Superadditivity of Krylov Complexity for Tensor Products

TL;DR

Addresses how Krylov (spread) complexity behaves when two quantum systems are combined via a tensor product. Uses an analytic proof based on a positive semidefinite excess operator $\hat{\Delta}_C$ and a Krylov-graph/diffusion framework that maps operator growth onto a two-dimensional lattice with diagonal shells and binomial weights. Finds a general superadditivity, $C_{12} \ge C_1 + C_2$, with equality only for synchronous evolution, and characterizes the excess by the diagonal/off-diagonal structure of $\hat{\Delta}_C$, illustrated explicitly in $su(2)$ examples. In the continuum limit, Krylov dynamics reduces to a two-dimensional diffusion with drift, and the excess is tied to the curvature of the Krylov geometry, explaining why tensor-product systems can exhibit enhanced but non-chaotic growth.

Abstract

We study Krylov complexity for quantum systems whose Hamiltonians factorise as tensor products. We prove that complexity is superadditive under tensor products, $C_{12}\ge C_1+C_2$, and identify a positive operator that quantifies the resulting excess complexity. The underlying mechanism is made transparent by introducing a Krylov graph representation in which tensor products generate a higher-dimensional lattice whose diagonal shells encode operator growth and binomial path multiplicities. In the continuum limit, Krylov dynamics reduces to diffusion on this graph, with superadditivity arising from geometric broadening across shells. Explicit examples illustrate how deviations from synchronous evolution generate bounded, oscillatory excess complexity.

Figures (5)

• Figure 1: The excess complexity of the time-evolved lowest weight state for $j_1 = j_2 = \frac{1}{2}$, $\alpha_1 = 1$ and various values of $\alpha_2$. The excess complexity is always positive, bounded and oscillatory.
• Figure 2: The excess complexity of the time-evolved lowest weight state for $j_1 = 1; j_2 = \frac{1}{2}$, $\alpha_1 = 1$ and various values of $\alpha_2$. The excess complexity is always positive, bounded and oscillatory.
• Figure 3: The excess complexity for $j_1 = j_2 = 2$, $\alpha_1 = 1$ and various values of $\alpha_2 \approx 1$. At small times the excess complexity remains small relative to the complexity (which has a maximal value of approximately $2(j_1 + j_2)$)
• Figure 4: The excess complexity for $j_1 = j_2 = 2$, $\alpha_1 = 1$ and various values of $\alpha_2 \approx 1$. At larger times the excess complexity becomes large for all cases away from the synchronous point, though the time-scale is longer for cases with evolution closer to synchrony.
• Figure 5: The grid of the tensor product of Krylov vectors. States with the same colour are eigenstates of $\hat{C}_1 + \hat{C_2}$ with the same eigenvalue.