bibkey: das2026krylovpurification
authors: Rathindra Nath Das, Takato Mori
year: 2026
title: "Krylov complexity of purification"
doi: 10.1103/qgcx-wxpd
url: https://arxiv.org/abs/2408.00826v4
claim: "For a density matrix ρ evolving as ρ(t) = U_t ρ U_t† with U_t = e^{−iHt}, the authors conjecture (Eq. (4)) that the Krylov operator complexity of the mixed state is bounded below by the spread complexity of the time-dependent purification |Ψ^{U*}_ρ(t)⟩ = (U_t ⊗ U_t^*)|Ψ_ρ⟩ and above by the operator complexity of the time-independent purification: C_S(|Ψ^{U*}_ρ(t)⟩) ≤ C_K(ρ(t)) ≤ C_K(|Ψ^I_ρ(t)⟩), with equality in the pure-state limit; they confirm it analytically for one-qubit states."
strata_touched:
  - D5/S3/Quantum/Dynamics/PurificationLowerBoundQutritRefutation
license: citation-only
triage: anchor

Das–Mori, Krylov complexity of purification

R. N. Das and T. Mori, Krylov complexity of purification, Phys. Rev. Lett. 136, 030201 (2026), DOI 10.1103/qgcx-wxpd; arXiv:2408.00826 [hep-th] (v4 21 January 2026).

Verified locator

DOI: 10.1103/qgcx-wxpd (Phys. Rev. Lett. 136, 030201, as listed on the arXiv abstract page). Primary version read: https://arxiv.org/abs/2408.00826v4 (the latest of versions v1–v4, dated 21 January 2026, the published version). Source: https://arxiv.org/pdf/2408.00826v4 (text read from the PDF):

• “Definitions”, Eqs. (1)–(3): the time-independent, time-dependent and instantaneous purifications;
• “Inequalities among complexities”, Eq. (4) and conjectures 1–3;
• “Examples”: the one-qubit confirmation and the Werner-state example, Eqs. (5)–(6);
• End Matter, Eqs. (15)–(16), and Supplemental Material, Eqs. (S.1)–(S.11): spread complexity and Krylov operator complexity.

Source statements

“Definitions”: “Given a density matrix ρ of a system S, its purification is generically written as |Ψ^V_ρ⟩{SA} = (ρ^{1/2} ⊗ V{R→A})|I⟩{SR}, where V is an isometry from R to A, i.e. V†V = 1_R, and |I⟩{SR} = Σ_{n=1}^{rank ρ} |n⟩_S ⊗ |n⟩_R is an unnormalized EPR state. For ρ evolving unitarily by U_t = e^{−iHt}, the purification evolves as |Ψ^V_ρ(t)⟩ = (U_t ρ^{1/2} ⊗ V(t))|I⟩ = (U_t ⊗ V(t))|Ψ_ρ⟩ … Choosing V(t) = U_t^, we define the time-dependent purification after time t by |Ψ^{U}_ρ(t)⟩ = (U_t ρ^{1/2} U_t† ⊗ I)|I⟩ = (U_t ⊗ U_t^)|Ψ_ρ⟩. (2)” Before Eq. (6): “For the time-independent and time-dependent purification, the evolution is governed by H ⊗ I and H ⊗ I − I ⊗ H^, respectively.”

“Inequalities among complexities”: “Based on the three types of purifications defined above, we conjecture inequalities for the complexities of mixed states and their purifications, with analytical and numerical evidence provided in the next section. 1. Krylov operator complexity of the mixed state is upper bounded by the operator complexity of the time-independent purification and is lower bounded by the state complexity of the time-dependent purification, C_S(|Ψ^{U*}_ρ(t)⟩) ≤ C_K(ρ(t)) ≤ C_K(|Ψ^I_ρ(t)⟩) (4) where the equality holds in the pure state limit.”

“Examples”: “A one-qubit mixed state under a generic unitary evolution is simple enough that the analytic expressions of its Krylov CoPs can be written down … the conjectures in the last section are analytically confirmed.”

Supplemental Material, Eq. (S.4): “the spread complexity of the state |ψ(t)⟩ is defined as C_S(t) ≡ Σ_{n=0}^{D−1} n|ψ_n(t)|²”, where ψ_n(t) are the amplitudes in the Lanczos basis |K_n⟩ of the Krylov sequence |ψ⟩, H|ψ⟩, H²|ψ⟩, …; Eqs. (S.5)–(S.7): the operator analogue, built from O, LO, L²O, … with the Liouvillian L = [H, ·] and the Hilbert–Schmidt inner product.

Reading used by the repository

For full-rank ρ the source’s |Ψ_ρ⟩ is the canonical purification Σ_{ij}(√ρ)_{ij}|i⟩|j⟩, and the time-dependent purification is generated by H ⊗ 1 − 1 ⊗ Hᵀ (Hᵀ = H^* for Hermitian H). Both complexities are the mean Lanczos index Σ_n n|⟨K_n, φ(t)⟩|² of the normalized initial vector evolved by the generator, with K_n the Gram–Schmidt orthonormalization of the Krylov sequence; the operator space carries the Hilbert–Schmidt inner product through row-major vectorization.