Koherence in Krylov Complexity

• Koherence is a basis-relative entropy diagnostic that compares two Krylov bases to measure how perturbations reorganize the basis structure.
• It is computed by evaluating the Shannon entropy of the squared overlap distribution between a reference and a perturbed Krylov basis.
• Koherence provides insights into dynamical sensitivity and chaos by revealing delocalization in basis representations beyond what spread complexity captures.

Searching arXiv for papers on Koherence and related Krylov-basis complexity. Koherence is a basis-relative entropy diagnostic introduced in the study of Krylov complexity to quantify how strongly a perturbation reorganizes the Krylov basis itself, rather than how far a single state has propagated within a fixed basis. In the formulation of “Variations on a Theme of Krylov” (Balasubramanian et al., 5 Nov 2025), one compares a reference Krylov basis with a perturbed Krylov basis and defines Koherence from the Shannon entropy of their overlap distribution. Equivalently, for each perturbed Krylov basis vector, Koherence is the relative entropy of coherence of that vector with respect to the reference Krylov basis (Balasubramanian et al., 5 Nov 2025). This places Koherence at the intersection of Krylov-basis methods, entropy-based coherence quantification, and perturbation sensitivity, with the specific purpose of diagnosing dynamical amplification of differences in initial conditions, Hamiltonians, or related structural deformations of a Krylov construction (Balasubramanian et al., 5 Nov 2025).

1. Definition and conceptual role

Koherence was introduced to complement spread complexity, which is defined for a single evolving state in its own Krylov basis (Balasubramanian et al., 5 Nov 2025). Spread complexity asks how far the wavefunction has moved along its Krylov chain, whereas Koherence asks how different two Krylov bases are when one changes the initial state, the Hamiltonian, or another structural input to the Krylov construction (Balasubramanian et al., 5 Nov 2025). In this sense, Koherence is not attached to one basis in isolation; it is pair-dependent and compares two Krylov organizations of Hilbert space (Balasubramanian et al., 5 Nov 2025).

The reference setting uses an unperturbed Krylov basis

$\{|K_n^{(0)}\rangle\}_{n=0}^{\mathcal K^{(0)}-1}$

and a perturbed Krylov basis

$\{|K_m\rangle\}_{m=0}^{\mathcal K-1}.$

Assuming $\mathcal K\le \mathcal K^{(0)}$, each perturbed basis vector is expanded in the reference basis as

$$|K_m\rangle = \sum_n \langle K_n^{(0)}|K_m\rangle\,|K_n^{(0)}\rangle.$$

For fixed $m$, the squared overlaps

$P_n^{\,m} \equiv |\langle K_n^{(0)}|K_m\rangle|^2$

form a probability distribution because

$\sum_n |\langle K_n^{(0)}|K_m\rangle|^2=1.$

Koherence is then defined as

$$S_K^{(m,0)} \equiv -\sum_n |\langle K_n^{(0)}|K_m\rangle|^2 \log |\langle K_n^{(0)}|K_m\rangle|^2.$$

The paper also defines the basis-averaged mean Koherence

$$\overline{S_K^{(0)}}= \frac{1}{\mathcal K}\sum_{m=0}^{\mathcal K-1} S_K^{(m,0)} = -\frac{1}{\mathcal K}\sum_{m=0}^{\mathcal K-1}\sum_n |\langle K_n^{(0)}|K_m\rangle|^2 \log |\langle K_n^{(0)}|K_m\rangle|^2,$$

while noting a sign typo in the typeset expression and clarifying that the intended quantity is the average of positive Shannon entropies (Balasubramanian et al., 5 Nov 2025).

The operational interpretation follows directly from this entropy. If a perturbed basis vector almost coincides with one reference basis vector, its Koherence is near zero. If it is distributed broadly across many reference basis vectors, its Koherence is large (Balasubramanian et al., 5 Nov 2025). High Koherence therefore signals strong delocalization of the perturbed Krylov basis in the unperturbed Krylov basis, and the paper presents this as a way to quantify dynamical amplification of small perturbations (Balasubramanian et al., 5 Nov 2025).

2. Krylov construction and distinction from spread complexity

Koherence is formulated within the standard Lanczos-based Krylov construction. Starting from

$|K_0\rangle \equiv |\psi_0\rangle,$

the Krylov basis is generated recursively by

$\{|K_m\rangle\}_{m=0}^{\mathcal K-1}.$0

with

$\{|K_m\rangle\}_{m=0}^{\mathcal K-1}.$1

In this basis the Hamiltonian is tridiagonal: $\{|K_m\rangle\}_{m=0}^{\mathcal K-1}.$2 A time-evolved state expands as

$\{|K_m\rangle\}_{m=0}^{\mathcal K-1}.$3

with Krylov-chain probabilities

$\{|K_m\rangle\}_{m=0}^{\mathcal K-1}.$4

Spread complexity is then

$\{|K_m\rangle\}_{m=0}^{\mathcal K-1}.$5

This quantity measures how much support the state has developed away from the initial site $\{|K_m\rangle\}_{m=0}^{\mathcal K-1}.$6 on its own Krylov chain (Balasubramanian et al., 5 Nov 2025).

Koherence addresses a different problem. The paper emphasizes that spread complexity alone does not reveal how sensitive the Krylov decomposition of Hilbert space is to perturbations (Balasubramanian et al., 5 Nov 2025). Two systems may display similar growth of spread complexity while having very different degrees of basis reorganization. Koherence is introduced precisely to isolate this structural sensitivity (Balasubramanian et al., 5 Nov 2025).

This comparison is naturally linked to broader coherence theory. In basis-dependent quantum coherence, one quantifies off-diagonal structure relative to a chosen basis, often using entropic constructions (Yu et al., 2016). Koherence specializes this logic to the case where the relevant basis is not fixed once and for all, but is itself a dynamical object generated by the Lanczos procedure (Balasubramanian et al., 5 Nov 2025). This suggests that Koherence is best understood not as a generic coherence monotone on states, but as a Krylov-basis-relative coherence functional on basis vectors.

3. Information-theoretic interpretation

A central observation of “Variations on a Theme of Krylov” is that Koherence is exactly a relative entropy of coherence in the sense of standard quantum information theory (Balasubramanian et al., 5 Nov 2025). For a density matrix $\{|K_m\rangle\}_{m=0}^{\mathcal K-1}.$7 and basis $\{|K_m\rangle\}_{m=0}^{\mathcal K-1}.$8, the relative entropy of coherence is

$\{|K_m\rangle\}_{m=0}^{\mathcal K-1}.$9

where

$\mathcal K\le \mathcal K^{(0)}$0

Identifying $\mathcal K\le \mathcal K^{(0)}$1 with the reference Krylov basis $\mathcal K\le \mathcal K^{(0)}$2 and taking

$\mathcal K\le \mathcal K^{(0)}$3

one obtains

$\mathcal K\le \mathcal K^{(0)}$4

Thus Koherence is exactly the relative entropy of coherence of a perturbed Krylov basis vector relative to the unperturbed Krylov basis (Balasubramanian et al., 5 Nov 2025).

This identification places Koherence in direct continuity with entropic coherence quantification. The relative entropy of coherence is a standard basis-dependent coherence measure in the Baumgratz–Cramer–Plenio setting and in related structural frameworks (Yu et al., 2016, Winter et al., 2015). In Koherence, however, the “reference basis” is itself the reference Krylov basis of another dynamical construction (Balasubramanian et al., 5 Nov 2025). This makes Koherence a comparative entropy of basis mismatch rather than a static measure of coherence relative to a laboratory basis.

The information-theoretic meaning is explicit in the paper: Koherence measures how distinguishable the pure state $\mathcal K\le \mathcal K^{(0)}$5 is from its decohered version in the old basis, or equivalently how much off-diagonal quantum coherence is required to represent the perturbed basis vector in the reference Krylov basis (Balasubramanian et al., 5 Nov 2025). A plausible implication is that Koherence translates a dynamical sensitivity problem into a resource-theoretic language in which perturbations are visible as off-diagonal delocalization across Krylov sectors.

4. Dynamical sensitivity and relation to spread-complexity variations

Koherence is not introduced merely as a static entropy. The paper derives a concrete relation between basis overlaps and the variation of Krylov amplitudes under perturbations (Balasubramanian et al., 5 Nov 2025). If

$\mathcal K\le \mathcal K^{(0)}$6

and

$\mathcal K\le \mathcal K^{(0)}$7

then, assuming the perturbed Krylov subspace lies within the original one,

$\mathcal K\le \mathcal K^{(0)}$8

The overlap coefficients $\mathcal K\le \mathcal K^{(0)}$9 are exactly the data that define Koherence, while the matrix elements $$|K_m\rangle = \sum_n \langle K_n^{(0)}|K_m\rangle\,|K_n^{(0)}\rangle.$$0 encode the unperturbed dynamics (Balasubramanian et al., 5 Nov 2025). The paper therefore interprets Koherence as the geometric ingredient governing how perturbations are processed dynamically.

The same logic appears in the “first law” of spread complexity discussed in the paper: $$|K_m\rangle = \sum_n \langle K_n^{(0)}|K_m\rangle\,|K_n^{(0)}\rangle.$$1 The authors show that overlaps between old and new Krylov vectors encode these moment variations already at low orders. For example,

$$|K_m\rangle = \sum_n \langle K_n^{(0)}|K_m\rangle\,|K_n^{(0)}\rangle.$$2

and

$$|K_m\rangle = \sum_n \langle K_n^{(0)}|K_m\rangle\,|K_n^{(0)}\rangle.$$3

This suggests that Koherence is sensitive to changes in the mean, variance, and higher moments of the energy distribution, not only to abstract basis delocalization (Balasubramanian et al., 5 Nov 2025). The paper explicitly uses this to justify Koherence as more than a formal entropy: it is tied to sensitivity of complexity growth (Balasubramanian et al., 5 Nov 2025).

This position differs from operational notions of coherence tied to dephasing sensitivity or interferometric visibility. For example, dephasing-based coherence quantifies the quantum Fisher information for estimating decoherence strength (Yadin et al., 2018), while multipath-interference work relates Hilbert–Schmidt coherence to visibility (Roy et al., 2021). Koherence instead concerns changes in the basis induced by perturbations of the dynamical generator or initial data (Balasubramanian et al., 5 Nov 2025).

5. Computation and representative models

In practical terms, Koherence is computed by constructing two Krylov bases, evaluating their overlap matrix, and then taking Shannon entropies of the corresponding overlap distributions (Balasubramanian et al., 5 Nov 2025). The paper presents the procedure in six steps:

1. choose two systems to compare;
2. construct both Krylov bases;
3. compute

$$|K_m\rangle = \sum_n \langle K_n^{(0)}|K_m\rangle\,|K_n^{(0)}\rangle.$$4

1. form

$$|K_m\rangle = \sum_n \langle K_n^{(0)}|K_m\rangle\,|K_n^{(0)}\rangle.$$5

2. compute

$$|K_m\rangle = \sum_n \langle K_n^{(0)}|K_m\rangle\,|K_n^{(0)}\rangle.$$6

3. optionally average over $$|K_m\rangle = \sum_n \langle K_n^{(0)}|K_m\rangle\,|K_n^{(0)}\rangle.$$7.

Because only $$|K_m\rangle = \sum_n \langle K_n^{(0)}|K_m\rangle\,|K_n^{(0)}\rangle.$$8 enters, phase conventions of basis vectors do not affect Koherence (Balasubramanian et al., 5 Nov 2025).

The paper analyzes several models in detail.

Model	Overlap/Koherence behavior	Interpretation in the paper
$$|K_m\rangle = \sum_n \langle K_n^{(0)}|K_m\rangle\,|K_n^{(0)}\rangle.$$9	Ballistic broadening; $m$0 grows linearly with Krylov index	Characteristic of chaotic or fast-scrambling dynamics (Balasubramanian et al., 5 Nov 2025)
$m$1	Overlaps remain near a tri-diagonal band; Koherence small and saturating	More integrable, bounded dynamics (Balasubramanian et al., 5 Nov 2025)
Heisenberg–Weyl	Intermediate spreading and slow growth without finite-size plateau	Intermediate between $m$2 and $m$3 (Balasubramanian et al., 5 Nov 2025)
Lattice model	Koherence quickly saturates despite linear spread complexity in continuum limit	Linear spread-complexity growth is not sufficient to diagnose chaos (Balasubramanian et al., 5 Nov 2025)

For $m$4, with Hamiltonian

$m$5

the paper gives explicit Lanczos coefficients

$m$6

with

$m$7

and an exact overlap formula in terms of Jacobi polynomials (Balasubramanian et al., 5 Nov 2025). The reported behavior is that perturbed basis vectors spread over a region of the old basis whose width grows roughly linearly with Krylov index, and correspondingly $m$8 grows linearly with index, so Koherence itself grows logarithmically (Balasubramanian et al., 5 Nov 2025).

For $m$9, with

$P_n^{\,m} \equiv |\langle K_n^{(0)}|K_m\rangle|^2$0

the Lanczos coefficients are

$P_n^{\,m} \equiv |\langle K_n^{(0)}|K_m\rangle|^2$1

where

$P_n^{\,m} \equiv |\langle K_n^{(0)}|K_m\rangle|^2$2

The Krylov dimension is finite, $P_n^{\,m} \equiv |\langle K_n^{(0)}|K_m\rangle|^2$3, and the overlap matrix stays concentrated near a narrow band, leading to small Koherence that plateaus at about half the Hilbert-space size in the displayed plot (Balasubramanian et al., 5 Nov 2025).

For the Heisenberg–Weyl case,

$P_n^{\,m} \equiv |\langle K_n^{(0)}|K_m\rangle|^2$4

with coherent initial states $P_n^{\,m} \equiv |\langle K_n^{(0)}|K_m\rangle|^2$5, the paper reports

$P_n^{\,m} \equiv |\langle K_n^{(0)}|K_m\rangle|^2$6

The overlap matrix is expressed through displacement-operator matrix elements and Laguerre polynomials, and the resulting Koherence is intermediate in growth behavior (Balasubramanian et al., 5 Nov 2025).

In the lattice model, comparing the localized state $P_n^{\,m} \equiv |\langle K_n^{(0)}|K_m\rangle|^2$7 with a “typical” delocalized pure state, the overlaps are computed analytically and shown not to broaden ballistically (Balasubramanian et al., 5 Nov 2025). The resulting Koherence saturates quickly, despite linear spread-complexity growth in the continuum limit. The paper treats this as conceptually significant because it shows that fast spread complexity need not imply large basis reorganization (Balasubramanian et al., 5 Nov 2025).

6. Relation to other coherence notions and broader significance

Koherence is explicitly compared with Relative Krylov Entropy (RKE), defined as

$P_n^{\,m} \equiv |\langle K_n^{(0)}|K_m\rangle|^2$8

RKE compares two probability distributions on Krylov chains, usually for the same initial state evolved with different Hamiltonians, while Koherence compares the Krylov bases themselves (Balasubramanian et al., 5 Nov 2025). The paper formulates the distinction as follows: Koherence answers how different the coordinate systems are, whereas RKE answers how distinguishable the resulting support distributions are (Balasubramanian et al., 5 Nov 2025).

This distinction helps situate Koherence relative to mainstream coherence theory. Standard coherence measures quantify superposition relative to a fixed basis (Yu et al., 2016, Winter et al., 2015). Some operational frameworks define coherence via dephasing sensitivity (Yadin et al., 2018), interference visibility (Roy et al., 2021), or even the security of quantum key distribution (Ma et al., 2018). Koherence does not replace these notions; rather, it repurposes relative entropy of coherence as a comparative diagnostic on dynamically generated bases (Balasubramanian et al., 5 Nov 2025).

The broader significance claimed in the paper is that Koherence reveals structure invisible to spread complexity alone (Balasubramanian et al., 5 Nov 2025). The group-manifold examples separate into:

• $P_n^{\,m} \equiv |\langle K_n^{(0)}|K_m\rangle|^2$9: strong growth, ballistic broadening, large Koherence;
• $\sum_n |\langle K_n^{(0)}|K_m\rangle|^2=1.$0: localized overlap structure, plateauing Koherence;
• Heisenberg–Weyl: intermediate behavior;
• lattice model: fast spread complexity but saturating Koherence.

This suggests that Koherence functions as a finer probe of scrambling or perturbation sensitivity than spread complexity by itself (Balasubramanian et al., 5 Nov 2025). A plausible implication is that Koherence should be read not as a universal chaos diagnostic, but as an entropy of Krylov-basis instability whose interpretation depends on how perturbations reorganize the accessible Hilbert-space geometry.

From the standpoint of encyclopedia classification, Koherence is therefore best described as a Krylov-complexity diagnostic defined by

$\sum_n |\langle K_n^{(0)}|K_m\rangle|^2=1.$1

with mean version

$\sum_n |\langle K_n^{(0)}|K_m\rangle|^2=1.$2

and with the exact interpretation of a relative entropy of coherence of one Krylov basis vector in another Krylov basis (Balasubramanian et al., 5 Nov 2025). Its purpose is not to quantify support spread of a state, but to quantify reorganization of the basis itself under perturbations (Balasubramanian et al., 5 Nov 2025).