Krylov complexity of purification

TL;DR

The paper develops a Krylov-framework for mixed-state complexity by purifying a density matrix into a pure state in an enlarged Hilbert space and studying three purification schemes (time-independent, time-dependent, instantaneous). It relates Krylov operator and state complexities of the purified system to the original mixed-state complexity and proves a set of inequalities, validated in one-qubit and two-qubit systems as well as eight-dimensional random ensembles. For arbitrary one-qubit mixed states, it derives closed-form CoPs and demonstrates key bounds, including $\mathcal{C}_K^{\mathbb{I}} \ge 2\mathcal{C}_S^{\mathbb{I}}$ and $\mathcal{C}_S \le \mathcal{C}_K$, with the ratios showing weak time dependence set by purity. The study extends to infinite-dimensional thermal states, where the spread CoP obeys a Lloyd-like bound and a quasiparticle (Rindler/TMSV) interpretation emerges; mutual Krylov complexity reveals subadditivity, contrasting holographic complexity proposals and suggesting CoPs as diagnostic tools for gravity duals. Overall, the work provides a cohesive framework linking mixed-state Krylov complexity, purification dynamics, and potential gravity duals, with practical diagnostics across finite and infinite-dimensional systems.

Abstract

In quantum systems, purification can map mixed states into pure states and a non-unitary evolution into a unitary one by enlarging the Hilbert space. We establish a connection between the complexities of mixed quantum states and their purification, proposing new inequalities among these complexities. By examining single qubits, two-qubit Werner states, eight-dimensional Gaussian random unitary ensembles, and infinite-dimensional systems, we demonstrate how these relationships manifest across a broad class of systems. We find that the spread complexity of purification of a vacuum state evolving into a thermal state equals the average number of Rindler particles. This complexity is also shown to adhere to the Lloyd-like bound, indicating a further relation to the quantum speed limit. Finally, using mutual Krylov complexity, we observe subadditivity of the Krylov complexities, which contrasts with known results from holographic volume complexity. We put forward Krylov mutual complexity as a diagnosis of a potential gravity dual of Krylov complexities.

Figures (4)

• Figure 1: The top left box shows a canonical purification $\ket{\psi_\rho}$ for a mixed state $\rho$. The central blue and orange boxes show three diferent purification schemes (time-independent, time-dependent, and instantaneous). The rightmost boxes highlight the conjectured bounds for mixed and purification complexities.
• Figure 2: Time dependence of the complexities of two states: [Row 1] Werner states \ref{['w_state']} evolved by the Hamiltonian \ref{['eq:Hamiltonian_werner']} with $r=4,q=15$; [Row 2] Eight-dimensional Gaussian random ensembles with the initial inverse temperature $\beta=1$ (solid line) and $\beta=3$ (dashed line). [Col. 1] CoPs $\mathcal{C}_K(|\Psi_{\rho}^{\mathbb{I}}(t)\rangle)$ (red), $\mathcal{C}_K(\rho(t))$ (blue), and $\mathcal{C}_S(|\Psi_{\rho}^{U^\ast}(t)\rangle)$ (green). In (a), $p=1/4$ corresponds to a solid curve and $p\rightarrow 1$ corresponds to dotted points. [Col. 2] Ratio between the operator complexity and CoPs: $\mathcal{C}_K(|\Psi_\rho^{\mathbb{I}}(t)\rangle)/\mathcal{C}_K(\rho(t))$ (dashed curve) and $\mathcal{C}_S(|\Psi_\rho^{U^\ast}(t)\rangle)/\mathcal{C}_K(\rho(t))$ (solid curve). In (b), the ratios with $p=1/4$ (blue), $p=1/3$ (red), $p=1/2$ (green), and $p\rightarrow 1$ (black) are plotted in log scale. [Col. 3] Ratio $\mathcal{C}_K/\mathcal{C}_S$ for time-independent (dashed) and time-dependent (solid) purifications. In (c), $p=1/4$ (blue), $p\to 1$ (red), with reference line at $2$ (black).
• Figure 3: Time dependence of the complexities of generic one-qubit mixed states \ref{['eq:notation-1q']} with different parameters $\Delta p=0.4,\theta=\pi/3$ (solid curve) and $\Delta p=0.98,\theta=\pi/4$ (dashed curve). (a) Three complexities $\mathcal{C}_K^{\mathbb{I}}(t)$ (red), $\mathcal{C}_K(t)$ (blue), and $\mathcal{C}_S^{U^\ast}(t)$ (green). (b) Ratio between the operator complexity and CoPs: $\mathcal{C}_K^{\mathbb{I}}((t))/\mathcal{C}_K(t)$ (blue) and $\mathcal{C}_S^{U^\ast}(t)/\mathcal{C}_K(t)$ (red) in log plot. (c) Ratio $\mathcal{C}_K/\mathcal{C}_S$ between the state and operator CoPs for the time-independent purification (blue) and the time-dependent purification (red) with a reference line at $2$ (black).
• Figure 4: In both panels, the red color signifies the Hamiltonian parameters of (6) with $r=4$ and $q=15$, while the blue color signifies the Hamiltonian parameters with $r=10$ and $q=50$. (a) The empty circles, in both red and blue, signify the purified state with $p\rightarrow 1$, and the solid lines signify $p=1/4$ for the initial state (5). (b) The solid triangles, in both red and blue, signify the purified state with $p\rightarrow 1$, and the solid lines signify $p=1/4$. For both the operator and the state complexity, the complexity overlaps for the different purity parameters with the same Hamiltonian but differs for different Hamiltonian parameters.