Shadow Tomography: A Game-Changer for Quantum State Estimation and Quantum Technologies

Shadow tomography is a crucial tool in quantum information processing, providing succinct descriptions of quantum states using randomized measurement bases known as classical shadows. The shadow tomography framework was recently applied to continuous variable (CV) systems, a significant step in quantum state estimation. This application allows for high accuracy with a minimum number of samples, a key factor in developing and implementing quantum technologies. The framework was also applied retroactively to experimental quantum-optical data, demonstrating its practical utility. The theoretical guarantees of shadow tomography improve upon previous results, offering a robust and efficient framework for quantum state estimation.

What is Shadow Tomography, and What Is It Important in Quantum Information Processing?

Shadow tomography is a framework used in quantum information processing to construct succinct descriptions of quantum states. These descriptions are created using randomized measurement bases, known as classical shadows. The primary goal of quantum state tomography is to estimate states accurately with as few measurements as possible. This is an indispensable tool in quantum information processing. State characterization is necessary for realizing quantum technologies in finite-dimensional tensor-product quantum systems governed by discrete variables (DVs) and in quantum systems governed by continuous variables (CVs) such as electromagnetic or mechanical modes.

The recent development of classical shadow tomography provides a concise way to learn about a DV quantum state through randomly chosen measurements. The learned information can later be used to predict the properties of the state. A key benefit of shadow tomography is that it comes with rigorously proven guarantees on the minimum number of samples required to achieve high accuracy with high probability. However, the best guarantees require structures from finite-dimensional DV spaces such as state and unitary designs or symmetric informationally complete positive operator-valued measures (SICPOVMs), thereby obscuring any practical extension to the intrinsically infinite-dimensional CV systems.

How is Shadow Tomography Applied to Continuous Variable Systems?

A recent study applied the shadow tomography framework to a large family of well-known and well-used CV tomographic protocols. The mathematical tools behind shadow guarantees were distilled to eliminate the dependence on strictly DV ingredients. This allowed for the reformulation of established CV protocols in the shadow framework. This reformulation yields accuracy guarantees for expectation values of local observables whose required number of samples scales polynomially with the number of participating CV modes and each mode’s maximum occupation (aka photon number).

CV tomography is a longstanding and well-developed field. Focusing on established protocols allows for boosting their credibility as opposed to developing new protocols that may be equally efficient theoretically but whose practical utility is left as an open question. The work focused on tomographic methods that can be easily implemented experimentally with existing quantum optical technology, which underpins fiber-based and free-space quantum communication and key distribution.

What are the Practical Applications of Shadow Tomography?

Homodyne detection was The first method recast in the shadow tomography framework. The number of samples needs to scale at most as the fifth power of the maximum occupation number, up to a logarithmic correction, to yield reliable homodyne shadow estimates of a single-mode state. Since a finite occupation number cutoff is required, this bound holds only for portions of states supported on finite-dimensional subspaces of the infinite-dimensional Fock space.

The theoretical guarantees apply equally well if the corresponding protocols are performed in other CV platforms, such as microwave cavities coupled to superconducting qubits, motional degrees of freedom of trapped ions, and optomechanical and nanoscale acoustic resonators. The framework was also applied retroactively to experimental quantum-optical data published in 2010.

How Does Shadow Tomography Improve Quantum State Estimation?

The theoretical guarantees of shadow tomography use several critical technical bounds proved in earlier work. While there have been studies of the statistical efficiency of homodyne tomography, the bounds provided by shadow tomography are an improvement over previous results. They allow for the sample complexity analysis to achieve the convergence of the operator norm of the estimated state to its ideal value. These improvements are made possible because of matrix concentration inequalities not used in past work on CV state tomography.

In conclusion, shadow tomography provides a robust and efficient framework for quantum state estimation. Applying this framework to continuous variable systems makes it possible to achieve high accuracy with a minimum number of samples. This has significant implications for the development and implementation of quantum technologies.