Continuous-variable quantum state designs: theory and applications

We generalize the notion of quantum state designs to infinite-dimensional spaces. We first prove that, under the definition of continuous-variable (CV) state $t$-designs from Comm. Math. Phys. 326, 755 (2014), no state designs exist for $t\ge 2$. Similarly, we prove that no CV unitary $t$-designs exist for $t\ge 2$. We propose an alternative definition for CV state designs, which we call rigged $t$-designs, and provide explicit constructions for $t=2$. As an application of rigged designs, we develop a design-based shadow-tomography protocol for CV states. Using energy-constrained versions of rigged designs, we define an average fidelity for CV quantum channels and relate this fidelity to the CV entanglement fidelity. As an additional result of independent interest, we establish a connection between torus $2$-designs and complete sets of mutually unbiased bases.