#### Friday Quantum Seminar

Quantum state and unitary $t$-designs play an important role in several applications, including tomography, randomized benchmarking, state discrimination, cryptography, sensing, and fundamental physics. In this work, we generalize the notion of state designs to infinite-dimensional, separable Hilbert spaces. We first prove that under the definition of continuous-variable (CV) state $t$-designs from [Comm. Math. Phys 326, 755-771 (2014)], no state designs exist for $t\geq2$. Similarly, we prove that no CV unitary $t$-designs exist for $t\geq 2$. 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 CV designs, we develop a protocol for the shadow tomography of CV states. We then regularize rigged $t$-designs and construct energy-constrained CV state designs. Using regularized-rigged designs, we develop the notion of the average fidelity of a CV quantum channel. We provide an explicit formula for the average fidelity between an ideal displacement operation and its experimental approximation. Moreover, analogous to qudits, we establish a relationship between the average gate fidelity to entanglement fidelity for CV operations.

(Pizza and refreshments will be served after the talk.)