Probabilistic theories stable under teleportation

Abstract

A long-standing problem in the foundations of quantum mechanics is to identify a physical principle that explains why algebraically maximal violations of Bell inequalities can generally not be achieved in Nature. One recently proposed approach considers iterated Bell tests, where a Bell test is performed on a state that has undergone several rounds of entanglement swapping. Obtaining large violations in this scenario is more demanding, because it requires a theory to have both highly entangled states and highly entangled measurements. It has been conjectured that the maximal quantum mechanical Clauser-Horne-Shimony-Holt (CHSH)-value of $2\sqrt2$ might be optimal for any probabilistic theory which, like quantum mechanics, maintains its CHSH-value after an arbitrary number of rounds of entanglement swapping. However, in a previous paper, we have exhibited a first example of a probabilistic theory that can sustain a CHSH value of $4$ in this setting. In this work, further investigating this property, we give a classification of all general probabilistic theories (GPTs) whose CHSH value is stable in the above sense. The problem reduces to a representation-theoretic condition that allows for exactly seven solutions. The GPT from our previous work showed some counter-intuitive features, e.g. that the local state space had a higher dimension than seemed necessary to realize CHSH tests. The classification shows that this is necessarily so. Along the way, we generalize the concept of self-testing to GPTs.

Figures (5)

• Figure 1: Diagrammatic representation of teleportation. The joint measurement $\{ \phi_k \}_k$ on a local state $\sigma$ and one-half of a bipartite state $\rho$, results in a local state $\sigma_k$, when conditioned on the measurement outcome corresponding to $\phi_k$.
• Figure 2: The possible ways to pair a bipartite state with a bipartite effect. (a) Depicts the usual pairing $\rho(\phi_k)$. In order to translate it into the trace pairing, we have to introduce a transpose to one of the two maps: $\mathop{\mathrm{tr}}\nolimits(\hat{\rho}^T\hat{\phi}_k)$. (b) Depicts the other way to pair the two, namely by "closing the loop". In order to translate this pairing into a trace we do not have to introduce a transpose: $\mathop{\mathrm{tr}}\nolimits(\hat{\rho}\hat{\phi}_k)$. Such pairings arise when the type of subsystems, which the "legs" of the states and effects correspond to, are restricted (see Sec. \ref{['subsec:elim_dofs']}).
• Figure 3: (a) The resulting (sub-normalized) state after the Bobs perform entanglement swapping. (b) The most general experiment that can be performed given the data specifying the iterated CHSH game. Here $\Phi_i \in \mathop{\mathrm{conv}}\nolimits(\mathcal{M} \cup \Omega_C \otimes \Omega_A)$, i.e., that can be either product or entangled (or a convex combination of both). The iterated CHSH game is a special case of this setup where e.g. $\Phi_1$ is chosen to be a product effect from $\Omega_C \otimes \Omega_A$ and the rest of the $\Phi_i$ are chosen from $\mathcal{M}$. This is because choosing one of the $\Phi_i$ to be a product effect allows us to "break the loop" and unravel it into an instance of the Iterated CHSH game.
• Figure 4: Diagrammatic representation of the teleportation map $R_k$.
• Figure 5: The two ways to read the result of entanglement swapping. One can either read it as first applying the map $\hat{\rho} : V_A \to V_C^*$ and then the map $R_{\vec{k}} : V_C^* \to V_C^*$ or first the map $L_{\vec{k}} : V_A \to V_A$ and then the map $\hat{\rho} : V_A \to V_C^*$.

Theorems & Definitions (25)

• Definition 1: The effective CHSH GPT
• Definition 2: GPT isomorphism
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof