A Dynamic Systems Approach to Fermions and Their Relation to Spins

Dynamic properties of fermionic systems, like contollability, reachability, and simulability, are investigated in a general Lie-theoretical frame for quantum systems theory. Observing the parity superselection rule, we treat the fully controllable and quasifree cases, as well as various translation-invariant and particle-number conserving cases. We determine the respective dynamic system Lie algebras to express reachable sets of pure (and mixed) states by explicit orbit manifolds.


I. INTRODUCTION
Over the last decade, there has been a considerable experimental progress in achieving coherent control of ultra-cold gases including fermionic systems [1][2][3][4][5][6]. This is also of great interest in view of quantum simulation (e.g., [7]) of quantum phase transitions [8,9], pairing phenomena [10], and in particular, for understanding phases in Hubbard models [11]. -Even earlier on, the simulation of fermionic systems on quantum computers had been in focus [12,13]. For either case, there are interesting algebraic aspects going beyond the standard textbook approach [14], some of which can be found in [15][16][17][18]. Here we set out for a unified picture of quantum systems theory in a Lie-algebraic frame following the lines of [19] to pave the way for optimal-control methods to be applied to fermionic systems.
It is generally recognized that optimal control algorithms are key tools needed for further advances in experimentally exploiting these quantum systems for simulation as well as for computation [20][21][22][23]. In the implementation of these algorithms it is crucial to know beforehand to which extent the system can be controlled. The usual scenario (in coherent control) is that we are given a drift Hamiltonian and a set of control Hamiltonians with tunable strengths. The achievable operations will be characterized by the system Lie algebra, while the reachable sets of states are given by the respective pure state orbits. Dynamic Lie algebras and reachability questions have been intensively studied in the literature for qudit systems [19,[24][25][26]. However, in the case of fermions these questions have to be reconsidered mainly due to the presence of the parity superselection rule. Hence in a broader sense the present work on fermions can be envisaged also as a step towards quantum control theory * zimboras@gmail.com † robert.zeier@ch.tum.de ‡ michael.keyl@tum.de § tosh@ch.tum.de for quantum simulation in the presence of superselection rules. Apart from discussing the implications of the parity superselection rule in the theory of dynamic Lie algebras and of pure-state orbits, we will also treat the case when one imposes translation-invariance or particlenumber conservation. Moreover, the experimentally relevant case of quasifree fermions (with and without translation invariance) is discussed in detail. Since we interrelate fermionic systems with the Lie-theoretical framework of quantum-dynamical systems, at times we will be somewhat more explicit and put known results into a new frame. The main results extend from general fermionic systems to the action of Hamiltonians with and without restrictions like quadratic interactions, translation invariance, reflection symmetry, or particle-number conservation.
The paper itself is structured as follows: In order to set a unified frame, we resume some basic concepts of Hamiltonian controllability of qudit systems in Sec. II, since for comparison these concepts will subsequently be translated to their fermionic counterparts, starting with the discussion of general fermionic systems in Sec. III.
Then the new results are presented in the following six sections: In Sec. IV we obtain the dynamic system algebra for general fermionic systems respecting the parity superselection rule (see Theorem 4 in Subsection IV A). An explicit example for a set of Hamiltonians that provides full controllability over the fermionic system is discussed in Subsection IV B. Some general results on the controllability of fermionic and spin systems, such as Theorem 51, are relegated to Appendix A. Following the same line, in Sec. V we wrap up some known results on quasifree fermionic systems in a general Lietheoretic frame by streamlining the derivation for the respective system algebra in Proposition 9 of Sec. V. Corollary 16 provides a most general controllability condition of quasifree fermionic systems building on the tensorsquare representation used in [19]. Furthermore, we develop methods for restricting the set of possible system algebras by analyzing their rank, see Theorem 13 as well as Appendices C and D. The structure and or-bits of pure states in quasifree fermionic systems are analyzed in Sec. VI leading to a complete characterization of pure-state controllability (Theorem 23). Sections VII and VIII are devoted to translation-invariant systems. For spin chains we give in Theorem 25 the first full characterization of the corresponding system algebras and strengthen in Theorem 27 earlier results on short-range interactions in [16]. The system algebras for general translation-invariant fermionic chains are given in Theorem 30 of Sec. VII C. We also identify translationinvariant fermionic Hamiltonians of bounded interaction length which cannot be generated from nearest-neighbor ones (see Theorem 33 of Sec. VII D). The particular case of quadratic interactions (see Sec. VIII A) is settled in Theorem 34. Corollary 35 considers systems which additionally carry a twisted reflection symmetry (or equivalently have no imaginary hopping terms) as discussed in [16]. Furthermore, we provide a complete classification of all pure quasifree state orbits in Theorem 39 of Sec. VIII B. This leads to Theorem 41 of Sec. VIII C presenting a bound on the scaling of the gap for a class of quadratic Hamiltonians which are translation-invariant. Section IX deals with fermionic systems conserving the number of particles. Their system algebras in the general case as well as in the quasifree case are derived in Proposition 42 and Proposition 43, respectively. Furthermore, a necessary and sufficient condition for quasifree pure-state controllability in the particle-number conserving setting is provided by Theorem 48. In Sec. X, we summarize the main results as given in Theorem 4, Corollary 16, as well as in Theorems 23,25,27,30,33,34,39,41,and 48. We conclude leaving a number of details and proofs to the Appendices in order to streamline the presentation.

II. BASIC QUANTUM SYSTEMS THEORY OF N -LEVEL SYSTEMS
As a starting point, consider the controlled Schrödinger (or Liouville) equatioṅ driven by the Hamiltonian H u := H 0 + m j=1 u j (t)H j and fulfilling the initial condition ρ 0 := ρ(0). Here the drift term H 0 describes the evolution of the unperturbed system, while the control terms {H j } represent coherent manipulations from outside. Equation (1) defines a bilinear control system Σ [27], as it is linear both in the density operator ρ(t) and in the control amplitudes u j (t) ∈ R.
For a N -level system, the natural representation as hermitian operators over C N relates the Hamiltonians as generators of unitary time evolutions to the Lie algebra u(N ) of skew-hermitian operators that generate the unitary group U(N ) of propagators. Let L := {iH 1 , iH 2 , . . . , iH m } be a subset of Hamiltonians seen as Lie-algebra elements. Then the smallest subalgebra (with respect to the commutator [A, B] := AB − BA) of u(N ) containing L is called the Lie closure of L written as iH 1 , iH 2 , . . . , iH m Lie . Moreover, for any element iH ∈ iH 1 , . . . , iH m Lie , there exist control amplitudes u j (t) ∈ R with j ∈ {1, . . . , m} such that where T denotes time-ordering. Now taking the Lie closure over the system Hamiltonian and all control Hamiltonians of a bilinear control system (Σ) defines the dynamic system Lie algebra (or system algebra for short) g Σ := iH 0 , iH j | j = 1, 2, . . . , m Lie . ( It is the key to characterize the differential geometry of a dynamic system in terms of its complete set of Hamiltonian directions forming the tangent space to the time evolutions. For instance, the condition for full controllability of bilinear systems can readily be adopted from classical systems [28][29][30][31] to the quantum realm such as to take the form of iH 0 , iH j | j = 1, 2, . . . , m Lie = u(N ) (4) saying that a N -level quantum system is fully controllable if and only if its system algebra is the full unitary algebra, which we will relax to su(N ) in a moment. This notion of controllability is also intuitive (recalling that the exponential map is surjective for compact connected Lie groups), as it requires that all Hamiltonian directions can be generated. So in fully controllable closed systems, to every initial state ρ 0 the reachable set is the entire unitary orbit reach full (ρ 0 ) := {U ρ 0 U † | U ∈ U(N )}. With density operators being hermitian, this means any final state ρ(t) can be reached from any initial state ρ 0 as long as both of them share the same spectrum of eigenvalues (including multiplicities). Thus the reachable set of ρ 0 equals the isospectral set of ρ 0 . Remark 1. Interestingly, this notion is stronger than the requirement that from any given (normalized) pure state one can reach any other (normalized) pure state, since it is well known [24][25][26] that for N being even, all rank-one projectors are already on the unitary symplectic orbit reach(|ψ 0 ψ 0 |) = {K|ψ 0 ψ 0 |K † | K ∈ Sp(N/2)} = {U |ψ 0 ψ 0 |U † | U ∈ SU(N )} (5) and Sp(N/2) is a proper subgroup of SU(N ).
In general, the reachable set to an initial state ρ 0 of a dynamic system (Σ) with system algebra g Σ is given by the orbit of the dynamic (sub)group G Σ := exp(g Σ ) ⊆ U (N ) generated by the system algebra reach Σ (ρ 0 ) := {Gρ 0 G † | G ∈ G Σ = exp(g Σ )}.
Thus the system algebra g Σ can be envisaged as the fingerprint encoding all the dynamic properties of a dynamic system Σ. Via the respective reachable sets (see, e.g., [19]) it is easy to see that a coherently controlled dynamic system Σ A can simulate the dynamics of another system Σ B if and only if the system algebra g ΣA of the simulating system Σ A encompasses the system algebra g ΣB of the simulated system Σ B , In [19], we have analyzed the possibility of quantum simulation with respect to the dynamic degrees of freedom and have given a number of illustrating worked examples. Next we describe dynamic symmetries of bilinear control systems whose Hamiltonians are given by m : is also a normal subalgebra or an ideal of u(N ) observing [cent(g Σ ), u(N )] ⊆ cent(g Σ ).
Likewise one can describe the symmetries to a given set ρ Σ of states by its centralizer where · R denotes the real span. Clearly, cent(ρ Σ ) ⊆ u(N ) generates the stabilizer group to the state space ρ Σ of the control system (Σ).
Since in the absence of other symmetries the identity is the only and trivial symmetry of both any state space ρ Σ as well as any set of Hamiltonians and their respective system algebra g Σ , one has cent(g Σ ) = cent(ρ Σ ) = {i λ½ N | λ ∈ R} =: u(1). So there is always a trivial stabilizer group U(1) := {e iφ ½ N | φ ∈ R}. This explains why the time evolutions generated by two Hamiltonians H 1 and H 2 coincide for the set of all density operators if (and without other symmetries only if) H 1 −H 2 = λ½. As is well known, by the same argument, in time evolutions following from Eq. (1), one may take U (t) := exp(−itH) equally well from U(N ) or SU(N ). Thus henceforth we will only consider special unitaries (of determinant +1) generated by traceless Hamiltonians iH ν ∈ su(N ), since for any HamiltonianH there exists an equivalent unique traceless Hamiltonian H :=H − 1 N tr(H)½ N generating a time evolution coinciding with the one ofH [32]. However, the above simple arguments are in fact much stronger, e.g., one readily gets the following statement: Lemma 2. Consider a bilinear control system with system algebra g Σ on a state space ρ Σ . Let iH 1 ∈ g Σ and iH 2 ∈ u(N ) while assuming that [H 1 , ρ Σ R ] ⊆ i ρ Σ R for all iH 1 ∈ g Σ , i.e., operations generated by g Σ map the set ρ Σ R into itself. Then the condition Proof. Using the formula e tA Be −tA = exp[ad tA (B)] = ∞ k=0 t k /k! ad k A (B) we show that Eq. (11) is equivalent to condition (a): ad k H1 (ρ) = ad k H1+H2 (ρ) for all nonnegative integer k and all ρ ∈ ρ Σ R . Moreover, (a) implies condition (b): (ad H2 • ad k H1 )(ρ) = 0 for all nonnegative integer k and all ρ ∈ ρ Σ R , as Therefore, let us consider a pair of Hamiltonians iH 1 , iH 3 ∈ g Σ (fulfilling the conditions of Lemma 2) as equivalent on the state space ρ Σ , if their difference iH 2 := i(H 1 − H 3 ) falls into the centralizer cent(ρ Σ ).

III. FERMIONIC QUANTUM SYSTEMS
In this section, we fix our notation by recalling basic notions for fermionic systems. In the first subsection, we discuss the Fock space and different operators acting on it as given by the creation and annihilation operators as well as the Majorana operators. We point out how the Lie algebra u(2 d ) of skew-hermitian matrices can be embedded as a real subspace in the set of the complex operators acting on the Fock space. In the second subsection, we focus on the parity superselection rule and how it structures a fermionic system.

A. The Fock Space and Majorana Monomials
The complex Hilbert space of a d-mode fermionic system with one-particle subspace C d is the Fock space Given an orthonormal basis {e i } d i=1 of C d , the Fock vacuum Ω := 1 (= 1 ⊕ 0 ⊕ · · · ⊕ 0) and the vectors of the form e i1 ∧ e i2 ∧ · · · ∧ e i k (with i 1 < i 2 < · · · < i k and The fermionic creation and annihilation operators, f † p and f p act on the Fock space in the following way: f † p Ω = e p , f p Ω = 0, f † p e q = e p ∧ e q , and f p e q = δ pq ; while in the general case of 1 ≤ ℓ ≤ d, their action is given by f † p (e q1 ∧ e q2 ∧ · · · ∧ e q ℓ ) = (e p ∧ e q1 ∧ e q2 ∧ · · · ∧ e q ℓ ) and f p (e q1 ∧e q2 ∧· · ·∧e q ℓ ) = n k=1 (−1) k δ pq k e q1 ∧· · ·∧e q (k−1) ∧ e q (k+1) ∧ · · · ∧ e q ℓ . By their definition, these operators satisfy the fermionic canonical anticommutation relations where {A, B} := AB + BA denotes the anticommutator. Moreover, every linear operator acting on F (C d ) can be be written as a complex polynomial in the creation and annihilation operators. Another set of polynomial generators acting on the Fock space is given by the 2d hermitian Majorana op- A product m q1 m q2 · · · m q k of k ≥ 0 Majorana operators is called a Majorana monomial. The ordered Majorana monomials with q 1 < q 2 < · · · < q k form a linearly independent basis of the complex operators acting on F (C d ).
Each Majorana monomial acting on d-mode fermionic system can be identified with a complex operator acting on a chain of d qubits via the Jordan-Wigner transformation [33][34][35][36] which is induced by where the following notation for the Pauli matrices X := where M denotes any ordered Majorana monomial and Similarly, one obtains a basis of su(2 d ) by excluding − i 2 ½.

B. Parity Superselection Rule
An additional fundamental ingredient in describing fermionic systems is the parity superselection rule. Superselection rules were originally introduced by Wick, Wightman, and Wigner [37] (see also [38,39]). These rules, in the finite-dimensional definition of Piron [40], describe the existence of non-trivial observables that commute with all physical observables. The existence of such a commuting observable in turn implies that a superposition of pure states from different blocks of a blockdiagonal decomposition w.r.t. the eigenspaces of this observable are equivalent to an incoherent classical mixture.
The parity superselection rule identifies among the operators acting on F (C d ) the physical observables H F as those that do commute with the parity operator where the adjoint action of P on a Majorana monomial is given as P m k1 m k2 · · · m k ℓ P −1 = (−1) ℓ m k1 m k2 · · · m k ℓ . These physical operators are also exactly the ones that can be written as a sum of products of an even number of Majorana operators (as P contains all Majorana operators whereof there exist an even number). They are therefore denoted as even operators for short. If the parity is the only non-trivial symmetry, we obtain H ′ where the bracket stands for the complex-linear span. Now we will discuss why the set of all physical fermionic states ρ F consists similarly of all density operators that commute with P , notably ρ ′ F = ½, P . As we will show, the parity superselection rule induces a decomposition into a direct sum of two irreducible statespace components exploiting H ′ F ∩ ρ ′ F = ½, P . Recall that P 2 = ½ and the eigenspaces to the eigenvalues +1 and −1 are indeed of equal dimension, as there are exactly 2 2d−1 even operators which map the vacuum state Ω into the +1 eigenspace of P . Note that P e q1 ∧ e q2 ∧ · · · ∧ e q ℓ = (−1) ℓ e q1 ∧ e q2 ∧ · · · ∧ e q ℓ . Thus the Fock space can be split up as a direct sum of two equal-dimensional eigenspaces of P , called the positive and negative parity subspaces (for clarity observe [41]): Now we may write P 2 = ½ = P + + P − with the orthogonal projections P + := 1 2 (½ + P ) and P − := 1 2 (½ − P ) projecting onto the respective subspaces. Any physical observable (i.e. even operator) A has a blockdiagonal structure with respect to the above splitting, i.e. A = P + AP + + P − AP − . This follows, as the requirement [A, P ] = 1 2 [A, P + ] = − 1 2 [A, P − ] = 0 enforces P + AP − = P − AP + = 0 for any operator A = P + AP + + P + AP − + P − AP + + P − AP − . We obtain Hence physical observables cannot distinguish between the density operator ρ and its block-diagonal projection to P + ρP + + P − ρP − (which is always an even density operator). In this sense, a physical linear combination (a formal superposition) of pure states from the positive and negative parity subspaces is equivalent to an incoherent classical mixture. Equation (15) also shows that without loss of generality we can restrict ourselves to even density operators and regard only those as physical.
Finally, we would like to recall three further aspects of the parity superselection rule. First, without the parity superselection rule, two noncommuting observables acting on two different and spatially-separated regions would exist which would allow for a violation of locality (e.g., by instantaneous signaling between the regions). Second, the parity superselection rule, of course, does not apply if one uses a spin system to simulate a fermionic system via the Jordan-Wigner transformation. This system respects locality, since the Majorana operators m k are-in this case-localized on the first [(k+1) div 2] spins; two noncommuting Majorana operators are therefore not acting on spatially-separated regions. Third, the parity superselection rule also affects the concept of entanglement as has been pointed out and studied in detail in [42,43].

IV. FULLY CONTROLLABLE FERMIONIC SYSTEMS
Here we derive a general controllability result for fermions obeying the parity superselection rule. We illustrate that full controllability for a fermionic system can be achieved with quadratic Hamiltonians and a single fourth-order interaction term. For example, in a system with d modes, the complete fermionic dynamical algebra Theorem 4) can be generated by a quartic interaction between the first two modes bined with three quadratic Hamiltonians which are: the nearest-neighbor hopping term the on-site potential of the first site ih 0 =i(2f † 1 f 1 −½) = m 1 m 2 , and a pairing-hopping term between the first two modes ih 12 Proposition 6). Finally, we provide a general discussion about when the commutant of a system algebra determines the algebra itself.

A. System Algebra
In the case of qubit systems mentioned in Sec. II, two Hamiltonians generate equivalent time evolutions if and only if they differ by a multiple of the identity. This condition can readily be modified for the fermionic case such as to match the parity-superselection rule as well. This also implies that for any physical fermionic Hamiltonian H, there exists a unique Hamiltoniañ that is traceless on both the positive and the negative parity subspaces, i.e., Remark 5. For Lie algebras, k 1 + k 2 will denote only an abstract direct sum without referring to any concrete realization. We reserve the notation k 1 ⊕ k 2 to specify a direct sum of Lie algebras which is (up to a change of basis) represented in a block-diagonal form k1 k2 .
Proof. It follows from Sec. III that F d commutes with P and that the matrix representation of F d splits into two blocks of dimension 2 d−1 corresponding to the + and − eigenspaces of P . As the center of F d is given . And indeed, all other system algebras are ruled out as the subalgebras acting on each of the two matrix blocks would have a smaller Lie-algebra dimension than su(2 d−1 ).

B. Examples and Discussion
We start out with an example realizing a fully controllable fermionic system by adding only one quartic operator to the set of quadratic Hamiltonians which will be discussed in Section V below (cf. Theorem 11): Proposition 6. Consider a fermionic quantum system with d > 2 modes. The system algebra L d = su(2 d−1 ) ⊕ su(2 d−1 ) of a fully controllable fermionic system can be generated using the operators w 1 := L(v 1 ), w 2 := L(v 2 ), w 3 := L(v 3 ), and w 4 := L(v 4 ) with the map L as defined in Eqs. (12) and (13), where Proof. It follows from the independent Theorem 11 (see Sec. V below) that w 1 , w 2 , and w 3 generate all quadratic Majorana monomials m p m q . Consider an even Majorana monomial s 1 := L( i∈I m i ) of degree 2d ′ , where s 2 is defined using the ordered index set I, and a quadratic operator s 2 := L(m p m q ) with p ∈ I and q / ∈ I. We can change any index p of s 1 into q of using L( k∈(I\{p})∪{q} m k ) = ±[s 1 , s 2 ]. Therefore, we get from w 4 and the quadratic operators all Majorana monomials of degree four.
Using the quartic Majorana monomials we can increase the degree of the monomials in steps of two: Consider the operators s 3 := L( i∈I m i ) and s 4 := L( j∈J m j ) which are defined using the ordered index sets I and J and have degrees 2d ′′ < 2(d − 1) and 4, respectively. Assuming that |I ∩ J | = 1, we can generate an operator L( k∈K m k ) = ±[s 3 , s 4 ] of degree |K| = 2(d ′′ + 1) < 2d where the corresponding ordered index set is given by K := (I ∪J )\(I ∩J ). By induction, we can now generate all even Majorana monomials except L( 2d q=1 m q ). Note that L( 2d q=1 m q ) cannot be obtained as I ∩ J K holds by construction. Thus, we get all elements of L d (see Subsection IV A) and the proposition follows.
The proof also implies that all the operators generated commute with 2d q=1 m q = P/i d [cf. Eq. (14)] (and the identity operator ½). In addition, all operators commuting simultaneously with all elements of L d can be written as a complex-linear combination of ½ and P . We thus obtain a partial characterization of full controllability in fermionic systems: Lemma 7. Consider a fermionic quantum system with d ≥ 2 modes. A necessary condition for full controllability of a given set of hermitian Hamiltonians H v is that One can expect that the condition of Lemma 7 is not sufficient under any reasonable assumption by applying counterexamples from spin systems in [19]. These counterexamples could be lifted to fermionic systems by providing the explicit form of the embeddings from su(2 d−1 ) to the first and second component of the direct sum We guide the discussion in a different direction by emphasizing that the property {iH v } ′ = ½, P does not determine the system algebra uniquely. We define the centralizer of a set B ⊆ su(k) in su(k) (e.g. k = 2 d ) as We consider the algebras where the latter algebra is isomorphic to su(2 d−1 ) + su(2 d−1 ) + u(1) and contains the additional (non-physical) generator L( . In particular, we have L d = cent su(k) (cent su(k) (L d )), and L d does not fulfill the double-centralizer property. A more general incarnation of this effect in line with a discussion of double centralizers is given in Appendix A. It leads in the case of irreducible subalgebras to the following maximality result: Corollary 8. Let g denote an irreducible subalgebra of su(k), i.e. cent su(k) (g) = {0}. Then one finds that cent su(k) (cent su(k) (g)) = g if and only if g = su(k).
To sum up, the symmetry properties of a Lie algebra g ⊆ su(k), as given by its commutant w.r.t. a representation of g, do not determine the Lie algebra g uniquely. Yet the commutant allows us to infer a unique maximal Lie algebra contained in su(k), which is (up to an identity matrix) equal to the double commutant of g, but in general not to g itself.

V. QUASIFREE FERMIONS
Here we present the dynamic system algebras for fermions with quadratic Hamiltonians. For illustration, also the relation to spin chains is worked out in detail. In this context, we show by free fermionic techniques that a Heisenberg-XX Hamiltonian of Eq. (24) combined with the one-site term ih 0 = iZ ⊗ I ⊗ · · · ⊗ I = m 1 m 2 and the two-site interaction ih 12 = iX ⊗ X ⊗ I ⊗ · · · ⊗ I = m 2 m 3 gives rise to the system algebra so(2d) (see Theorem 11), while the first two operators generate only the subalgebra u(d) (see Theorem 13). Further results along this line are presented in Appendix C.
Finally, we arrive at a very useful general result: In order to decide if a set of operators generates the full quadratic algebra for d modes, we characterize quadratic operators by a real skew-symmetric matrix T whose entries are given via − 1 2 2d k,ℓ T kℓ m k m ℓ (see Eq. (22)). Adapting our tensor-square criterion for full controllability from spin systems [19] to quasifree fermionic systems, a set of operators T ν generates the full quadratic algebra so(2d) if and only if the joint commutant of the operators T ν ⊗½ 2d +½ 2d ⊗T ν has dimension three (see Corollary 16).

A. Quadratic Hamiltonians
A general quadratic Hamiltonian of a fermionic system can be written as (cf. [16,[44][45][46][47]) where the coupling coefficients A pq and B pq are complex entries of the d × d-matrices A and B, respectively. The canonical anticommutation relations and the hermiticity of H require that A is hermitian and B is (complex) skew-symmetric. The terms corresponding to the nonzero matrix entries of A and B are usually referred to as hopping and pairing terms, respectively. Related parameterizations for quadratic Hamiltonians are discussed in Appendix B.
In the Majorana monomial basis, the quadratic Hamil-tonian H can be rewritten such that The properties of A and B directly imply that the matrix T is real and skew-symmetric. Using the formula one can easily verify that the space of quadratic Hamiltonians is closed under the commutator. To sum up, we have established the well-known Lie homomorphism from the system algebra generated by a set of quadratic Hamiltonians (whose control functions are given by the matrix entries of A and B) onto the system algebra so(2d) represented by the entries of T (cf. pp. 183-184 of [36]): Proposition 9. The maximal system algebra for a system of quasifree fermions with d modes is given by so(2d).
Proof. Let the map h transform the Majorana monomial − 1 2 (m p m q −m q m p ) into the skew-symmetric matrix e pq −e qp where e pq has the matrix entries [e pq ] uv := δ pu δ qv . We show that h is a Lie-homomorphism assuming p = q and r = s in the following, while the case of p = q or r = s holds trivially. Note that 1 2

B. Examples and Explicit Realizations
We start by showing that the full system algebra so(2d) of quasifree fermions can be generated using only three quadratic operators, namely The Jordan-Wigner transformation maps these generators respectively to the Heisenberg-XXterm − i 2 Z 1 , and − i 2 X 1 X 2 , where operators as (e.g.) Z 1 are defined as Z ⊗ I ⊗ · · · ⊗ I. Lemma 10. Consider a fermionic quantum system with d ≥ 2 modes. The system algebras k 1 and k 2 generated by the set of Lie generators {w 1 , w 2 } and {w 1 , w 2 , w 3 } contain the elements L(a p ) with a p := m 2p−1 m 2p for all p ∈ {1, . . . , d} as well as Note that the elements L(a p ), L(b p ), and L(c p ) are mapped by the Jordan-Wigner transformation to the Proof. We compute the commutators We can now reduce the problem from d to d−1 by subtracting w 5 from w 1 . The cases of d ∈ {2, 3, 4} can be verified directly and the proof is completed by induction.
This proof also yields an explicit realization for the algebra so(2d) while providing a more direct line of reasoning as compared to our proof of Theorem 32 in [19].
Theorem 11. Consider a fermionic quantum system with d ≥ 2 modes. The system Lie algebra k 2 generated by {w 1 , w 2 , w 3 } is given by so(2d).
Proof. The cases of d ∈ {2, 3, 4} can be verified directly. We build on Lemma 10 and remark that k 2 ⊆ so(2d) as it is generated only by quadratic operators (see Proposition 9). We compute in the Jordan-Wigner picture . This shows by induction that so(2d) ⊇ k 2 u(1) + so(2d − 2). As u(1) + so(2d − 2) is a maximal subalgebra of so(2d) (see p. 219 of [48] or Sec. 8.4 of [49]), one obtains that k 2 = so(2d). Alternatively, one can explicitly show that k 2 consists of all quadratic Majorana operators, which combined with Proposition 9 would also complete the proof.
Note that the generators w 1 , w 2 , and w 3 can be described using the Hamiltonian of Eq. (21) while keeping the control functions given by the matrix entries A pq and B pq in the real range, see Appendix B for details. This also provides a simplified approach to Theorem 32 in [19], where only the real case was considered: Corollary 12 (see Theorem 32 in [19]). Consider a control system given by the Hamiltonian components of Eq. (21). The control functions are specified by the matrix entries A pq and B pq which are assumed to be real. The resulting system algebra is so(2d).
The relations between quasifree fermions and spin systems will be analyzed in Appendix C. -Next we treat the case of the algebra u(d).
Theorem 13 (see Lem. 36 in [19]). Consider a fermionic quantum system with d ≥ 2 modes. The system Lie algebra k 1 generated by {w 1 , w 2 } is given by u(d).
Here we just sketch ideas for the proof of Theorem 13 while leaving the full details to Appendix D. Our methods exploit the detailed structure of the appearing Majorana operators while being more explicit than in [19] and avoiding obstacles of the spin picture. Building on the notation of Lemma 10, we show that the elements L(a p ) with 1 ≤ p ≤ d together with the elements L(b Furthermore, the elements L(a p ) form a maximal abelian subalgebra and the rank of k 1 is equal to d [50]. We limit the possible cases further by showing that k 1 is a direct sum of a simple and a one-dimensional Lie algebra. A complete enumeration of all possible cases completes the proof.
Remark 14. A spin chain equivalent to the fermionic system in Theorem 13 was also considered in [51], where it was shown how to swap pairs of fermions using the given Hamiltonians. As a consequence of Theorem 13, the Lie algebra in the spin chain of [51] can be identified as u(d).
Clearly, its size grows only linearly with the number of modes d. However, the addition of controlled-Z gates, as discussed in [51], already allows for scalable quantum computation.

C. Tensor-Square Criterion
Consider a control system of quasifree fermions which is represented by matrices T ν in the form of Eq. (22). For more than two modes (i.e. d ≥ 3), we can efficiently decide if the system algebra is equal to so(2d). Recall that the alternating square Alt 2 (φ) and the symmetric square Sym 2 (φ) of a representation φ are defined as restrictions to the alternating and symmetric subspace of the tensor Theorem 15. Assume that k is a subalgebra of so(2d) with d ≥ 3 and denote by Φ the standard representation of so(2d). Then, the following are equivalent: (2) The restriction of Alt 2 Φ to the subalgebra k is irreducible and the restriction of Sym 2 Φ to k splits into two irreducible components. Each irreducible component occurs only once.
(3) The commutant of all complex matrices commuting with the tensor square (Φ| k ) ⊗2 of k has dimension three.
Proof. Assuming (1), condition (2) follows from the formulas for the alternating and symmetric square of so(2d) with d ≥ 3 given in its standard representation φ (1,0,...,0) [where (1, 0, . . . , 0) denotes the corresponding highest weight]: The alternating square is given as Alt 2 φ (1,0,0) = φ (0,1,1) for so (6) and Alt 2 φ (1,0,0,...,0) = φ (0,1,0,...,0) for so(2d) in the case of d > 3 (cf. Table 6 in [52] or Table X in [19]). The symmetric square Sym 2 φ (1,0,...,0) = φ (2,0,...,0) ⊕ φ (0,0,...,0) for so(2d) and d ≥ 3 can be computed using Example 19.21 of Ref. [53]. We verify the dimension of the commutant and show that (3) is a consequence of (2) by applying Proposition 50 which says that the dimension of the commutant of a representation φ is given by i m 2 i where the m i are the multiplicities of the irreducible components of φ. For the rest of the proof we assume that condition (3) holds. We remark that the representation Φ| k is irreducible as otherwise the dimension of the commutant would be larger than three. Thus, we obtain that k is semisimple. The dimension of the commutant allows only two possibilities: one of the restrictions (Alt 2 Φ)| k or (Sym 2 Φ)| k to the subalgebra k has to be irreducible. We emphasize that k is given in an orthogonal representation (i.e. a representation of real type) of even dimension, as k is given in an irreducible representation obtained by restricting the standard representation of so(2d). Therefore, we can use the list of all irreducible representations which are orthogonal or symplectic (i.e. of quaternionic type) and whose alternating or symmetric square is irreducible (Theorem 4.5 as well as Tables 7a  and 7b  Describing the matrices in the tensor square more explicitly along the lines of Ref. [19], we present a necessary and sufficient condition for full controllability in systems of quasifree fermions. Corollary 16. Consider a set of matrices {T ν | ν ∈ {0; 1, . . . , m}} as given by Eq. (22) generating the system algebra k ⊆ so(2d) with d ≥ 3. We obtain k = so(2d) if and only if the joint commutant of Along the lines of Eq. (22), one can apply Corollary 16 to the matrices T corresponding to the generators of so(2d) of Theorem 11. For d ≥ 3 one can verify that the commutant of the tensor square has dimension three. But for d = 2 one computes a dimension of four as so(4) = su(2) + su(2) is not simple.
For illustration, note that two elements in the commutant are trivial, to wit the identity and the generator for the swap-operation between the two tensor copies. The third element does not yet occur in the unitary case described in [19]: it is the projector P S onto the totally anti-symmetric state. To see this, recall that Ref. [54] implies that if the Hamiltonians {iH ν | ν ∈ {0; 1, . . . , m}} generate a system algebra of orthogonal type, then there is an operator S ∈ SL(N ) satisfying jointly for all ν ∈ {0; 1, . . . , m} as in [19]. Using Kronecker products and writing |S := vec (S) [55], one sees that |S is in the intersection of all the kernels of the tensor squares, so and thus P S := |S S| ∈ (H ν ⊗ ½ + ½ ⊗ H ν ) ′ holds jointly for all ν ∈ {0; 1, . . . , m}; 0 N denotes the zero matrix of degree N .

VI. PURE-STATE CONTROLLABILITY FOR QUASIFREE SYSTEMS
In this section, we present a straightforward criterion for pure-state controllability of quasifree fermionic systems with d modes. A fermionic state is called quasifree if Majorana operators of odd degree map it to zero and even-degree ones map it to states which factorize into the Wick expansion form (see below). We obtain that quadratic Hamiltonians act transitively on pure quasifree states, i.e., every pure quasifree state can be transformed into any other pure quasifree state using only quadratic Hamiltonians (see Theorem 20).
In particular, an algebra isomorphic to u(d) is left invariant by quadratic Hamiltonians and pure quasifree states can be related to an homogeneous space of type SU(2d)/U(d). At first glance, this might suggest that for full pure-state controllability the system algebra has to be isomorphic to so(2d). However, the central result of this section shows that this is in general not necessary: a quasifree fermionic system (with d > 4 or d = 3) is fully pure-state controllable iff its system algebra is isomorphic to so(2d) or so(2d − 1), see Theorem 23.

A. Quasifree States
A fermionic state ρ on F (C d ) is called quasifree or Gaussian if it vanishes on odd monomials of Majorana operators and factorizes on even monomials into the Wick expansion form Here the sum runs over all pairings of [1, . . . , 2d], i.e., over all permutations π of [1, . . . , 2d] satisfying π(2q − 1) < π(2q) and π(2q − 1) < π(2q + 1) for all q. The covariance matrix of ρ is defined as the 2d × 2d skew-symmetric matrix with real entries Due to the Wick expansion property, a quasifree state is uniquely characterized by its covariance matrix. (General references for this section include [17,[56][57][58][59].) The following proposition resumes a known result on these covariance matrices (see, e.g., Lemma 2.1 and Theorem 2.3 in [57]), which will be useful in the later development: The singular values of the covariance matrix of a d−mode fermionic state must lie between 0 and 1. Conversely, for any 2d × 2d skew-symmetric matrix G ρ with singular values between 0 and 1 there exist a quasifree state that has G ρ as a covariance matrix.

B. Orbits and Stabilizers of Quasifree States under the Action of Quadratic Hamiltonians
The action of the time-evolution unitaries generated by quadratic Hamiltonians on quasifree states can be described by the following proposition (see Lemma 2.6 in [57]): Proposition 18. Consider a quasifree state ρ a corresponding to the (skew-symmetric) covariance matrix G a . The quadratic Hamiltonian is defined using the skew-symmetric matrix T and generates the time-evolution of ρ a . The time-evolved state (at unit time), ρ b = e −iH ρ a e iH is again a quasifree state with a (skew-symmetric) covariance matrix Any skew-symmetric matrix G can be brought into its canonical form denotes the singular values of G. This means that a quasifree state can be reached from another one by the action of quadratic Hamiltonians if their covariance matrices share the same singular values (including multiplicities). Let us now recall another result related to the singular values of the covariance matrices of pure quasifree states (Theorem 6.2 in [57], and Lemma 1 in [59]): Proposition 19. A quasifree state ρ is pure iff the following (equivalent) conditions hold for its covariance matrix G ρ : (a) The rows (and columns) of G ρ are real unit vectors which are pairwise orthogonal to each other. (b) The singular values of G ρ are all 1.
Applying this result together with Proposition 18, we obtain the following theorem: Theorem 20. The set of quadratic Hamiltonians act transitively on pure quasifree states, and the corresponding stabilizer algebras are isomorphic to u(d).
Proof. We have already shown that the singular values of the covariance matrices (with multiplicities) form a separating set of invariants for the orbits generated by quadratic Hamiltonians over the set of quasifree states. This means, according to Proposition 19, that the pure quasifree states form a single orbit.
As the set of quadratic Hamiltonians generate a transitive action over the pure quasifree states, the corresponding stabilizer subalgebras are isomorphic to each other. Consider a quadratic Hamiltonian H with the coefficient matrices A and B as given in Eq. (21) and the Fock state ρ Ω , which is the projection onto the Fock vacuum vector Ω. The state ρ Ω is left invariant under the time evolution generated by By noting that Ω and f † p f † q Ω (with p < q) are linearly independent vectors, we can conclude that a quadratic Hamiltonian H leaves the Fock vacuum invariant iff (a) via its adjoint action the group K acts transitively on the set of all skew-symmetric covariance matrices of pure quasifree states (whose singular values are all 1); (b) the quasifree system is pure-state controllable if its system algebra is conjugate under SO(2d) to k.
Proof. We prove the statement (a) by showing that all pure quasifree states can be transformed under Kconjugation to the same pure state. We employ an induction on d. The base case d = 2 can be directly verified. It follows from Proposition 19(b) that the skew-symmetric covariance matrix of a pure quasifree state can be written as where v 1 denotes a normalized (2d−1)-dimensional vector and A 1 denotes a (2d−1) × (2d−1)-dimensional skewsymmetric matrix. We consider the action of a general orthogonal transformation Since any (2d−1)-dimensional vector v 1 with unit length can be transformed by an orthogonal transformation where v 2 is a 2d − 3 dimensional unit real vector and A 2 is a (2d − 3) × (2d − 3) skew-symmetric matrix. Now the proof of (a) follows using the induction hypothesis. The statement (b) is a consequence of (a).
In most cases the so(k − 1)-subalgebras of so(k) are conjugate to each other. More precisely, Lemma 7 of [64] states that for 3 ≤ k / ∈ {4, 8} all subalgebras of so(k) whose dimension is equal to (k−1)(k−2)/2 are conjugate to each other under the action of the group SO(k). In particular, it follows in these cases that all subalgebras of so(k) with dimension (k − 1)(k − 2)/2 are isomorphic to so(k − 1). Interestingly, the last statement holds also for k ∈ {4, 8} (see Lemma 3 of [64]); however not all of these subalgebras of so(k) are conjugate. We obtain the following theorem providing a necessary and sufficient condition for full quasifree pure-state controllability in the case of d > 4 or d = 3 modes: Theorem 23. A quasifree fermionic system with d > 4 or d = 3 modes is fully pure-state controllable iff its system algebra is isomorphic to so(2d) or so(2d − 1).
Proof. "⇒": Note that Theorem 20 identifies the space of pure quasifree states with the homogeneous space SO(2d)/U(d). Assuming d ≥ 3, we summarized above that a group acting transitively on this homogeneous space is isomorphic either to SO(2d) or SO(2d − 1). Thus only the full quasifree system algebra so(2d) or a system algebra isomorphic to so(2d−1) can generate a transitive action on the space of pure quasifree states. "⇐": As discussed, all so(2d − 1)-subalgebras are conjugate to each other for d > 4 and d = 3. Lemma 22(b) then implies that any set of Hamiltonians generating a system algebra isomorphic to so(2d − 1) will allow for full quasifree pure-state controllability.
Note that the cases d = 2 and d = 4 are well-known pathological exceptions. The algebra so(4) breaks up into a direct sum of two so(3)-algebras which hence cannot be conjugate to each other. For d = 4, there are three classes of non-conjugate subalgebras of type so (7) in so (8) where two classes are given by irreducible embeddings and the third one is conjugate to the reducible standard embedding fixing the first coordinate [65].
On a more general level, Theorem 23 can be seen as a fermionic variant of the pure-state controllability criterion for spin systems [24][25][26]. We note here that the result for spin systems has been recently generalized from the transitivity over a set of one-dimensional projections (i.e. pure states) to the transitivity over a set of projections of arbitrary fixed rank (i.e., over Grassmannian spaces) [63]. We will use exactly this generalization in Section IX C in order to find a necessary and sufficient pure-state controllability condition for particleconserving quasifree systems.

VII. TRANSLATION-INVARIANT SYSTEMS
We study system algebras generated by translationinvariant Hamiltonians of the type which arises approximately in experimental settings of, e.g., optical lattices. As the naturally occurring interactions are usually shortranged, we pay particular attention to the case of Hamiltonians with restricted interaction length. For example, consider a d-site fermionic chain with Hamiltonians which are translation-invariant and are composed of nearestneighbor (plus on-site) terms. All elements in its dynamic algebra can be written as linear combinations of six types of terms: the chemical potential the real and complex hopping Hamiltonians the real and complex pairing terms as well as a local density-density-type interaction The corresponding dynamic system algebras (given in Table I) were computed with the help of the computer algebra system magma [66] for up to six modes while distinguishing nearest-neighbor interactions from arbitrary translation-invariant ones.
In this context, two sets of natural questions arise: (a) How does the dimension of these dynamic system algebras scale with the number of modes? (b) How do the system algebras generated by the nearest-neighbor terms differ from the general translation-invariant ones? Can one characterize those elements that are translationinvariant yet not generated by nearest-neighbor Hamiltonians? Are there, for example, next-nearest-neighbor interactions of this type? In this section, we will answer these questions partially. We determine the system algebra for general translation-invariant fermionic Hamiltonians, and conclude that its dimension scales exponentially with the the number of modes. We also provide translation-invariant fermionic Hamiltonians of bounded interaction length which cannot be generated by nearestneighbor ones.
The structure of this section is the following: As the structure of system algebras for translation-invariant systems has only been studied sparsely even for simple scenarios of spin models, we start by examining this case first. In Secs. VII A and VII B, we determine the system algebras of all translation invariant spin-chain Hamiltonians with L qubits. In particular, we simplify and generalize results of [16] concerning finite-ranged interactions. Finally, we present the corresponding results for the fermionic case in Sec. VII C and VII D.

A. Translation-Invariant Spin Chains
Consider a chain of L qubits with Hilbert space The translation unitary U T is defined by its action on the canonical basis vectors as U T |n 1 , n 2 , . . . , n L = |n L , n 1 , . . . , n L−1 (34) where n i ∈ {0, 1}. We will determine the translationinvariant system algebra which is defined as the maximal Lie algebra of skew-hermitian matrices commuting with the translation unitary U T .
Lemma 24. The translation unitary can be spectrally decomposed as U T = L−1 ℓ=0 exp(2πiℓ/L)P ℓ , and the rank r ℓ of the spectral projection P ℓ is given by the Fourier transform where gcd(L, k) denotes the greatest common divisor of L and k.
D ℓ (g k ) = exp(2πikℓ/L). Therefore, we determine the rank of a projection P ℓ by computing the multiplicity of D ℓ using the character scalar product It follows that the number of U k T -invariant basis vectors and-hence-the trace of D T (g k ) = U k T is equal to 2 gcd(L,k) . Thus, the multiplicities of D ℓ are given accordingly by r ℓ = 1 L L−1 k=0 2 gcd(L,k) exp(−2πikℓ/L). Note that a Hamiltonian commutes with U T iff it commutes with all spectral projections P ℓ of U T . Combining this fact with Theorem 51 we obtain a characterization of the system algebra for translation-invariant spin systems: In complete analogy one can show that for a chain consisting of L systems with N levels, the system algebra is equal to s[⊕ L−1 ℓ=0 u(r N,ℓ )], where r N,ℓ denotes the Fourier transform of the function N gcd(L,k) .

B. Short-Ranged Spin-Chain Hamiltonians
In many physical scenarios, we may only have direct control over translation-invariant Hamiltonians of limited interaction range. We will investigate in this section how the limitations on the interaction range constrain the set of reachable operations. In particular, we provide upper bounds for the system algebras with finite interaction range.
Let us denote the Lie algebra corresponding to Hamiltonians of interaction length less than M by t M (L), or t M for short. In other words, t M (L) is the Lie subalgebra of t(L) generated by the skew-hermitian operators for all combinations of Q p ∈ {½ 2 , X, Y, Z} apart from the case when Q 1 = ½ 2 . In this way, t 1 (L) corresponds to the translation-invariant on-site Hamiltonians, while t 2 (L) is generated by the on-site terms and the nearest-neighbor interactions, and so on. Finally, we have t L (L) = t L .
We computed all the algebras t M (L) for 1 ≤ L ≤ 6 and 1 ≤ M ≤ L using the computer algebra system magma [66]. The results, shown in Table II, suggest that for certain restrictions on the interaction length (e.g., nearestneighbor terms), there will be some translation-invariant interactions that cannot be generated. This is in accordance with the result of Kraus et al. [16]. Building partly on their work, we analyze the properties of the algebras t M (L) for general M and L values, and then compare our theorems with Table II. We first mention a central proposition whose proof can be found in Appendix E: Applying Proposition 26, we can present upper bounds for the system algebras with restricted interaction length. Proof. (a) Since M is a divisor of L, the equation holds for any integer q. One can invert the equation as If ih ∈ t M , we obtain by applying Proposition 26 that holds for ℓ ′ ∈ {0, 1, . . . , R − 1}. It follows that t M is a subalgebra of the Lie algebra f which consists of all skewhermitian matrices satisfying the condition in Eq. (36).
The maximal Lie algebra consisting of skew-hermitian matrices which satisfy the condition in Eq. (37) is isomorphic to [ Obviously, ih ∈ t M+1 holds. Using the formula for F (1, M +1), we obtain that tr(U qM T ih) = i2L holds for every integer q. Hence, ih / ∈ t M .
In particular, this theorem implies that the algebra t(L) = t L (L) of all translation-invariant Hamiltonians cannot be generated from the subclass of nearestneighbor Hamiltonians, cp. also [16]. More precisely, one finds: Let us now compare our upper bounds with the results of Table II. Theorem 27 restricts the possibilities for the M -local algebras t M (L) only by some central elements u(1) when compared to the corresponding full translation-invariant algebra t(L). One can indeed identify in Table II some missing u(1)-parts for L ∈ {3, . . . , 6}. In general, the dimensions of the Mlocal algebras t M (L) can be even smaller than predicted by the upper bounds of Theorem 27 as can be seen in Table II for L = 4. Theorem 27 and Table II suggest that the prime decomposition of the chain length L will have strong implications on the dimension of t M (L).

C. Translation-Invariant Fermionic Systems
To determine the system algebra generated by all translation-invariant Hamiltonians of a fermionic chain, we can follow similar lines as in Sec. VII A. Here, however, we additionally have to consider the parity superselection rule. We define the fermionic translationinvariant system algebra as the maximal Lie subalgebra of su(2 d−1 ) ⊕ su(2 d−1 ) [see Theorem (4)] which contains only skew-hermitian matrices commuting with the fermionic translation unitary U, which is defined below such that it commutes with the parity operator P (see Eq. (14)). The standard orthonormal basis in the Fock space for a chain of d fermionic modes is given by with n i ∈ {0, 1}. Note that for the purpose of unambiguously defining this basis, we order the operators (f † i ) ni in Eq. (38) with respect to their site index i. The fermionic translation unitary U is defined by its action on the standard basis. The adjoint action of U on the creation operators f † ℓ is then given by The superselection rule for fermions splits the spectral decomposition of the translation unitary into two blocks corresponding to the positive and negative parity subspace. The translation unitary U commutes with the parity operator P , and hence U = U + + U − is blockdiagonal in the eigenbasis of P where U + := P + UP + and U − := P − UP − . The following lemma gives the spectral decomposition of the operators U ± : Lemma 29. The unitary operators U ± can be spectrally decomposed as U ± = d−1 ℓ=0 e 2πiℓ/d P ± ℓ , where the rank r ℓ of the spectral projection P ± ℓ is given by the Fourier transformr Proof. We determine the spectral decomposition of U + and U − along the lines of Lemma 24. Let F + (C d ) denote the subspace spanned by those basis vectors of Eq. (38) for whichn = d i=1 n i is even. Likewise, F − (C d ) corresponds to the case of oddn. As (U ± ) d = ½ F±(C d ) , the eigenvalues of U ± are of the form exp(2πiℓ/d) with ℓ ∈ {0, . . . , d−1}. Hence, the spectral decomposition is given by U ± = d−1 ℓ=0 exp(2πiℓ/d)P ± ℓ . We define representations D ± of the cyclic group Z d which map the k-th power of the generator g ∈ Z d of degree d to D ± (g k ) := U k ± . Note that D ± splits up into a direct sum D ± ∼ = ⊕ ℓ∈{0,...,L−1} (D ℓ ) ⊕ dim(P ℓ ) containing dim(P ± ℓ ) copies of the one-dimensional representations satisfying D ℓ (g k ) = exp(2πikℓ/d). The rank r ± k of the projection P ± ℓ is equal to the multiplicity of D ℓ in the decomposition of the reducible representation D ± . This multiplicity can be computed as the character scalar product tr[D ± (g k )] exp(−2πiℓk/d).
In the standard basis, all matrix entries of D ± (g k ) = U k ± are elements of the set {0, 1, −1}. It follows by repeated applications of Eq. (39) that U k maps the basis vectors |n 1 , n 2 , . . . , n d to s|n π(1) , n π(2) , . . . , n π(d) where π is a cyclic shift by k positions and the sign s is given by Recall from the proof of Lemma 24 that a bit string (n 1 , n 2 , . . . , n N ) is left invariant under a cyclic shift by k positions iff it is of the form (n 1 , n 2 , . . . , n gcd(d,k) , . . . , n 1 , n 2 , . . . , n gcd(d,k) ) .
If d/ gcd(d, k) is even, the sumn = d i=1 n i is even for all of the 2 gcd(d,k) bit strings invariant under a cyclic shift by k positions. It follows that all the diagonal entries of U k − are zero, while U k + has 2 gcd(d,k) non-zero diagonal entries. The non-zero diagonal entries of U k + are given by the number s of Eq. (41). Note that s is +1 if d−k j=1 n j is even; and −1 otherwise. Hence the frequencies of +1 and −1 in the set of diagonal entries are equal. In summary, tr(U k ± ) = 0 if d/ gcd(d, k) is even. Assume now that d/ gcd(d, k) is odd. The sumn is odd for half of the 2 gcd(d,k) bit strings and even for the other half. Applying again Eq. (41), we obtain always a positive sign. Hence, both traces tr(U k ± ) are equal to 2 gcd(d,k)−1 . This completes the proof.
Lemma 29 together with Theorem 51 implies the following characterization of the system algebra for a translation-invariant fermionic system: Theorem 30. Consider the translation-invariant Hamiltonians acting on a fermionic system with d modes. The corresponding system algebra t f is given by where the numbersr ℓ are defined in Eq. (40).
Remark 32. Assuming that the number of modes is given by a prime number p, we can explicitly determine the numbersr ℓ from Eq. (40). The corresponding system algebras are where F p = (2 p−1 − 1)/p is guaranteed to be an integer by Fermat's little theorem.

D. Fermionic Nearest-Neighbor Hamiltonians
For spin systems (see Section VII B) we verified that the translation-invariant nearest-neighbor interactions together with the on-site elements will never generate all translation-invariant operators, i.e. t L = t 2 (if the number of spins L is greater than two). This means that there exist certain translation-invariant elements which cannot be generated by nearest-neighbor interactions and onsite elements, but we could not identify the explicit form of these translation-invariant elements for general L. In particular, it would be interesting to know if t M = t 2 holds for interaction lengths less than M (2 < M < L), where M is independent of L.
In the case of fermionic systems, we can provide a result in this direction due to the restriction imposed by the parity superselection rule, which strongly limits the set of nearest-neighbor Hamiltonians. As we have discussed at the beginning of this section, the fermionic translation-invariant Hamiltonians of nearest-neighbor type are spanned by only six elements: h 0 , h rh , h ch , h rp , h cp , and h int as defined in Eqs. (28)- (33). We can show that there exist next-nearest-neighbor or third-neighbor interactions for odd d ≥ 5 which cannot be generated by these six Hamiltonians, while for even d ≥ 6 we provide a fourth-neighbor element.
Let t f M denote the subalgebra of t f (see Theorem 30) which is generated by all elements of interaction length less than M . In particular, t f 2 is generated by nearestneighbor and on-site elements. The result of this subsection is summarized in the following theorem:

Theorem 33. Let us consider the Hamiltonian
and fourth-neighbor Hamiltonian The generator ih o ∈ t f 4 is not contained in the system algebra t f 2 generated by nearest-neighbor interactions and on-site Note that the Hamiltonian h o of Theorem 33 is a thirdneighbor Hamiltonian for d ≥ 7 and a next-nearestneighbor Hamiltonian for d = 5. The proof of Theorem 33 is rather involved. The proof for even d is given in Appendix F, while Appendix G contains the proof for odd d.

VIII. QUASIFREE FERMIONIC SYSTEMS SATISFYING TRANSLATION-INVARIANCE
We continue the discussion of translation-invariant fermionic systems from Sec. VII by narrowing the scope to quadratic Hamiltonians. In Sec. VIII A, we derive the dynamic algebras for systems with and without (twisted) reflection symmetry. Both of these cases are summarized for quasifree fermionic systems in Table III: the system algebras were computed using the computer algebra system magma [66] for cases with low number of modes, while the complete picture is provided by Theorem 34 and Corollary 35. Sec. VIII B yields a classification of the orbit structure of pure translation-invariant quasifree states. This allows us to present an application to manybody physics in Sec. VIII C, where we bound the scaling of the gap for a class of quadratic Hamiltonians.

A. Translation-Invariant Quadratic Hamiltonians
. To study such Hamiltonians, it is useful to rewrite them in terms of the Fourier-transformed annihilation and creation operators with k ∈ {0, 1, . . . , d−1}, which satisfy again the canonical anticommutation relations A Hamiltonian from Eq. (21) with cyclic A and B can now be rewritten as applying the definitionsÃ k := where one has the following definitions with k ∈ {1, . . . , ⌊(d − 1)/2⌋} as well as Note that the operators ℓ Z d/2 (for d even), ℓ Z 0 , ℓ Z k , ℓ ½ k , ℓ X k , and ℓ Y k are linearly independent and span the (⌊d−1⌋+d)dimensional space of all translation-invariant quadratic Hamiltonians. For notational convenience we also introduce the dummy operators ℓ Q d/2 := 0 (assuming d is even) and ℓ Q 0 := 0 for Q ∈ {½, X, Y}. With these stipulations, we can characterize the system algebra: Theorem 34. Let q d denote the system algebra on a fermionic system with d modes which corresponds to the set of Hamiltonians that are translation-invariant and quadratic. Then the Lie algebra q d is isomorphic to [ and iℓ Z 0 can be partitioned into m pairwisecommuting sets, which each span linear subspaces as Moreover, L 0 is one-dimensional and forms a u(1)algebra. Using Eq. (44), the relations . Substituting k and d − k into a and b in the previous formula, one can verify directly that the correspondence provides an explicit Lie isomorphism between L k and u(2). If d = 2m is even, the system algebra consists of the above-described generators supplemented with the element iℓ Z d/2 . This additional element commutes with all the other generators and-therefore-provides an additional u(1).
The isomorphism between L k and u(2) as given in the proof leads to a compact formula for the time evolution (in the Heisenberg picture) of the elements of L k . Since the operators ℓ X k , ℓ Y k , ℓ Z k , and ℓ ½ k (with k ∈ {1, . . . , ⌊(d − 1)/2⌋}) satisfy the same commutation relations as the Pauli matrices X, Y, Z, and ½, their timeevolution generated by the Hamiltonian H in Eq. (46) can be straightforwardly related to a qubit time-evolution where H s = Re(Ã k )Z + Re(B k )X/2 + Im(B k )Y/2. The twisted reflection symmetry plays an important role in translation-invariant quasifree fermionic systems. It is defined by the unitary whose adjoint action on creation operators and their Fourier transforms are specified as A given translation-invariant quasifree Hamiltonian is Rsymmetric (i.e. [R, H] = 0) iff the coefficient matrix is restricted to be real. In our language, these Hamiltonians are exactly the ones for which Im(Ã k ) = 0, i.e., the corresponding generators are spanned by the operators iℓ Z d/2 (for d even), iℓ Z 0 , iℓ Z k , iℓ X k , and iℓ Y k . From the proof of Theorem 34 one can immediately deduce the corresponding system algebra: Corollary 35. Consider a fermionic system with d modes and the set of quadratic Hamiltonians which are translation-invariant and R-symmetric. The corresponding system algebra q R d is isomorphic to [ Given the system algebras q d and q R d , we investigate the subalgebras generated by short-range Hamiltonians. It will be useful to introduce for p ∈ {1, . . . , ⌊(d − 1)/2⌋} the Hamiltonians as well as the additional ones (h Z d/2 only for even d) In these definition we used cyclic indices, e.g. f d+a = f a . The operators h Z d/2 (for d even), h Z 0 , h Z p , h ½ p , h X p , and h Y p span q d linearly. Using the identities above, the commutation relations of the ℓ Q k operators, and some trigonometric identities, we obtain for a, b ∈ {0, . . . , ⌊d/2⌋}. In [16] it was shown that already the nearest-neighbor Hamiltonians of q R d generate the whole q R d . Now we are in the position to provide a more systematic proof of their result: Lemma 36. The system algebra q R d can be generated using the one-site-local operator ih Z 0 and a nearest-neighbor with α i ∈ R assuming that α 2 = 0 or α 3 = 0 for odd d and additionally requiring α 1 = 0 for even d.
it follows that one can generate ih Z 0 , ih Z 1 , ih X 1 , and ih Y 1 . Hence according to observation (1), the whole q R d is generated. (3) Suppose now that α 1 = 0, d is odd, and α 2 2 +α 2 (53) it follows that these generators in turn generate all ih Z 2p mod d . Since d is odd, ih Z 1 is also generated. Hence we obtain ih Z 0 , ih Z 1 , ih X 1 , and ih Y 1 , and according to (1), the algebra q R d is generated.
For the more general q d , we obtain a slightly larger system algebra when we do not assume R-symmetry: Proposition 37. The elements of q d with interaction length less than M (where 2 ≤ M ≤ ⌈d/2⌉ and d ≥ 3) generate a system algebra which is isomorphic to [ We illustrate Lemma 36 and Proposition 37 with a fermionic ring of d = 6 modes. Suppose that the drift Hamiltonian of this system is the nearest-neighbor hopping Hamiltonian ih Z 1 = i , and that one can additionally control the on-site po- , and the magnetic flux in the ring. Lemma 36 implies that the first three Hamiltonians generate the Lie algebra q R 6 of all Hamiltonians which are simultaneously R-invariant, translation-invariant, and quadratic. The magnetic flux term ih ½ 1 commutes with all elements of q R 6 and contributes only an additional u(1) to the system algebra. Thus, the system algebra generated by all nearest-neighbor quadratic Hamiltoinans that are translation-invariant is given by q R 6 +u(1) ∼ = su(2)+su(2)+u(1)+u(1)+u (1). In order to achieve full controllability for a translation-invariant and quasifree fermionic system (which corresponds to the Lie algebra q 6 ∼ = su(2)+su(2)+u(1)+u(1)+u(1)+u (1)), one has to add a next-nearest neighbor Hamiltonian as ih

B. Orbits of Pure Translation-Invariant Quasifree States
We characterize now the orbits of pure translationinvariant quasifree states under the action of translationinvariant quadratic Hamiltonians. Since the opera- Let us recall that a quasifree state is fully characterized by its Majorana covariance matrix, defined in Eq. (27). The translation unitary U acts on the Majorana operators by conjugation as U m p U † = m (p+2 mod 2d) . It follows that a quasifree state ρ is translation-invariant (i.e. [ρ, U] = 0) iff its covariance matrix G pq is doublycyclic, i.e. G pq = G (p+2 mod 2d),(q+2 mod 2d) . The doublecyclicity of G implies that it can be expressed as a block-Fourier transform of a block-diagonal matrix, i.e.
where U F := ( 1 0 0 1 ) ⊗ W with W pq := exp(2πi/d) q−p and G = ⊕ d−1 k=0 ig(k) withg(k) being 2 × 2-matrices. The matricesg(k) can be calculated by the inverse block-Fourier transform The fact that G is skew-symmetric and real implies Moreover, due to Eq. (54) the set of eigenvalues of all the matricesg(k) equals the one of −iG (including multiplicities). Combining these observations with Proposition 17 and Proposition 19, we obtain the following characterization of pure translation-invariant quasifree states: The entries ofg(k) and the expectation values of the ℓ k operators defined in Eq. (47) can be related by using Eq. (55) and the definitions for ℓ ½ k , ℓ X k , ℓ Y k , and ℓ Z k . Now we can prove the main theorem of this subsection: Theorem 39. Two pure quasifree states ρ 1 and ρ 2 can be connected through the action of a translation-invariant quadratic Hamiltonian if and only if tr(ρ 1 ℓ ½ k ) = tr(ρ 2 ℓ ½ k ) holds for all ℓ ½ k with k ∈ {0, . . . , ⌊(d−1)/2⌋}. Proof. First, we consider the 'if'-case: Let H be a translation-invariant quadratic Hamiltonians for which ρ 1 = e −iHt ρ 2 e iHt holds. Since the operators ℓ ½ k commute with any translation-invariant Hamiltonian, we have that . Second, we treat the 'only if'-case: Letg 1 (k) andg 2 (k) denote the Fourier-transformed Majorana twopoint functions (defined as in Eq. (55)) of ρ 1 and ρ 2 , respectively. The action of a translation-invariant Hamiltonian, ρ a → e −iH ρ a e iH is represented by the map Using Eq. (57), we obtain tr(ρ a ℓ ½ k ) = i tr[g a (k)] for a ∈ {1, 2}. These expectation values have to be in the set {−2, 0, 2}, since the eigenvalues ofg 1 (k) andg 2 (k) are in the set {−1, 1}. Then, it follows from tr(ρ 1 ℓ ½ k ) = tr(ρ 2 ℓ ½ k ) that the expectation values ofg 1 (k) andg 2 (k) coincide. Thus, we obtain from Eq. (58) that ρ 1 and ρ 2 can be transformed into each other.
Finally, we turn to the R-symmetric setting, as introduced in Sec. VIII A, and determine the orbit structure of quasifree pure states which are translation-invariant and R-symmetric under the action of operators in q R d . Proposition 40. The unitaries generated by the Lie algebra q R d act transitively on the set of quasifree pure states which are translation-invariant and R-symmetric.
Proof. Since Rℓ ½ k R −1 = −ℓ ½ k , the expectation value of these operators in R-symmetric states must vanish as Moreover, by Theorem 39 we know that two pure translation-invariant states are on the same q d -orbit iff the expectation values of the ℓ ½ k operators coincide for all k ∈ {0, . . . , ⌊(d − 1)/2⌋}. Hence the translation-invariant R−symmetric states lie on the same q d -orbit. As Eq. (58) implies that the q d -orbits are equivalent to q R d -orbits, we have proved the proposition.

C. An Application to Many-Body Physics
In many-body physics, one of the important characteristics of quantum criticality is the closing of the gap. This means that the energy difference between the ground state and the first excited state goes to zero in the thermodynamic limit, when the number of spins or fermionic modes goes to infinity. Quasifree fermionic models can display both gapped and gapless behavior. Using the techniques developed in the previous subsections, we will prove that the gap always disappears (i.e. closes) for translation-invariant quasifree models if the coefficient matrix A of Eq. (21) is purely imaginary while B is an arbitrary, complex skew-symmetric matrix.
To formalize this statement, let us consider a set a r of fixed (finite) real numbers with r ∈ {1, . . . , M −1} and a set b r of fixed complex numbers (of finite modulus) with r ∈ {1, . . . , M −1}. With these stipulations, we define for any d ≥ 2M the cyclic d × d matrices A d and B d (or A and B for short) by specifying their entries and By applying these definitions to Eq.
where A and B are defined in Eqs. (59) and (60). Assume that H d has a unique ground state. Then the gap ∆ d of i.e. the gap closes algebraically in the thermodynamic limit of d going to infinity.
Proof. Since H d is translation-invariant and its coefficient matrix is imaginary, it can be decomposed in terms of the operators ℓ Q k with Q ∈ {½, X, Y} and k ∈ {1, . . . , ⌊(d−1)/2⌋} as Let ρ d be a pure quasifree state, and letg d (k) denote its Fourier-transformed Majorana two-point functions (see Eq. (55)). From Eq. (49) we know that ρ d is an eigenstate of The eigenvalue of H d corresponding to this state is given by Let us emphasize that the proof builds on the fact that M is fixed and finite, while d goes to infinity in the thermodynamic limit. Among the eigenstates of H d , consider the (unique) ground state ρ d gs , whose Fourier-transformed Majorana two-point functions (see Eq. (55)) will be denoted byg d gs (k). From this ground state let us construct another quasifree state ρ d e which is defined through its Majorana two-point functions This completes the proof of the theorem.

IX. PARTICLE-NUMBER CONSERVING SYSTEMS
Finally, we treat fermionic systems whose particlenumber is conserved. The corresponding system algebras are given both in the general case as well as in the quasifree case. Furthermore, a necessary and sufficient condition for quasifree pure-state controllability in this setting is provided.
Proposition 45. Consider a particle-number conserving quasifree state ρ F of a fermionic system, and let M denote its one-particle density matrix. The following statements hold: (a) The eigenvalues of M lie between 0 and 1. (b) ρ F is pure iff M is a projection. (c) If ρ F is pure, then tr(M ) = n is an integer, and ρ F is supported on the n-particle subspace ∧ n C d of the Fock space, i.e.
The dynamics of particle-number conserving quasifree fermions can also be represented using the one-particle density matrices (see [12,57]): Proposition 46. Consider a particle-number conserving quasifree state ρ a corresponding to the one-particle density matrix M a . Assume that the quadratic Hamiltonian which is defined by the Hermitian matrix A, generates the time-evolution of ρ a . The time-evolved state (at unit time), ρ b = e −iH ρ a e iH is again a number-conserving quasifree state with a one-particle density matrix . A particle-number conserving pure quasifree state ρ F with tr(M ) = n is sometimes called an n-particle pure quasifree state, since according to Proposition 45 its state is supported on the n-particle subspace ∧ n C d . We will denote the set of such quasifree pure states by QF n . A system of number-conserving quadratic Hamiltonians S = {ih 1 , . . . , ih ℓ } is said to provide quasifree pure-state controllability for a fixed particle number n if there exists an iH ∈ S Lie for any ρ a , ρ b ∈ QF n such that ρ b = e −iH ρ a e iH . To find a necessary and sufficient conditions for this type of controllability, let us invoke a Theorem 4.1 of Ref. [63]: Theorem 47. Consider the Lie algebra s Σ generated by the traceless d×d skew-Hermitian matrices iB 1 , . . . , iB ℓ and let P(d, n) denote the set of all projections acting on C d whose rank n lies between 1 and d − 1. The Lie group corresponding to s Σ acts naturally via the adjoint action on P(d, n) . This action is transitive if and only if either (a) s Σ is isomorphic to su(d) or (b) d is even, n ∈ {1, d−1}, and s Σ is isomorphic to sp(d/2).
The theorem implies the following necessary and sufficient condition: Theorem 48. Consider the set S = {ih 1 , . . . , ih ℓ } corresponding to number-conserving quadratic Hamiltonians of a fermionic system with d ≥ 2 modes. The set S generates a particle-number conserving system giving rise to full quasifree pure-state controllability on the n-particle subspace with 1 ≤ n ≤ d−1, iff either (a) d is odd and S Lie is isomorphic to u(d) or su(d) or (b) d is even, n ∈ {1, d−1} and S Lie is isomorphic to u(d), su(d), u(1) + sp(d/2), or sp(d/2).
Proof. We consider the set A = {iA (1) , iA (2) , . . . , iA (ℓ) } of skew-Hermitian matrices which correspond to the gen- We apply Remark 44 and obtain that S Lie is isomorphic to A Lie . We combine this result with Propositions 45 and 46: There exists an ih ab ∈ S Lie for each pair ρ a , ρ b ∈ QF n such that e −ih ab ρ a e ih ab = ρ b , iff there exists an iA ab ∈ A Lie for each pair M a , M b ∈ P(d, n) such that e −iA ab M a e iA ab = M b . Thus we have to find necessary and sufficient conditions under which A Lie generates a transitive action on P(d, n) Hence we can infer that A Lie generates a transitive action iff the system algebra generated by the set A ′ := {i(A (1) − tr(A (1) )½/d), . . . , i(A (ℓ) − tr(A (ℓ) )½/d)} also gives rise to a transitive action. Since A ′ contains only traceless skew-Hermitian operators, we know from Theorem 47 that it can act transitively on P(d, n) if and only if either A ′ Lie is isomorphic to su(d), or d is even, n ∈ {1, d−1}, and A ′ Lie is isomorphic to sp(d/2).
On the other hand, if A ′ Lie = su(d) or A ′ Lie = sp(d/2) then A ′ Lie is a simple irreducible Lie subalgebra of su(d). It follows that A Lie is either isomorphic to A ′ Lie if tr(A (k) ) = 0 for all k ∈ {1, . . . , ℓ} or to u(1) + A ′ Lie if there exists a k such that tr(A (k) ) = 0. This proves the theorem.

X. CONCLUSION
We have put dynamic systems theory of coherently controlled fermions into a Lie-algebraic frame in order to answer problems of controllability, reachability, and simulability in a unified picture. As summarized in Tab. IV, to this end we have determined the dynamic system Lie algebras in a comprehensive number of cases, illustrated by examples, with and without confinement to quadratic interactions (quasifree particles) as well as with and without symmetries such as translation invariance, twisted reflection symmetry, or particle-number conservation. Once having established the system algebras, the group orbits of a given (pure or mixed) initial quantum state determine the respective reachable sets of all states a system can be driven into by coherent control. In this respect, different types of pure-state reachability and its relation to coset spaces has been treated with particular attention.
There are illuminating analogies and differences between spin and fermionic systems. For quasifree systems, this was discussed in Sec. V and in Appendix C, while the translation-invariant case is addressed in Sec. VII. In particular, translation-invariant Hamiltonians which cannot be generated from nearest-neighbor ones appear both for spin systems (Sec. VII B) and for fermionic systems (Sec. VII D). Moreover, for fermionic systems some of these Hamiltonians have bounded interaction length.
It is an open question if the same also holds for spin systems.
On a general scale, the system algebras determined serve as a dynamic fingerprint. Their application to quantum simulation has been elucidated in a plethora of paradigmatic settings. Hence we anticipate the comprehensive findings presented here will find a broad scope of use. {T}, d odd a besides parity superselection rule P: T = translation-invariance, R = twisted reflection symmetry, N = particle-no. conservation b the orthogonal algebra is represented as direct sum of two equal copies given as irreducible blocks of dimension 2 d−1 ; the system algebra so(2d) itself was determined already, e.g., in Ref. [36].

ACKNOWLEDGMENTS
This work was supported in part by the eu through the programs coquit, q-essence, chist-era quasar, siqs and the erc grant gedentqopt, by the Bavarian Excellence Network enb via the international doctorate programme of excellence Quantum Computing, Control, and Communication (qccc), by Deutsche Forschungsgemeinschaft (dfg) in the collaborative research centre sfb 631 as well as the international research group for 1482 through the grant schu 1374/2-1.

Appendix A: Discussion of Double Centralizers
Motivated by Sec. IV B, in this appendix we discuss how the form of the double centralizer of a Lie algebra g ⊂ su(k) limits the possibilities for g: Proposition 49. Let g denote a subalgebra of su(k). There exists a set A ⊂ su(k) such that g = cent su(k) (A), if and only if cent su(k) (cent su(k) (g)) = g.
To further analyze the influence of symmetry properties on the system algebra, we recall some elementary representation theory (see, e.g., Theorem 1.5 of [69]): Proposition 50. Consider a completely reducible complex matrix representation Φ(g) of a group G, where k is the degree of Φ. Let comm(Φ) = Φ ′ denote the commutant algebra of all complex k × k-matrices simultaneously commuting with Φ(g) for g ∈ G. Then, Φ(g) is equivalent to where φ j denote for j ∈ {1, . . . , w} distinct inequivalent irreducible complex matrix representations of G with degree k j , occurring with multiplicity e j in Φ. In particular, Obviously, the same is true for representations of a compact Lie group or its Lie algebra. Given a subalgebra g of su(k) (or respectively of u(k)) and a representation Φ of g with degree k, we discuss the easiest case of Proposition 50 where w = 1 and e 1 = 1. Hence, Φ is irreducible and g is an irreducible subalgebra of su(k) (or respectively of u(k)). But g is not necessarily equal to su(k) (or respectively to u(k)). Irreducible simple subalgebras of su(k) were studied extensively in this regard in Ref. [19]. Note that the irreducible subalgebras of u(k) are of the form g or g+u(1) where g denotes any irreducible subalgebra of su(k) (cf. pp. 27-28 and p. 321 of [49]). -A slight generalization is given by the case of an abelian commutant algebra, i.e. dim comm(Φ) = dim center(comm(Φ)) and e j = 1 for all j ∈ {1, . . . , w}. One may thus apply the spectral theorem (see, e.g., [70][71][72]) simultaneously to all the elements of the commutant algebra: Theorem 51. Consider a Lie algebra g ⊆ su(k) and its representation Φ of degree k. Assume that the corresponding commutant algebra C = comm(Φ) is abelian. One obtains that g is a subalgebra of s[⊕ dim C j=1 u(k j )] and it is equivalent to s[⊕ dim C j=1 g j ], where k = dim C j=1 k j and g j are irreducible subalgebras of u(k j ). Furthermore, one finds k j = dim(P j ), where P j are the orthogonal projection operators given by the joint spectral decomposition of C with dim C j=1 P j = ½ k and P i P j = 0 for i = j. If g is the maximal Lie algebra with these properties, then . Using Proposition 50 one can directly characterize a maximal Lie algebra g contained in su(k) which is defined by all its symmetries including cases where the commutant to g is not necessarily abelian. Observe the notation of Remark 5 and the one of Proposition 50.
Theorem 52. Consider a Lie algebra g ⊆ su(k) and its representation Φ of degree k. Let C = comm(Φ) denote the commutant of g. If g is the maximal Lie algebra with these properties, then g = s[ ω j=1 u(k j )] where ω = dim[center(C)] and ω j=1 k j ≤ k. Proof. Using Proposition 50 (and its notation) one obtains that g is equivalent to ⊕ w j=1 ½ ej ⊗ φ j (g) . Therefore, g is a subalgebra of s[ ω j=1 u(k j )] with ω j=1 k j ≤ k. The maximality of g completes the proof.
In a dual approach, one could start from a set S of symmetries of g. Due to the maximality of g, the set S has to comprise all symmetries of g. Next, one can apply Proposition 50 to the subalgebra of su(k) generated by the linear span intersected with su(k), i.e. S ∩ su(k). The theorem then follows directly using Schur's lemma and the maximality of g.
The reader familiar with the double-commutant theorem in algebraic quantum mechanics will wonder about the different power of symmetries for characterizing algebras of observables on the one hand and Lie algebras on the other: a von-Neumann algebra A is entirely determined by its commutant A ′ , since A ′′ = A [73,74]. In this sense, there is a duality between the algebra A and its commutant A ′ encapsulating all symmetries. On the other hand, consider the illustrative case of an irreducible Lie subalgebra g of su(k) [75], where the centralizer cent su(k) (g) is trivial, i.e. zero. This centralizer is shared with all irreducible Lie subalgebras of su(k). So in turn, the double centralizer in su(k) to all these subalgebras is su(k) itself. We thus obtain the following corollary to Proposition 49 and Theorem 51, where the double centralizer gives a maximality criterion ensuring that an irreducible subalgebra g of su(k) is in fact fulfilling g = su(k) [76]: Corollary 8. Let g denote an irreducible subalgebra of su(k), i.e. cent su(k) (g) = {0}. Then one finds that cent su(k) (cent su(k) (g)) = g if and only if g = su(k) Note that Corollary 8 can be readily generalized: Let g, h denote two irreducible subalgebras of su(k) with g ⊆ h ⊆ su(k) so that cent h (g) = {0} = cent h (h). Then one finds cent h (cent h (g)) = g if and only if g = h.
Summarizing the general case, the symmetry properties of a Lie algebra g ⊆ su(k), as given by its commutant w.r.t. a representation of g, do not determine the Lie algebra g uniquely. Yet the commutant allows us to infer a unique maximal Lie algebra contained in su(k), which is (up to an identity matrix) equal to the double commutant of g, but in general not to g itself. Although all representations of compact Lie algebras, such as su(k) and its semisimple subalgebras, are completely reducible, the situation for Lie algebras also differs from the case of associative algebras: here complete reducibility of a representation implies the double-commutant theorem (see Theorem (3.5.D) of [77] or Theorem 4.1.13 of [78]), whereas the double-commutant theorem does not apply to Lie algebras as discussed above.
in Sec. V. We start with the parametrization and Im(B) = −Im(B) t which is a consequence of A = A † and B = −B t . We rewrite the Hamiltonian using Majorana operators such that By applying the Jordan-Wigner transformation we obtain the Hamiltonian for the corresponding spin system (for better readability, the tensor-product symbol is omitted, e.g., IXY := I ⊗ X ⊗ Y): Here we take new fermionic approaches to exhaustively prove and improve some results of Ref. [19], where some proofs were still sketchy-thereby also filling a desideratum voiced in [79]. 1. A Spin System with System Algebra so(2n + 1) Proposition 53 (see Proposition 27 in [19]). Consider a Heisenberg-XX chain with the drift Hamiltonian H d = XX · · II + YY · · II + · · · + II · · XX + II · · YY on n spin-1 2 qubits with n ≥ 2. Assume that one end qubit is individually locally controllable. The system algebra is given as the subalgebra so(2n + 1) which is irreducibly embedded in su(2 n ).
Proof. We use the fermionic picture where the number of modes d equals the number of spins n. The generators are given by to show that all degree-one operators can be generated. This immediately gives all quadratic operators as well, while operators of higher degree are not attainable. Therefore, the dimension of the system algebra is 2d 2 + d. Note that the operators L(m 2p−1 m 2p ) form a maximal abelian subalgebra a which proves that the system algebra has rank d. In the spin picture, we can directly verify that a = −iZ 1 /2, . . . , −iZ n /2 Lie by computing the centralizer c a := − i 2 j∈S Z j | {} = S ⊂ {1, . . . , n} Lie of a in su(2 n ). Let us compute the centralizer c b of b = m 2p m 2p+1 , −m 2p−1 m 2p+2 | p ∈ {1, . . . , d−1} Lie in su(2 n ). Note that the generators of b are given in the spin picture by −iX p X p+1 /2 and −iY p Y p+1 /2. One can readily show by induction that It follows that the centralizer c of the full system algebra in su(2 n ) has to be contained in c a ∩ c b = − i 2 n j=1 Z j Lie . One can now easily prove that the centralizer of the full system algebra in su(2 n ) is trivial and that the system algebra is irreducibly embedded in su(2 d ). As the coupling graph of the spin system is connected, we conclude with Theorem 6 of [19] that the system algebra is simple. Listing all simple (and compact) Lie algebras with the correct dimension and rank, we obtain (a) so(2d + 1) for d ≥ 1, (b) sp(d) for d ≥ 1, (c) su(2) ∼ = so(3) for d = 1, and (d) e 6 for d = 6. As the system algebra contains also all quadratic operators, it has a subalgebra so(2d) which is of maximal rank. This rules out the cases (b) and (d) (see p. 219 of [48] or Sec. 8.4 of [49]) for d = 2. But the case (b) agrees with (a) for d = 2. For d = 1, the cases (a) and (c) coincide. This completes the proof.
Note that with our fermionic approach one can readily determine the dimension and rank of the system algebra. Likewise, we establish that all fermionic operators act irreducibly from which we can infer that the system algebra is simple. The rest of the proof follows by an exhaustive enumeration.-In more general terms, as in Theorem 34 and Corollary 35 of [19], we connect a spin system with a fictitious fermionic system: Corollary 54. Consider a fictitious fermionic system with d modes which consists of all linear and quadratic operators and whose generators can, e.g., be chosen as all Majorana operators of type L(m 2p−1 ) combined with the Hamiltonian from Eq. (21) where the control functions A pq and B pq can be assumed to be real. This fictitious fermionic system and the spin system of Proposition 53 with n = d spins can simulate each other. In particular, both can simulate a general quasifree fermionic system with d modes and system algebra so(2d) as presented in Proposition 9 and Theorem 11.

A Spin System with System Algebra so(2n + 2)
Proposition 55 (see Proposition 28 in [19]). Consider a Heisenberg-XX chain with the drift Hamiltonian H d = XX · · II + YY · · II + · · · + II · · XX + II · · YY on n spin-1 2 qubits with n ≥ 2. Assume that each of the two end qubits is individually locally controllable. The system algebra is given as the subalgebra so(2n+2) which is irreducibly embedded in su(2 n ).
Proof. We switch to a fermionic picture where the number of modes d equals the number of spins n. The generators are and L(m 2d d−1 p=1 m 2p−1 m 2p ). One can verify by explicit computations that exactly all Majorana operators of degree one, two, 2d−1, and 2d can be generated. Therefore, the dimension of the system algebra is 2d 2 +3d+1. Using a similar argument as in the proof of Proposition 53, we conclude that the operators L(m 2p−1 m 2p ) together with the operator L( d p=1 m 2p−1 m p ) form a maximal abelian subalgebra which proves that the system algebra has rank d + 1. One can also show that the system algebra is irreducibly embedded in su(2 d ). As the coupling graph of the spin system is connected, we conclude with Theorem 6 of [19] that the system algebra is simple. The proof is completed by listing all simple (and compact) Lie algebras with the correct dimension and rank. We obtain (a) so(2d + 2) for d ≥ 1 and (b) su(4) ∼ = so(6) for d = 2.
Principle Remark: Now we have established a setting that allows for exploiting the powerful general results of [64] on the structure of orthogonal groups that provide a second avenue to Proposition 53 assuming we have already established Proposition 55: Lemmata 3 and 4 of [64] show that for k ≥ 3 any subalgebra of so(k) with dimension (k − 1)(k − 2)/2 is isomorphic to so(k − 1); moreover so(k − 1) is a maximal subalgebra of so(k). Thus, by proving that the system algebra has dimension 2d 2 + d with d ≥ 1, it can be identified as the subalgebra so(2d + 1) of so(2d + 2). We emphasize that this particular proof technique should be widely applicable in quantum systems theory.
Relying on the proof of Proposition 55 and building on Theorem 32 as well as Corollary 34 of [19], we obtain connections between a spin system, a quasifree fermionic system, and a fictitious fermionic system: Corollary 56. The following control systems all have the system algebra so(2k + 2) and can simulate each other: (a) the spin system of Proposition 55 with k spins, (b) the quasifree fermionic system with k + 1 modes as presented in Proposition 9 and Theorem 11, and (c) a fictitious fermionic system with k modes which contains all Majorana operators of degree one, two, 2k−1, and 2k, and whose generating Hamiltonian can be chosen from Eq. (21) where the control functions A pq and B pq can be assumed to be real. 1. Generalizing a Key Observation of Ref. [16] Proposition 57. The trace of the product of U −K T with a tensor product of Pauli operators Q i ∈ {½ 2 , X, Y, Z} can be computed as Proof. To simplify our calculations, let us introduce the notation v(ℓ) = (K + ℓ) mod L, note that (v • v)(ℓ) = v(v(ℓ)) = (2K + ℓ) mod L, or more generally v •p (ℓ) = (pK + ℓ) mod L. We can now write the action of U −K T on an arbitrary standard basis vector as Without loss of generality we can confine the discussion to the case where K ≤ L. We complete the proof, by evaluating the trace in Eq. (E1) as which can be further simplified to The Lie algebras t M and t M+1 are generated by elements of the form of respectively. Here, we consider all combinations of Q p ∈ {½ 2 , X, Y, Z} apart from the case when Q 1 = ½ 2 . We where a ∈ {1, −1} and W ∈ {M, M +1}. Using Proposition 57, we compute the formulas It follows that the respective statements in the proposition hold for the generators of t M and t M+1 . Now we prove this consequence also for any element in t M (or t M+1 ). First, let us note that the elements generated must be contained in . Second, since all elements in t M+1 (and hence in t M ) commute with U qM T , we have that tr(U qM T ih) = 0 holds for any element ih ∈ [t M+1 , t M+1 ], as Thus Proposition 26 follows.
Appendix F: Proof of Theorem 33 for d Even Let us introduce the notation N 2 , which corresponds to the linear space spanned by the nearest-neighbor (and on-site) operators. Note that N 2 forms only a linear space and is in general not equal to the Lie algebra t f 2 generated by its elements. We first prove a fermionic generalization of Lemma 26.
Lemma 58. Consider a fermionic system for which the number d ≥ 6 of modes is even. For any ih ∈ N 2 the condition tr(ih U −2 ) = 0 holds if d mod 4 = 2, while tr(ih U −4 ) = 0 holds if d mod 4 = 0.
Proof. By definition, any element ih ∈ N 2 can be written as ih = d−1 n=0 U n ih 12 U −n , where ih 12 is a traceless skew-Hermitian operator acting only on the first two modes of the fermionic system. Therefore, ih 12 is a linear combination of the elements i m 1 m 2 m 3 m 4 and m a m b where a, b ∈ {1, 2, 3, 4} and a = b. We obtain that tr(ih (F1) In the sum given above, the basis vectors are orthogonal and thus most of the terms are zero. The only terms with non-zero contributions can occur in the cases of n 1 = n 2ℓ−1 and n 2 = n 2ℓ with ℓ ∈ {1, . . . , d/2}. In particular, we have κ(n 1 , n 2 , n 1 , n 2 , . . . , n 1 , n 2 ) = 1 as d/2 is an odd number if d mod 4 = 2. Hence we obtain that n 1 , n 2 , n 1 , n 2 , . . . |ih 12 |n 1 , n 2 , n 1 , n 2 , . . . Lemma 59. Consider a fermionic system for which the number d ≥ 6 of modes is even. The properties tr(ih e U −2 ) = 0 and tr(ih e U −4 ) = 0 hold for the operator ih e of Theorem 33.
Using χ = n 1 , n 2 , n 3 , n 4 , n 1 , n 2 , n 3 , n 4 , . . . , n 4 we can simplify the trace to Now we can prove Theorem 33 for even d as summarized in the following proposition: Proposition 60. Consider a fermionic system for which the number d ≥ 6 of modes is even. The fourth-neighbor element ih e ∈ t f 5 of Theorem 33 is not contained in the system algebra t f 2 of nearest-neighbor interactions. Proof. We introduce the operator It follows from Lemma 58 that the equality tr(ih C d ) = 0 holds for any ih ∈ N 2 . Since C d commutes with all elements of t f 2 and t f 2 = span(N 2 , [t f 2 , t f 2 ]), we have tr(C d [ih 1 , ih 2 ]) = tr(C d ih 1 ih 2 ) − tr(C d ih 2 ih 1 ) = tr(C d ih 1 ih 2 ) − tr(ih 1 C d ih 2 ) = tr(C d ih 1 ih 2 ) − tr(C d ih 1 ih 2 ) = 0.
This means that tr(C d ik) = 0 for all ik ∈ t f 2 . On the other hand, we know from Lemma 59 that tr(ih e C d ) = 0. This means that ih e / ∈ t f 2 .
Appendix G: Proof of Theorem 33 for d Odd which implies that Applying the formulasf k |0 = 0 and [f † kfk ,f † k ′f k ′ ] = 0, we conclude that exp[−i In the next step, we provide a polynomial of U which multiplied by any nearest-neighbor Hamiltonian gives an operator with zero trace (if the system is composed of an odd number of modes). One key observation is that the action of the twisted reflection operator on the translation unitary is which follows directly form the definition of R, see Eq. (51). Using this equation and Lemma 61, one can prove the following statement: Lemma 62. Consider a fermionic system for which the number d ≥ 5 of modes is odd and introduce the operator The equality tr(ih C ′ d ) = 0 holds for any ih ∈ t f 2 .
Proof. We will first prove that tr(v C ′ d ) = 0 holds for all v ∈ N 2 , where N 2 denotes the linear space spanned by the nearest-neighbor interactions (as in Appendix F). The equation R C ′ d R † = −C ′ d follows from Eq. (G7). On the other hand, Eq. (51) implies that R ih R † = ih holds for any ih ∈ {ih 0 , ih rh , ih rp , ih cp , ih int }, hence tr(ih C ′ d ) = tr(R ih R −1 R C ′ d R −1 ) = − tr(ih C ′ d ) = 0. In order to calculate tr(ih ch C ′ d ), we first note that using Eq. (52a) the operator ih ch can be written as Next, let us expand U 2 using Lemma 61: where M 1 is a linear combination of Majorana monomials of degree greater than two and λ 1 := d−1 k=0 cos 2πk d . Similarly, let us expand U 4 : where M 2 is a linear combination of Majorana monomials of degree greater than two. We employed that We note that all monomials of Fourier-transformed Majorana operators have zero trace and determine the traces tr(U 2 ih ch ) and tr(U 4 ih ch ) by calculating the coefficient of ½ in U 2 ih ch and U 4 ih ch : tr(U 2 ih ch ) = 2 d λ 1 Note that tr(ih ch U −ℓ ) = tr(R ih ch R † R U −ℓ R † ) = − tr(ih ch U ℓ ), which allows us to conclude tr(ih ch C ′ d ) = 2(−1) ⌊d/4⌋ tr(ih ch U 2 ) − 2(−1) d tr(ih ch U 4 ).
This implies that tr(C ′ d ih ch ) = 0, and thus tr(v C ′ d ) = 0 holds for all v ∈ N 2 . As C ′ d commutes with all elements of t f 2 , it also follows that tr(ih C ′ d ) = 0 for any ih ∈ t f 2 . After these preparations we can prove Theorem 33 for odd d as summarized in the following proposition: Observe that tr(ih o U −ℓ ) = tr(R ih o R † R U −ℓ R † ) = − tr(ih o U ℓ ) and conclude that the formula tr(ih o C ′ d ) = 2(−1) ⌊d/4⌋ tr(ih o U 2 ) − 2(−1) d tr(ih o U 4 ) holds. Now, the expansion of U given by Eq. (G10) allows us to calculate the trace of ih o C ′ d : On the other hand, we know from Lemma 62 that the equality tr(C ′ d ih) = 0 holds for any ih ∈ t f 2 . Therefore, ih o / ∈ t f 2 .