Jonathan SteinbergPhD student
See also arxiv
Hamza Jnane, Jonathan Steinberg, Zhenyu Cai, H. Chau Nguyen, Bálint Koczor
Quantum Error Mitigated Classical Shadows
Classical shadows enable us to learn many properties of a quantum state ρ with very few measurements. However, near-term and early fault-tolerant quantum computers will only be able to prepare noisy quantum states ρ and it is thus a considerable challenge to efficiently learn properties of an ideal, noise free state ρid. We consider error mitigation techniques, such as Probabilistic Error Cancellation (PEC), Zero Noise Extrapolation (ZNE) and Symmetry Verification (SV) which have been developed for mitigating errors in single expected value measurements and generalise them for mitigating errors in classical shadows. We find that PEC is the most natural candidate and thus develop a thorough theoretical framework for PEC shadows with the following rigorous theoretical guarantees: PEC shadows are an unbiased estimator for the ideal quantum state ρid; the sample complexity for simultaneously predicting many linear properties of ρid is identical to that of the conventional shadows approach up to a multiplicative factor which is the sample overhead due to error mitigation. Due to efficient post-processing of shadows, this overhead does not depend directly on the number of qubits but rather grows exponentially with the number of noisy gates. The broad set of tools introduced in this work may be instrumental in exploiting near-term and early fault-tolerant quantum computers: We demonstrate in detailed numerical simulations a range of practical applications of quantum computers that will significantly benefit from our techniques.
Jonathan Steinberg, H. Chau Nguyen, Matthias Kleinmann
Certifying activation of quantum correlations with finite data
Quantum theory allows for different classes of correlations, such as entanglement, steerability or Bell-nonlocality. Experimental demonstrations of the preparation of quantum states within specific classes and their subsequent interconversion have been carried out; however, rigorous statements on the statistical significance are not available. Behind this are two difficulties: the lack of a method to derive a suitable confidence region from the measured data and an efficient technique to classify the quantum correlations for every state in the confidence region. In this work, we show how both of these problems can be addressed. Specifically, we introduce a confidence polytope in the form of a hyperoctahedron and provide a computationally efficient method to verify whether a quantum state admits a local hidden state model, thus being unsteerable and, consequently, Bell-local. We illustrate how our methods can be used to analyse the activation of quantum correlations by local filtering, specifically for Bell-nonlocality and quantum steerability.
Jonathan Steinberg, Otfried Gühne
Maximizing the geometric measure of entanglement
The characterization of the maximally achievable entanglement in a given physical system is relevant, as entanglement is known to be a resource for various quantum information tasks. This holds especially for pure multiparticle quantum states, where the problem of maximal entanglement is not only of physical interest, but also closely related to fundamental mathematical problems in multilinear algebra and tensor analysis. We propose an algorithmic method to find maximally entangled states of several particles in terms of the geometric measure of entanglement. Besides identifying physically interesting states our results deliver insights to the problem of absolutely maximally entangled states; moreover, our methods can be generalized to identify maximally entangled subspaces.
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.
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.
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.
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.
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)