Universal method for optimized robustness in self-testing of quantum resources

Abstract

Self-testing is a phenomenon where the use of specific quantum states or measurements can be inferred solely from the correlations they generate. We introduce a universal method for conducting robustness analysis in the self-testing of various quantum resources. Unlike previous numerical approaches, which rely on selecting specific isometries, our method optimizes over equivalence transformations, thereby leading to tighter robustness bounds. This optimization employs the well-established technique of semidefinite programming relaxations for non-commuting polynomial optimization. Our method can be universally applied to diverse self-testing settings, including steerable assemblages in the Bell scenario, constellations of quantum states in the prepare-and-measure scenario, and entangled states in the steering scenario. We demonstrate the method's capability to surpass previously reported robustness bounds across a range of concrete examples.

Figures (4)

• Figure 1: The considered self-testing settings: (a) steerable assemblages $\Set{\sigma_{a\vert x}}_{a,x}$ in the Bell scenario, (b) constellations of states $\Set{\rho_x}_x$ in the prepare-and-measure scenario, (c) entangled states ${\rho_\mathrm{AB}}$ in the steering scenario. In all the considered two-party scenarios, $x\in [n_x]$, $a\in [n_a]$ and $y\in [n_y]$, $b\in [n_b]$, are Alice's and Bob's inputs and outputs, respectively. The central question in self-testing, is whether from the observed data $\Delta$, namely $P(a,b\vert x,y)$ in (a), $P(b\vert x,y)$ in (b), and $\sigma_{a\vert x}$ in (c), it follows that there exists an equivalence transformation $\Lambda$ that maps the reference resource to the experimental one.
• Figure 2: Robust self-testing of a steerable assemblage in the CHSH scenario. In this example, we assume $P(a\vert x)=1/2$. The level of hierarchy we implement is $\ell=3$. The horizontal line with the value of $0.8536$ represents the classical fidelity.
• Figure 3: Robust self-testing of a constellation of states in $2\to 1$RAC with shared randomness. $l\in\{1,2,3\}$ stands for the level of the SDP hierarchy, defined similarly as in Refs. Navascues15prlNavascues15.
• Figure 4: Robust self-testing of a two-qubit entangled state in the steering scenario. $l\in\{1,2,3\}$ stands for the level of the SDP hierarchy, defined similarly as in Refs. NPANPA2008.