Hamiltonian purification Available to Purchase

Actually one should work in an interaction picture, in which the system undergoes a spontaneous unitary evolution given by a free Hamiltonian H_free, to which an interaction is added as prescribed by the experimenter.¹⁰ To simplify the discussion, we will assume that this step has already been factored out, i.e., we imagine that H_free = 0.

As a matter of fact, since global phases are irrelevant in quantum mechanics, it would be sufficient to focus on the algebra 𝔰𝔲(d) formed by the traceless self-adjoint d × d complex matrices.

More precisely, P = I_d ⊕ 0 is a rank-d orthogonal projection acting on the Hilbert space $ℋdE=ℋd⊕ℋd⊥$⁠, and Eq. (3) should read h_j ⊕ 0 = PH_jP. By abuse of notation, we will use the same symbol for h_j on ℋ_d and for its extension h_j ⊕ 0 on ℋ_dE. More generally, we can consider a Hamiltonian purification in a space $ℋdE=ℋ˜d⊕ℋ˜d⊥$⁠, where $ℋ˜d$ is isomorphic to ℋ_d. Again, in the following, we will not be pedantic in distinguishing isomorphic spaces and will commit the sin of denoting them with the same symbols.

A normal completion of a complex matrix M ∈ ℂ^d×d is a normal matrix M_E ∈ ℂ^d_E×d_E such that M = PM_EP, with P an orthogonal projector from ℂ^d_E to ℂ^d. There is a one to one correspondence between complex matrices M and ordered pairs of Hermitian matrices (h₁, h₂), given by M ≡ h₁ + ih₂. It is straightforward to verify that M is normal if and only if h₁ and h₂ commute. Thus, there is a one to one correspondence among purifications of pairs of Hermitian matrices and normal completions of complex matrices. Therefore, all the results on normal completions given in Refs. 17–19 apply also to purifications of two Hamiltonians.

Notice that this provides a Hamiltonian purification up to the identification $ℋd≅ℋd⊗C0$⁠, with the subspace $ℋd⊗C0⊂ℋdE$⁠, and the extension of h_j in ℋ_dE is $hj⊗00$⁠. See Ref. 26.

We assume that maxσ(h₁) is not degenerated, but it is not a restrictive assumption: if it is degenerated, one can purify the operators in a Hilbert space with a smaller dimension d_E.

Consider a Hermitian traceless operator A. This is always unitarily equivalent to a Hermitian operator with zeros on the diagonal. Hence we assume, without loss of generality, that A has zero diagonal. Take h₁ diagonal with real and distinct diagonal entries λ₁, …, λ_d. The equation A = i[h₁, h₂] then is equivalent to $i(λi−λj)hij(2)=aij$⁠, which has a solution $hij(2)=aij/[i(λi−λj)]$⁠.

Denote with $λk*$ the eigenvalue with highest multiplicity of h_k, and with $gk(max)$ its multiplicity. Then, with the replacement $hk→hk−Idλk*$⁠, we can assume h_k to be of rank $d−gk(max)$⁠.

In general, h_i is of rank $d−gi(max)$⁠, so $Ui′$ can be taken as a $d×(d−gi(max))$ matrix and $D˜i∈Diag(d−gi(max))$⁠.

Identity (A5) clarifies that the $fPU(uˆii)$ cannot be all mutually orthogonal with respect to the Hilbert-Schmidt scalar product. Indeed there can be at most d such matrices which are orthogonal. To see this simply observe that $Tr[fPU(uˆii)fPU†(uˆjj)]=Xj†⋅Xi2$ and remember that the Xⁱ are d² column vectors of ℂ^d.

To see this simply observe that for n < m, the matrices $en†×en±em†×em$⁠, $en,m(+)†×en,m(+)−en†×en−em†×em$ and $en,m(−)†×en,m(−)−en†×en−em†×em$ form the set of generalized Pauli operators in the subspace spanned by the vectors e_n and e_m.