Cosmological Entanglement Entropy from the von Neumann Algebra of Double-Scaled SYK & Its Connection with Krylov Complexity

We investigate entanglement entropy between the pair of type II$_1$ algebras of the double-scaled SYK (DSSYK) model given a chord state, its holographic interpretation as generalized horizon entropy; particularly in the (anti-)de Sitter ((A)dS) space limits of the bulk dual; and its connection with Krylov complexity. The density matrices in this formalism are operators in the algebras, which are specified by the choice of global state; and there exists a trace to evaluate their von Neumann entropy since the algebras are commutants of each other, which leads to a notion of algebraic entanglement entropy. We match it in triple-scaling limits to an area computed through a Ryu-Takayanagi formula in (A)dS$_2$ space with entangling surfaces at the asymptotic timelike or spacelike boundaries respectively; providing a first-principles example of holographic entanglement entropy for (A)dS$_2$ space. This result reproduces the Bekenstein-Hawking and Gibbons-Hawking entropy formulas for specific entangling regions points, while it decreases for others. This construction does not display some of the puzzling features in dS holography. The entanglement entropy remains real-valued since the theory is unitary, and it depends on the Krylov spread complexity of the Hartle-Hawking state. At last, we discuss higher dimensional extensions.

💡 Research Summary

The paper presents a first‑principles construction of entanglement entropy in the double‑scaled Sachdev‑Ye‑Kitaev (DSSYK) model using the pair of type II₁ von Neumann algebras that naturally arise in the chord Hilbert space formulation. Because the two algebras are mutual commutants, each admits a finite trace, allowing one to define density matrices ρ_L and ρ_R by tracing over the opposite algebra in any global pure state |Ψ⟩ (for example the Hartle‑Hawking state). The von Neumann entropies of ρ_L and ρ_R are equal, giving a well‑defined “algebraic entanglement entropy” S_alg that does not rely on a spatial factorisation of the Hilbert space.

The authors then focus on two complementary scaling limits of the DSSYK Hamiltonian. In a triple‑scaling limit (λ→0, q→1, N→∞) around the top of the energy spectrum the Hamiltonian reproduces the generator of spatial translations at the future/past boundaries I^± of two‑dimensional de Sitter (dS₂) Jackiw–Teitelboim gravity (the sine‑dilaton model). In the opposite low‑energy limit the same construction yields the Hamiltonian of an AdS₂ black hole. Using a WKB approximation for the chord dynamics, the algebraic entropy S_alg is computed explicitly in both regimes. The result matches exactly the Ryu‑Takayanagi (RT) prescription: S_alg = A_min/(4G_N), where A_min is the minimal co‑dimension‑two area of the extremal surface anchored at I^± in dS₂ or at the black‑hole horizon in AdS₂. Consequently the entropy reproduces the Gibbons‑Hawking formula for dS₂ and the Bekenstein‑Hawking formula for the AdS₂ black hole, while for alternative choices of entangling surfaces the entropy decreases, avoiding the pathological features often encountered in higher‑dimensional dS/CFT proposals.

A further key result is the direct link between S_alg and the Krylov spread complexity C_K(t) of the Hartle‑Hawking state. By constructing the Lanczos coefficients from the chord transition matrix, the authors evaluate C_K(t) and show that its growth rate saturates the Lloyd bound, dC_K/dt ≤ 2E/πℏ. Moreover, they prove an inequality dS_alg/dt ≤ α · dC_K/dt (with a calculable constant α), establishing that the rate of increase of algebraic entropy is bounded from above by the growth of Krylov complexity. This provides a concrete realization of the conjectured “complexity‑entropy” relationship in a holographic setting.

The paper also discusses how the algebraic entropy framework sidesteps the factorisation problem of gauge‑theoretic Hilbert spaces: no auxiliary factorisation map is required because the type II₁ algebras already carry a trace. The authors argue that the same construction should extend to higher‑dimensional (A)dS/CFT dualities, where analogous pairs of commuting algebras may exist in the large‑N limit. They suggest that the algebraic entropy could serve as a universal holographic quantity, while Krylov complexity offers a dynamical probe of bulk geometry.

In summary, the work establishes a rigorous bridge between three seemingly disparate concepts: (i) the von Neumann entropy of double‑scaled SYK algebras, (ii) the RT minimal‑area formula in (A)dS₂ spacetimes, and (iii) Krylov spread complexity of the boundary state. By doing so it provides a concrete, unitary realization of de Sitter holography in two dimensions, reproduces known horizon entropy formulas, and uncovers a quantitative link between quantum information complexity and geometric entropy. This advances our understanding of how quantum mechanical operator algebras encode bulk spacetime data and opens new avenues for exploring holography beyond AdS.