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Inequalities for the Ranks of Quantum States

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If you have a question about this talk, please contact Mustapha Amrani.

Mathematical Challenges in Quantum Information

We investigate relations between the ranks of marginals of multipartite quantum states. These are the Schmidt ranks across all possible bipartitions and constitute a natural quantification of multipartite entanglement dimensionality. We show that there exist inequalities constraining the possible distribution of ranks. This is analogous to the case of von Neumann entropy (lpha-R’enyi entropy for lpha=1), where nontrivial inequalities constraining the distribution of entropies (such as e.g. strong subadditivity) are known. It was also recently discovered that all other lpha-R’enyi entropies for $lphain(0,1)p(1,infty)$ satisfy only one trivial linear inequality (non-negativity) and the distribution of entropies for $lphain(0,1)$ is completely unconstrained beyond non-negativity. Our result resolves an important open question by showing that also the case of lpha=0 (logarithm of the rank) is restricted by nontrivial linear relations and thus the cases of von Neumann entropy (i.e., lpha=1) and 0-R’enyi entropy are exceptionally interesting measures of entanglement in the multipartite setting.

This talk is part of the Isaac Newton Institute Seminar Series series.

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