QLunch: Lasse Wolff

Speaker: Lasse Wolff from QMATH

Title: The Quantum Entropy Cone near its Apex

Abstract: Motivated by the search for von Neumann entropy inequalities beyond strong subadditivity, Andreas Winter has conjectured that 4-party entropy-vectors of the form: c(1,1,1,1,2,2,2,2,2,2,1,1,1,1,0), with sufficiently small non-zero c, cannot be realized by quantum states. This would be interesting since the vector is realizable with c=1. 

I will show this conjecture to be true by presenting a proof that the above entropy-vector is not realizable if: 0 < c < 0.65. The techniques behind this proof can further be used to derive new constrained non-linear entropy inequalities and new unconstrained entropy inequalities for states of fixed maximal dimension, with interesting consequences. 

The main idea behind the proofs is to relate certain vector-component associated with total multi-party pure states to the largest eigenvalues of associated single-party density matrices. Products of these vector components must satisfy some Cauchy-Schwarz-related inequalities due to their sums defining the relevant reduced states. With these tools, it is possible to construct quadratic inequalities for the mentioned maximal eigenvalues, which will lead to non-trivial entropy restrictions near the apex of the quantum entropy cone.