Self-Testing of Physical Theories, or, Is Quantum Theory Optimal with Respect to Some Information-Processing Task?

Self-testing usually refers to the task of taking a given set of observed correlations that are assumed to arise via a process that is accurately described by quantum theory, and trying to infer the quantum state and measurements. In other words it is concerned with the question of whether we can tell what quantum black-box devices are doing by looking only at their input-output behavior and is known to be possible in several cases. Here we introduce a more general question: is it possible to self-test a theory, and, in particular, quantum theory? More precisely, we ask whether within a particular causal structure there are tasks that can only be performed in theories that have the same correlations as quantum mechanics in any scenario. We present a candidate task for such a correlation self-test and analyze it in a range of generalized probabilistic theories (GPTs), showing that none of these perform better than quantum theory. A generalization of our results showing that all nonquantum GPTs are strictly inferior to quantum mechanics for this task would point to a new way to axiomatize quantum theory, and enable an experimental test that simultaneously rules out such GPTs.

Figure 1

The bipartite Bell causal structure and CHSH game. A referee asks question $R_A$ to Alice and $R_B$ to Bob, choosing these uniformly. Their respective answers to these questions are $A$ and $B$. These may be formed by measurement on part of a shared bipartite state $S_{AB}$, whose most general form depends on the theory under consideration (in quantum theory it can be any shared quantum state, for instance). In the case where $A$, $B$, $R_A$, and $R_B$ are binary, the CHSH game is said to be won if $A\oplus B=R_A·R_B$ , where $\oplus$ denotes addition modulo 2. Note that in this causal structure the measurements only act on single subsystems of the shared state.

Figure 2

Causal structure of the adaptive CHSH game. The values of $A$, $B$, and $C$ as well as the referee’s questions $R_A$ and $R_C$ determine whether the game is won. Resources are shared between $A$ and $B$, and between $B$ and $C$, but there are no shared tripartite resources. Although $A$ and $C$ can be formed by measurements on single subsystems, the value of $B$ can be obtained by jointly measuring the $B$ part of $S_{AB}$ and $B^{\prime }$ part of $S_{B^{\prime }C}$.

Figure 3

Maximal winning probability in the CHSH game achievable in the maximal tensor product of self-dualized polygon systems. Along the horizontal axis we display the number of extremal vertices $n$ of the local state space, while on the vertical axis we display the respective winning probability. The points are the values obtained in our optimizations while the curves depict respective formulas these values follow. The colors are used to group points that appear to follow the same analytic formula.