Symmetrizing Evolutions

The importance of the notion of symmetry in quantum theory cannot be overstimated [1]. The associated statespace decomposition into dynamically invariant sectors is a highly desirable property in that it can strongly simplify the analysis of the system evolution. Suppose that on the state-space H of a quantum system S acts a group G via a representation ρ. In general the Hamiltonian H of S is not G-invariant i.e., [H, ρ(G)] 6= 0. The goal of this letter is to present a quantum procedure for generating an effective dynamics on H ruled by an operator H̃ that is the G-invariant component of H. It amounts to a sort of generalized Fourier transform is which one discards all the non-zero (i.e., non-translation invariant) components. We first discuss a procedure that involves frequently iterated measurements. The key idea is very simple: introducing an auxiliary space and resorting to the intrinsic parallelism of quantum dynamics one can simultaneously evolve all the group-rotated copies of an initial state. Then by repeated measurements one singles out the G-invariant component of the dynamics. After discussing several applications to state preparation, decoherence avoiding/suppression and constrained dynamics, we show that symmetrization can be achieved by purely unitary means and without additional space resources. This formulation will make apparent that the recently proposed schemes for decoherence control [2], [3] in quantum computers [4] are nothing but special cases of this general group-theoretic idea. For the sake of clarity in this letter we will concentrate on physical examples mostly suggested by quantum computation. A deeper analysis of the algebraic structures involved along with further applications will be presented elsewhere. Let us begin by a simple example aimed to give a first hint about the possible use of G-symmetrization for noise suppression. Let S be a single two-level system (qubit) dissipatively coupled with an environment E. H = C ⊗HE , and H = H0 +H1, where H0 = ε σz ⊗ 1 + 1 ⊗HE , H1 = σ ⊗ E + σ− ⊗ E†. (1) Here HE (HE) is the environment Hamiltonian (statespace) and E, E† operators associated to the creation/annihilation of elementary excitations of E. On the total space acts the group {g0 = 1 , g1 = σ ⊗ 1 } ∼= Z2. The operators transform according the adjoint action: X 7→ g† αX gα, (α = 0, 1). It is immediate to check that whereas the first two terms in H (the self-Hamiltonians) are invariant under the action of σ , the interaction part changes sign (σ σ± σ = −σ±). Therefore by “ averaging over the group ” H one finds H̃ = 2−1 ∑ α g † αH gα = H0. This tell us that if one, in some way, were able to make the system evolving according H̃ the interaction with the environment would be washed out. Invariant subspaces. Now we set the general framework and recall the relevant group/representation-theoretic notions [1]. The general situation can be abstractly defined in terms of the data (H, H, G, ρ) where i) H is a finite dimensional Hilbert space, ii) H an hermitian operator (Hamiltonian) over H, iii) G a finite group of order |G|, iv) ρ : g ∈ G 7→ ρg = exp(i hg), a unitary representation of G in H [ ρgh = ρg ρh, ρg−1 = ρg]. The representation ρ is irreducible (irrep) if it does not admit non-trivial invariant subspaces in H. The space H splits according the G-irreps: H = ⊕JnJ HJ where nJ is the multiplicity of invariant subspace HJ associated to the J-th irrep of G. For instance the abelian (additive) group Z2 = {0, 1} has two (1-d) irreps ρJ(α) = e J , the identical (J = 0) and the antisymmetric one (J = 1).