QUANTUM HAMILTONIAN AND DIPOLE MOMENT IDENTIFICATION IN PRESENCE OF LARGE CONTROL PERTURBATIONS

DOI:10.1051/cocv/2016026 www.esaim-cocv.org

QUANTUM HAMILTONIAN AND DIPOLE MOMENT IDENTIFICATION IN PRESENCE OF LARGE CONTROL PERTURBATIONS

Ying Fu

and Gabriel Turinici

Abstract.The problem of recovering the Hamiltonian and dipole moment is considered in a bilinear quantum control framework. The process uses as inputs some measurable quantities (observables) for each admissible control. If the implementation of the control is noisy the data available is only in the form of probability laws of the measured observable. Nevertheless it is proved that the inversion process still has unique solutions (up to phase factors). Both additive and multiplicative noises are considered.

Numerical illustrations support the theoretical results.

Mathematics Subject Classification. 93-XX, 49-XX, 81Q93.

Received September 26, 2014. Revised January 20, 2016. Accepted May 17, 2016.

1. Introduction and motivation

Successful manipulation of quantum dynamics (see [5] and references therein for a recent review) leads to interesting perspectives among which is the possibility to identify the system through measurements of control- dependent observations. This technique, called quantum identification or quantum inversion, was documented both theoretically [1,4,16,23] and numerically [8,11,18]. However although the numerical implementations show interesting robustness of the identification process with respect to noise, there is less theoretical guidance to explain this fact. Two fundamental questions concerning the well-posedness of this problem arise: the existence and the uniqueness of the Hamiltonian, and/or the dipole moment, compatible with the given measurements.

In this work we only study the uniqueness.

More specifically we start from the setting in [16] which treats the case without noise. After some technical preliminaries in Section3 we address the noise-free case in Section4 and relax many of the assumptions used in the previous work. Then in Section5we introduce the possibility that the control is subject at each time to unknown perturbations. We consider both additive and multiplicative noise. Since the actual control that acts on the system is unknown, only the probability laws of the observations are available. We explain which are the properties of the set of measurements required to determine uniquely (up to phase factors) the free Hamiltonian and dipole moment.

Then a numerical implementation is presented in Section6. Some closing remarks are the object of Section7.

Keywords and phrases.Quantum control, quantum identification.

1 Universit´e Paris-Dauphine, PSL Research University, CNRS, Ceremade, 75016 Paris, France.[email protected]

2 Institut Universitaire de France, 1 Rue Descartes, 75231 Paris cedex 05, France.[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2017

1.1. Notations

We introduce the following notations

• L_M1,M2,...,Mm is the Lie algebra spanned by the matricesM₁, M₂, . . . , M_m;

• for any matrix or vectorXwe denote byXits conjugate (the matrix whose entries are the complex conjugates of the entries ofX) and byX^∗ its adjoint (the transpose conjugate);

• HN is the set of all Hermitian matricesHN ={X∈C^N×N|X^∗=X};

• SN is the unit sphere ofC^N:SN ={v∈C^N|v= 1};

• Ψ(t, H, u(·), μ, Ψ₀) is the solution of the equation (2.1) below; to simplify the notation, when there is no ambiguity, we denote itΨ(t);

• λ_k(X), k = 1, . . . , N are the eigenvalues of X ∈ HN taken in increasing order; we also introduce φ_k(X) k= 1, . . . , N to be eigenvectors ofX (forming an orthonormal basis ofC^N) corresponding to eigenvalues λ_k(X); note that Span{φ_k(X)}may not be unique;

• SU(N) is the special unitary group of degree N, which is the group of N ×N unitary matrices with determinant 1;

• su(N) is the Lie Algebra of skew-Hermitian matrices (the Lie algebra ofSU(N));

2. The model

We present the mathematical framework following closely the notations of the previous work [16].

Consider a controlled quantum system with time-dependent wave-function Ψ(t) satisfying the Schr¨odinger

equation:

iΨ˙(t, H, u(·), μ, Ψ0) = (H+u(t)μ)Ψ(t, H, u(·), μ, Ψ0)

Ψ(0, H, u(·), μ, Ψ₀) =Ψ₀, (2.1)

whereH is the internal (“free”) Hamiltonian andμthe coupling operator between the controlu(t)∈L¹_loc(R+;R) and the system. We work in a finite dimensional framework, thereforeH, μ∈ HN for someN ∈N^∗. The goal is to determine the matrix entries ofH and μfrom laboratory measurements of some observables depending on Ψ(t). The controlu(t) can be changed in order to gather enough information on the system.

However, contrary to [16], we allow in this work some time independent perturbations to appear in the control u(t). That is, when the control is implemented in practice the nominal control intensity required by the experimentalist, denoted (t), is perturbed byY which means that u=u(t, (·), Y); here Y is a discrete random variable with possible outcomesy₁, y₂, . . .. We assume that the law of the random variable Y is time independent. A first example is the additive perturbation u(t) = (t) +Y. Such perturbation models have already been used in the quantum computing literature under the name of “fixed systematic errors”, see section VI.A. equation (40) of [14] or “systematic control error”, see [15]. In [19] the authors use a noise model called

“low frequency noise” (see Sect. IV.C. of [12]): it is defined as the portion of the (control) amplitude noise that has a correlation time that is long (up to 10³times) compared to the timescale of the dynamics therefore it can be considered as constant in time. Additional noise models (additive or multiplicative) are presented in [24].

The perturbation Y is unknown and thus Ψ(t) is a random variable, as are all measurements depending on Ψ(t). Repeating the control experiment several times the experimentalist will only learn the law of the measurements. From now on we will denote byLYZthe law of the random variableZ (that is measurable with respect to the sigma-algebra generated byY).

Two different settings are considered depending on which parameters are to be identified and the nature of the information available:

• Setting(S1): The HamiltonianH is known and the goal is to identify the dipole momentμ.

• Setting(S2): Both the HamiltonianH and the dipole momentμare unknown.

The measurements are of the form OΨ(T, H, u, μ, Ψ0), Ψ(T, H, u, μ, Ψ₀) with O ∈ HN a member of a list of possible measurements. Often, the experimentalist only measures one observable in a list (but can repeat the

experiment many times). This means that for general O₁, O₂ ∈ HN no information is available on the joint distribution of the values O₁Ψ(T, H, u, μ, Ψ₀), Ψ(T, H, u, μ, Ψ₀) and O₂Ψ(T, H, u, μ, Ψ₀), Ψ(T, H, u, μ, Ψ₀) of these two observables.

3. Some technical preliminaries

3.1. Complete sets of commuting observables

We recall in this section several facts about complete sets of commuting observables (hereafter abbreviated CSCO). We refer the reader to ([6], p. 146) for details.

First, recall that an observable is a self-adjoint operator onC^N. Once a basis ofC^N is chosen the observable can be represented as a matrixO∈ HN.

A set of observables O = {O1, . . . , O_K} is called set of commuting observables (named SCO hereafter) if [O_k, O] = 0,∀k, ∈ {1, . . . , K}.

When all observables in the SCO are multiples of the identity operator the SCO is said to be trivial; unless specified otherwise, we only work with non-trivial SCO.

All observables in the SCO Ocan be diagonalized simultaneously i.e., there exists at least an orthonormal basisΦ={φ1, . . . , φ_N} ofC^N such that anyO ∈ O is diagonal in the basisΦ. This means that in particular any φ is an eigenvector of any observable O ∈ O. In general the basis Φ is not unique because of possible degeneracies in the spectrum of the observables inO. By definition a SCO is called acomplete set of commuting observables (CSCO) if the orthonormal basis that diagonalizes the SCO is unique up to phase factors and permutations,i.e., if{ϕ₁, . . . , ϕ_N}is another orthonormal basis rendering allO∈ Odiagonal then there exists a permutationσof{1, . . . , N}and phasesβ₁, . . . , β_N ∈Rsuch thatϕ_k= e^iβkφ_σ(k) for allk= 1, . . . , N.

Examples:

(1) Let H be a Hamiltonian with all eigenvaluesλ(H) of multiplicity 1. ThenO={H}is a CSCO.

(2) Let {v1, . . . , v_N} be an orthonormal basis of C^N. Then defining P_k to be the projection on v_k (that is P_k =v_kv_k^∗) the setO={P_k,1≤k≤N}is a CSCO. In this caseP_k are called populations of the statesv_k. (3) ConsiderN = 3 andO={Od} with:

O_d=

⎛

⎝−1 0 0 0 1/2 0 0 0 1/2

⎞

⎠. (3.1)

Because the eigenspace corresponding to the eigenvalue 1/2 is of dimension 2 O is not a CSCO.

In this case both the canonical base of C³: {(1,0,0)^T,(0,1,0)^T,(0,0,1)^T} and the orthonormal basis {(1,0,0)^T,(0,1/2,−√

3/2)^T,(0,√

3/2,1/2)^T}renderO_d diagonal.

(4) Consider the truncated spin-less Hydrogen atom whose eigenstates can be labeled by a set of three indexes φ_n,l,mwithn= 1,2, . . . , N_t,l= 0,1, . . . , n−1,m=−l,−l+ 1, . . . , l−1, l. HereN_t∈Nis a fixed truncation threshold. A CSCO is given by the operators H (Hamiltonian), L² (square of the angular momentum operator),L_z(thezcomponent of the angular momentum operator) which act on the eigenstateφ_n,l,mas:

Hφ_n,l,m=C_H

n²φ_n,l,m, L²φ_n,l,m=l(l+ 1)²φ_n,l,m, L_zφ_n,l,m=mφn,l,m, (3.2) withC_H an universal constant andthe Plank constant. Herenis called principal quantum number,lthe angular momentum quantum number andmthe magnetic quantum number. Note that in this case{H, L²} is a SCO but not a CSCO.

Measuring simultaneously all observables in a CSCO is in principle possible as it is compatible with the Heisenberg uncertainty principle since all observables in a CSCO commute two by two; therefore the values of those observables may be simultaneously computed with infinite precision.

The following characterization of a CSCO will be used in the following sections:

Lemma 3.1. Let O={O1, . . . , O_K} be a SCO. ThenOis a CSCO iff there existγ₁, . . . , γ_K∈Rsuch that all eigenvalues of _K

k=1γ_kO_k have multiplicity one.

Proof. We prove first the direct implication. Consider a basisΦ ={φ1, . . . , φ_N} of C^N that renders all O_k diagonal and denote (O_k)_jthejth eigenvalue ofO_k, that isO_kφ_j= (O_k)_jφ_j. Suppose now by contradiction that for any γ = (γ₁, . . . , γ_K) ∈R^K there exists i(γ) =j(γ)≤N, such that_K

k=1γ_k(O_k)_i(γ) =_K

k=1γ_k(O_k)_j(γ). Define the functions g^1,2 :R^K →Rby g^1,2(γ₁, . . . , γ_K) =_K

k=1γ_k[(O_k)₁−(O_k)₂] and let A^1,2 ={γ ∈ R^K;g^1,2(γ) = 0}. We obtain that ∪_1≤1<2≤NA^1,2 =R^K. By the Baire’s theorem at least a couple (^∗₁, ^∗₂) exists such that A^∗1,∗2 has non empty interior. Therefore the analytic function g^∗1,∗2 is null on a non empty open set hence it is null everywhere.

But this means that (O_k)^∗₁ = (O_k)^∗₂ for all k = 1, . . . , K. Therefore for all O_k ∈ O the ^∗₁th eigenvalue is of multiplicity 2 (the ^∗₁th eigenvector and the ^∗₂th eigenvector are associated to the same eigenvalue) which contradicts the uniqueness of the basis that diagonalizes O, and hence we obtain a contradiction with the definition of a CSCO.

The reverse implication is more straightforward. Any basisΦ={φ1, . . . , φ_N}that renders allO_k ∈ Odiagonal will also render_K

k=1γ_kO_k diagonal. But by hypothesis all eigenvalues of_K

k=1γ_kO_k are distinct and therefore the basisΦis unique (up to permutation and phases) and henceOis a CSCO.

3.2. Background on controllability results

Let L ∈ N^∗ and G₁, . . . , G_L be L finite dimensional, connected, compact and simple Lie groups with the identity elementId. LetA, B∈g for all= 1, . . . , L whereg is the Lie algebra ofG.

Definition 3.2. ConsiderLbilinear systems on the Lie groupsG: dX(t)

dt = (A+u(t)B)X(t),

X(0) =Id. (3.3)

The systems are calledsimultaneously controllable (or ensemble controllable)if there existsT_{A1,...,AL,B1,...,BL} >0 such that for all T ≥ T_{A1,...,AL,B1,...,BL} and for all V ∈ G, = 1, . . . , L arbitrary, there exists a control u∈L¹([0, T],R) withX(T) =V,∀= 1, . . . , L.

LetA=A₁ . . .

A_L ∈_L

=1g andB=B₁ . . .

B_L∈_L

=1g. The following simultaneous control- lability results are proved in ([21], Thms. 1 & 2 and [3], Lem. 3, p. 29).

Theorem 3.3. The collection (3.3)of L bilinear systems is simultaneously controllable if and only if L_A,B = _L

=1g or equivalently dim_RLA,B=_L

=1dim_Rg.

Lemma 3.4. We suppose thatL_A,B =g, for all= 1, . . . , L. ThenL_A,B =_L

=1gif and only if there exist , ∈ {1, . . . , L},= and an isomorphism f :g→g such that f(A) =A andf(B) =B.

Theorem 3.5. LetGbe a finite dimensional, connected, compact and simple Lie group andgbe its Lie algebra.

Let A, B ∈gsuch that LA,B=gandα₁,. . . ,α_L ∈Rbe real constants, α_i =α_j ∀i =j. Consider the collection of control systems on G:

dX(t)

dt ={A+ (u(t) +α)B}X(t),

X(0) =Id. (3.4)

Then the collection of systems (3.4)is simultaneously controllable.

Remark 3.6. Although the Theorems 3.3 and 3.5 are formulated on a Lie group, this is enough to obtain controllability for the wave-function; recall that if X(t, H, u(·), μ) : R× H_N ×L¹_loc(R₊;R)× H_N → SU(N) satisfies the following equation:

iX˙(t, H, u(·), μ) = (H+u(t)μ)X(t, H, u(·), μ),

X(0, H, u(·), μ) =Id, (3.5)

thenΨ(t, H, u(·), μ, Ψ₀) =X(t, H, u(·), μ)Ψ₀is the solution of (2.1). SinceSU(N) is transitive on the sphereS_N (see [7], p. 88), if the control system is controllable on the Lie groupSU(N) then it will also be controllable in the wave-function formulation.

4. Inversion without noise

Theorem 4.1 (Setting (S1)). Let H, μ₁, μ₂ ∈ H_N, H diagonal, Ψ₀¹, Ψ₀² ∈ S_N and denote for a = 1,2 and ∈L¹_loc(R+,R):Ψ_a(t, ) =Ψ(t, H, (·), μ_a, Ψ₀^a). Let O be a (non-trivial) SCO. We suppose thatN ≥3 and:

• (A1):LiH,iμ1 =LiH,iμ2 =su(N).

• (A2):tr(H) =tr(μ₁) =tr(μ₂) = 0.

• the eigenvalues ofH are all of multiplicity one.

Then there existsT >0 such that if:

OΨ₁(T, ), Ψ₁(T, )=OΨ₂(T, ), Ψ₂(T, ) ∀∈L¹([0, T];R), ∀O∈ O, (4.1) then for some(α_i)^N_i=1∈R^N:

(μ₁)_jk= e^i(αj−αk)(μ₂)_jk, ∀j, k≤N. (4.2) Remark 4.2.

(1) Assumption(A1)is required for the simultaneous controllability, see Theorem3.3and Lemma 3.4.

(2) The assumption (A2) can be made without loss of generality according to [16]. In fact, changing the HamiltonianH and/ or dipole momentμby adding a multiple of the identity operatorId, does not change the observations. In this case, the state Ψ(t) is replaced by e^iϕΨ(t) with the phase ϕ∈ R depending on tr(H),tr(μ) and on the controlu.

Remark 4.3. The proof also shows that the values of any additional observable commuting withHare identical for both systems, in particular all populations are always identical.

Moreover, whenμ₁ andμ₂are matrices of dipole operators (i.e., have the form of real potentials) truncated to dimensionN, thenμ₁andμ₂are real symmetric matrices; the Theorem implies (μ₁)_jk=±(μ₂)_jkfor allj, k.

In general this is not enough to conclude thatμ₁=μ₂as it can be seen from the counter-example 1 from ([16], p. 381) whereN = 3,Ψ₀¹=Ψ₀²= (1,0,0)^T:

H =

⎛

⎝E₁ 0 0 0 E₂ 0 0 0 E₂

⎞

⎠, μ₁=

⎛

⎝ 0 −μα 0

−μα 0 μ_β 0 μ_β 0

⎞

⎠, μ₂=

⎛

⎝0 μ_α 0 μ_α 0 μ_β

0 μ_β 0

⎞

⎠, E₁, E₂, μ_α, μ_β∈R(arbitrary). (4.3)

In this case all control fields give rise to identical populations for both systems. This under-determination can be mitigated under additional hypothesis as in Remark4.8.

Remark 4.4. When eigenvalues ofH are degenerate butOis a CSCO the Theorem4.6below should be used instead.

Proof. Consider the collection of two systems (H, μ₁) and (H, μ₂) seen as a control system onSU(N) SU(N) with operatorsiH

iH,iμ₁

iμ₂∈su(N)

su(N). This collection can either be controllable or not. Denote Rt={(Ψ1(t, ), Ψ₂(t, ))|∈L¹([0, t];R)},R∞=∪t≥0Rt. It is known (see [13], Thm. 6.5 item (ii) p. 322) that there existsT such that RT =R_∞.

SinceOis a non-trivial SCO it contains at least an observable, denotedO, that is not multiple of the identity.

For this observable there exist Ψ_x, Ψ_y ∈ SN such that OΨ_x, Ψ_x = OΨ_y, Ψ_y. But the condition (4.1) shows that no controlexists that drivesΨ₀¹toΨ_xandΨ₀²toΨ_y; therefore the joint systemiH

iH,iμ₁

iμ₂is not controllable simultaneously. Then it exists an automorphism ofsu(N) that sendsiHtoiHandiμ₁toiμ₂. But the automorphisms ofsu(N) are of the formX ∈su(N)→W XW⁻¹∈su(N) orX ∈su(N)→W XW⁻¹∈su(N) for someW ∈SU(N). Recall that the matrixH is real (because it is diagonal and inHN). Consider first that one can findW ∈SU(N) such thatH=W HW⁻¹andμ₂=W μ₁W⁻¹. The first identity shows that [H, W] = 0 and thereforeW is diagonal with the diagonal containing entries of the form e^iα,≤N; the conclusion follows from the second identity. Consider now that there existsW ∈SU(N) such thatiH=W iHW⁻¹; then [H², W] = 0, thusW diagonal and thereforeH =−H, impossible.

Remark 4.5. The result is stronger than the Theorem 1 in ([16], p. 380) which requires:

• A stronger condition on the spectrum ofH (the non-degenerate transition condition); recall that the transi- tions ofH are called non-degenerate if the eigenvaluesλ_k(H) ofH satisfyλ_i(H)−λ_j(H) =λ_a(H)−λ_b(H) for all (a, b) = (i, j). Here we only ask that the eigenvalues have multiplicity one.

• That observables in O are the populations (thus in particular O is a CSCO). Here a single non-trivial observable is enough.

• That the equality (4.1) take place at all timesT ≥0. Here only one time (large enough) is required.

Theorem 4.6 (Setting (S2)). Let μ₁, μ₂, H₁, H₂ ∈ HN, Ψ₀¹, Ψ₀² ∈ SN and denote for a = 1,2 and ∈ L¹_loc(R+,R):Ψ_a(t, ) =Ψ(t, H_a, (·), μa, Ψ₀^a). We suppose thatN ≥3 and the following assumptions hold true:

(A1):LiH1,iμ1 =LiH2,iμ2 =su(N);

(A2):tr(H₁) =tr(H₂) =tr(μ₁) =tr(μ₂) = 0;

Let O={O1, . . . , O_K} be a CSCO andΦ={φ1, . . . , φ_N} an orthonormal basis that diagonalizes O.

Then there existsT >0 such that if:

OkΨ₁(T, ), Ψ₁(T, )=OkΨ₂(T, ), Ψ₂(T, ) ∀∈L¹([0, T];R), ∀k= 1, . . . , K, (4.4) then there exist(α_i)^N_i=1∈R^N andθ∈Rsuch that for all j, k≤N either

μ1φ_j, φ_k= e^i(αj−αk)μ2φ_j, φ_k, H1φ_j, φ_k= e^i(αj−αk)H2φ_j, φ_k, Ψ₀¹, φ_j= e^i(θ−αj)Ψ₀², φ_j, (4.5) or

μ₁φ_j, φ_k=−e^i(αj−αk)μ₂φ_j, φ_k, H₁φ_j, φ_k=−e^i(αj−αk)H₂φ_j, φ_k, Ψ₀¹, φ_j= e^i(θ−αj)Ψ₀², φ_j. (4.6) Remark 4.7. When O is not a CSCO, the same proof allows only to obtain that an isomorphism of Lie algebras exists that sendsiH₁toiH₂andiμ₁toiμ₂. In general it is not possible to obtain more than a general isomorphism as shown by the following counter-example:

H₁=

⎛

⎝1 0 0 0−1/2 0 0 0 −1/2

⎞

⎠, W =

⎛

⎝1 0 0

0 1/2 √

3/2 0−√

3/2 1/2

⎞

⎠, μ₁=

⎛

⎝0 1 2 1 0 0 2 0 0

⎞

⎠, (4.7)

H₂=W H₁W⁻¹=H₁, μ₂=W μ₁W⁻¹=

⎛

⎝ 0 √

3 + 1/2 1−√

√ 3/2

3 + 1/2 0 0

1−√

3/2 0 0

⎞

⎠, Ψ₀²=W Ψ₀¹,O={O_d}. (4.8)

It is immediate to see that O_dψ, ψ= 1/2−3/2|ψ,(1,0,0)^T|² and that W(1,0,0)^T = (1,0,0)^T. When the controlon the first system realizes the transformationX the observable is 1/2−3/2|XΨ₀¹,(1,0,0)^T|²; at the same time the control realizes the transformationW XW⁻¹ on the second system giving the observable 1/2− 3/2|W XW⁻¹W Ψ₀¹,(1,0,0)^T|². But W XW⁻¹W Ψ₀¹,(1,0,0)^T = XΨ₀¹, W⁻¹(1,0,0)^T = XΨ₀¹,(1,0,0)^T. Therefore it is not possible to distinguish between the couple (H₁, μ₁) and (H₂, μ₂) (at least for this initial data).

Remark 4.8. If, for physical reasons, we know the initial state of the system, thenΨ₀¹=Ψ₀²; when this initial state has non-zero components along every element of the basis,i.e.Ψ₀¹, φ_k = 0, for allk≥1 the equation (4.5) implies e^iαj = e^iθ for allj= 1, . . . , N which means H₁=H₂ andμ₁=μ₂. When some coefficientsΨ₀¹, φ_kare zero, further symmetries may occur and one can have, for instance,μ₁ =μ₂: see counter-example 1 from ([16], p. 381) presented in Remark4.3.

On the other hand, in this case, the conclusion (4.6) can be written more conveniently in the adapted basis {v1= e^−iα1/2φ₁, . . . , v_N = e^−iαN/2φ_k}:

μ₁v_j, v_k=−μ₂v_j, v_k, H₁v_j, v_k=−H₂v_j, v_k, Ψ₀¹=Ψ₀²= e^iθ/2

N

=1

ςv, ς∈R. (4.9)

Proof. Denote byT the time at which the couple of systems (H₁, μ₁), (H₂, μ₂), seen as a control system on SU(N)

SU(N) with operatorsiH

iH,iμ₁

iμ₂ ∈su(N)su(N) reaches all attainable states. Since a CSCO is a non-trivial SCO it follows as in the Theorem 4.1that there exists an isomorphism of f :su(N)→ su(N) such thatiH₂=f(iH₁),iμ₂=f(iμ₁).

All isomorphisms of su(N) are of the formX∈ su(N)→ WXW⁻¹ ∈su(N) or X ∈su(N) →WXW⁻¹ ∈ su(N) for someW ∈SU(N). We only treat here the “exotic” casef(X) =WXW⁻¹ as the second alternative is similar. ThusH₂=−W H1W⁻¹ andμ₂=−W μ1W⁻¹. With the notations in the equation (3.5) we write:

X(t, H₂, u(·), μ2) =X(t,−W H1W⁻¹, u(·),−W μ1W⁻¹) =W X(t, H₁, u(·), μ1)W⁻¹. (4.10) As the first system is controllable then every stateX ∈SU(N) can be reached by some controlu(·) thus

O_kXΨ₀¹, XΨ₀¹=O_kW XW⁻¹Ψ₀², W XW⁻¹Ψ₀²,∀X ∈SU(N),∀k≤K. (4.11) Note that (4.11) also holds for any linear combination of observables inO. We invoke the Lemma3.1and obtain the existence of an observableO, diagonal in the basisΦand with all eigenvalues distinct, such that

OXΨ₀¹, XΨ₀¹=OW XW⁻¹Ψ₀², W XW⁻¹Ψ₀²,∀X ∈SU(N). (4.12) The vectors φ_k are eigenvectors of O and denote as λ_k(O) the corresponding eigenvalues. In particular O=_N

k=1λ_k(O)φ_kφ^∗_k. We can suppose thatλ₁(O)< λ₂(O)< . . . < λ_N(O) (otherwise re-index the vectors).

Let us writeW⁻¹Ψ₀²=xΨ₀¹+yv withx, y∈C,|x|²+|y|²= 1,v∈ SN,v⊥Ψ₀¹.

Suppose y = 0; then there exists X ∈ SU(N) such that XΨ₀¹ = φ_N and Xv ∈ Span{W⁻¹φ_N, φ_N}^⊥. Then OXΨ₀¹, XΨ₀¹ = λ_N(O) = OW XW⁻¹Ψ₀², W XW⁻¹Ψ₀². Since λ_N(O) is the maximum possible value forO and all eigenspaces of O are of dimension 1 it follows thatW XW⁻¹Ψ₀² ∈Span{φN} henceXW⁻¹Ψ₀² ∈ Span{W⁻¹φ_N}. Then:

1 =|XW⁻¹Ψ₀², W⁻¹φ_N|=|X(xΨ₀¹+yv), W⁻¹φ_N|=|xφN, W⁻¹φ_N|=|x||φN, W⁻¹φ_N|. (4.13)

It follows y = 0, |x| = 1; therefore W⁻¹Ψ₀² ∈ Span{Ψ₀¹} which means that there exists θ ∈ R such that Ψ₀²= e^iθW Ψ₀¹; after trivial simplifications the equality (4.12) can be written

OXΨ₀¹, XΨ₀¹=OW XΨ₀¹, W XΨ₀¹,∀X ∈SU(N). (4.14) ButX ∈SU(N) means thatXΨ₀¹ can be chosen arbitrary inSN; we have therefore:

∀w∈ S_N : Ow, w=OW w, W w=W^∗OW w, w=W^∗OW w, w. (4.15) But this impliesO=W^∗OW and thus:

N

k=1

λ_k(O)φ_kφ^∗_k=O=W^∗OW =W^∗ _N

k=1

λ_k(O)φ_kφ^∗_k

W =

N

k=1

λ_k(O)(W^∗φ_k)(W^∗φ_k)^∗. (4.16)

Since all eigenvalues ofO are non-degenerate the representationO=_N

k=1λ_k(O)φ_kφ^∗_k is unique up to phases.

Therefore there existα_k ∈Rsuch that W^∗φ_k = e^−iαkφ_k or, equivalently,W^∗φ_k = e^iαkφ_k.

The conclusion follows from the relationshipsH₂=−W H1W^∗,μ₂=−W μ1W^∗ andΨ₀²= e^iθW Ψ₀¹.

5. Inversion in presence of noise

Let (Ω,F,P) be a discrete probability space, V = {y ∈ R^d| ∈ I ⊂ N} a set of values in R^d (possibly infinite) and letY :Ω→ V be a random variable. We can suppose that for ally∈ V,P(Y =y)>0 (otherwise we eliminate all y such that P(Y =y) = 0). Moreover after re-indexing I we can suppose that I = N^∗ or I ={1, . . . , L0} for someL₀∈N^∗. Denoteξ_k=P(Y =y_k),∀k∈ I.

We can suppose that (ξ)_≥1 is a decreasing sequence (re-indexing if necessary).

5.1. Technical preliminaries: a correspondence lemma

LetJ_a :C^N×N →R,a= 1,2 andh:R^d+1→Rbe real analytic functions withJ_a bounded.

Lemma 5.1. Let A_a, B_a∈su(N),T >0,∈L¹([0, T],R)and denote byX_a(t, y, )the solution of dXa(t,y, )

dt = (A_a+h((t), y)B_a)X_a(t, y, )

X_a(0, y, ) =Id, (5.1)

for a= 1,2 and any∈ I. Suppose that the following equality in law holds

LY(J₁(X₁(T, Y, ))) =LY(J₂(X₂(T, Y, ))) ∀∈L¹([0, T],R). (5.2) Then for any ∈ I, there existsn₀(, ξ₁, . . . , ξ_n, . . .)andκ()∈ I,κ()≤n₀(, ξ₁, . . . , ξ_n, . . .)such that

J₁(X₁(T, y, )) =J₂(X₂(T, y_κ(), )) ∀∈L¹([0, T],R). (5.3) Proof. Let∈ I. The proof is divided in several steps.

Step 1:

Fix a control. We introduce the notation:

v_a^k=J_a(X_a(T, y_k, )), a= 1,2 and k∈ I.

According to the assumption (5.2) we know thatJ₁(X₁(T, Y, )) andJ₂(X₂(T, Y, )) follow the same law. Thus P(J₁(X₁(T, Y, )) =v₁) =P(J₂(X₂(T, Y, )) =v₁). Then

k∈I/v₂^k=v₁

ξ_k =

k∈I/v^k₁=v₁

ξ_k≥ξ>0. (5.4)

Therefore {k ∈ I/v₂^k = v₁} = ∅. In addition, there exists a n₀() ∈ I such that

k>n0(),k∈Iξ_k < ξ and

k>n0()−1,k∈Iξ_k ≥ξ (by convention a sum over an empty set of indexes is zero). So we have {k ∈ I/k ≤ n₀(), v^k₂ =v₁} =∅. The indexn₀() depends only on the law ofY and the index.

Letting vary inL¹([0, T];R) we obtain a functionκ₁:L¹([0, T];R)→ {k∈ I, k≤n₀()}such that

J₁(X₁(T, y, )) =J₂(X₂(T, y_{κ1( )}, )). (5.5) Note: when the indexκ₁() with the property (5.5) is not unique, any compatible value in the set{k∈ I, k≤ n₀()}can be chosen.

Step 2:

Let n∈N^∗. We consider the spacePn of piecewise constant controls Pn ={f : [0, T]→ R| f =α₁1_[0,T n]+ α₂1_]T

n,^2T_n]+. . .+α_n1

]⁽ⁿ⁻¹⁾_n T,T], α₁, . . . , α_n ∈R}. Denoteα= (α₁, . . . , α_n). Therefore for anyk∈ I, we can define the functionsg_k fromRⁿ toRby

g_k(α) =J₂(X₂(T, y_k, _α))−J₁(X₁(T, y, _α)), with_α=α₁1_[0,T

n]+α₂1_]T

n,^2T_n]+. . .+α_n1

]⁽ⁿ⁻¹⁾_n T,T]. We know that

X_a(T, y_k, _α) = e^{(Aa+h(αn,yk)Ba)Tn}e^{(Aa+h(αn−1,yk)Ba)Tn}. . .e^{(Aa+h(α1,yk)Ba)Tn} a= 1,2. (5.6) Therefore the functions X_a are analytic in α (recall that the function h is analytic in α), and since J_a are analytic, the functionsg_k are analytic. We denoteA_k ={α∈Rⁿ/g_k(α) = 0}. EachA_k is closed becauseg_k is continuous. In Step 1, it is proved that

∃κ^P :Rⁿ→ {k∈ I, k≤n₀()} such that ∀α∈Rⁿ g_κP(α)(α) = 0. (5.7) So

k∈I,k≤n0()A_k=Rⁿ. By the Baire’s theorem, it exists aksuch thatA_k has an interior point. This means thatg_k is analytic and identically zero on a not empty open set. Therefore,g_k ≡0. So∀n,∃κ2(n)∈ {k∈ I, k≤ n₀()}such thatg_κ2(n)() = 0, for any control∈ Pn.

Step 3:

Take q ∈N and denote B_q ={k ∈ I, k ≤n₀()}/gk() = 0, ∀ ∈ P2^q}. In Step 2 it is proved that for any q ∈ N the set B_q is not empty. Obviously (B_q)_q∈N is a decreasing sequence and B_q becomes constant from a certain term, thus B_∞ =

q≥0B_q =∅. This means that there exists κ() ∈ {k ∈ I, k ≤ n₀()} such that g_κ()() = 0,∀∈ P2^q for allq. Yet,_∞

q=0P2^q is dense inL¹([0, T];R). So we haveg_κ()() = 0, for any control in L¹([0, T];R).

5.2. Main results

Theorem 5.2. Consider the same setting and assumptions as in the Theorem 4.6 with the exception of the relation (4.4). Then there existsT >0 such that if:

LYOkΨ₁(T, +Y), Ψ₁(T, +Y)=LYOkΨ₂(T, +Y), Ψ₂(T, +Y) ∀∈L¹([0, T];R), ∀k= 1, . . . , K, (5.8) then either the conclusion (4.5)or the conclusion (4.6)of the Theorem 4.6holds(see also Rem.4.8).

Remark 5.3. WhenOis not a CSCO, the same proof allows only to obtain that an isomorphism of Lie algebras exists that sendsiH₁ toiH₂and iμ₁to iμ₂.