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bibkey: caputadigiuliolo2026complexity authors: Pawel Caputa, Giuseppe Di Giulio, Tran Quang Loc year: 2026 title: “Complexity Inequalities for Quantum Subsystems” doi: 10.48550/arXiv.2606.20790 url: https://arxiv.org/abs/2606.20790v2 claim: ‘Based on all the examples discussed in this manuscript, we conjecture that ${\left(b_n^{(A)}\right)}^2>0$ for every $n$, and hence that all the coefficients $b_n^{(A)}$ are real.’ strata_touched:

• D5/S3/Quantum/Dynamics/SubsystemLanczosPositivityRefutation license: citation-only triage: anchor

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DOI: 10.48550/arXiv.2606.20790

DOI URL: https://doi.org/10.48550/arXiv.2606.20790

Source: https://arxiv.org/abs/2606.20790v2

PDF: https://arxiv.org/pdf/2606.20790v2

The conjecture appears in §3.1, printed p. 15 (PDF page 16), in v2. The source TeX is SubsystemComplexity_Draft_v0.tex.

At present, however, we are unable to establish the sign of ${\left(b_n^{(A)}\right)}^2$ in full generality. Based on all the examples discussed in this manuscript, we conjecture that ${\left(b_n^{(A)}\right)}^2>0$ for every $n$, and hence that all the coefficients $b_n^{(A)}$ are real.

The normalized subsystem overlap is defined by

$$R_A(t)=\frac{\text{Tr}\left({\rho }_A(t){\rho }_A(0)\right)}{\text{Tr}\left({\rho }_A^2(0)\right)}.$$

The moment convention and the first squared coefficient are

$${\mu }_n^{(A)}={\partial }_t^n{R}_A(t){|}_{t=0},{\left(b_1^{(A)}\right)}^2={\left({\mu }_1^{(A)}\right)}^2-{\mu }_2^{(A)}.$$

Their TeX labels are generaldef_RL, subsystem moments and b1_generalprocedure; the v2 PDF numbers them (3.2), (3.3) and (3.5), on printed pp. 14–15. The full pure state evolves as $|\psi (t)⟩=e^{-\mathrm{i}Ht}|\psi (0)⟩$ under a time-independent Hermitian Hamiltonian, and ${\rho }_A(t)$ is its partial trace over $B$.

Appendix C.1.1 establishes the restricted nonnegative conclusion ${\left(b_1^{(A)}\right)}^2\ge 0$ when $[{\rho }_A(0)\otimes 1_B,\rho (0)]=0$, including product initial states. This is distinct from strict positivity for all states and Hamiltonians.