Information-theoretic quantities of quantum channels with partition quantum group symmetries

Abstract

In the field of quantum information theory, it is well-known that the purely quantum phenomenon called quantum entanglement can boost the capacity of a quantum channel, which is called the superadditivity of the capacity. Shor showed in his breakthrough paper on the equivalence of additivity conjectures that this superadditivity is equivalent to the subadditivity of the minimum output entropy of quantum channels, and Hastings gave a high-dimensional and probabilistic counterexample to the minimum output entropy additivity conjecture. Since then, researchers have endeavored to classify which quantum channels have subadditive minimum output entropy and to find deterministically constructed quantum channels with subadditive minimum output entropy. It is well-known that determining the minimum output entropy of an arbitrary quantum channel is a hard problem (in fact, it has been shown to be NP-complete), which in turn makes it difficult to determine whether a quantum channel has subadditive minimum output entropy.

In this thesis, we utilize the representation theory of compact (partition) quantum groups to construct so-called covariant Clebsch-Gordan quantum channels and analyze their minimum output entropy. We review the work of Brannan and Collins in the case of the free orthogonal quantum groups $O_N^+$, and introduce a similar analysis for the quantum permutation group $S_N^+$. Afterwards, we specialize to the subfamily of lowest-weight Clebsch-Gordan channels, and we show that, in the case where one embeds the fundamental representation of $O_N^+$ in the tensor product of two irreducible representations of $O_N^+$, the associated lowest-weight quantum channels have sufficient additional structure to analytically compute their minimum output entropy in terms of a recurrence relation.

Lastly, we present an analysis of certain numerical methods that can be utilized to bound the minimum output entropy from below and from above in low dimensions: we use epsilon-covers for lower bounds, and we use a modified version of the derivative-free optimization method called Particle Swarm Optimization over the unit sphere, hybridized with gradient descent, to find upper bounds. We show a proof-of-concept by applying the epsilon-cover to three $S_N^+$ channels with small input dimensions, but also note that the exponential scaling of epsilon-covers with the input dimension makes them intractable with higher input dimensions. We benchmark the Particle Swarm Optimization enriched with gradient-descent on the highest-weight Clebsch-Gordan channels for which it can be shown that the minimum output entropy is zero, and we see that the optimization technique performs adequately. We also apply the optimization scheme to the tensor product of two $S_N^+$-channels, but do not find any violation of the minimum output entropy additivity conjecture.