Quantum validity of the Zhang–Yeung inequality

Problem

Does the four-party Zhang–Yeung inequality hold for the von Neumann entropies of every finite-dimensional quantum state?

For a state \(\rho_{ABCD}\), define

\begin{equation} \begin{aligned} S(R)&=-\operatorname{Tr}(\rho_R\log_2\rho_R),\\ I(A:B)&=S(A)+S(B)-S(AB),\\ I(A:B\mid C)&=S(AC)+S(BC)-S(C)-S(ABC). \end{aligned} \tag{1} \end{equation}

Using the quantities in Eq. (1), the proposed inequality is

\begin{equation} 2I(C:D)\leq I(A:B)+I(A:CD)+3I(C:D\mid A)+I(C:D\mid B) \tag{2} \end{equation}

Does Eq. (2) hold for every \(\rho_{ABCD}\), without independence, separability, stabilizer, or holographic assumptions?

Source

Bao, Cao, Walter, and Wang discuss this exact unrestricted four-party candidate in Section 4.2 [Bao15]. The statement is rewritten here to make its hypotheses and success criterion self-contained.

Progress

For classical random variables, inequality (2) is the Zhang–Yeung non-Shannon inequality. It was proved by Zhang and Yeung and later treated systematically by Dougherty, Freiling, and Zeger [Zhang98][DFZ11].
The 2025 Quantum Entropy Prover paper still identifies the search for additional unconstrained quantum entropy inequalities as open. Its Section V.A proves a related five-party inequality. Classically, an auxiliary-copy construction turns that precursor into a four-party non-Shannon inequality; the five-party quantum proof does not establish the required four-party quantum extension. [Huang25]
A second important distinction is between failure to derive an inequality from strong subadditivity and an actual violating density operator. A linear-programming witness outside the basic entropy cone need not be realizable by a quantum state. Likewise, the known constrained quantum inequalities, which assume specified conditional mutual informations vanish, do not settle this unconstrained question. [Huang25]

Comment

Retained as unresolved. Bao, Cao, Walter, and Wang provide an explicit source for the candidate, while the later entropy-cone work documents the broader gap in unconstrained quantum entropy inequalities.

Useful research target: Either prove the displayed inequality for unrestricted density operators or exhibit a finite-dimensional state with a rigorously certified negative difference between its right- and left-hand sides. Numerical minimization alone would be a search tool, not a proof of validity.

References

[Zhang98]: Z. Zhang and R. W. Yeung, On Characterization of Entropy Function via Information Inequalities, IEEE Transactions on Information Theory 44(4), 1440–1452 (1998).DOI
[DFZ11]: R. Dougherty, C. Freiling, and K. Zeger, Non-Shannon Information Inequalities in Four Random Variables, arXiv:1104.3602 (2011).arXiv
[Bao15]: N. Bao, C. Cao, M. Walter, and Z. Wang, Holographic entropy inequalities and gapped phases of matter, arXiv:1507.05650v2 (2015), Section 4.2. This is the source for the unrestricted quantum candidate and the Ingleton distinction. Full text.link
[Huang25]: S.-L. Huang, T. Rippchen, and M. Berta, Quantum Entropy Prover, arXiv:2501.16025v1, January 27, 2025, Sections V.A and V.C. Full text.link

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