Jonathan Steinberg

PhD student

Room: B-106

Phone:

Preprints

Publications

Adam Czaplinski, Thorsten Raasch, Jonathan Steinberg
Real eigenstructure of regular simplex tensors
Adv. Appl. Math. 148, 102521 (2023), arXiv:2203.01865

We are concerned with the real eigenstructure of symmetric tensors. As in the matrix case, normalized tensor eigenvectors are fixed points of the tensor power iteration map. However, unless the given tensor is orthogonally decomposable, some of these fixed points may be repelling and therefore be undetectable by any numerical scheme. In this paper, we consider the case of regular simplex tensors whose symmetric decomposition is induced by an overcomplete, equiangular set of n+1 vectors from Rn. We discuss the full real eigenstructure of such tensors, including the robustness analysis of all normalized eigenvectors. As it turns out, regular simplex tensors have robust as well as non-robust eigenvectors which, moreover, only partly coincide with the generators from the symmetric tensor decomposition.

Zhen-Peng Xu, Jonathan Steinberg, Jaskaran Singh, Antonio J. López-Tarrida, José R. Portillo, and Adán Cabello
Graph-theoretic approach to Bell experiments with low detection efficiency
Quantum 7, 922 (2023), arXiv:2205.05098

Bell inequality tests where the detection efficiency is below a certain threshold can be simulated with local hidden-variable models. Here, we introduce a method to identify Bell tests requiring low detection efficiency and relatively low dimension of the local quantum systems. The method has two steps. First, we show a family of bipartite Bell inequalities for which, for correlations produced by maximally entangled states, the detection efficiency can be upper bounded by a function of some invariants of graphs, and use it to identify correlations that require small detection efficiency. The second step is based on the observation that, using the initial state and measurement settings identified in the first step, we can construct Bell inequalities with smaller detection efficiency and better noise robustness. For that, we use a modified version of Gilbert's algorithm that takes advantage of the automorphisms of the graphs used in the first step. We illustrate its power by explicitly developing an example in which the tools presented here may allow for developing high-dimensional loophole-free Bell tests and loophole-free Bell nonlocality over long distances.

Zhen-Peng Xu, Jonathan Steinberg, H. Chau Nguyen, and Otfried Gühne
No-go theorem based on incomplete information of Wigner about his friend
Phys. Rev. A. 107, 022424 (2023), arXiv:2111.15010

The notion of measurements is central to many debates in quantum mechanics. One critical point is whether a measurement can be regarded as an absolute event, giving the same result for any observer in an irreversible manner. Using ideas from the gedanken experiment of Wigner's friend, it has been argued that, when combined with the assumptions of locality and no superdeterminism, regarding a measurement as an absolute event is incompatible with the universal validity of quantum mechanics. We consider a weaker assumption: Is the measurement event realized relatively to the observer when he only partially observed the outcome? We proposed a protocol to show that this assumption in conjunction with the natural assumptions of no superdeterminism and locality is also not compatible with the universal validity of quantum mechanics.

H. Chau Nguyen, Jan Lennart Bönsel, Jonathan Steinberg, Otfried Gühne
Optimising shadow tomography with generalised measurements
Phys. Rev. Lett. 129, 220502 (2022), arXiv:2205.08990

Advances in quantum technology require scalable techniques to efficiently extract information from a quantum system. Traditional tomography is limited to a handful of qubits, and shadow tomography has been suggested as a scalable replacement for larger systems. Shadow tomography is conventionally analyzed based on outcomes of ideal projective measurements on the system upon application of randomized unitaries. Here, we suggest that shadow tomography can be much more straightforwardly formulated for generalized measurements, or positive operator valued measures. Based on the idea of the least-square estimator shadow tomography with generalized measurements is both more general and simpler than the traditional formulation with randomization of unitaries. In particular, this formulation allows us to analyze theoretical aspects of shadow tomography in detail. For example, we provide a detailed study of the implication of symmetries in shadow tomography. Moreover, with this generalization we also demonstrate how the optimization of measurements for shadow tomography tailored toward a particular set of observables can be carried out.

Jonathan Steinberg, H. Chau Nguyen, Matthias Kleinmann
Minimal scheme for certifying three-outcome qubit measurements in the prepare-and-measure scenario
Phys. Rev. A 104, 062431 (2021), arXiv:2105.09925

The number of outcomes is a defining property of a quantum measurement, in particular, if the measurement cannot be decomposed into simpler measurements with fewer outcomes. Importantly, the number of outcomes of a quantum measurement can be irreducibly higher than the dimension of the system. The certification of this property is possible in a semi-device-independent way either based on a Bell-like scenario or by utilizing the simpler prepare-and-measure scenario. Here we show that in the latter scenario the minimal scheme for a certifying an irreducible three-outcome qubit measurement requires three state preparations and only two measurements and we provide experimentally feasible examples for this minimal certification scheme. We also discuss the dimension assumption characteristic to the semi-device-independent approach and to which extend it can be mitigated.

Jonathan Steinberg
Extensions and Restrictions of Generalized Probabilistic Theories
Springer Spektrum BestMaster, ISBN:978-3-658-37580-5 (2021)

Generalized probabilistic theories (GPTs) allow us to write quantum theory in a purely operational language and enable us to formulate other, vastly different theories. As it turns out, there is no canonical way to integrate the notion of subsystems within the framework of convex operational theories. Sections can be seen as generalization of subsystems and describe situations where not all possible observables can be implemented. Jonathan Steinberg discusses the mathematical foundations of GPTs using the language of Archimedean order unit spaces and investigates the algebraic nature of sections. This includes an analysis of the category theoretic structure and the transformation properties of the state space. Since the Hilbert space formulation of quantum mechanics uses tensor products to describe subsystems, he shows how one can interpret the tensor product as a special type of a section. In addition he applies this concept to quantum theory and compares it with the formulation in the algebraic approach. Afterwards he gives a complete characterization of low dimensional sections of arbitrary quantum systems using the theory of matrix pencils.

Jonathan Steinberg, H. Chau Nguyen, Matthias Kleinmann
Quaternionic quantum theory admits universal dynamics only for two-level systems
J. Phys. A: Math. Theor. 53, 375304 (2020), arXiv:2001.05482

We revisit the formulation of quantum mechanics over the quaternions and investigate the dynamical structure within this framework. Similar to standard complex quantum mechanics, time evolution is then mediated by a unitary which can be written as the exponential of the generator of time shifts. By imposing physical assumptions on the correspondence between the energy observable and the generator of time shifts, we prove that quaternionic quantum theory admits a time evolution for systems with a quaternionic dimension of at most two. Applying the same strategy to standard complex quantum theory, we reproduce that the correspondence dictated by the Schr\"odinger equation is the only possible choice, up to a shift of the global phase.

The Kadison-Singer conjecture (Bachelor Thesis in math)

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