Krylov Complexity and Entropy of Composite Quantum Systems

arXiv:2610.31394 · hep-th, quant-ph · Submitted 2026-10-22 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Krylov Complexity and Entropy of Composite Quantum Systems".

Mira: Here's the gist of the paper. The gist The leading excess complexity and Krylov entropy are complementary probes of collective dynamics in large quantum systems.

Kai: Excess Complexity The Krylov basis is determined by a choice of Hamiltonian, H, and a reference state K0⟩.

Mira: In terms of the Krylov basis, a generic hermitian Hamiltonian can be thus be decomposed as H = L− + L0 + L+ (1) The Krylov complexity operator is then defined (see

1, 2, 16: ) as Kˆ = Xn nKn⟩⟨Kn and complexity of any state in the Krylov subspace, ψ⟩ is given by ⟨ψKˆψ⟩.

Kai: In particular, the spread complexity of the seed state K0⟩ is defined as C(t) = ⟨K0e itH Keˆ −itHK0⟩ (2) The operator ∆ˆ measures the excess complexity of the tensor product as compared to the subsystems.

Title and authors: Mira: So let's look at the title of this paper again, "Krylov Complexity and Entropy of Composite Quantum Systems." It sounds very comprehensive, almost like they're trying to build a full map for how these systems behave when you combine them.

Kai: I think it’s accurate because they are taking something that used to be just about one subsystem—its own Krylov dynamics—and scaling it up to look at the whole tensor product structure.

Lev: But what's the actual mechanism here? Is this just a fancy way of saying we need more resources to simulate the full system?

Mira: No, it’s more specific than that. They show exactly how you can decompose a generic hermitian Hamiltonian into parts H = L- + L zero + L+ where L+ relates to the Krylov basis expansion <ref:2610.31394#pg2>.

Kai: So they are mapping the dynamics of an arbitrary system onto a nearest-neighbor spin chain structure, which is the Krylov chain, using those Lanczos coefficients a n and b n.

Lev: Mapping it to a chain is helpful because chains are easier to simulate on classical hardware or even some quantum devices, but how does that mapping affect the fidelity of the dynamics we care about?

Mira: The paper shows that the spread complexity, which measures how far the initial state spreads through Krylov space, can be characterized by this semi-infinite Krylov chain structure twelve <ref:2610.31394#pg2>.

Kai: And they show a specific expansion for the spread complexity up to fourth order in time t, which is given as C(t) - X i C i(t) = one over six [] t four + O(t six), which is related to the Lanczos coefficients <ref:2610.31394#pg3,up to fourth order in>.

Lev: That expansion shows that for commuting operators, this quantity is always positive, which supports the idea that we get a positive excess complexity in most cases.

The paper's summary: Kai: The summary points out that Krylov entropy is really powerful because it can disambiguate two things: the information just from how you composed the subsystems versus what comes from their actual dynamical mismatch.

Mira: That’s a big deal because in our experimental setup, we often have to worry about whether a signal we see is due to the way we set up the interaction or because two parts are just evolving differently.

Lev: So if Krylov entropy can tell us that it’s the latter, that implies it’s a better tool for diagnosing real physical problems than just looking at complexity alone.

Kai: And they show that this mismatch-dependent entropy contribution is proportional to the excess complexity we discussed earlier, specifically delta a S K(t) = two C(t) one/t + O(t four) <ref:2610.31394#pg3>.

Mira: That relationship is the most concrete part—it directly links the two measures of dynamical heterogeneity together through that same weighted variance of the Lanczos data.

Lev: So, if we track how that variance evolves over time, we should be able to predict both complexity and entropy contributions simultaneously based on that single value.

Kai: And they use a specific example with the Kitaev Chain model, where each sector is a two-level Krylov system, giving us concrete results for q k(t).

The paper's improvements: Lev: What about the limitations of this approach? The authors have to be honest about what this method doesn't cover when we try to apply it to something more complex than a simple chain.

Mira: They explicitly state that the analysis using these simplifications is for commuting operators, which means they can’t directly apply this framework to non-commuting systems without some extra work.

Kai: And they mention that while they show how the excess complexity remains significant even as the number of tensor products is increased, there are still regimes where we haven't fully captured all the emergent collective behavior.

Lev: So, the method works well for characterizing certain features in many-body systems, but it might miss other types of collective effects that don't fit this Lanczos expansion framework.

Mira: It really seems like they’ve created a tool that lets us distinguish between composition effects and dynamical mismatch using these two complementary probes.

Conclusion: Kai: To wrap up, the main point of "Krylov Complexity and Entropy of Composite Quantum Systems" is that we have two distinct sources of non-additive behavior in composite quantum systems.

Mira: The first source is due purely to subsystem composition—the way you build the system—and the second is genuinely dynamical mismatch, which arises when subsystems explore their Krylov spaces differently.

Lev: And the paper shows these two sources are linked by that weighted variance of Lanczos data controlling both leading excess complexity and the mismatch-dependent entropy contribution to show how they are not unrelated effects.

Kai: So we have a precise way to detect synchronous evolution by looking for zero excess complexity, and a more nuanced way to measure the dynamical mismatch using Krylov entropy.

Mira: It’s about using these probes together to get a better picture of how collective dynamics emerge in large quantum systems without getting lost in the details of every single term.

Lev: For me, it’s important that we know where this method stops working—it doesn't cover all possible types of non-trivial dynamics, but it gives us a very clear boundary for what this specific mathematical framework can capture.

Kai: So that’s the summary of "Krylov Complexity and Entropy of Composite Quantum Systems." It’s a solid piece of work connecting the math to how we actually look at collective dynamics in these systems.

Jeff Murugana, Victor Van Aswegena, Hendrik J R Van Zyla

Laboratory for Quantum Gravity & Strings · National Institute of Theoretical and Computational Sciences

hep-th, quant-ph

Submitted: 2026-10-22

Updated: 2026-10-22

Comments: 8+2 pages

License: http://creativecommons.org/licenses/by/4.0/

The gist: The gist The leading excess complexity and Krylov entropy are complementary probes of collective dynamics in large quantum systems.

Key concepts

Excess Complexity: This measures the deviation of the total system's Krylov dynamics from the sum of individual subsystem complexities. If this value is zero, it means all subsystems evolve synchronously, indicating perfect matching in their decomposition.
Krylov Entropy: This quantifies information related to how a state is distributed across different Krylov basis states. It can be decomposed into parts related to subsystem composition and dynamical mismatch, helping distinguish between these two sources of non-additive behavior.
Synchronization Detection: The leading excess complexity is zero if and only if the subsystems evolve synchronously. This provides a precise mathematical test to determine if the constituent systems are evolving in perfect unison, a key feature of collective dynamics.

Terminology

Krylov Complexity — 1× in this paper · explained in 3 episodes

Summary

The gist The leading excess complexity and Krylov entropy are complementary probes of collective dynamics in large quantum systems.

Excess Complexity

** The Krylov basis is determined by a choice of Hamiltonian, H, and a reference state K0⟩. In terms of the Krylov basis, a generic hermitian Hamiltonian can be thus be decomposed as H = L− + L0 + L+ (1) **

** The Krylov complexity operator is then defined (see [1, 2, 16]) as Kˆ = Xn nKn⟩⟨Kn and complexity of any state in the Krylov subspace, ψ⟩ is given by ⟨ψKˆψ⟩. In particular, the spread complexity of the seed state K0⟩ is defined as C(t) = ⟨K0e itH Keˆ −itHK0⟩ (2) **

** The operator ∆ˆ measures the excess complexity of the tensor product as compared to the subsystems. In particular, the semi-positive definiteness implies that the excess complexity of the time-evolved reference state ∆(t) = C123···n(t) − Xn j=1 Cj(t) is a non-negative quantity for all times. It was furthermore shown that the excess complexity is zero if and only if the subsystems evolve synchronously. By this it is meant that the decomposition of the combined system (1) is matched perfectly with the decomposition of the subsystem as1 L+ = ⊕ n j=1 Lj,+. **

** The excess complexity, up to fourth order in t, expands as C(t) − Xn i Ci(t) = - (1/6) [3/2 ⟨K0H 3K0⟩ squared ⟨K0H 2K0⟩ - 3/2 Xn i ⟨K0H 3iK0⟩ squared ⟨K0H 2iK0⟩] t4 + O(t6) **

** Written in terms of the Lanczos coefficients, this reads C(t) − Xn i Ci(t) = Xn i j b2 1,i b2 1, j 8 Pk b2 1,k a1,i − a1, j 2 t4 + O(t6) **

** The coefficient of the leading excess complexity is just a weighted variance of the first diagonal Lanczos coefficients across the subsystems. If all the subsystems have a1,i = a1, then Varw(a1) = 0 and at this order, their Krylov dynamics are synchronized; after the first Krylov step they all see the same diagonal Lanczos coefficient. **

** If however the a1,i’s differ, the subsystems begin to distinguish themselves dynamically at this order, and Varw(a1) > 0. It is then precisely this mismatch that generates a positive excess complexity. **

Krylov Entropy

** For a state ψ(t)⟩ = Pn ϕn(t)Kn⟩, we can define the Krylov entropy (see [27]) S K(t) = - Xn pn(t) log pn(t), pn(t) = ϕn(t)2. **

** The product Krylov basis is labelled by n = (n1,..., nN) with probability distribution Pn(t) = Yi p (i) ni(t). Its entropy is therefore exactly additive, S prod K(t) = - Xn Pn log Pn = Xi S(i) K(t). **

** A useful decomposition of this entropy which makes its relation to the combined Krylov chain transparent is S prod K= S∥ + S ⊥, where S∥ = - Xr Pr log Pr (16) is the entropy associated with spreading between radial shells, while S ⊥ = Xr Pr [- Xn: P i ni=r P(nr) log P(nr)] measures the information contained in the distribution within each shell. **

** The leading contribution to S ⊥ is present even when all subsystems evolve synchronously. It is kinematic rather than a measure of dynamical mismatch in the sense that it simply counts the information associated with the different subsystem directions within the first Krylov shell. **

** The first contribution that is sensitive to dynamical mismatch enters at the next order. The first diagonal coefficient of the combined chain is αcomb = α¯ = Xi wiαi, (26) **

** Its second hopping coefficient is b2 2,comb = Varw(α) + Xi wiyi + 2 [B − Xi wixi], (27) **

** The mismatch-dependent entropy contribution is δa∆S K(t) = B 2 Varw(α)t4 log 1/t + O(t4). **

** The excess complexity found in the previous section may be written in the explicitly shift-invariant form ∆C(t) = B 4 Varw(α)t4 + O(t6). **

** Consequently, δa∆S K(t) = 2∆C(t) log 1/t + O(t4). **

The Kitaev Chain

** The model consists of a scalable 1D system of Majorana fermions that together exhibit striking global features such as quantum phase transitions [30]. Because of this, it serves as a strong mathematical probe of emergent behaviour from simple constituents. **

** The momentum-space decomposition of the Kitaev chain also allows the Krylov entropy to be studied particularly explicitly. For s = 1/2, each independent (k, −k) sector is a two-level Krylov system. **

** The sector Krylov entropy is therefore simply the binary entropy S K,k(t) = h2(qk) = -qk log qk − (1 − qk) log(1 − qk), where qk is the probability of occupying the first Krylov state of the kth sector. **

** The return amplitude in Eq. (35) implies that the probability of occupying the first Krylov state of the kth sector is qk(t) = d2 k E2 k sin 2(Ekt). **

** At early times, qk(t) = d2 k t2 - d2 k E2 k 3 t4 + O(t6), so that S K,k(t) = d2 k t2 h1 − log d2 k t2 i + t4 [d squared k E squared k 3 log d squared k t squared − d4 k 2] + O(t6 log t). **

** The leading excess complexity (37) and, from Eq. (30), the mismatch-dependent entropy contribution δa∆S K(t) = ∆ 2 J squared L 8 t4 log 1/t + O(Lt4) = 2∆C(t) log 1/t + O(Lt4). **

Discussion

** A central outcome of our analysis is that there are two distinct sources of non-additive behaviour. The first is associated purely with subsystem composition. Even perfectly synchronous subsystems possess distinguishable directions in the product Krylov space, and this information is not retained when the dynamics is represented on the one-dimensional Krylov chain of the full Hamiltonian. **

** The second is genuinely dynamical and arises when the constituent systems explore their respective Krylov spaces differently. The fact that the same weighted variance of the Lanczos data controls both the leading excess complexity and the first mismatch-dependent contribution to the entropy shows that these are not unrelated effects.

Improvements for AI systems

Precision Characterization of Dynamical Heterogeneity: The AI system can quantify dynamical heterogeneity in large quantum systems by calculating excess complexity, which is defined as the departure of the combined Krylov dynamics from the sum of the individual subsystem complexities.
Disambiguation of Composition Effects: The system can distinguish between information associated with subsystem composition and information generated by dynamical mismatch through Krylov entropy, as Krylov entropy is able to disambiguate the information associated with subsystems composition from that generated by dynamical mismatch.
Enhanced Synchronization Detection: The AI can detect when subsystems evolve synchronously by checking if the leading excess complexity is zero, which occurs if and only if the subsystems evolve synchronously, providing a precise measure of this collective behavior.
Predictive Modeling of Collective Dynamics: By analyzing the scaling of excess complexity with system size, the system can predict how collective behavior emerges, as the excess complexity remains significant even as the number of tensor products is increased in certain regimes.
Optimized Resource Allocation in Composite Systems: The AI can use Krylov entropy to assess the loss of subsystem-resolved information versus dynamical cost of asynchronous evolution, allowing for a more nuanced understanding of how information is retained or lost when moving from subsystem descriptions to the full system description.

Abstract

We investigate the behaviour of Krylov complexity as the number of degrees of freedom of a quantum system increases. Specifically, we show that for tensor products of commuting subsystems, the leading excess complexity is controlled by a weighted variance of the subsystem Lanczos coefficients. This provides a quantitative measure of dynamical heterogeneity and a precise characterization of synchronous evolution. We also show that Krylov entropy is able to disambiguate the information associated with subsystems composition from that generated by dynamical mismatch, and that the same Lanczos variance controls the leading mismatch contribution to both complexity and the entropy. Our results establish excess complexity and Krylov entropy as complementary probes of the emergence of collective dynamics in large quantum systems.