First, as we have seen above, the smallest contributions to the Schmidt spectrum are left away, the state being faithfully represented up to: ϵ ϵ ( D ) = 1 − − ∏ ∏ n = 1 N − − 1 ( 1 − − ϵ ϵ n ) where ϵ ϵ n = ∑ ∑ α α = χ χ c χ χ ( λ λ α α [ n ] ) 2 is the sum of all the discarded eigenvalues of the reduced density matrix, at the bond n . The state | ψ ψ ⟩ ⟩ is, at a given bond n , described by the Schmidt decomposition: | ψ ψ ⟩ ⟩ = 1 − − ϵ ϵ n | ψ ψ D ⟩ ⟩ + ϵ ϵ n | ψ ψ D ⊥ ⊥ ⟩ ⟩ where | ψ ψ D ⟩ ⟩ = 1 1 − − ϵ ϵ n ∑ ∑ α α n = 1 χ χ c λ λ α α n [ n ] | Φ Φ α α n [ 1.. n ] ⟩ ⟩ | Φ Φ α α n [ n + 1.. N ] ⟩ ⟩ is the state kept after the truncation and | ψ ψ D ⊥ ⊥ ⟩ ⟩ = 1 ϵ ϵ n ∑ ∑ α α n = χ χ c χ χ λ λ α α n [ n ] | Φ Φ α α n [ 1.. n ] ⟩ ⟩ | Φ Φ α α n [ n + 1.. N ] ⟩ ⟩ is the state formed by the eigenfunctions corresponding to the smallest, irrelevant Schmidt coefficients, which are neglected. Now, ⟨ ⟨ ψ ψ D ⊥ ⊥ | ψ ψ D ⟩ ⟩ = 0 because they are spanned by vectors corresponding to orthogonal spaces. Using the same argument as for the Trotter expansion, the error after the truncation is: ϵ ϵ n = 1 − − | ⟨ ⟨ ψ ψ | ψ ψ D ⟩ ⟩ | 2 = ∑ ∑ α α = χ χ c χ χ ( λ λ α α [ n ] ) 2