Unitary channel approximation).**

*Consider a subset $R\subseteq\mathcal{U}(\mathcal{H})$ of the unitary group of $\mathcal{H}$ such that $\overline{R}^{s}=\mathcal{U}(\mathcal{H})$, i.e. its strong closure relative to $\mathcal{U}(\mathcal{H})$ yields the full group. Furthermore, let $\rho\in\mathbb{D}(\mathcal{H})$ and $U\in\mathcal{U}(\mathcal{H})$.

Then for all $\varepsilon>0$ one can find $\tilde{U}\in R$ such that $\|U\rho\,U^{\dagger}-\tilde{U}\rho\,\tilde{U}^{\dagger}\|_{1}<\varepsilon\,.$*

Now in our control setting we do not have direct access to the “pure” noise generator $-\mathbf{\Gamma}_{V}$. However, we may use the Lie-Trotter product formula (cf. [36, Thm. VIII.29]) to approximate the noise dynamics $(\exp(-t\mathbf{\Gamma}_{V}))_{t\in\mathbb{R}_{+}}$:

Lemma 7** (Trotter trick).**

For $u_{1}(t)=\ldots=u_{m}(t)=0$ and $\gamma(t)=1$ (i.e. noise only) the operator solution of (2) reads $(\exp(-it\mathbf{H}_{0}-t\mathbf{\Gamma}_{V}))_{t\in\mathbb{R}_{0}^{+}}$. Then for $t\geq 0$ (uniformly on bounded intervals) one has

[TABLE]

Thus given a time $t\geq 0$ and precision $\varepsilon>0$, to “simulate” $\exp(-t\mathbf{\Gamma}_{V})$ within this precision it suffices to apply the noisy evolution as well as the unitary channel $\exp(it\mathbf{H}_{0}/N)$ to the system—in an alternating manner, $N$ times (for sufficiently large $N\in\mathbb{N}$). — Now we are ready to outline the proof of Thm. 1.

Sketch of the proof of Theorem 1.

“$\subseteq$”: As $V$ is assumed to be normal one has $(i\mathbf{H}(t)+\gamma(t)\mathbf{\Gamma}_{V})(\mathbf{1})=0$ at all times so the operator solution of Eq. (2) is in $\mathbb{S}(\mathcal{H})$, i.e. a bi-stochastic quantum map and one can never leave the set of states majorized by $\rho_{0}$ (cf. Lemma 1). By Lemma 5, the $\|\cdot\|_{1}$-closure yields

[TABLE]

“$\supseteq$”: As $V\in\mathcal{K}(\mathcal{H})$ is normal we can diagonalize it (see Appendix Appendix A: Notation and Basics) with orthonormal eigenbasis $(e_{j})_{j\in\mathbb{N}}$. Now, let $\varepsilon>0$ and $\rho\in\mathbb{D}(\mathcal{H})$ with $\rho\prec\rho_{0}$ be given. We have to find $\rho_{F}\in\mathfrak{reach}_{\Sigma_{V}}(\rho_{0})$ such that $\|\rho-\rho_{F}\|_{1}<\varepsilon$. By assumption there exist $x,y\in\ell^{1}_{+}(\mathbb{N})$, $x,y\neq 0$ as well as $W_{1},W_{2}\in\mathcal{U}(\mathcal{H})$ such that $\rho=W_{1}\operatorname{diag}(x)W_{1}^{\dagger}$, $\rho_{0}=W_{2}\operatorname{diag}(y)W_{2}^{\dagger}$ with $x\prec y$. Here, $\operatorname{diag}$ refers to the above eigenbasis of $V$. Applying Prop. 1 to $x,y$ gives us unitary $U\in\mathcal{B}(\mathcal{H})$ such that $U\operatorname{diag}(y)U^{\dagger}$ has diagonal entries $(x_{n})_{n\in\mathbb{N}}$. The proof roughly consists of three steps shown here:

[TABLE]

Step 1 and 3 merely apply a unitary channel; assuming strong operator controllability, we may use unitary channels giving the target state with arbitrary precision (cf. Lemma 6). Step 2 is about getting rid of all off-diagonal elements of $U\operatorname{diag}(y)U^{\dagger}$ to reach $\operatorname{diag}(x)$ by applying pure noise $\exp(-t\mathbf{\Gamma}_{V})$ in the limit $t\to\infty$ (cf. Lemma 4). As expected there are a few delicate issues:

• •

We have no access to pure noise, as in our setting we cannot switch off $H_{0}$. Yet by a Trotter-type argument we can approximate the desired noise with arbitrary precision, cf. Lemma 7 and Lemma 14.

• •

If the eigenvalues of $V$ are not pairwise different, there are some “matrix” elements left untouched by the noise as a consequence of (9). So one may need permutation channels (which in particular are unitary) to rearrange those elements into “spots” where the noise affects them.

• •

As in Step 1 and 3 we have to approximate these permutation channels. Here we use the approximation property of the trace class (cf. Lemma 9), i.e. we invoke decoherence on a sufficiently large but finite “block” of the density operator so we only need finitely many permutations.

• •

Applying Prop. 1 requires that $\rho_{0}$, $\rho$ are unitarily diagonalized so that the original and the modified eigenvalue sequences of these states coïncide (which either means the states are finite-rank or have trivial kernel)—else the zeros that have to be added for the modified eigenvalue sequence prevent this. In the latter case we can proceed to states $\rho^{\prime}$, $\rho_{0}^{\prime}$ which satisfy the assumptions of Prop. 1 and which are close (in trace norm) to the original states, and execute the scheme of Eq. (10).

Altogether this is enough to perform the scheme suggested in Eq. (10) with arbitrary precision. So $\rho$ $\prec\rho_{0}$ is in the $\|\cdot\|_{1}$-closure of the reachable set. The full proof with all detail is in Appendices Appendix C: Proof of the Main Theorem for Bounded $H_{0}$ and Appendix D: Theorem 1 for Unbounded Drift $H_{0}$. ∎

4. Conclusions and Outlook

For the first time, here we have derived sufficient conditions under which a quantum dynamical system can actually reach (in the closure) all quantum states majorized by the respective initial state in an infinite dimensional quantum system following a controlled Markovian master equation. To this end, we have extended the standard unital gksl master equation to an infinite dimensional bilinear control system $\Sigma_{V}$ the unitary part of which has to be operator controllable and the dissipative part (generated by a single normal compact noise term $V$) has to be bang-bang switchable. This takes recent results on finite dimensional systems [4, 39] to infinite dimensions. — While the generalization from a single such $V$ to several commuting compact noise terms $V_{k}$ is obvious, a generalization beyond compact $V$ seems challenging. One may also relax considerations to weak-$*$ continuity of the semigroup, which goes beyond the standard gksl-equation, as pursued by Carbone, Fagnola [9] and more recently by Siemon, Holevo and Werner [40].

For applying the results to broader classes of physical systems, one may think of further generalizations. The current setup restricts us to (possibly unbounded) system Hamiltonians $H_{0}$ with discrete spectrum such as bound systems where particles are trapped within an unbounded potential (e.g., harmonic oscillators). To look at more interesting setups where processes like ionization, tunneling and evaporation play a role, we have to use operators with continuous spectrum. However, in this area even coherent control is not understood well enough (if at all). Closing this gap is therefore an obvious (yet non-trivial !) next step.

Thus the spirit of Sudarshan still promises insightful results to come.

Acknowledgements

This work was supported by the Bavarian excellence network enb via the International PhD Programme of Excellence Exploring Quantum Matter (exqm) and by Deutsche Forschungsgemeinschaft (dfg, German Research Foundation) under Germany’s Excellence Strategy exc-2111–390814868.

Appendix A: Notation and Basics

For a comprehensive introduction to infinite-dimensional separable Hilbert spaces and Schatten-class operators we refer to, e.g., [3, 32, 36]. As we will encounter compact normal operators repeatedly, let us first recap the well-known diagonalization result.

Lemma 8** ([3], Thm. VIII.4.6).**

Let $T\in\mathcal{K}(\mathcal{H})$ be normal, i.e. $T^{\dagger}T=TT^{\dagger}$. Then there exists an orthonormal basis $(e_{n})_{n\in\mathbb{N}}$ of $\mathcal{H}$ such that

[TABLE]

where $(\tau_{j})_{j\in\mathbb{N}}$ is the modified eigenvalue sequence (cf. footnote 2a) of $T$.

Moreover, recall that the set of trace-class operators is given by

[TABLE]

which forms a Banach space under the trace norm $\|\cdot\|_{1}$ and constitutes a two-sided ideal in the $C^{*}$-algebra of all bounded operators $\mathcal{B}(\mathcal{H})$. Important properties of the trace are

[TABLE]

for all $x,y\in\mathcal{H}$, $C\in\mathcal{B}^{1}(\mathcal{H})$ and $T\in\mathcal{B}(\mathcal{H})$. Furthermore, the trace class has the approximation property:

Lemma 9** ([15], Lemma 3.2).**

Let $C\in\mathcal{B}^{1}(\mathcal{H})$ and let $(e_{n})_{n\in\mathbb{N}}$ be any orthonormal basis of $\mathcal{H}$. For arbitrary $k\in\mathbb{N}$, let $\Pi_{k}:=\sum_{j=1}^{k}\langle e_{j},\cdot\rangle e_{j}$ denote the orthogonal projection onto $\operatorname{span}\{e_{1},\ldots,e_{k}\}$. Then the sequence of “block approximations” $(\Pi_{n}C\Pi_{n})_{n\in\mathbb{N}}$ converges (in trace norm) to $C$, i.e.

[TABLE]

Definition 4**.**

• (a)

A linear map $T:\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{G})$ is trace-preserving if $T(\mathcal{B}^{1}(\mathcal{H}))\subseteq\mathcal{B}^{1}(\mathcal{G})$ with $\operatorname{tr}(T(A))=\operatorname{tr}(A)$ for all $A\in\mathcal{B}^{1}(\mathcal{H})$.

• (b)

A bi-stochastic quantum map is a linear, ultraweakly continuous4a4a4aThe ultraweak topology is the weak-$*$ topology on $\mathcal{B}(\mathcal{H})$ inherited by the isometrically isomorphic map $\psi:\mathcal{B}(\mathcal{H})\to(\mathcal{B}^{1}(\mathcal{H}))^{\prime}$, $B\mapsto\operatorname{tr}(B(\cdot))$., completely positive, unital (identity-preserving) and trace-preserving map $T:\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{G})$. We define

[TABLE]

and $\mathbb{S}(\mathcal{H}):=\mathbb{S}(\mathcal{H},\mathcal{H})$.

Thus using the terminology of [44, Def. 2], a bi-stochastic quantum map is a Heisenberg quantum channel which also is trace-preserving and its restriction to the trace class is a Schrödinger quantum channel. Using [44, Prop. 2] this directly implies the following.

Lemma 10**.**

Let $T\in\mathbb{S}(\mathcal{H},\mathcal{G})$ and consider the restricted (well-defined) map $T_{\mathcal{B}^{1}}:\mathcal{B}^{1}(\mathcal{H})\to\mathcal{B}^{1}(\mathcal{G})$. Then $\|T\|_{\text{op}}=\|T_{\mathcal{B}^{1}}\|_{\text{op}}=1$ so

[TABLE]

for all $B\in\mathcal{B}(\mathcal{H})$, $A\in\mathcal{B}^{1}(\mathcal{H})$.

Appendix B: Proof of von Neumann Type of Trace Inequality

The following von Neumann type of trace (in-)equality was used in Sec. 2.2.

Corollary 1. Let $C\in\mathcal{B}^{1}(\mathcal{H})$ and $T\in\mathcal{K}(\mathcal{H})$ both be selfadjoint. Then

[TABLE]

where $\lambda_{j}^{\downarrow}(C^{+})$, $\lambda_{j}^{\downarrow}(T^{+})$ and $\lambda_{j}^{\downarrow}(C^{-})$, $\lambda_{j}^{\downarrow}(T^{-})$ denote the decreasing eigenvalue sequences of the positive semi-definite operators $C^{+}$, $T^{+}$ and $C^{-}$, $T^{-}$, respectively, where $C=C^{+}-C^{-}$ and $T=T^{+}-T^{-}$ as usual.

Below, we provide a proof of the above statement, which to the best of our knowledge is new. To this end, we need the notion of set convergence using the Hausdorff metric on compact subsets (of $\mathbb{C}$) and the associated notion of convergence, see, e.g., [34]. The distance between $z\in\mathbb{C}$ and any non-empty compact subset $A\subseteq\mathbb{C}$ is defined by

[TABLE]

Based on (12) the Hausdorff metric $\Delta$ on the set of all non-empty compact subsets of $\mathbb{C}$ is given by

[TABLE]

The following characterization of the Hausdorff metric is readily verified.

Lemma 11**.**

Let $A,B\subset\mathbb{C}$ be two non-empty compact sets and let $\varepsilon>0$. Then $\Delta(A,B)\leq\varepsilon$ if and only if for all $z\in A$, there exists $w\in B$ with $d(z,w)\leq\varepsilon$ and vice versa.

With this metric one can introduce the notion of convergence for sequences $(A_{n})_{n\in\mathbb{N}}$ of non-empty compact subsets of $\mathbb{C}$ such that the maximum-operator is continuous in the following sense.

Lemma 12**.**

Let $(A_{n})_{n\in\mathbb{N}}$ be a bounded sequence of non-empty, compact subsets of $\mathbb{R}$ which converges to $A\subset\mathbb{R}$. Then the sequence of real numbers $(\max A_{n})_{n\in\mathbb{N}}$ is convergent with

[TABLE]

Proof.

Let $\varepsilon>0$. By assumption, there exists $N\in\mathbb{N}$ such that $\Delta(A_{n},A)<\varepsilon$ for all $n\geq N$. Hence, by Lemma 11, there exists $a_{n}\in A_{n}$ such that $|\max A-a_{n}|<\varepsilon$ and thus

[TABLE]

Similarly, there exists $a\in A$ such that $|\max A_{n}-a|<\varepsilon$ and thus

[TABLE]

Combining both estimates, we get $|\max A-\max A_{n}|<\varepsilon$. ∎

Just like [15, Thm. 3.1] one can show the following:

Lemma 13**.**

Let $C\in\mathcal{B}^{1}(\mathcal{H}),T\in\mathcal{B}(\mathcal{H})$ and $(C_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{B}^{1}(\mathcal{H})$ which converges to $C$ w.r.t. $\|\cdot\|_{1}$. Then

[TABLE]

Moreover, if $T$ is compact as well, then

[TABLE]

where $\Pi_{k}$ is the orthogonal projection onto the span of the first $k$ elements of an arbitrary orthonormal basis $(e_{n})_{n\in\mathbb{N}}$ of $\mathcal{H}$.

Proof of Corollary 1.

Let $C\in\mathcal{B}^{1}(\mathcal{H})$ and $T\in\mathcal{K}(\mathcal{H})$ both be selfadjoint and let us first assume that $T$ has at most $k\in\mathbb{N}$ non-zero eigenvalues. Then

[TABLE]

with $P_{C}(T)$ the $C$-spectrum of $T$ (cf. Lemma 3), is straightforward to show.

Now let us address the general case. Choose any orthonormal eigenbasis $(e_{n})_{n\in\mathbb{N}}$ of $T$ with icorresponding modified eigenvalue sequence (Lemma 8). Moreover, let $\Pi_{k}=\sum\nolimits_{j=1}^{k}\langle e_{j},\cdot\rangle e_{j}$ the projection onto the span of the first $k$ eigenvectors of $T$. Then $\Pi_{k}T\Pi_{k}$ has at most $k$ non-zero eigenvalues and our preliminary considerations combined with Lemma 3, 12 and 13 readily imply

[TABLE]

where we used the identity $(\Pi_{k}T\Pi_{k})^{\pm}=\Pi_{k}T^{\pm}\Pi_{k}$. This yields the result. ∎

Appendix C: Proof of the Main Theorem for Bounded $H_{0}$

Lemma 5. Let $\rho_{0}\in\mathbb{D}(\mathcal{H})$. Then the set $\{\rho\in\mathbb{D}(\mathcal{H})\,|\,\rho\prec\rho_{0}\}$ is closed w.r.t. the trace norm $\|\cdot\|_{1}$.

Proof.

For given $\omega\in\overline{\{\rho\in\mathbb{D}(\mathcal{H})\,|\,\rho\prec\rho_{0}\}}^{1}{}$ there exists a sequence $(\rho_{n})_{n\in\mathbb{N}}$ in $\{\rho\in\mathbb{D}(\mathcal{H})\,|\,\rho\prec\rho_{0}\}\subseteq\mathbb{D}(\mathcal{H})$ such that $\|\omega-\rho_{n}\|_{1}\to 0$ as $n\to\infty$. Obviously,

[TABLE]

for all $x\in\mathcal{H}$ so $\omega\in\mathbb{D}(\mathcal{H})$. Now let $T\in\mathcal{K}(\mathcal{H})$ be arbitrary but selfadjoint. Then Lemma 12 and 13 implies

[TABLE]

On the other hand, due to Prop. 2 and $\rho_{n}\prec\omega$ for all $n\in\mathbb{N}$, one has

[TABLE]

which again by Prop. 2 (as $T$ was chosen arbitrarily) implies $\omega\prec\rho_{0}$. ∎

Lemma 6 (Unitary channel approximation)**. ** Consider a subset $R\subseteq\mathcal{U}(\mathcal{H})$ of the unitary group on $\mathcal{H}$ such that $\overline{R}^{s}=\mathcal{U}(\mathcal{H})$, i.e. its strong closure relative to $\mathcal{U}(\mathcal{H})$ yields the full group. Furthermore let $\rho\in\mathbb{D}(\mathcal{H})$ and $U\in\mathcal{U}(\mathcal{H})$. Then for all $\varepsilon>0$ one can find $\tilde{U}\in R$ such that $\|U\rho U^{\dagger}-\tilde{U}\rho\tilde{U}^{\dagger}\|_{1}<\varepsilon\,.$

Proof.

Due to Lemma 8 there exists a modified eigenvalue sequence $(r_{n})_{n\in\mathbb{N}}\in\ell^{1}_{+}(\mathbb{N})$ and an orthonormal basis $(e_{n})_{n\in\mathbb{N}}$ of $\mathcal{H}$ such that $\rho=\sum_{j=1}^{\infty}r_{j}\langle e_{j},\cdot\rangle e_{j}$. Then one also finds $N\in\mathbb{N}$ such that the “tail” of $\rho$ is sufficiently small, i.e.

[TABLE]

By assumption there is $\tilde{U}\in R\subseteq\mathcal{U}(\mathcal{H})$ such that

[TABLE]

Moreover, the triangle inequality, non-negativity of $r_{n}$ and the trace norm identity $\|\langle x,\cdot\rangle y\|_{1}=\|x\|\|y\|$ for all $x,y\in\mathcal{H}$ imply

[TABLE]

Splitting the sum at $N$ and using Eq. (14) and (15) finally yields the estimate $\|U\rho U^{\dagger}-\tilde{U}\rho\tilde{U}^{\dagger}\|_{1}\leq 2\sum_{j=1}^{N}r_{j}\|Ue_{j}-\tilde{U}e_{j}\|+4\sum_{j=N+1}^{\infty}r_{j}<\frac{\varepsilon}{3}+\frac{2\varepsilon}{3}=\varepsilon\;.$ ∎

Next, let us refine Lemma 7 in terms of precision as follows:

Lemma 14**.**

Let $V\in\mathcal{K}(\mathcal{H})\setminus\{0\}$ be normal, $H_{0}\in\mathcal{B}(\mathcal{H})$ be selfadjoint, $\rho\in\mathbb{D}(\mathcal{H})$ be arbitrary and $[0,T]\subset\mathbb{R}_{0}^{+}$ be given. Furthermore let $R\subseteq\mathcal{U}(\mathcal{H})$ with $\overline{R}^{s}=\mathcal{U}(\mathcal{H})$, where the closure is taken in $\mathcal{U}(\mathcal{H})$. Then for all $\varepsilon>0$ there exists $m\in\mathbb{N}$ and $U_{1},\ldots,U_{m}\in R$ such that for all $s\in[0,T]$

[TABLE]

Proof.

By Lemma 7 there exists $m\in\mathbb{N}$ with

[TABLE]

for all $s\in[0,T]$, where $F$ is a unitary channel and $G$ is unital (because $V$ is normal) and reflects the operator solution of (2) with $u_{1}(s)=\dots=u_{m}(s)=0$ and $\gamma(s)=1$, i.e. the noisy but uncontrolled evolution of the system.

For convenience define $\rho_{j}:=(G\circ(F\circ G)^{m-j})(\rho)$ for $j=1,\ldots,m$. Then, Lemma 6 yields $U_{j}\in R\subseteq\mathcal{U}(\mathcal{H})$ with $\|F(\rho_{j})-U_{j}\rho_{j}U_{j}^{\dagger}\|_{1}<\frac{\varepsilon}{2m}$ for all4b4b4bNote that $\rho_{j}=\rho_{j}(s)$ depends on $s\in[0,T]$ as $F$ and $G$ do so. Moreover, the set $\{\rho_{j}(s)\,|\,s\in[0,T]\}$ is compact as $F$ and $G$ are continuous in $s$ and hence the proof of Lemma 6 can be easily modified to obtain the desired result. $s\in[0,T]$. Finally, Lemma 10 and Lemma 15 (below) imply

[TABLE]

for all $s\in[0,T]$. ∎

A simple and readily verified induction argument shows:

Lemma 15**.**

Let $m\in\mathbb{N}$ and $A_{1},\ldots,A_{m}$, $B_{1},\ldots,B_{m}:D\to D$ be arbitrary maps acting on some common domain $D$ be given. Then

[TABLE]

Here and henceforth, the order of the “product” $\prod_{j=1}^{m}\,A_{j}$ shall be fixed by $A_{1}\circ\cdots\circ A_{m}$.

Proof of Theorem 1.

“$\subseteq$”: Obviously, $\rho_{0}\in\{\rho\in\mathbb{D}(\mathcal{H})\,|\,\rho\prec\rho_{0}\}$ and by assumption of $V$ being normal, $\mathbf{\Gamma}(\mathbf{1})=-VV^{\dagger}+\frac{1}{2}(V^{\dagger}V+V^{\dagger}V)=0$. Thus the operator solution of (2) remains in $\mathbb{S}(\mathcal{H})$ for $t\geq 0$, and by Lemma 1 the set $\{\rho\in\mathbb{D}(\mathcal{H})\,|\,\rho\prec\rho_{0}\}$ is forward invariant, i.e. solutions of the given control problem can never leave the set of states majorized by $\rho_{0}$. Taking the $\|\cdot\|_{1}$-closure by Lemma 5 yields

[TABLE]

“$\supseteq$”: As $V\in\mathcal{K}(\mathcal{H})$ is normal, by Lemma 8 there exists an orthonormal basis $(f_{j})_{j\in\mathbb{N}}$ of $\mathcal{H}$ such that $V=\sum_{j=1}^{\infty}v_{j}\langle f_{j},\cdot\rangle f_{j}$ with modified eigenvalue sequence $(v_{j})_{j\in\mathbb{N}}$. Whenever we use the term “diagonal” or “diag” in the following, it always refers to $(f_{j})_{j\in\mathbb{N}}$.

Let $\varepsilon>0$ and $\rho\in\mathbb{D}(\mathcal{H})$ with $\rho\prec\rho_{0}$ be given. We now have to find $\rho_{F}\in\mathfrak{reach}_{\Sigma_{V}}(\rho_{0})$ such that $\|\rho-\rho_{F}\|_{1}<\varepsilon$. As seen before there exist $x,y\in\ell^{1}_{+}(\mathbb{N})$, $x,y\neq 0$ as well as unitary $W_{1},W_{2}\in\mathcal{B}(\mathcal{H})$ such that

[TABLE]

with $x\prec y$ (so $x$ and $y$ denote the modified eigenvalue sequence of $\rho$ and $\rho_{0}$, respectively, see also Remark 1).

First assume that the original and the modified eigenvalue sequence of $\rho$ as well as $\rho_{0}$ coïncide, i.e. $x=x^{\downarrow}$, $y=y^{\downarrow}$ from the start (necessary to apply Prop. 1). The subsequent steps of the proof were sketched in the main text on page 10, where Step 1 & 3 are the mere application of a suitable unitary channel whereas Step 2 is about (approximately) getting rid of all “off-diagonal” elements $\langle f_{j},UW_{2}^{\dagger}\rho_{0}W_{2}U^{\dagger}f_{k}\rangle$ of $UW_{2}^{\dagger}\rho_{0}W_{2}U^{\dagger}=U\operatorname{diag}(y)U^{\dagger}$.

Step 1: By assumption $\Sigma_{0}$ is strongly operator controllable so we have the unitary orbit of $\rho_{0}$ in the closure of $\mathfrak{reach}_{\Sigma_{V}}(\rho_{0})$. Although we may not have access to $\operatorname{Ad}_{UW_{2}^{\dagger}}=UW_{2}^{\dagger}(\cdot)W_{2}U^{\dagger}$ directly, by Lemma 6 we find $\tilde{U}\in\mathcal{B}(\mathcal{H})$ unitary such that $\tilde{U}\rho_{0}\tilde{U}^{\dagger}\in\mathfrak{reach}_{\Sigma_{V}}(\rho_{0})$ with

[TABLE]

Step 2: By Lemma 4 the pure noise generator $\mathbf{\Gamma}_{V}$ acts like

[TABLE]

on arbitrary $X\in\mathcal{B}(\mathcal{H})$ for all $j,k\in\mathbb{N}$ and $t\in\mathbb{R}_{0}^{+}$. Evidently,

[TABLE]

If we assume $v_{j}\neq v_{k}$ for all $j\neq k$, all the off-diagonal terms of $X$ vanish in the limit $t\to\infty$ and one is left with $\sum_{j=1}^{\infty}\langle f_{j},Xf_{j}\rangle\langle f_{j},\cdot\rangle f_{j}=:P(X)$. Note that this projection map has Kraus operators $(\langle f_{j},\cdot\rangle f_{j})_{j=1}^{\infty}$ so $P\in\mathbb{S}(\mathcal{H})$. Since we want to approximate a density operator in the trace norm, we only have to care about a sufficiently large upper left block of the matrix representation $(\langle f_{j},Xf_{k}\rangle)_{j,k\in\mathbb{N}}$ as the rest is “already small” in the trace norm. More formally, by Lemma 9 there exists $N_{1}\in\mathbb{N}$ such that

[TABLE]

for all $n\geq N_{1}$, where $\Pi_{n}:=\sum_{j=1}^{n}\langle f_{j},\cdot\rangle f_{j}$ for all $n\in\mathbb{N}$.

Of course, there is no reason for the eigenvalues of $V$ to be pairwise different. Therefore we have to make sure that the upper left block is large enough such that it corresponds to at least two different eigenvalues of $V$—thus we have access to partial decoherence, which we then may spread anywhere needed via permutation channels.

Due to $V\neq 0$ and $v_{j}\to 0$ as $j\to\infty$ (compactness of $V$), there exists $M\in\mathbb{N}$ such that $v_{1}\neq v_{M}$. On the other hand (19) still holds so we define $N:=\max\{N_{1},M\}$. Then, by construction and (18), we know that $\langle f_{1},Xf_{M}\rangle$ (and $\langle f_{M},Xf_{1}\rangle$) tend to zero when pure noise is applied.

Thus we find $\alpha\in\mathbb{N}_{0}$, $\alpha\leq N(N-1)/2$ (number of matrix elements above the diagonal), permutation operators $\sigma_{1},\ldots,\sigma_{\alpha}\in\mathcal{U}(\mathcal{H})$ (in abuse of notation we write $f_{j}\mapsto\sigma_{l}f_{j}=f_{\sigma^{-1}_{l}(j)}$, yet the explicit form of $\sigma_{j}$ is not that important) and relaxation times $s_{1},\ldots,s_{\alpha}\in\mathbb{R}_{0}^{+}$ such that

• •

the permutations only operate non-trivially on the $N\times N$-block, i.e. for all $l=1,\ldots,\alpha$ and $k>N$ one has $\sigma_{l}f_{k}=f_{k}$.

• •

for every matrix element $\langle f_{k},\cdot\rangle f_{j}$ with $j,k=1,\ldots,N$, $j\neq k$ there exists a permutation $\sigma_{l}$ with $1\leq l\leq\alpha$ such that $\langle f_{k},\cdot\rangle f_{j}$ sits in the “relaxation” spot (i.e. $\langle f_{1},\cdot\rangle f_{M}$ or $\langle f_{M},\cdot\rangle f_{1}$). More precisely,

[TABLE]

• •

after having successively applied all operations from (20), every matrix element $\langle f_{k},\cdot\rangle f_{j}$ is in its original spot because all $\langle f_{k},\cdot\rangle f_{j}$ are eigenvectors of $\exp(-s_{l}\mathbf{\Gamma}_{V})$.

Now, using linearity of the involved maps, the estimate in question reads

[TABLE]

The first summand is smaller than $2\cdot\frac{\varepsilon}{24}=\frac{\varepsilon}{12}$ by Lemma 10 and (19). For the second one notice that

[TABLE]

for all $j\in\mathbb{N}$. Now, $P(\langle f_{k},\cdot\rangle f_{j})=0$ whenever $j\neq k$. Moreover,

[TABLE]

by (18) and (20). Putting together gives the estimate

[TABLE]

This leaves us with two problems:

We have to approximate all permutation channels.

• 2.

We do not have access to pure noise $(\exp(-t\mathbf{\Gamma}_{V}))_{t\in\mathbb{R}_{0}^{+}}$ within the given control problem.

For solving the first problem we exploit that we can strongly approximate every unitary channel. First, to simplify the upcoming computations, let us assume w.l.o.g. that $\sigma_{\alpha}$ is the identity and let us introduce the notation $\pi_{l}:=\sigma_{l}\circ\sigma_{l-1}^{\dagger}$ for $l\in\{2,\ldots,\alpha\}$ and $\pi_{1}:=\sigma_{1}$. Moreover, define

[TABLE]

for every $l\in\{1,\ldots,\alpha\}$ as then by Lemma 6 we find $\tilde{\pi}_{l}\in\mathcal{U}(\mathcal{H})$ which we have access to within $\mathfrak{reach}_{\Sigma_{0}}$ (and thus $\mathfrak{reach}_{\Sigma_{V}}$) such that

[TABLE]

Then a telescope argument (cf. Lemma 15) yields the estimate

[TABLE]

where in the last step we once again used Lemma 10.

For the second problem we luckily may approximate the pure noise as precisely as needed using Lemma 14. For every $l=1,\ldots,\alpha$ define

[TABLE]

Then by Lemma 14 there exists a cptp map $F_{l}$ which we have access to such that $\|\exp(-s_{l}\Gamma_{V})(\rho_{l})-F_{l}(\rho_{l})\|_{1}<\frac{\varepsilon}{12\alpha}$. Just as before

[TABLE]

Step 3: The current state \tilde{\rho}:=\prod_{m=1}^{\alpha}\big{(}\operatorname{Ad}_{\tilde{\pi}_{m}^{\dagger}}{}\!\circ\,F_{m}\big{)}(\tilde{U}\rho_{0}\tilde{U}^{\dagger}) of the system is “close to $\operatorname{diag}(x)$” in the trace distance as we saw before. Now we want to apply the unitary channel generated by $W_{1}$ so again by Lemma 6 one finds unitary $\tilde{W}\in\mathcal{B}(\mathcal{H})$ such that $\|W_{1}\tilde{\rho}W_{1}^{\dagger}-\tilde{W}\tilde{\rho}\tilde{W}^{\dagger}\|_{1}<\frac{\varepsilon}{3}\,.$ Then one has \rho_{F}=\operatorname{Ad}_{\tilde{W}}\circ\prod_{m=1}^{\alpha}\big{(}\operatorname{Ad}_{\tilde{\pi}_{m}^{\dagger}}{}\!\circ\,F_{m}\big{)}(\tilde{U}\rho_{0}\tilde{U}^{\dagger})\in\mathfrak{reach}_{\Sigma_{V}}(\rho_{0}) and by (17)

[TABLE]

As all channels involved are in $\mathbb{S}(\mathcal{H})$, by Lemma 10 we ultimately obtain

[TABLE]

Now what happens if we cannot apply Prop. 1 directly, i.e. if the original and the modified eigenvalue sequence of $\rho=W_{1}\operatorname{diag}(x)W_{1}^{\dagger}$ or $\rho_{0}=W_{2}\operatorname{diag}(y)W_{2}^{\dagger}$ do not coïncide? Given $\varepsilon>0$, we first of all find $N\in\mathbb{N}$ such that

[TABLE]

Take unitaries $X$, $Y\in\mathcal{B}(\mathcal{H})$ so that $X\rho X^{\dagger}=\operatorname{diag}(x_{1}^{\downarrow},\ldots,x_{N}^{\downarrow},*,*,\ldots)$ and $Y\rho_{0}Y^{\dagger}=\operatorname{diag}(y_{1}^{\downarrow},\ldots,y_{N}^{\downarrow},*,*,\ldots)$ where the diagonal entries differ from the original only by a permutation on a finite block. As the tail of these new diagonals is “already small” we may change these elements within the realm of approximation. Given $\sum_{j=1}^{N}x_{j}^{\downarrow}\leq\sum_{j=1}^{N}y_{j}^{\downarrow}$ (because $\rho\prec\rho_{0}$) where this inequality may or may not be strict, we want to fill up $X\rho X^{\dagger}$ with small entries such that the traces match. Define $\varphi:=\sum_{j=1}^{N}(y_{j}^{\downarrow}-x_{j}^{\downarrow})$ where $0\leq\varphi<\frac{\varepsilon}{12}$ due to (22) and $\rho\geq 0$, as well as $m:=\lceil\varphi/x_{k}^{\downarrow}\rceil\in\mathbb{N}$. Here $k\in\{1,\ldots,N\}$ is chosen such that $x_{k}^{\downarrow}$ is the smallest non-zero entry of $(x_{1}^{\downarrow},\ldots,x_{N}^{\downarrow})$. The new (eigenvalue) sequences then are $\hat{x}:=(x_{1}^{\downarrow},\ldots,x_{k}^{\downarrow},\tfrac{\varphi}{m},\ldots,\tfrac{\varphi}{m},0,0,\ldots)$ (where $\varphi/m$ occurs $m$ times) and $\hat{y}:=(y_{1}^{\downarrow},\ldots,y_{N}^{\downarrow},0,0,\ldots)\,.$ These sequences satisfy $\hat{x}^{\downarrow}=\hat{x}$, $\hat{y}^{\downarrow}=\hat{y}$ and $\hat{x}\prec\hat{y}$ (for this note that if $k<N$ then majorization forces $\sum_{j=1}^{k}x_{j}^{\downarrow}=\sum_{j=1}^{k}y_{j}^{\downarrow}=1$ and thus $\varphi=0$) so we could apply Prop. 1 to them. Now to

[TABLE]

which are both in $\mathbb{D}(\mathcal{H})$, we can apply the original scheme which yields a cptp map $f$ on $\mathcal{H}$ such that $f(\omega_{0})=\omega_{F}\in\mathfrak{reach}_{\Sigma_{V}}(\omega_{0})$ and $\|\omega-\omega_{F}\|_{1}<\frac{\varepsilon}{6}$. Of course linearity implies $\|\operatorname{diag}(\hat{x})-f(\operatorname{diag}(\hat{y}))\|_{1}<\frac{\varepsilon}{6}$. The final scheme goes as follows:

[TABLE]

More precisely, by Lemma 6 we find unitaries $\tilde{X},\tilde{Y}\in\mathcal{B}(\mathcal{H})$ such that

[TABLE]

and $\rho_{F}:=(\operatorname{Ad}_{\tilde{X}^{\dagger}}\circ f\circ\operatorname{Ad}_{\tilde{Y}})(\rho_{0})\in\mathfrak{reach}_{\Sigma_{V}}(\rho_{0})$. Putting things together,

[TABLE]

so $\rho\in\overline{\mathfrak{reach}_{\Sigma_{V}}(\rho_{0})}^{1}$, which concludes the proof. ∎

Appendix D: Theorem 1 for Unbounded Drift $H_{0}$

The physically relevant case of an unbounded system Hamiltonian $H_{0}$ is a bit more intricate. To show that Eq. (2) is well-defined in this general setting, we have to resort to some basic results from the theory of strongly continuous one-parameter semigroups as presented in [12, Ch. 1.9 & Ch. 5.5].

To begin with, for selfadjoint $H_{0},\ldots,H_{m}\in\mathcal{B}(\mathcal{H})$ and $u_{1},\dots,u_{m}\in\mathbb{R}$ and setting $H:=H_{0}+\sum_{j=0}^{m}u_{j}H_{j}$ and $\mathbf{H}:=\mathrm{ad}_{H}$, the solution of

[TABLE]

is obviously given by applying the corresponding unitary channel

[TABLE]

for all $t\in\mathbb{R}_{+}$ (even for all $t\in\mathbb{R}$). The control case $H(t)$ with piecewise constant control amplitudes $u_{j}(t)$ is solved by compositions of such solutions.

Now let us assume that $H_{0}$ is unbounded and defined on some dense domain $D(H_{0})\subset\mathcal{H}$. Then $H:=H_{0}+\sum_{j=0}^{m}u_{j}H_{j}$ is selfadjoint with dense domain $D(H)=D(H_{0})$ and Stone’s Theorem implies that $-iH$ is the infinitesimal generator of a strongly continuous group $U(t)=e^{-itH}$ of unitary operators. The corresponding one-parameter group $\mathbf{U}(t):=\operatorname{Ad}_{U(t)}:=e^{-itH}(\cdot)\,e^{itH}$ of unitary channels (isometries !) is strongly continuous on the trace class $\mathcal{B}^{1}(\mathcal{H})$ and hence it is generated via the densely defined, closed operator $-i\mathbf{H}$. More precisely, one has the following result4c4c4cNB: Davies [12] proved the sequel on the Banach space of all selfadjoint trace-class operators. As selfadjointness is neither used nor necessary, we extend the results to $\mathcal{B}^{1}(\mathcal{H})$. which allows us to identify $-i\mathbf{H}$ with $=-i\mathrm{ad}_{H}$.

Lemma 16** ([12], Ch. 5, Lemma 5.1).**

The domain $D(\mathbf{H})$ of $\mathbf{H}$ is the set of all $\rho\in\mathcal{B}^{1}(\mathcal{H})$ such that $\rho(D(H))\subset D(H)$ and such that the operator $H\rho-\rho H$ on $D(H)$ is norm bounded with an extension to a trace class operator on $\mathcal{H}$. Moreover, one has the identity $\mathbf{H}(\rho)=H\rho-\rho H=:\mathrm{ad}_{H}(\rho)$.

In the sequel, the explicit form of $D(\mathbf{H})$ is irrelevant as is the explicit construction of $\mathbf{U}_{t}$, e.g., via the Post-Widder inversion formula

[TABLE]

for all $t\geq 0$ (even for all $t\in\mathbb{R}$) and $\rho\in\mathcal{B}^{1}(\mathcal{H})$. Recall that the ODE (23) has a classical solution only for a dense set of initial values, namely for $\rho_{0}\in D(\mathbf{H})\subset\mathcal{B}^{1}(\mathcal{H})$. Nevertheless, we will write $U(t)=e^{-it\mathbf{H}}$ although this only holds formally.

Lemma 17**.**

Let any $V\in\mathcal{B}(\mathcal{H})$. The operator $(-i\mathbf{H}-\mathbf{\Gamma}_{V})$ on $\mathcal{B}^{1}(\mathcal{H})$ with domain equal to that of $D(\mathbf{H})$ is the generator of a strongly continuous, positive, trace-preserving semigroup on $\mathcal{B}^{1}(\mathcal{H})$, formally denoted by $(e^{-it\mathbf{H}-t\mathbf{\Gamma}_{V}})_{t\in\mathbb{R}_{+}}$.

Proof.

Apply [12, Thm. 5.2] with $\mathscr{J}:=V(\cdot)V^{\dagger}$ so $\mathscr{J}^{*}(\mathbf{1})=V^{\dagger}V$. ∎

The corresponding semigroup is completely positive by its gksl form, yet positivity and trace-preservation would suffice for what follows.

Thus even if $H_{0}$ is unbounded, for every choice of constant controls Eq. (2) gives rise to a strongly continuous one-parameter semigroup $(e^{-it\mathbf{H}-t\mathbf{\Gamma}_{V}})_{t\in\mathbb{R}_{+}}$ and therefore we obtain a well-defined reachable sets $\mathfrak{reach}_{\Sigma_{V}}(\rho_{0})$ in the sense of Eq. (5) for all initial values $\rho_{0}\in\mathbb{D}(\mathcal{H})$. The fact, that the ODE (2) allows classical solutions only on a dense domain of initial values, may be neglect when specifying $\mathfrak{reach}_{\Sigma_{V}}(\rho_{0})$.

With the stage being set, we only need the Trotter product formula for contraction semigroups on Banach spaces before we can highlight how the proof of Theorem 1 changes to the new frame.

Lemma 18** ([37], Thm. X.51).**

Let $\mathbf{A}_{1}$ and $\mathbf{A}_{2}$ be generators of contraction semigroups on $\mathcal{B}^{1}(\mathcal{H})$, i.e. strongly continuous semigroups of operator norm less or equal one for all $t\in\mathbb{R}_{+}$. Suppose that the closure of $(\mathbf{A}_{1}+\mathbf{A}_{2})$ generates a contraction semigroup on $\mathcal{B}^{1}(\mathcal{H})$ and denote by $\overline{(\mathbf{A}_{1}+\mathbf{A}_{2})}$ its closure. Then for all $\rho\in\mathcal{B}^{1}(\mathcal{H})$ and all (fixed) $t\geq 0$

[TABLE]

With all these ingredients we are prepared to

Generalizing the proof of Thm. 1 to unbounded $H_{0}$.

“$\subseteq$”: As $V$ is assumed to be normal, one has $\mathbf{\Gamma}_{V}(\mathbf{1})=0$ and so the corresponding one-parameter semigroup is in $\mathbb{S}(\mathcal{H})$, i.e. it consists of bi-stochastic quantum maps. To see that $e^{-it\mathbf{H}-t\mathbf{\Gamma}_{V}}(\rho)\prec\rho$ for all $\rho\in\mathbb{D}(\mathcal{H})$ and $t\in\mathbb{R}_{+}$ we note

• •

$e^{-it\mathbf{H}}(\rho)\prec\rho$ as unitary channels do not change the eigenvalues. Thus, majorization cannot increase if the noise $\mathbf{\Gamma}_{V}$ is switched off.

• •

$e^{-t\mathbf{\Gamma}_{V}}(\rho)\prec\rho$ by Lemma 1.

Therefore and since $\prec$ is a preorder (so in particular transitive), one has

[TABLE]

for all $n\in\mathbb{N}_{0}$. Now apply Lemma 18 to conclude that $(e^{-it\mathbf{H}/n}e^{-t\mathbf{\Gamma}_{V}/n})^{n}(\rho)$ converges to $e^{-it\mathbf{H}-t\mathbf{\Gamma}_{V}}(\rho)$ in trace norm for all $t\in\mathbb{R}_{+}$ and $n\to\infty$, Then, by Eq. (24) in combination with Lemma 5 (the set of majorized states is trace-norm closed), we conclude $e^{-it\mathbf{H}-t\mathbf{\Gamma}_{V}}(\rho)\prec\rho$.

We saw earlier that $(e^{-it\mathbf{H}})_{t\in\mathbb{R}_{+}}$ is a contractive semigroup. The same holds for $(e^{-t\mathbf{\Gamma}_{V}})_{t\in\mathbb{R}_{+}}$ by Lemma 10 as well as $(e^{-it\mathbf{H}-t\mathbf{\Gamma}_{V}})_{t\in\mathbb{R}_{+}}$ by Lemma 17 & [44, Prop. 2] 4d4d4dThe proof only uses positivity and trace-preservation, so we apply the respective result.. The respective generators are all densely defined (on at least $D(\mathbf{H})$) and closed, so Lemma 18 yields

[TABLE]

for all $\rho\in\mathcal{B}^{1}(\mathcal{H})$, which shows the inclusion in question.

“$\supseteq$”: Generalizing this inclusion to unbounded $H_{0}$ is easier as we only have to modify Lemma 7. By the same line of reasoning, Lemma 18 gives

[TABLE]

for all $\rho\in\mathbb{D}(\mathcal{H})$ and $t\geq 0$. With this, the original proof of “$\supseteq$” holds without further changes—since we have never used the Trotter product formula explicitly in the uniform or norm topology, but only in the strong topology, i.e. when applied to some density or trace-class operator, see also Lemma 14.

Thereby all instances of Theorem 1 are finally proven. ∎