5 research outputs found
Measurement entropy in Generalized Non-Signalling Theory cannot detect bipartite non-locality
We consider entropy in Generalized Non-Signalling Theory (also known as box
world) where the most common definition of entropy is the measurement entropy.
In this setting, we completely characterize the set of allowed entropies for a
bipartite state. We find that the only inequalities amongst these entropies are
subadditivity and non-negativity. What is surprising is that non-locality does
not play a role - in fact any bipartite entropy vector can be achieved by
separable states of the theory. This is in stark contrast to the case of the
von Neumann entropy in quantum theory, where only entangled states satisfy
S(AB)<S(A).Comment: 14 pages, includes minor corrections from v
Infinitely many constrained inequalities for the von Neumann entropy
We exhibit infinitely many new, constrained inequalities for the von Neumann
entropy, and show that they are independent of each other and the known
inequalities obeyed by the von Neumann entropy (basically strong
subadditivity). The new inequalities were proved originally by Makarychev et
al. [Commun. Inf. Syst., 2(2):147-166, 2002] for the Shannon entropy, using
properties of probability distributions. Our approach extends the proof of the
inequalities to the quantum domain, and includes their independence for the
quantum and also the classical cases.Comment: 11 page
Inequalities for the Ranks of Quantum States
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
(\alpha-R\'enyi entropy for \alpha=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 \alpha-R\'enyi
entropies for satisfy only one trivial linear
inequality (non-negativity) and the distribution of entropies for
is completely unconstrained beyond non-negativity. Our result
resolves an important open question by showing that also the case of \alpha=0
(logarithm of the rank) is restricted by nontrivial linear relations and thus
the cases of von Neumann entropy (i.e., \alpha=1) and 0-R\'enyi entropy are
exceptionally interesting measures of entanglement in the multipartite setting