All rank inequalities for reduced states of quadripartite quantum states

Cite this problem as Problem 41.

Problem

Given any tripartite density matrix $\rho_{ABC}$ and let $r_{AB}=\text{rank}(\text{Tr}_C(\rho_{ABC}))$ denote the rank of the respective marginals, prove or find a counterexample to the following hypothesis [1]:

(1)   \begin{equation*} r_{AB}\leq r_{AC}\,r_{BC} \end{equation*}

Background

The reduced states of a multi-partite quantum state are not independent, and so in particular the reduced state entropies satisfy certain universal relations. Following seminal results in quantum entropy inequalities, most notable strong subadditivity [2], it was later proven in [3] that Renyi- $\alpha$ -entropies $S_\alpha(\rho):=\frac{1}{1-\alpha}\log_2(\text{Tr}(\rho^\alpha))$ do not satisfy any non-trivial inequalities for $0 \textless \alpha<1$ and no non-trivial linear inequalities for $\alpha>1$ .

The case $\alpha=1$ reproduces the von Neumann entropy, which satisfies several nontrivial inequalities [4], and whose further inequalities are of vital interest to quantum and classical information theory, cf. [5,6]. For $\alpha=0$ , $S_0(\rho) = \log \operatorname{rank}\rho$ .

In [1], several inequalities are introduced for the ranks of quantum marginals. These are equivalent to linear inequalities for the Renyi-0-entropy (which is just the logarithm of the rank). Since the set of possible log-rank vectors is asymptotically close to a cone in $\mathbb{R}^{2^{n}-1}$ , all linear inequalities define facets of this cone. Via constructing quantum states whose marginal ranks correspond to the extremal rays of that cone, one can check for (asymptotic) completeness of the set of obtained entropy inequalities for a given number of parties.

In [1], three entropy inequalities for $n=4$ were introduced, yet some corresponding extremal rays eluded construction. The set would be complete, however, if the above inequality were true for all quantum states.

References

[1] J. Cadney, M. Huber, N. Linden and A. Winter, Inequalities for the Ranks of Quantum States. Linear Algebra and Applications 452, pp. 153-171 (2014). https://arxiv.org/abs/1308.0539

[2] E.H. Lieb and M.-B. Ruskai. Proof of the strong subadditivity of quantum-mechanical entropy. J. Math.Phys. 14:1938-1941 (1973).

[3] N. Linden, M. Mosonyi and A. Winter. The structure of Renyi entropic inequalities. Proc. R. Soc. A 469(2158):20120737 (2013). https://arxiv.org/abs/1212.0248

[4] N. Pippenger. The inequalities of quantum information theory. IEEE Trans. Inf. Theory 49(4):773-789 (2003).

[5] R.W. Yeung and Z. Zhang. On Characterization of Entropy Function via Information Inequalities. IEEE Trans. Inf. Theory 44(4):1440-1452 (1998).

[6] N. Linden and A. Winter. A New Inequality for the von Neumann Entropy. Commun. Math. Phys. 259:129-138 (2005). https://arxiv.org/abs/quant-ph/0406162