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 [Blume-Kohout et al., Commun. 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.

Popular Summary

Physical quantum systems are often described by an infinite number of degrees of freedom. Such systems often offer advantages over finite-dimensional systems for various applications in quantum communication and computation. However, since finite-dimensional systems are typically much simpler to analyze, many tools and techniques have been developed for finite-dimensional quantum information that have not yet been applied to infinite-dimensional systems. In this work, we study some examples. In particular, we study the theory and applications of quantum $t$-designs in infinite-dimensional quantum-information theory and experiment.

Quantum $t$-designs are ensembles of quantum states or operations that satisfy certain properties that closely mimic uniform averaging. Whereas many $t$-designs have been constructed and used in finite-dimensional systems, constructing $t$-designs for an infinite-dimensional system has been a long-standing problem.

We present a systematic study of infinite-dimensional designs, answering many previously open questions about their existence and relevant applications. In particular, we prove that infinite-dimensional $t$-designs do not exist. Fortunately, not all hope is lost, as we introduce the concept of rigged $t$-designs and provide various constructions for these. A notable application of rigged $t$-designs is to record classical snapshots, known as shadows, of an infinite-dimensional quantum state, efficiently characterizing many of its properties. Another long-standing question we address is to define the notion of the average fidelity for an infinite-dimensional quantum channel, which characterizes how well a desired quantum operation is realized in experiments.

A practical next step is to construct more examples of rigged $t$-designs along with their experimental realization. Another intriguing avenue is extending the concept of infinite-dimensional $t$-designs to more general function spaces, such as those inherent in quantum field theories.

Figure 1

Sketch of definitions of finite-dimensional designs, CV designs, and rigged designs. The key point is the absence of the middle block in the middle and right columns. A generalization of the middle block to the continuous-variable case is ill defined, as discussed in Sec. 3. Therefore, to define CV designs, we simply skip the middle step, as discussed in Definition 1. An alternative characterization or definition of CV and rigged designs is described in Appendix pp4-s5.

Figure 2

Various fidelity benchmarks for the pure-loss channel ${\mathcal{L}}^{\kappa }$ plotted vs the channel’s transmissivity $\kappa$, with the energy-constrained parameter $\overline{n}=4$, and all other fidelity parameters being functions of $\overline{n}$ according to Eq. (50). (a) Comparison of fidelities that utilize a reference mode: the CV entanglement fidelity $F_e$ (45) with soft- and hard-energy constraints (19) as well as the minimum energy-constrained entanglement fidelity $F_{\min }$ (51). (b) Comparison of our three average-fidelity quantities—the soft-energy-constrained average fidelities ${\overline{F}}_1^{(R_{\beta })}$ (46a) and ${\overline{F}}_2^{(R_{\beta /2})}$ (46b) as well as the hard-energy-constrained case ${\overline{F}}_{1,2}^{(P_d)}$—with the fidelity ${\overline{F}}_{\mathrm{coh}}^{\overline{n}}$ (52) calculated by averaging over an ensemble of coherent states. The qualitatively different behavior of the coherent-state fidelity suggests that it may not be a good approximation to averages over CV states.

Figure 3

An outline of the proof of the non-existence of continuous-variable $t$-designs for $t\ge 2$.