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bibkey: fullwood2025dynamical authors: James Fullwood and Arthur J. Parzygnat year: 2025 title: “On Dynamical Measures of Quantum Information” doi: 10.3390/e27040331 url: https://doi.org/10.3390/e27040331 claim: “Remark 1 conjectures that the entropy functional S(X) = -tr(X log|X|), restricted to quantum states over time (half the anticommutator of the input state tensor the identity with the Jamiołkowski matrix of a channel), is subadditive; Section 5 states the conjecture as non-negativity of the dynamical mutual information. Fullwood and Yang (arXiv:2608.28946) prove it for single-qubit inputs and state it open in general.” strata_touched:

• D5/S3/Quantum/Information/FullwoodParzygnatSubadditivityRefutation license: citation-only triage: anchor

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DOI: 10.3390/e27040331

Source: https://doi.org/10.3390/e27040331 (Entropy 27(4), 331, 2025; arXiv:2306.01831). The full text was read through Europe PMC (PMC12025998). Equation (15) and Remark 1 are in Section 3; the dynamical mutual information is discussed in Section 5.

Equation (15) and the definition of a quantum state over time read:

${\varrho }_{AB}$ is a quantum state over time if and only if there exists a completely positive trace-preserving map $\mathcal{E}:A\to B$ such that ${\varrho }_{AB}=\frac{1}{2}\{{\rho }_A\otimes 1,\mathcal{J}[\mathcal{E}]\}$.

Remark 1 ends:

At present, we do not know of any examples of quantum states over time that violate subadditivity, which leads us to conjecture that the entropy functional S restricted to quantum states over time does in fact satisfy subadditivity.

Section 5 reads:

From the perspective of dynamical measures of quantum information, such a conjecture is equivalent to the conjecture that the dynamical mutual information $I(\mathcal{E},\rho )=S(\rho )+S(\mathcal{E}(\rho ))-S(\mathcal{E},\rho )$ is non-negative for all processes $(\mathcal{E},\rho )$.

J. Fullwood and B. Yang, On the entropy of a pseudo-density matrix, https://arxiv.org/abs/2608.28946v1 (28 August 2026), fixes the conventions used in the formal statement: Section 2 gives ${\varrho }_{AB}=\frac{1}{2}\{{\rho }_A\otimes 1_B,\mathcal{J}[\mathcal{E}]\}$ with $\mathcal{J}[\mathcal{E}]={\sum }_{i,j}|i⟩⟨j|\otimes \mathcal{E}(|j⟩⟨i|)$, “the Jamiołkowski matrix”, which “is not to be confused with the Choi matrix”; Theorem 3.1 characterizes $S(X)=-\mathrm{tr}(X\log |X|)$ on unit-trace Hermitian matrices; Theorem 4.1 proves subadditivity when the input is a single qubit and the output dimension is arbitrary. Section 6 reads:

it is presently unknown if PDM entropy is subadditive for 2-time pseudo-density matrices. Although we proved subadditivity for 2-time pseudo-density matrices whose initial state is that of a single qubit, numerically generated examples suggest that subadditivity may fail in higher dimensions. However, we still do not know of a simple, physically motivated example where subadditivity fails.