# Jonathan Steinberg

PhD studentRoom: B-106

Phone:

## Preprints

See also arxiv

H. Chau Nguyen, Jan Lennart Bönsel, Jonathan Steinberg, Otfried Gühne
*
Optimising shadow tomography with generalised measurements
*

arXiv:2205.08990

Advances in quantum technology require scalable techniques to efficiently extract information from a quantum system, such as expectation values of observables or its entropy. 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 analysed based on outcomes of ideal projective measurements on the system upon application of randomised unitaries. Here, we suggest that shadow tomography can be much more straightforwardly formulated for generalised measurements, or positive operator valued measures. Based on the idea of the least-square estimator, shadow tomography with generalised measurements is both more general and simpler than the traditional formulation with randomisation of unitaries. In particular, this formulation allows us to analyse theoretical aspects of shadow tomography in detail. For example, we provide a detailed study of the implication of symmetries in shadow tomography. Shadow tomography with generalised measurements is also indispensable in realistic implementation of quantum mechanical measurements, when noise is unavoidable. Moreover, we also demonstrate how the optimisation of measurements for shadow tomography tailored toward a particular set of observables can be carried out.

Zhen-Peng Xu, Jonathan Steinberg, Jaskaran Singh, Antonio J. López-Tarrida, José R. Portillo, Adán Cabello
*
Graph-theoretic approach to Bell experiments with low detection efficiency
*

arXiv:2205.05098

Bell inequality tests where the detection efficiency is below a certain threshold $\eta_{\rm{crit}}$ can be simulated with local hidden-variable models. For the Clauser-Horne-Shimony-Holt Bell inequality and maximally entangled states, $\eta_{\rm{crit}} = 0.828$. Massar noticed that $\eta_{\rm{crit}}$ can tend to zero as the dimension $d$ of the local quantum systems grows, but found no advantage except for $d>1600$. V\'ertesi {\em et al.} lowered $\eta_{\rm{crit}}$ down to $0.770$ for maximally entangled states using $d=4$. M\'arton {\em et al.} studied the case of $N$ copies of the two-qubit maximally entangled state and obtained an upper bound of $0.693$ for $N=4$ (which is equivalent to $d=16$). Recently, Miklin {\em et al.} have presented a strategy that allows us to reduce $\eta_{\rm{crit}}$ down to $0.469$ for $d=512$. Here, we introduce a method to identify Bell tests requiring low $\eta_{\rm{crit}}$ and relatively low $d$. The method has two steps. First, we show a family of bipartite Bell inequalities for which $\eta_{\rm{crit}}$ is a function of some invariants of a certain type of graphs, and use it to identify correlations that require small $\eta_{\rm{crit}}$ for maximally entangled states. We present examples in which $\eta_{\rm{crit}} = 0.421$ for $d=16$, $\eta_{\rm{crit}} = 0.285$ for $d=28$, and $\eta_{\rm{crit}} = 0.192$ for $d=32$. We also show evidence that $\eta_{\rm{crit}}$ can be lowered down to $0.294$ for $d=16$ and present a method to make $\eta_{\rm{crit}}$ arbitrarily small by increasing the dimension and the number of settings. All these values for $\eta_{\rm{crit}}$ are valid (as it is the case in the literature) assuming no noise. 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 $\eta_{\rm{crit}}$ 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 $\eta_{\rm{crit}}$ is $12\%$ lower and the required visibility is $14.6\%$ lower than the ones required after the first step. The tools presented here pave the way for high-dimensional loophole-free Bell tests and loophole-free Bell nonlocality over long distances.

Adam Czaplinski, Thorsten Raasch, Jonathan Steinberg
*
Real eigenstructure of regular simplex tensors
*

arXiv:2203.01865

We are concerned with the eigenstructure of supersymmetric tensors. Like 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 exhibit 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, H. Chau Nguyen, Otfried Gühne
*
No-go theorem based on incomplete information of Wigner about his friend
*

arXiv:2111.15010

The notion of measurements is central for 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 gedankenexperiment 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 show that already partial information about a measurement carried out by Wigner's friend leads to the same conclusion. Our approach uses strictly less assumptions than previous approaches; moreover, our protocol allows to ensure that the measurement process did take place, without merely trusting the honesty of the actors or the functionality of the devices.

## Publications

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.