Paolo Forni
email: paolo.forni@inria.fr

Timothée Launay
email: timothee.launay@mines-paristech.fr

Alain Sarlette

Pierre Rouchon
email: pierre.rouchon@mines-paristech.fr.

A palette of approaches for adiabatic elimination in bipartite open quantum systems with Hamiltonian dynamics on target

Adiabatic elimination is a model reduction technique commonly used by physicists to eliminate quickly dissipating components from quantum physics equations. We revisit this technique when the target non-dissipating component is driven by Hamiltonian actuation at a fast timescale. Following center manifold theory, we can still write reduced dynamics for the target component, but there may be new conditions to ensure that it takes the standard structure of quantum dynamics, i.e. evolution equations of Lindblad type and coordinate changes in Kraus map form. We here propose various approaches to recover a Lindblad form up to third-order terms: without conditions for finite-dimensional systems, and under some finite set conditions for infinite-dimensional systems.

I. INTRODUCTION

The Lindblad master equation is the quantum equivalent of dissipative systems, where in principle Hamiltonian interactions with the large outside world are summarized by dissipation operators (Born-Markov approximation, [START_REF] Breuer | The Theory of Open Quantum Systems[END_REF]). When engineering quantum systems, dissipative components are interconnected with Hamiltonian ones and the behavior of interest is the one after the fast dissipative transient has stabilized. We study the basic building block consisting of a target system weakly coupled with a strongly dissipative ancillary system. By exploiting a time-scale separation between the fast uncoupled dynamics and the weak interaction dynamics, the full system can be decoupled with high accuracy into a fast decaying component, approximately corresponding to the ancillary system, and a remaining component of lower dimension corresponding closely to the quantum system of interest. Physicists have routinely considered the dominant term of this approximation as "adiabatic elimination" of the fast dynamics.

This picture can be made more precise in a system theoretic context on the basis of geometric singular perturbation theory and center manifold theory, as introduced in [START_REF] Azouit | Towards generic adiabatic elimination for bipartite open quantum systems[END_REF], [START_REF] Azouit | Adiabatic elimination for open quantum systems with effective lindblad master equations[END_REF]. From a mathematical viewpoint an asymptotic series expansion at higher orders is clear since the Lindblad equation is linear -we are thus approximating the slowly decaying eigenvalues and eigenspaces of a linear system. However, to preserve an intrinsic physical meaning for the reduced system, the elimination should provide again a Lindblad form Zwijnaarde, Belgium. alain.sarlette@inria.fr.

This work has been supported by the ANR-JCJC project HAMROQS in the reduced equations; and to easily view it as a component of the full world, the coordinate change mapping the reduced model into the complete one should be a completely positive trace preserving map, or Kraus map.

In [START_REF] Azouit | Towards generic adiabatic elimination for bipartite open quantum systems[END_REF], [START_REF] Azouit | Adiabatic elimination for open quantum systems with effective lindblad master equations[END_REF] general formulas are given that solve the series expansion, with Lindblad and Kraus form, up to second order included, when the center manifold ( target quantum system) corresponds to a set of steady states at zero order. However, quantum technology definitely aims to act on the target systems, possibly at fast time scales. This fast Hamiltonian action makes the center manifold correspond to the eigenspace of imaginary eigenvalues at zero order, and changes the computations towards recovering Lindblad form reduced dynamics. A first generalization in [START_REF] Forni | Adiabatic elimination for multi-partite open quantum systems with non-trivial zero-order dynamics[END_REF] takes into account fast Hamiltonian dynamics on target, under a precise condition linking the actuation and interaction Hamiltonians.

In the present paper we remove this condition: the actuation Hamiltonian, as well as the interaction Hamiltonian, are arbitrary. We study how the actuation influences the reduced model dynamics, and find an appropriate Kraus map.

The paper is organized as follows. Section II describes the abstract problem setting and Section III provides the adiabatic elimination formulas that can be used in presence of fast actuation. Section IV completes the presentation with two applications in ongoing quantum experiments. The technical proofs are given in appendix.

II. SETTING A. Bipartite system

Open quantum systems are typically described by differential equations on the manifold of density operators ρ ∈ D(H), namely the set of linear Hermitian nonnegative operators of trace one on a Hilbert space H. We consider the composite Hilbert space H = H A ⊗ H B , where H A is the auxiliary space and H B is the target space. The evolution on H is then described by the Lindblad master equation [START_REF] Breuer | The Theory of Open Quantum Systems[END_REF]:

dρ dt = L A (ρ) + εL int (ρ) -i[H B , ρ], (1)
where is a small parameter emphasizing the time-scale separation; H B is a Hermitian operator acting on H B only; L A is a Lindbladian super-operator acting on H A only:

L A (ρ) := -i [H A , ρ] + µ D[L A,µ ](ρ),
with L A,µ a decoherence operator on H A and D[.](.) the decoherence super-operator:

D[X](ρ) := XρX † - 1 2 X † Xρ + ρX † X ;
L int is a Lindbladian super-operator that captures the interaction between H A and H B , and here we assume that this interaction is Hamiltonian:

L int (ρ) = -i k A k ⊗ B † k , ρ . (2)
where A k and B k respectively are non-necessarily-Hermitian operators acting on H A and H B only. We assume that L A is strongly dissipative and that there exists a unique density operator ρA such that L A (ρ A ) = 0. For ε = 0, the system is uncoupled and the solutions stay separable into H A and H B at all times.

B. Asymptotic expansion

Adiabatic elimination of the strongly dissipative component L A can be approached via center manifold theory by observing that, for ε = 0, there exists an asymptotically stable center manifold M 0 of same dimension as H B . Therefore, by virtue of Fenichel's Invariant Manifold Theorem [START_REF] Fenichel | Geometric singular perturbation theory for ordinary differential equations[END_REF], the existence of an asymptotically stable center manifold M ε is persistent for all small enough ε = 0. An approximation of M ε can be computed up to arbitrary precision (see Carr [START_REF] Carr | Applications of Centre Manifold Theory[END_REF]). The fact that (1) is linear in ρ further facilitates the obtention of guarantees and bounds.

The quantum particularity to adiabatic elimination is to preserve the structure of the reduced-order model, i.e. we aim to find:

• a Kraus 1 map K ε from ρ s ∈ D(H s ) the reduced space with dim[H s ] = dim[H B ] to the complete space ρ ∈ D(H), such that K(D[H s ]
) yields M ε . Such map ensures that the reduced-order model ρ s still corresponds to density operators in the full physical space, thus ρ s retains physical meaning as directly describing measurement probabilities; • the reduced dynamics on H s in Lindblad form, i.e. d dt ρ s = L s,ε (ρ s ) for some Lindbladian superoperator L s,ε . Invariance of M ε is then captured by the equation:

LA(Kε(ρs)) + εLint(Kε(ρs)) -i[HB, Kε(ρs))] = Kε(Ls,ε(ρs)). (3)
Carr's result [START_REF] Carr | Applications of Centre Manifold Theory[END_REF] considers K , L s, as infinite series in :

K (ρ s ) = +∞ =0 K (ρ s ), L s, (ρ s ) = +∞ =0 L s, (ρ s ) . ( 4
)
By plugging (4) into (3) and identifying the terms of same order, an invariance relation is obtained at all orders. At zeroorder, we have:

L A (K 0 (ρ s )) -i[H B , K 0 (ρ s )] = K 0 (L s,0 (ρ s )). (5)
At first-order invariance, we get:

L A (K 1 (ρ s )) + L int (K 0 (ρ s )) -i[H B , K 1 (ρ s )] = K 0 (L s,1 (ρ s )) + K 1 (L s,0 (ρ s )) , (6)
1 A Kraus map takes the form K(ρs) = l M l ρsM † l for some operators M l , to ensure complete positivity [START_REF] Choi | Completely positive linear maps on complex matrices[END_REF], with l M † l M l =identity for trace preservation. and at second-order:

L A (K 2 (ρ s )) + L int (K 1 (ρ s )) -i[H B , K 2 (ρ s )] = K 0 (L s,2 (ρ s )) + K 1 (L s,1 (ρ s )) + K 2 (L s,0 (ρ s )) . (7)

III. REDUCED MODEL FORMULAS

In this section we provide solutions for the zero-, firstand second-order invariance equations, and consequently formulas for the reduced-order model of (1) up to thirdorder terms, in quantum form. The presence of H B will be resolved in two possible ways. A first solution is proposed for systems where the spectrum of H B is finite, thus in particular for any finite-dimensional systems. A second solution is proposed, possibly for infinite-dimensional systems, if some commutation relations end after a finite number of steps.

A. Case of finite spectrum of H B

In this section we make the following assumption. Assumption 1: The spectrum of H B consists of a finite set. Assumption 1, together with hermiticity, implies the existence of an orthonormal basis of eigenvectors {x i } i∈I of H B and its associated set of real eigenvalues {λ i } i∈I , for some index set I. It is then straightforward to show that the family of operators {x i x † j } i,j∈I represents an orthonormal basis for the Liouville space

End[H B ]. Moreover, each C † (i,j) := x i x † j is an eigenoperator of [H B , .
] with associated eigenvalue c (i,j) := λ i -λ j , in other words:

[H B , C † (i,j) ] = c (i,j) C † (i,j) , C (i,j) C † (i ,j ) = δ i,i C † (j,j )
. where δ h,j denotes the Kronecker delta. For brevity, we will equivalently index the sets over one index:

C † (i,j) = C † k and c (i,j) = c k .
Without loss of generality, we can still consider a Hamiltonian interaction in form (2) with the operators B k now being (decomposed on the basis of) the eigenoperators of [H B , •] (, with possible weights factored into the A k ). The zero-order ( 5) is immediately solved by setting:

L s,0 (ρ s ) = -i[H B , ρ s ], K 0 (ρ s ) = ρA ⊗ ρ s . (8)
At first order, let the Kraus map have the following structure:

K(ρs) =K0(ρs) + K1(ρs) + O( 2 ) (9)
:=(I -i k M k )(ρA ⊗ ρs)(I -i k M k ) † + O( 2 ),
where the M k are operators to be defined in Theorem 1.

Theorem 1: Consider model ( 1)-(2). Let Assumption 1 hold and, without loss of generality, B k = C k . Let H 1 be an arbitrary Hermitian operator in End[H B ], and (f k ) its coordinates in the basis {C † k }. Then the first order invariance equation ( 6) is satisfied by the Lindbladian:

L s,1 (ρ s ) = -i k Tr(A k ρa )B † k -i[H B , H 1 ], ρ s
and by a map of the form [START_REF] Zk Minev | To catch and reverse a quantum jump mid-flight[END_REF] where

M k := F k ⊗ B † k and F k is the unique solution of: L A (F k ρa ) + S (-ic k F k ρa + A k ρa ) = 0 , (10)
Tr(F k ρa ) = f k ,
where S(X) := X -Tr[X]ρ A . Proof: See Appendix A. To provide formulas for the second-order adiabatic elimination of model ( 1), we will build on a decomposition similar to [START_REF] Azouit | Adiabatic elimination for open quantum systems with effective lindblad master equations[END_REF]. Let X, Y , Z be the matrices whose entries are defined by (here denotes complex conjugate):

x k ,k :=z k ,k + z k,k (11a)
y k ,k := 1 2i (z k ,k -z k,k ) (11b) z k ,k := Tr(P k ,k ρa ) - i 2 (c k -c k ) Tr(F k ρa F † k ) (11c) P k ,k =A † k F k -(Tr(A k ρa ) -ic k f k )Fk (11d) k :=(j, i) when k = (i, j) , (11e)
and the F k are solutions to [START_REF] Mirrahimi | Dynamically protected cat-qubits: a new paradigm for universal quantum computation[END_REF]. It is possible to show along the lines of [3, Lemma 7] that:

x k ,k = Tr µ [L A,µ , F k ]ρ A [L A,µ , F k ] † , (12)
and thus X 0 which implies that there exists

Λ ∈ C K×K such that X = ΛΛ † , i.e. x k ,k = p Λ k ,p (Λ k,p ) .
Theorem 2: Consider model ( 1)-( 2). Let Assumption 1 hold and, without loss of generality, B k = C k . Select L s,1 and K 1 according to Theorem 1. Let H 2 be an arbitrary Hermitian operator in End[H B ]. Then the second order invariance equation is satisfied by the Lindbladian:

L s,2 (ρ s ) = -i [H s,2 -i[H B , H 2 ] , ρ s ] + p D[L p ](ρ s ), where H s,2 := k ,k y k ,k B k B † k and L p := k Λ k,p B † k .
Furthermore, there exists an associated map K 2 such that K(ρ s ) = K 0 (ρ s ) + K 1 (ρ s ) + 2 K 2 (ρ s ) is CPTP up to thirdorder terms.

Proof: See Appendix B. Remark 1: When solving the invariance equations, the choice of coordinate mapping between H s and H is not unique. This offers various possibilities to be leveraged in jointly determining the final forms of K ε and of the reduced dynamics L s,ε , and similarly in the series expansion. The Hamiltonians H 1 and H 2 in the above theorems explicitly express some possibilities offered by this gauge degree of freedom. We will by default take them zero in the below applications.

B. Case of vanishing commutators

Assumption 1 is often not feasible with infinitedimensional systems, like the quantum harmonic oscillator which is nevertheless a canonical system of interest. In these cases, one typically resorts to operator properties, typically commutation relations, and we propose to do the same for adiabatic elimination. In [START_REF] Forni | Adiabatic elimination for multi-partite open quantum systems with non-trivial zero-order dynamics[END_REF] we have exploited the particular property ∀k ∃c

k : [H B , B † k ] = c k B † k .
In this section, we propose a complementary approach motivated by the example in Section IV-B. Let C h (•, •) denote the repeated commutator:

C h (X 1 , X 2 ) := [X 1 , [X 1 , [. . . , X 2 ]]] h times .
Assumption 2: There exists a finite h ∈ N such that C h (H B , B k ) = 0 for all k ∈ {1, . . . K}. For clarity of presentation, we will focus on the case h = 3. Assumption 2 induces the following definitions:

C k := [H B , B k ] and D k := [H B , C k ],
and for brevity (h)

B k := C h (H B , B k ), e.g. D k = (2) B k .
Theorem 3: Consider model ( 1)-( 2). Let Assumption 2 hold with h = 3. Then, the first-order invariance equation ( 6) is satisfied by the Lindbladian

L s,1 (ρ s ) = -i [H s,1 , ρ s ] , (13)
with H s,1 := k Tr(A k ρa )B † k , and by a map K 1 of the form [START_REF] Zk Minev | To catch and reverse a quantum jump mid-flight[END_REF] where

M k := h F h k ⊗ (h) B † k and F h k is the unique solution of: LA(F k ρA) + δ ,0 S(A k ρA) + i(1 -δ ,0 )F -1 k ρA =0 , (14) Tr[F k ρA] =0
for = 0, 1, ..., h. Furthermore, K(ρ s ) is a CPTP map up to second-order terms.

Proof: See Appendix C. Consider now the matrix X ∈ C Kh×Kh defined as:

X k ,k := Tr [L A,µ , F k ]ρ A [L A,µ , F k ] † =: x kk ,
for all k = 1, ..., K and l = 1, ..., h. It is possible to show along the lines of [3, Lemma 7] that X 0 and thus there exists Λ ∈ C Kh×Kh such that X = ΛΛ † , i.e. X k ,k = p Λ k ,p (Λ k ,p ) . Remark 2: By virtue of [3, Lemma 6] and equation ( 14), matrix elements x kk may be also computed as follows: 1)-( 2). Let Assumption 2 hold with h = 3. Then, the second-order invariance equation ( 7) is satisfied by Lindbladian:

x 00 kk = Tr[F 0 k ρA A † k + A k ρA F 0 † k ] x 11 kk =i Tr[F 0 k ρA F 1 † k -F 1 k ρA F 0 † k ] x 22 kk =i Tr[F 1 k ρA F 2 † k -F 2 k ρA F 1 † k ] x 01 kk =i Tr[F 0 k ρA F 0 † k ] + Tr[A k ρA F 1 † k ] x 02 kk =i Tr[F 0 k ρA F 1 † k ] + Tr[A k ρA F 2 † k ] x 12 kk =i Tr F 0 k ρA F 2 † k -F 1 k ρA F 1 † k . Theorem 4: Consider model (
L s (ρ s ) = -i[H B + εH s,1 + ε 2 H s,2 , ρ s ] + ε 2 p D[L p ](ρ s ) + O(ε 3 ), with L p := k Λ k0,p B † k + Λ k1,p C † k + Λ k2,p D † k and Hs,2 := 1 2i k,k , Tr[A † k F k ρA]Bk ( ) B † k -herm. conj. ,
with H s,1 and the F k computed as in Theorem 3. There exists an associated map K 2 , obtained from formula (28), such that K(ρ s ) = (K 0 + εK 1 + εK 2 )(ρ s ) is a CP map up to third-order terms and TP up to second-order terms.

Proof: See Appendix D.

IV. APPLICATIONS

We now present two examples of physical relevance that can be treated with our new formulas.

A. Case of finite spectrum of H B

The following example comes from [START_REF] Zk Minev | To catch and reverse a quantum jump mid-flight[END_REF], where a drive is applied to a so-called V three-level system and its transitions are monitored indirectly by dispersive interaction with a strongly dissipating harmonic oscillator ("cavity" or "resonator").

Let H A and H B respectively be the Hilbert space of the resonator and three-level system, respectively considered as the auxiliary and target component. The three levels are subsequently denoted as |G ground, |D dark, |B bright. The so-called dark state is decoupled from its environment, and a small drive Ω DG allows the state to transition from |G to |D and back. At the same time, the possible presence on |G is monitored by a strong drive Ω BG on its transition to |B , which is dispersively coupled with strength χ B to a strongly dissipative resonator. In this way the resonator, probed at a frequency detuned by χ B from its resonance, responds resonantly when the system is on the "bright" state. In the appropriate rotating frame and after rotating wave approximation, this setting corresponds to

HB :=   0 ΩDG ΩBG ΩDG 0 0 ΩBG 0 0   , Hint := a † a(|G G| + |D D|)
with the matrix components ordered as |G , |D , |B ; the Lindblad dynamics writes

dρ dt = -i u(a † + a) + HB + χBHint, ρ + κD[a](ρ) (15)
with u the strength of the coherent drive on the resonator mode, κ the relaxation rate of the resonator and a the photon loss operator on the resonator. A well-resolved observation of being on |B B|, corresponds to χ m > κ. In contrast, our formulas allow to check what happens when this is not satisfied, i.e. when χ m κ (in practice a factor of 2 or 5 is sufficient to apply our formulas).

The commutator of (|G G|+|D D|) with H B does not allow us to apply the result of [START_REF] Forni | Adiabatic elimination for multi-partite open quantum systems with non-trivial zero-order dynamics[END_REF], but since H B has finite dimension we can apply the results of Section III-A. We have ρA = |α α| a coherent state, satisfying a |α = α |α , with α = -2iu/κ. This already gives

H s,1 = χ B 4u 2 κ 2 (|G G| + |D D|) -χ B 4u 2
κ 2 |B B| at first order, to be added to

H s,0 = H B .
To get the second order, the decomposition of H B into eigenvectors can be easily done by hand. Furthermore, we can observe that (10) is solved by

F k = αχ B ic k +κ/2 (a -α) † , with f k = 0.
From there straightforward (yet tedious) computations lead to the solution. Alternatively, we can use the formula (12) to directly obtain X = ΛΛ † with Λ a vector. This shows that we have a single decoherence channel, which we can analyze numerically. When taking small drives, our results are similar to those of previous papers, e.g. with Ω DG = 2•10 -5 κ and Ω BG = 2•10 -4 κ we have a dissipation operator

L 1 = 2|α| κ   1 0 0 0 1 0 0 0 0   .
Physically, this resembles the decoherence implied by a QND measurement distinguishing |B from the two other states. When taking large drives, e.g. Ω DG = 0.2κ and Ω BG = 2κ, our formulas are necessary and they indicate a switch in the measurement behavior:

L 1 = 2|α| κ   0.512 -0.011i 0.062i 0.011i 0.986 -0.043 -0.062i -0.043 0.501   .
Now, in agreement with the intuitive measurement principle, we are rather distinguishing whether the state is in |D , or oscillating somewhere between the two other states. In both cases, the measurement principle works according to this model, even if the spectral lines are not well resolved.

B. Case of vanishing commutators

The continuous-variable approach for protecting quantum information encodes a logical qubit (two-level system) in a quantum resonator (infinite number of levels). In particular, taking advantage of this redundancy via a reservoir engineering scheme, so-called cat qubits are an experimentally promising candidate [START_REF] Mirrahimi | Dynamically protected cat-qubits: a new paradigm for universal quantum computation[END_REF]. More precisely, the target resonator is coupled via a two-photon process to an environment resonator such that the overall Lindbladian stabilizes the subspace span(|β , |-β ) where |β denotes the coherent state of amplitude β, satisfying a |β = β |β . Hamiltonians must then be added to apply gates on this encoded information. Our formulas allow to investigate what happens when we push the speed of these gates to be as fast as the protection process.

Let H a and H b respectively be the Hilbert spaces of two resonators that are respectively modeling a highly dissipating component -used to evacuate entropy in the stabilization process -and the target "memory" component. Denote with a and b the annihilation operators in the first and second resonator mode respectively. By setting up an appropriate coupling and a constant drive on H a [10], the coupled system can be engineered to follow:

d dt ρ = κD[a](ρ)-ig[a(b 2 -β 2 ) † +a † (b 2 -β 2 ), ρ]-i[v b+vb † , ρ] (16)
where v is a drive meant to apply gates on the target system. The standard situation corresponds to g, v κ. In this case all the small dynamics acts on H a and we can apply the result in [START_REF] Azouit | Towards generic adiabatic elimination for bipartite open quantum systems[END_REF], yielding the reduced model:

d dt ρ s = 4g 2 κ D[b 2 -β 2 ](ρ s ) -i[v b + vb † , ρ s ] . (17)
When v 4g 2 /κ, these dynamics effectively operate a gate [START_REF] Azouit | Convergence and adiabatic elimination for a driven dissipative quantum harmonic oscillator[END_REF] that rotates the interference fringes on the subspace The model satisfies the hypothesis of Theorem 4 since system (16)-( 2) is in the form (1) with, taking units such that κ is of order 1:

L A := κD[a](ρ), ε := g 1, H B := v b+vb † , ρA := |0 0|, A 1 := a, A 2 := a † , B 1 := b 2 -β 2 , B 2 := (b 2 -β 2 ) † , C 1 := -2vb, C 2 := 2v b † , D 1 := 2v
F 0 1 = F 1 1 = F 2 1 = 0, F 0 2 = 2 κ |1 0| , F 1 2 = i 4 κ 2 |1 0| , F 2 2 = - 8 κ 3 |1 0| .
Matrix X is computed via Remark 2 as follows: By Theorem 4 and after regrouping terms, the reduced model is:

Ls(ρs) = -i 1 + 32g 2 |v| 2 κ 4 v b + vb † , ρs -i 4g 2 κ 2 [Hs,2,c, ρs] + 4g 2 κ D[b 2 -β 2 ](ρs) + 32g 2 |v| 2 κ 3 D[b](ρs) + O(g 3 ), (18)
with

H s,2,c = (v(b 2 -β 2 ) † b + v b † (b 2 -β 2 )) + 2i κ (v 2 b † 2 - v 2 b 2
). When v κ, neglecting higher-order terms gives back (17). For v of the same order as κ, we see not only a modification of the Hamiltonian, which is reversible and thus in principle could be countered, but also a significant new decoherence channel in b. Since b |±β = ±β |±β , this contributes to further blurring the interference fringes, as illustrated on Figure 1: model (18) converges to a fixed point without fringes in about 0.36 microseconds, while model (17) erroneously predicts that we would still have significant fringes at this time, and only lose them much later. We thus conclude that pushing with v as strongly as κ destroys the interference fringes very quickly, resulting in information loss.

V. CONCLUSIONS

We have provided formulas for adiabatic elimination of a fast dissipating component connected to a target component, when the target component is subject to maybe slow dissipation but fast Hamiltonian actuation. Formulas are obtained for two cases: when actuation has a finite spectrum, and when it satisfies some iterated commutation relation. We have illustrated those methods on two examples, showing that they capture significant effects which should be taken into account in experiment design.

Fig. 1 .

 1 Fig. 1. Section IV-B: (a-d) Wigner functions of the state of resonator model (17) captured at four different times t = 0 µs, 0.36 µs, 1.08 µs2.16 µs, showing convergence to a fixed point; (e) Wigner function of the state of model (18) captured at t = 0.36 µs, showing that interference fringes have been lost. In both simulations, the initial density operator is: ρs(0) = C + β C + β with C + β := (|β + |-β )/N + where N + is a normalization constant and β = 2; the parameter values are: κ = 1 MHz, g = 200 kHz, v = 500 kHz.

  2 , D 2 := 2v 2 , and thus [H B , D k ] = 0. The associated F operators are computed via (14) as follows:

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APPENDIX

A. Proof of Theorem 1

From (9), the map K 1 would take the form:

Existence and uniqueness of solutions F k is proved along the lines of [START_REF] Azouit | Towards generic adiabatic elimination for bipartite open quantum systems[END_REF]Lemma 4] by observing that L A (•) -ic k Id(•) is still a bijection on traceless operators. For a superoperator

Super-operator V satisfies the property:

By considering the zero-order solution [START_REF] Forni | Adiabatic elimination for multi-partite open quantum systems with non-trivial zero-order dynamics[END_REF] and K 1 from (19), and by making use of formulas [START_REF] Mirrahimi | Dynamically protected cat-qubits: a new paradigm for universal quantum computation[END_REF] and (20), the first-order invariance equation ( 6) simplifies to:

By taking the partial trace over H A over both sides of (21), we obtain the L s,1 in the statement of Theorem 1.

It is immediate to see from ( 9) that K 0 (ρ s ) + εK 1 (ρ s ) is a completely positive map, as long as one can neglect the terms of order ε 2 . It is also straightforward to see from (19) and property f k ∈ R that Tr[K 1 (ρ s )] = 0 for any ρ s , thus K 0 (ρ s ) + εK 1 (ρ s ) is TP up to second-order terms.

B. Proof of Theorem 2

First, we define the map K 2 (ρ s ) and show that satisfies the second-order invariance equation [START_REF] Fenichel | Geometric singular perturbation theory for ordinary differential equations[END_REF]. Let:

where S(X) := X -Tr[X]ρ A and δ is the Kronecker delta.

Using this identity, it is possible to show along the lines of [8, Claim 2] that there exists τ > 0 large enough such that K Q 2 (•) is a completely positive map and the following property holds true :

. Now let H 2 be an arbitrary Hermitian operator in H B and {g k } k ∈ R its coordinates in the basis {B † k } k , and define : u k ,k = u (i ,j ),(i,j) := δ i ,0 δ i,0 g (j ,j) .

(

Scalars u k ,k satisfy the property:

k ρ s where U k ,k is the unique operator satisfying:

Consider the CP map:

We can verify that:

where we used the cycle property of the trace and then the definition of U k ,k and the hermiticity of H 2 . Hence we proved K is trace preserving up to third order terms. Moreover, it is possible to show that, by making use of maps K 0 , K 1 , K 2 as in (25), maps L s,0 , L s,1 as in [START_REF] Azouit | Convergence and adiabatic elimination for a driven dissipative quantum harmonic oscillator[END_REF], and [3, Lemma 6], the second-order invariance equation ( 7) is satisfied with:

By making use of decomposition x k ,k = p Λ k ,p Λ k,p , the previous expression simplifies into:

which readily gives our expression for L s,2 in Theorem 2.

C. Proof of Theorem 3

From the definitions of (h) B k , C k , and D k , we have:

. By plugging the zero-order solution (8) and map K 1 (ρ s ) from ( 9) into the first order invariance equation ( 6), and by making use of (14), we obtain:

By considering Tr[F h k ρA ] = 0 and by taking the partial trace over H A in (27), we immediately obtain L s,1 (ρ s ) as in the Theorem 3 and thus ( 6) is satisfied. By taking the trace of K 1 (ρ s ) from ( 9) we also get:

which proves that K is TP up to second-order terms.

D. Proof of Theorem 4

First, we define the map K 2 (ρ s ) and show that it satisfies the second-order invariance equation [START_REF] Fenichel | Geometric singular perturbation theory for ordinary differential equations[END_REF]. Let:

It is possible to show along the lines of [8, Claim 2] that there exists T > 0 and τ > 0 large enough such that K Q 2 (•) is a completely positive map and the following property holds true:

Now let E k be the possibly non-unique operator such that:

. Define the operator:

where U k,k ,w and V k ,w are the unique operators satisfying:

and where:

Consider the CP map:

. By making use of K 0 , K 1 , K 2 as in (28),L s,0 , L s,1 as in (3), and [3, Lemma 6], the second-order invariance equation [START_REF] Fenichel | Geometric singular perturbation theory for ordinary differential equations[END_REF] reads as:


Paolo Forni

Alain Sarlette

Minimizing decoherence on target in bipartite open quantum systems

We consider a target quantum system, coupled to an auxiliary quantum system which dissipates rapidly at somewhat adjustable rates. The goal is to minimize the dissipation induced on the target system by this coupling. We use explicit model reduction formulas to express this as a quadratic optimization problem. We prove that maybe counterintuitively, when the auxiliary system dissipates along Hermitian (entropy-increasing) channels, the minimum induced dissipation is reached by maximizing the dissipation rate of the auxiliary system.

I. INTRODUCTION

The development of second-generation quantum technologies relies on an accurate control and stabilization of single quantum degrees of freedom, which encode quantum information. Considering such quantum systems as isolated from their environment is an idealized picture. Much of the difficulty in developing quantum processing hardware lies in achieving such controlled isolation in practice, at the very high accuracies necessary for obtaining the effects useful in applications.

Accordingly, one major aspect of building quantum hardware is to design a specific dominant behavior of the target system by smartly engineering some features of its environment. Belonging to this framework are two approaches. First, regarding the unavoidable coupling of a target system to some environment, one may focus on minimizing the perturbations, or decoherence, induced by this coupling on the information stored in the target system. This is the standard effort on isolating a system from perturbations. Second, by viewing part of the environment as a stabilizing controller, one would wish to maximize a particular stabilizing effect induced on the target by its coupling to the controller. This stabilization, usually called "reservoir engineering" [START_REF] Poyatos | Quantum reservoir engineering with laser cooled trapped ions[END_REF], should be understood in the framework of information protection as irreversible operations associated to error-correcting codes [START_REF] Cohen | Autonomous quantum error correction with superconducting qubits[END_REF]. Designing a system, out of any realistically available components, which would optimally process quantum information is the aim of building a viable quantum computer.

In the present paper we address a minimal part of this aim. Namely, we assume that a target system is given, with a fixed unavoidable coupling to a given dissipative environment, but the strength of the dissipation acting on the environment 9052 Zwijnaarde, Belgium. alain.sarlette@inria.fr.

This work has been supported by the ANR project HAMROQS can be adjusted. As a goal, we restrict ourselves to the first case, i.e. minimizing the impact of the environment on the target system. At first sight this may appear as a somewhat trivial situation: if adjustable, would you not just set all dissipations on the environment to their minimal value?

The main motivation for this paper is that the answer to this question is no. Not only can some dissipation channels have a stabilizing effect on the environment, but even when channels encourage mixing in the environment, it can be better for the target system to make them stronger. In rough practical words, if your target system is weakly coupled to an environment that is subject to some unavoidable noise, then you can get a better system behavior by further increasing this noise -something that should not be too hard in practice.

Effects induced by the environment onto a target quantum system can be summarized in approximate reduced models, thanks to the timescale separation between the (weak) coupling of the environment to the target, and the rest of the dynamics. In particular, adiabatic elimination has been routinely used by physicists to eliminate fast dissipating components in composite quantum systems, and our analysis is based on the system-theoretic model reduction formulas of e.g. [START_REF] Azouit | Towards generic adiabatic elimination for bipartite open quantum systems[END_REF]. These assume that dissipation in the environment is much stronger than the coupling, and our conclusions thus remain valid when the dissipation can be adjusted within these limitsin particular, it is perfectly compatible to require a maximal dissipation. Based on these adiabatic elimination expressions, we formulate the objective as a quadratic optimization problem. We then provide several analytic results about the optimal solution. In the case where all decoherence channels are Hermitian, which implies among others that they each tend to maximize entropy in the environment towards the fully mixed state, we prove that induced decoherence on the target is minimized when the strength of each decoherence channel on the environment is maximized. In other cases the situation can be more complicated, with some channels being maximized and others minimized, as we illustrate with simulation results on a standard qubit example.

The conclusion is linked to the fact that the decoherence channels do not act directly on the target system, but only through the dynamics of an environment. Apparently, inducing faster mixing in this environment somehow uncouples the dissipative effect from the target system. Thinking of rotating frames, this may be related to observations of dynamical decoupling [START_REF] Viola | Dynamical decoupling of open quantum systems[END_REF], where a Hamiltonian actuation is set up with the explicit goal to decouple via averaging the target system from its environment.

II. SETTING

A. Bipartite open quantum systems

The object of our study is systems on the composite Hilbert space H :" H A bH B , composed of a target quantum system on H B and its environment on H A . The dynamics on H is assumed to satisfy a time scale separation:

dρ dt " L A pρq `εL int pρq, (1)
where ε is a small positive parameter. L A is a Lindbladian super-operator acting exclusively on H A . It comprises a Hamiltonian H A and K g decoherence channels, whose strengths can be tuned by appropriately selecting some gain vector ph, gq :" rh, g 1 , . . . , g Kg s J P R Kg`1 ě0

; plus K e decoherence channels whose strengths are fixed. Explicitly:

L h,g A pρq " ´irhH A , ρs`K g ÿ k"1 g k DrL k spρq`K e ÿ k"1 e k DrM k spρq (2) with DrD k spρq :" D k ρD : k ´1 2 pD : k D k ρ `ρ D : k D k q
, and the L k , M k are given (not necessarily Hermitian) operators acting on H A . The interaction superoperator L int is here assumed of Hamiltonian form and expressed as:

L int pρq :" ´i « J ÿ j"1 A j b B : j , ρ ff , (3)
where A j and B j are (not necessarily Hermitian) operators acting respectively on H A and on H B only. We will specify in Section III-A that the gain vector ph, gq takes values that always make L A strongly dissipative with respect to the ε timescale. Furthermore, L A is assumed to admit a unique steady state ρA such that L A pρ A q " 0. For ε " 0, the system is uncoupled and the solutions stay separable in H A and H B at all times.

B. Adiabatic elimination formulas

In this section, we recall a system-theoretic approach to adiabatic elimination and associated formulas for the class of systems (1)-(3), see [START_REF] Azouit | Towards generic adiabatic elimination for bipartite open quantum systems[END_REF] for more explanation. For ε " 0, the system features an asymptotically stable center manifold M 0 , actually a subspace, of the same dimension as H B . By Fenichel's Invariant Manifold Theorem [START_REF] Fenichel | Geometric singular perturbation theory for ordinary differential equations[END_REF], for 0 ă ε ! 1 small enough, there persists an asymptotically stable manifold M ε which is ε-close to M 0 and on which the dynamics is slow. Identifying this manifold and the dynamics on it is a systematic approach to eliminating the (uninteresting) fast dynamics of the environment, routinely called adiabatic elimination by physicists.

Let the manifold M ε be the image of a parameter space EndrH s s of dimension dim EndrH B s through a linear map K ε : EndrH s s Ñ EndrHs. The dynamics on M ε would be described by some linear operator L s,ε on EndrHs. Invariance of M ε is then characterized by the following equation:

L A pK ε pρ s qq `εL int pK ε pρ s qq " K ε pL s,ε pρ s qq. (4) Following Carr's approach [START_REF] Carr | Applications of Centre Manifold Theory[END_REF], operators K ε and L s,ε can be parameterized as infinite series K pρ s q " `8 ÿ "0 K pρ s q, L s, pρ s q " `8 ÿ "0 L s, pρ s q. (5) By plugging (5) into (4) and identifying the terms of same order, an invariance relation is obtained at all orders.

To preserve the quantum structure of the reduced model, the authors in [START_REF] Azouit | Towards generic adiabatic elimination for bipartite open quantum systems[END_REF] conjectured that K ε and L s,ε pρ s q respectively take Kraus 1 and Lindblad form at all orders of approximation. In particular, the main result in [START_REF] Azouit | Towards generic adiabatic elimination for bipartite open quantum systems[END_REF] proves that the conjecture is valid up to third-order terms.

We now recall the main result from [START_REF] Azouit | Towards generic adiabatic elimination for bipartite open quantum systems[END_REF] that is adopted in the remainder of this paper. Let F j be a family of operators such that: L A pF j ρA q `Aj ρA ´TrrA j ρA sρ A " 0 , TrrF j ρA s " 0 .

(6) Existence and uniqueness of solutions of ( 6) can be proved along the lines of [1, Lemma 4]. Furthermore, let:

X k,j " TrrF j ρA A : k `Aj ρA F : k s, (7a)
Y k,j " 1 2i TrrF j ρA A : k ´Aj ρA F : k s. (7b)
One shows (see [START_REF] Azouit | Towards generic adiabatic elimination for bipartite open quantum systems[END_REF]Lemma 7]) that matrix X is Hermitian and positive-definite, thus there exists Λ such that X " ΛΛ : . Furthermore, matrix Y is Hermitian. Theorem 1: [1, Section 3]: Consider model ( 1)-( 3). The invariance condition resulting from (4),( 5) is satisfied up to second order by the Lindblad reduced dynamics:

L s,ε " ´irεH s,1 `ε2 H s,2 , ρ s s `ε2 ÿ p DrL p spρ s q `Opε 3 q, (8)
with H s,1 :" andL p " ř j Λ ‹ j,p B : j ; this dynamics is associated to a completely positive map K ε pρ s q which is trace-preserving up to third-order terms. Note that in this result, the decoherence channels induced on the target system come from coupling channels in the original system.

ř k TrrA k ρA sB : k , H s,2 :" ř k,j Y k,j B k B : j ,
Remark 1: Theorem 1 shows that the Hamiltonian coupling to the environment H A induces decoherence on the target H B only at second order. Because matrix X is positive semidefinite, the induced decoherence will be bounded by the diagonal elements of X, i.e. by TrrA : j R j `R: j A j s for j P t1, . . . Ju, where R j :" F j ρA . We will thus use the diagonal of X as a measure to summarize the induced decoherence.

Remark 2: Model (1) assumes no dissipation on H B . If such dissipation L B was present with strength ε 2 , all our results still hold, with the induced dissipation just adding to L B . If L B was of order ε, the same results would hold provided a Hamiltonian control is added to cancel H s,1 to first order; we cannot claim though that this would imply the best system protection; information encoding will have to be investigated explicitly as L B would be dominating the induced dissipation by an order of magnitude 1{ε. For a similar reason, we did not investigate the case where L B is of the same order as L A .

III. GENERAL RESULTS

A. Formulation as optimization and time rescaling

The aim of this work is to minimize the induced decoherence on the reduced quantum system according to L s,ε from Theorem 1. We will assume that the coupling strength ε is fixed, the decoherence channels acting on the environment are fixed too, but their strengths can be adapted within some range -in particular, the noise acting on the environment can be increased (at little practical cost).

In this section we set up the basic elements. We first reformulate decoherence minimization as a standard optimization problem and then observe the most trivial effect, namely overall rescaling.

To this aim, let the gains h and g k for k P t1, . . . , K g u take values on a polytope Ω which is the product of strictly positive closed intervals, i.e.

Ω :" rh, hs ˆKg ą k"1 rg k , ḡk s,
with g k ą 0 for all k P t1, . . . , K g u. Most importantly, we must assume that g :" rg 1 , . . . , g Kg s and coefficients e k , k P t1, . . . , K e u together are large enough to guarantee the timescale separation of L h,g A with the environment, i.e. such that the spectral gap λ sp of L h,g A satisfies |λ sp | " ε. Minimizing the decoherence induced on the target system, according to L s,ε from Theorem 1, can be reformulated as the following quadratic optimization problem:

min x t (9a) s. t. Tr " A : j R j `R: j A j ı ď t (9b) TrrR j s " 0 (9c)
L h,g A pR j q `Aj ρA ´TrrA j ρA sρ A " 0 @j P t1, . . . , Ju (9d)

Trrρ A s " 1 (9e) L h,g A pρ A q " 0 (9f) h ď h ď h (9g) g ď g ď ḡ, (9h)
where x :" tR 1 , . . . , R J , ρA , h, g, tu represents the optimization variables. Statement (9a)-(9b) corresponds to the objective of minimizing the diagonal elements of X, as mentioned in Remark 1. Equality constraints (9d)-(9e) correspond to [START_REF] Forni | Adiabatic elimination for multi-partite open quantum systems with non-trivial zero-order dynamics[END_REF], whereas (9e)-(9f) express that ρA is the steadystate density operator of L g A . Note that the factorization of A j b B : j is only defined up to a scalar for each j. Therefore, when assigning the same bound for each j in (9b), we implicitly assume that these scalars have been chosen such that a given decoherence speed takes the same practical importance for each channel j. A more complicated treatment of the relation between different channels is of course possible along the same principles, including the off-diagonals of X as well; we just take this form (9b) for concreteness. In further analysis we will assume a single coupling channel, so these questions will not appear.

A first observation on ( 9) is the following result. Proposition 1: Consider problem (9) and assume that K e " 0. Then, for the optimizer x ‹ " tR ‹ 1 , . . . , R ‹ J , ρ‹ A , h ‹ , g ‹ , t ‹ u, either we have h ‹ " h or there exists at least an index k such that g ‹ k " ḡk. Proof: Assume by contradiction that g ‹ ă ḡ and h ‹ ă h. Then there exists a constant α :" max tr ě 0 s. t. rg ‹ ď ḡ and αh ‹ ď hu ą 1 .

One checks that tα ´1R ‹ 1 , . . . , α ´1R ‹ J ρ‹ A , αh ‹ , αg ‹ , α ´1t ‹ u is a better minimizer than x ‹ for (9). Remark 3: α expresses time-rescaling. If K e ‰ 0, the solely feasible time-rescaling is the trivial one, α " 1.

Proposition 1 provides the maybe counter-intuitive result that adiabatic elimination reverses the time scaling: the faster the environment dissipates, the slower the reduced-order model decoheres, and viceversa. We will briefly discuss the interpretation of this effect in the conclusion.

B. Single interaction channel and mixing decoherence

We have so far only looked at rescaling the speed of motion of the environment treated as a whole. A more interesting question is how we may want to tune each individual decoherence channel. Quantum decoherence channels can be of different types. A channel like L k " σ ´" |g e| for a qubit, expressing spontaneous photon emission, is in fact stabilizing the state towards a well-defined value (zero entropy). Once this would be done, the target system only sees a well-defined environment state, where the decoherence dynamics annihilates, and it would not be surprising that the target system thus behaves well.

In contrast, a Hermitian channel like L k " |g g| ´|e e| for a qubit, expressing dephasing between energy levels, always pushes the environment towards a more entropic state, or in other words "mixes" its state. In this section, we obtain results for such mixing environments, whose entropy can only increase in time, under the technical simplifying assumption of a single Hamiltonian interaction term. We show (see Corollary 1) that, maybe surprisingly, decoherence induced on the target system again is minimized when the strength of each mixing channel in the environment is maximized.

Let J " 1 and thus:

L int pρq " ´i rA b B, ρs , (10)
where A and B respectively are Hermitian operators acting on H A and H B only. The class of environments in consideration is characterized by the following assumptions.

Assumption 1: The tunable decoherence operators in L g A are Hermitian, i.e. L : k " L k for all k P t1, . . . , K g u, and h " 0.

Assumption 2: For some ĝ P rg, ḡs, the unique steadystate featured by L ĝ A is ρA " I{n where n :" dimrH A s. Assumption 2 involves the K e fixed decoherence channels. It is satisfied e.g. for a qubit subject to relaxation in a thermal environment in the limit where the average number n th of thermal photons is infinite: having two fixed channels M 1 " a γp1 `nth qσ ´and M 2 " ? γn th pσ ´q: , taking the limit n th Ñ `8 with γn th finite we have ρA " I{2.

Under Assumptions 1 and 2, ρA " I{n remains the unique steady-state of L g A for any g ď g ď ḡ. Therefore the optimization problem (9) simplifies to:

min x Tr rAF s (11a) s. t. TrrF s " 0 (11b) L g A pF q `A ´1 n TrrAsI " 0 (11c) g ď g ď ḡ (11d)
where x :" tF , gu represents the optimization variables.

The Lagrangian for (11) reads as:

Lp xq " Tr rAF s `Kg ÿ k"1 µ k p´g k `gq `μ k pg k ´ḡq `λtrace TrrF s `Tr " Λ inv ˆLg A pF q `A ´1 n TrrAsI ˙ (12)
where x :" tF , g, λ trace , Λ inv , µ, μu while λ trace , Λ inv , µ, and μ respectively are a scalar, a matrix, and two vectors of Karush-Kuhn-Tucker multipliers on the equality and inequality constraints.

The Karush-Kuhn-Tucker necessary conditions for the optimizer x‹ of (11) read as:

0 "A `λ‹ trace I `Kg k"1 g ‹ k DrL k spΛ ‹ inv q (13a) 0 " ´µ‹ `μ ‹ `¨T rrΛ ‹ inv DrL 1 spF ‹ qs . . . TrrΛ ‹ inv DrL K spF ‹ qs ‹ ' (13b) 0 " μ‹ d pg ‹ ´ḡq (13c) 0 "µ ‹ d p´g ‹ `gq (13d) 0 ď μ‹ (13e) 0 ďµ ‹ , (13f)
where d denotes element-wise multiplication.

Proposition 2: If x‹ 1 " tF ‹ , g ‹ , λ ‹ trace , Λ ‹ inv , µ ‹ ,
μ‹ u is a minimizer for (11) satisfying g ‹ k P pgk, ḡkq for some k, then x‹ 2 :" tF ‹ , h, λ ‹ trace , Λ ‹ inv , µ ‹ , μ‹ u with h k " g ‹ k for all k ‰ k and any hk P rgk, ḡks is also a minimizer for (11) attaining the same minimum of the cost function.

Proof: Slackness conditions (13c)-(13f) immediately imply that µ ‹ k " μ‹ k " 0 and thus 0 " μ‹ d ph ´ḡq " µ ‹ d p´h `gq and μ‹ , µ ‹ ě 0. By (13b), we then have TrrΛ ‹ inv DrLkspF ‹ qs " 0. By Lemmas 1 and 2 in appendix, we then have DrLkspF ‹ q " 0 and thus

L g ‹ A pF ‹ q "L g ‹ A pΛ inv ‹ q " ÿ k‰ k g ‹ k DrL k spΛ ‹ inv q "L h A pΛ inv ‹ q " L h A pF ‹ q " ´A ´λ‹
trace I, We also have TrrF ‹ As " ´řk‰ k g ‹ k TrrF ‹ DrL k spF ‹ qs, which concludes the proof.

Proposition 3: Let pF ‹ , gq satisfy the constraints of (11). Fix any k P t1, . . . , K g u. Let h be the gain vector characterized by h k " g k for all k ‰ k and hk " gk `δg for some infinitesimal δg ą 0. Let F h be the traceless solution of L h

A pF h q `A ´TrrAsI " 0. Then TrrAF h s ď TrrAF ‹ s.

Proof: Equality constraints (11b)-(11c) imply that F is a smooth function of the gain vector, thus we can consider the linearization F h " F ‹ `δF for some differential δF . They also imply that TrrδF s " 0, δF is Hermitian, and that the following equalities hold:

ÿ k g k DrL k spF ‹ q " 1 n TrrAsI ´A (14) ÿ k h k DrL k spF ‹ `δF q " 1 n TrrAsI ´A (15)
Subtracting ( 14) from (15) yields: L g A pδF q " ´δg DrLkspF ‹ `δF q. Premultiplying both sides of this expression by F ‹ and taking the trace yields:

TrrF ‹ L g A pδF qs " ´TrrδgF ‹ DrLksF ‹ s ´Trrδg F ‹ DrLkspδF qs . ( 16
)
By using identity TrrX 1 DrL k spX 2 qs " TrrX 2 DrL k spX 1 qs for any matrices X 1 , X 2 on the left-hand side of (16), and identity TrrXDrL k spXqs " ´TrrrX, L k srX, L k s : s{2 for any Hermitian matrix X on the right-hand side of (16), and by observing that TrrδF L g A pF ‹ qs " ´TrrδF As, it follows that:

TrrAδF s " ´δg 2 Tr " rF ‹ , LksrF ‹ , Lks : ‰ `Trrδg F ‹ DrLkspδF qs.
For rF ‹ , Lks ‰ 0, choosing δg small enough allows to dominate the term in δg δF , we then have TrrA δF s ă 0 implying TrrAF h s ă TrrAF ‹ s. For rF ‹ , Lks " 0 and recalling that both F ‹ and Lk are Hermitian, the term in δg δF annihilates as well and we have TrrAF h s " TrrAF ‹ s.

Altogether, we get the following result. Corollary 1: Let Assumptions 1, 2 hold. Then the decoherence minimization problem (11) is solved by g ‹ " ḡ.

Proof: From Proposition 2 and continuity of L g A , either the value of g k does not matter or the optimum corresponds to g k taking its minimum or its maximum value. Proposition 3 further implies that having g k at its lower bound cannot correspond to an isolated minimum of the cost function. Thus the only remaining possibilities are that either the value of g k does not matter, or it must take its maximum value.

A possible physical justification of this result is discussed in the conclusion.

IV. A QUBIT AS ENVIRONMENT

The qubit is the most basic quantum system. Nevertheless, it is an ubiquitous model for expressing the presence of defects in an environment, see e.g. [START_REF] Lisenfeld | Observation of directly interacting coherent two-level systems in an amorphous material[END_REF], [START_REF] Gao | A semiempirical model for two-level system noise in superconducting microresonators[END_REF]. Furthermore, the effects of several such independent environment qubits just add up in the decoherence induced on a target system [START_REF] Forni | Adiabatic elimination for multi-partite open quantum systems with non-trivial zero-order dynamics[END_REF]. This motivates to investigate the effect of adjusting decoherence rates on a qubit as environment. The low dimensionality will also allow us to illustrate the case not covered so far, namely the effect of adjusting non-Hermitian decoherence channels individually.

A. Analysis

We consider the special case of an environment consisting of a qubit H A " C 2 rotating around the z axis frequency h z , subject to relaxation in a thermal environment and to the three Pauli Hermitian decoherence channels: L A pρq " ´irh z σ z , ρs `γ´D rσ ´spρq `γ`D rσ `spρq `γx Drσ x spρq `γy Drσ y spρq `γz Drσ z spρq, (17) where we assume that the gain vector g :" pγ `, γ ´, γ x , γ y , γ z q J can be arbitrarily tuned within a compact and strictly positive interval Ω :" rg, ḡs, so as to guarantee time-scale separation of the dynamics (1). The qubit is weakly interacting with a target system via a single interaction Hamiltonian [START_REF] Viola | Dynamical decoupling of open quantum systems[END_REF], and we will specifically consider two separate cases: A " a x σ x `ay σ y and A " a z σ z . In both cases the related operator B is left arbitrary.

Proposition 4: Consider a qubit auxiliary system (17) and single interaction Hamiltonian [START_REF] Viola | Dynamical decoupling of open quantum systems[END_REF]. Let Ω " rg, ḡs be a compact and strictly positive interval of gains and denote with Ω the set of vertices of Ω, i.e. where each gain takes an extremal value. Then the decoherence on target is minimized by g ‹ P Ω in both cases A " a x σ x `ay σ y or A " a z σ z . Furthermore, if h z " 0, at least one gain takes its maximum at the optimum. In particular:

(i) if A " a x σ x , then pγ ‹ x , γ ‹ y q " pγ x , γ y q; (ii) if A " a y σ y , then pγ ‹

x , γ ‹ y q " pγ x , γy q; (iii) if A " a x σ x `ay σ y and h z " 0, then g ‹ " ḡ.

Proof: Define:

A 11 pgq :" γ ``γ 4 `γy `γz

A 22 pgq :" γ ``γ 4 `γx `γz A 33 pgq :" γ ``γ 2
`γx `γy .

The fixed point ρA :" pI `ρ x σ x `ρ y σ y `ρ z σ z q{2 of L A p¨q has coordinates ρx " ρy " 0 and:

ρz pgq " γ `´γ 2A 33 pgq
.

In order to minimize the decoherence on the target, we aim to minimize rTrrF ρA Ass which reads as:

rTrrF ρA Ass " A 11 a 2 x `A22 a 2 y h 2 z `A11 A 22 `a2 z p1 ´ρ 2 z q A 33 , (18)
where we drop the dependency on g for brevity and we took into account the fact that L A pF ρA q `Aρ A " TrrAρ A sρ A and TrrF ρA s " 0.

(Case A " a x σ x `ay σ y ). Let s :" pγ ``γ ´q{4 `γz . Then (18) reads as:

f pgq " 1 Dpgq `ps `γy qa 2 x `ps `γx qa 2 y ˘, (19)
with Dpgq :" h z `ps `γx qps `γy q. First, the partial derivative:

df ds " ´1 Dpgq 2 " pa 2
x `a2 y qs 2 `2pγxa 2 y `γya

2 x qs `pγx `γyqpγxa 2 y `γya 2 x q ´h2 z pa 2 x `a2 y q ı ( 20
)
is either non-positive for all s P R or is positive only in the interval ps ´, s `q, with s ´being its negative root s ´ă ´pγ x a 2 y `γy a 2 x q{pa 2

x `a2 y q and s `being the other one. Since s must be chosen in a strictly positive interval, we can thus observe that, for any fixed γ x , γ y , the optimal s must be at the extrema of its interval. Second, the partial derivative:

df dγ y " h 2 z a 2 x ´sa 2 y ps `γx q Dpgq 2 (21)
is sign-definite for any fixed γ x and s, and thus the optimal γ y must be at the extrema of its interval. Statement g ‹ P Ω then follows from symmetry of the γ x and γ y cases.

(Case A " a z σ z ). We want to minimize (18) over g. To this end, we treat γ x `γy as a single optimization variable γ xy , and we introduce two new optimization variables c `, γ such that γ `" γp1 `c`q and γ ´" γ. Without loss of generality, it holds c `ě 0. Then (18) reads as:

f pgq " a 2 z Dpgq 3 `γ`γ´`γ 2 xy `pγ´`γ`qγxy " a 2 z Dpgq 3 `γ2 xy `γp2 `c`qγxy `γ2 p1 `c`q ˘, (22)
with Dpgq :" γ xy `pγ ´`γ `q{2. First, by considering the partial derivative:

df dγxy " ´a2 z Dpgq 4 ˆγ2 xy `γp2 `c`qγxy `γ2 ˆ1 `c`´c 2 2 ˙˙,
and computing its roots γ xy,˘" γ 2 p´2 ´p1 ¯?3qc `q, since the local minimum is achieved at γ xy,´ă 0 and the local maximum is achieved at γ xy,`ě γ xy,´a nd γ xy must be chosen in a strictly positive interval, we observe that for any fixed γ `, γ ´, the optimal γ xy is at the extrema of that interval. Second, by observing the sign of the partial derivative: df dγ`" ´a2 z γ´`γxy 2Dpgq 4 p2γ``pγxy ´γ´qq, we observe that, for any fixed γ xy , γ ´, the optimal γ `must be at one extrema of its interval. Statement g ‹ P Ω then follows from symmetry of the γ ´and γ `cases.

Furthermore, if h z " 0, Proposition 1 implies that at least one optimal gain is at its maximum.

Case (i). From ( 22) with a y " 0, it is straightforward to compute: df {dγ y " h z a 2

x {D 2 ě 0, which implies that, for any fixed g and γ x , the optimal γ y is γ y ; df {dγ x " ´pps `γy qa x {Dq 2 ď 0, which implies that, for any fixed g and γ y , the optimal γ x is γx .

Case (ii) follows by symmetry with (i).

Case (iii). If h z " 0, then f pgq " a 2 x {ps `γx q `a2 y {ps γy q. By monotonicity of f pgq with respect to γ x , γ y , s, it follows that g ‹ "

B. Simulations

Section III-B states that, at least for a single Hamiltonian coupling, the gain on Hermitian decoherence channels must be maximized in order to minimize induced decoherence on target, while Proposition 4 cases (i),(ii) show that in presence of non-Hermitian decoherence channels this is not always true anymore. In this section, we directly investigate how the choice of optimal gains for the qubit example can depend on the tuning interval.

First, we consider the case of A " a x σ x `ay σ y and h z ‰ 0. According to the computations above, the induced decoherence f pγ x , γ y , sq only depends on γ x , γ y and s " pγ ``γ ´q{4 `γz . On Figure 1 (a) we show how the optimal gains depend on the lower bound for the tuning interval. When very low values are allowed for all the gains, the optimum is to put all gains at their minimum value; however, as soon as one of the decoherence channels is confined to larger values, the optimal situation is with all the gains at their maximum. The threshold between the two is a function of h z . On Figure 1 (c), we investigate what happens when s has a fixed value. For s ă s T HR , the optimal values of γ x and γ y are their minimum ones, while for s ą s T HR they should be maximum. The value s T HR is a function of a x {a y and of the upper bounds on the tuning intervals. Figure 1 (b) shows how the induced decoherence depends on γ x and γ y , for values of s above and below this threshold.

Second, we consider the case of A " a z σ z . Let γ xy :" γ x `γy . Then the induced decoherence on target (18) simplifies to the function f pγ ´, γ `, γ xy q in (22). On Figure 2 (a) we again show how the optimal gains depend on the lower bound for the tuning interval. The situation now is somewhat more complicated but can be understood on intuitive grounds. If γ ´is allowed to reach a very small value and γ xy is not constrained to too high values, then the best is to just push with γ `, making the environment qubit converge essentially towards the pure state |e e|. A symmetric situation is obtained when γ `is allowed to take very small values. In all other cases, we should push all decoherence gains to the maximum. This behavior in the optimal gain selection is mirrored in Figure 2 (b), where we show how the optimum depends on the upper bound of the tuning interval. As soon as we can push γ `at a value significantly larger than γ ´, we just do this and take γ maximal and γ ´minimal; and conversely when the upper bound on γ ´is larger than the one on γ `. When both γ ´and γ `are restricted to a small interval of values, we are back to the situation where the optimum corresponds to making them all maximal. Finally, on Figure 2 (c) we show how the fixed value of γ xy influences the optimal choice. For γ xy below a threshold value that depends on the upper bounds of Ω, the dominating effect of having only γ ´or only γ ìs most beneficial, driving the qubit very close to a pure state |g g| or |e e| respectively; which one to choose is perfectly symmetric in this case. However, once γ xy exceeds the threshold value, the purifying effect appears to be not beneficial enough and the optimum instead corresponds to putting all decoherence gains to their maximum.

Note that all these conclusions are in agreement with the general statement of Proposition 1.

V. CONCLUDING DISCUSSION

We have studied the decoherence induced on a target quantum system by its weak coupling to a strongly dissipative environment. Under mild assumptions, this setting allows for a model reduction expressing the dominating decoherence induced on the target system by algebraic formulas. According to these formulas, the way to obtain as little as possible decoherence on the target system depends nontrivially on the environment parameters. In particular, we have investigated how it depends on the gains of various fixed decoherence channels acting on the environment. It appears, maybe counterintuitively, that having stronger mixing in the environment can induce less dissipation on the coupled target system; while in general the optimization of decoherence gains is a nontrivial problem depending on the available tuning intervals.

After reformulating the problem of minimizing the decoherence induced on the target system as a quadratic optimization problem on the variables associated with adiabatic elimination, we have proved two precise results. First, as a simple general observation, adiabatic elimination reverses the time scaling: the stronger the environment dissipates, the weaker the reduced-order model decoheres. Second, having only Hermitian decoherence channels in the environment, which thus cause its entropy to increase: decoherence induced on target is minimized by pushing each of these entropy-increasing channels in the environment to their maximal strength. Finally, we have explored a case with non-Hermitian decoherence channels for an environment consisting of a single qubit. In this case a transition is observed between two behaviors: when gain choices are available which would stabilize the environment close to a pure state, this choice is taken; while if the environment steady state cannot be made sufficiently pure, then it is preferred again to push all gains to the maximum.

In all these results, we have never encountered a case where some decoherence gain should be chosen inside the optimization interval. This may be a property of the considered cases, or maybe more general. Therefore ongoing work is further focusing on studying non-Hermitian decoherence channels for general open quantum systems and interaction Hamiltonians consisting of multiple terms. Another major x , γ LB y q of the optimization interval, for a fixed ḡ " r2, 2, 2s. For each point g in the graph with coordinates ps LB , γ LB x , γ LB y q, the optimal gain g ‹ P rg, ḡs is evaluated and color-coded according to the legend; (b) two plots of f pγx, γy, sq as in ( 19) respectively evaluated at s 1 " 1 (in red) and s 2 " 1.4 (in blue), together with their associated minima (in black circles); (c) for any given s P r0.1, 2s, the optimal gains γ OP T ´, γ LB `q of the optimization interval, for a fixed ḡ " r2, 2, 2s. For each point g in the graph with coordinates pγ LB xy , γ LB ´, γ LB `q, the optimal gain g ‹ P rg, ḡs is evaluated and color-coded according to the legend; (b) Showing how the optimal decoherence gains depend on the upper bound pγ U B xy , γ U B

´, γ U B `q of the optimization interval, for a fixed g " r0.02, 0.02, 0.02s. For each point ḡ in the graph with coordinates pγ U B xy , γ U B

´, γ U B `q, the optimal gain g ‹ P rg, ḡs is evaluated and color-coded according to the legend; (c) for any given γxy P r0.1, 2s, the optimal gains γ OP T `, γ OP T ´are computed within the interval r0.1, 2s ˆr0.1, 2s.

point for future work is to consider a different cost function, in a reservoir engineering context, where the goal on the target subsystem is to stabilize as strongly as possible a target subspace ("error-correcting-code-space") while minimizing any other decoherence effects.

We cannot close the paper without a tentative discussion of the somewhat counterintuitive conclusions that, the faster the environment dissipates, the slower the reduced-order model decoheres. One may want to connect this observation to classical mechanical systems. Indeed, for e.g. flexible links or the pendulum-on-cart system, by weakly coupling two degrees of freedom, it is often observed that the larger the dissipation on the first link is designed, the smaller the dissipation on the second is obtained. This analogy may be valid for quantum decoherence channels of the stabilizing type, e.g. L k " |g e|. However, for L k Hermitian, decoherence towards a state ρA of full rank means convergence towards a mixed, stochastic and possibly very dynamic situation whose expectation is given by ρA . This dynamic stochastic situation does not appear to be as clearly linked to highly dissipative mechanical systems. One may thus rather have to view our results as saying that faster mixing in the environment leads to less uncertainty on the target system. This would be more reminiscent of an averaging-type interpretation, with fast motion of the environment weakening its coupling to the target system. In the same sense, the dynamical decoupling approach [START_REF] Viola | Dynamical decoupling of open quantum systems[END_REF] explicitly aims at filtering out unwanted system-environment interactions by adding appropriate Hamiltonian drives that make the system "average out" the coupling Hamiltonian; the observations in our paper might be viewed as the stochastic equivalent of this.

APPENDIX

A. Technical lemmas

Lemma 1: If x‹ :" tF ‹ , g ‹ , λ ‹ trace , Λ ‹ inv , µ ‹ , μ‹ u is a minimizer for (11), then:

(i) Λ ‹ inv " F ‹ `f ‹ I for some f ‹ P C; (ii) TrrΛ ‹ inv DrL k spF ‹ qs ď 0 for all k P t1, . . . , Ku; (iii) TrrF ‹ As " ´řk g ‹ k TrrF ‹ DrL k spF ‹ qs. Proof: Taking the trace over equation (13a) yields λ ‹ trace " ´TrrAs{n, which implies that (13a) and (11c) are the same equation, and thus (i). It follows that:

TrrΛ ‹ inv DrL k spF ‹ qs " TrrF ‹ DrL k spF ‹ qs " ´1 2
TrrS : k S k s ď 0, @k P t1, . . . Ku,

with S k :" rF ‹ , L k s. Furthermore, by multiplying both sides of (13a) by F ‹ and taking the trace again, we get (iii).

Lemma 2: If x‹ :" tF ‹ , g ‹ , λ ‹ trace , Λ ‹ inv , µ ‹ , μ‹ u is a minimizer for (11) and TrrF ‹ DrLkspF ‹ qs " 0 for some k, then: (i) rLk, F ‹ s " 0; (ii) DrLkspF ‹ q " 0; (iii) TrrF ‹ As " ´řk‰ k g ‹ k TrrF ‹ DrL k spF ‹ qs. Proof:

By making use of (23), we have TrrF ‹ DrLkspF ‹ qs " ´Tr " rF ‹ , Lks : rF ‹ , Lks ‰ {2 " 0 which implies (i). It thus follows that DrLkspF ‹ q " 0 which, together with statement (iii) in Lemma 1, implies (iii).

Fig. 1 .

 1 Fig.1. Section IV, Proposition 4, case A " axσx `ayσy and hz ‰ 0: (a) Showing how the optimal decoherence gains depend on the lower bound ps LB , γ LB x , γ LB y q of the optimization interval, for a fixed ḡ " r2, 2, 2s. For each point g in the graph with coordinates ps LB , γ LB x , γ LB y q, the optimal gain g ‹ P rg, ḡs is evaluated and color-coded according to the legend; (b) two plots of f pγx, γy, sq as in (19) respectively evaluated at s 1 " 1 (in red) and s 2 " 1.4 (in blue), together with their associated minima (in black circles); (c) for any given s P r0.1, 2s, the optimal gains γ OP T

  the interval r0.1, 2s ˆr0.1, 2s. Parameter values for simulations are: ax " 1, ay " 1.4, hz " 2.

Fig. 2 .

 2 Fig. 2. Section IV, Proposition 4, case A " azσz: (a) Showing how the optimal decoherence gains depend on the lower bound pγ LB xy , γ LB´, γ LB `q of the optimization interval, for a fixed ḡ " r2, 2, 2s. For each point g in the graph with coordinates pγ LB xy , γ LB ´, γ LB `q, the optimal gain g ‹ P rg, ḡs is evaluated and color-coded according to the legend; (b) Showing how the optimal decoherence gains depend on the upper bound pγ U B xy , γ U B´, γ U B `q of the optimization interval, for a fixed g " r0.02, 0.02, 0.02s. For each point ḡ in the graph with coordinates pγ U B xy , γ U B´, γ U B `q, the optimal gain g ‹ P rg, ḡs is evaluated and color-coded according to the legend; (c) for any given γxy P r0.1, 2s, the optimal gains γ OP T `, γ OP T ´are computed within the interval r0.1, 2s ˆr0.1, 2s.

Centre Automatique et Systèmes, Mines-ParisTech, PSL Research University. 60 Bd Saint-Michel, 75006 Paris, France; and INRIA Paris

, 2 rue Simone Iff, 75012 Paris, France. paolo.forni@inria.fr. 2 INRIA Paris, 2 rue Simone Iff, 75012 Paris, France; and Ghent University / Electronics and Information Systems (ELIS), Technologiepark 914,

A Kraus map takes the form Kpρsq " ř l M l ρsM : l for some operators M l , to ensure complete positivity[START_REF] Choi | Completely positive linear maps on complex matrices[END_REF], with ř l M : l M l " identity for trace preservation.