Focus on quantum tomography

The ability to completely characterize the state and dynamics of a quantum system through physical measurements is an essential element in the emerging field of quantum technologies. Owing to extensive early research on the reconstruction of Wigner functions from their projections along a collection of quadratures [1, 2], this task is now commonly known as quantum tomography. Theoretical work on this problem dates back at least to the 1970s, and experimental implementations are routinely carried out in a wide variety of systems—in this collection alone, the reader will encounter characterizations of single-photon and continuous-variable states and polarization transformations [3–9], cavity fields [10], atomic ensembles [11–13], trapped ions [14] and of optical detectors [15–17].

Despite this success, many fundamental questions remain unanswered. What is more, a variety of new experimental and theoretical developments have given rise to a substantial surge of research activity in this area in recent years. The most obvious development is the tremendous progress in controlling large, highly accessible quantum systems composed, for example, of trapped ions [18]. In these experimental platforms, measurements of arbitrary observables on individual systems can be performed with great accuracy. As a consequence, the bottleneck limiting further progress in estimating the states of such systems has shifted from physical controllability to the problem of handling the massive quantity of data resulting from the exponential scaling of the number of parameters describing quantum many-body states. This curse of dimensionality renders any naive approach to quantum tomography manifestly impossible, even for moderately large systems. Ultimately, this scaling problem cannot be overcome, it is a necessary feature of quantum processing devices outperforming the classical. It turns out, however, that the boundary where classical methods fail can be pushed significantly if non-trivial structural information on the quantum systems under consideration is utilized. We list a few examples: due to either physical reasons (low temperature) or 'engineering' reasons (in which one aims to prepare a pure state, which is commonly the case), states encountered in the laboratory are often fairly pure in the sense that their effective rank is small. This allows for state reconstruction with a square root improvement [14, 19–22]. The state under consideration may be the ground or thermal state of a local Hamiltonian, which are known to be well approximated by matrix product states and operators [23], allowing for a reconstruction that may be achieved with linearly many measurement settings [24–27]. A state may have been prepared such that it is permutationally invariant, which allows for its reconstruction using polynomial resources [28, 29]. Turning from states to processes, a structural feature of channels may be that they are given by a network of gates. Under suitable assumptions on the figure of merit, it turns out that independent measurements on the components are optimal [30]. This illustrates the more general challenge of building estimation schemes that utilize structure and symmetry of the underlying system—be it physically or operationally motivated. We believe that there is both the potential and the urgent need to explore this line of research further.

New impetus for quantum tomography developments has repeatedly come from novel non-trivial developments in classical machine-learning theory. Indeed, the problem of turning huge and noisy data sets into meaningful information is by no means unique to the quantum laboratory. In the classical world, the ubiquity of the internet and the availability of cheap sensors in areas as diverse as life sciences and industrial applications has given rise to the paradigm of big data. We have recently seen several instances—compressed sensing being a prominent example—where ideas have flowed in both directions between researchers working on classical high-dimensional data analysis on the one hand, and quantum physicists thinking about new theoretical models for tomography on the other.

We would also like to draw attention to the fact that new directions in tomography have recently been driven by the unique needs of quantum cryptography. Here, the need for absolutely rigorous statements concerning the uncertainties of the available resources is particularly acute. This has led to a new perspective on the concept of region estimators for quantum problems and to the revisiting of intrinsic challenges such as incomplete measurements, imperfectly characterized detectors, finite data and other error sources [31–41]. This development also serves as a reminder that generic, off-the-shelf methods are often insufficient for highly specialized applications.

The editors agree that the appeal of the subject lies largely in the breadth of techniques that must be brought together in order to fully understand the problem. To live up to the highest standards, it is essential to have a thorough understanding of the particular experiment generating the data; one needs a solid grasp of theoretical physics to understand the uniquely quantum mechanical aspects; a rigorous error analysis requires knowledge of mathematical statistics; and lastly, non-trivial problems in numerical analysis need to be solved. We believe that more communication between researchers working in these very different fields is crucial for further progress. It is our hope that this 'focus on' collection provides a platform for facilitating this necessary exchange of ideas.