Enabling Computation of Correlation Bounds for Finite-Dimensional Quantum Systems via Symmetrization

We present a technique for reducing the computational requirements by several orders of magnitude in the evaluation of semidefinite relaxations for bounding the set of quantum correlations arising from finite-dimensional Hilbert spaces. The technique, which we make publicly available through a user-friendly software package, relies on the exploitation of symmetries present in the optimization problem to reduce the number of variables and the block sizes in semidefinite relaxations. It is widely applicable in problems encountered in quantum information theory and enables computations that were previously too demanding. We demonstrate its advantages and general applicability in several physical problems. In particular, we use it to robustly certify the nonprojectiveness of high-dimensional measurements in a black-box scenario based on self-tests of $d$-dimensional symmetric informationally complete positive-operator-valued measurements.

Figure 1

Illustration of the CCP [Eq. (5)]. Bob has ($\genfrac{}{}{0ex}{}{N}{2}$) settings labeled by $(y,y^{\prime })$ and one additional setting labeled $povm$. Alice and Bob aim to satisfy the following relations: $o=x$ for the setting $povm$, and $b=0$ when $x=y$ and $b=1$ when $x=y^{\prime }$, respectively, for the settings $(y,y^{\prime })$.