We introduce new methods and tools to study and characterise classical and quantum correlations emerging from prepare-and-measure experiments with informationally restricted communication. We consider the most general kind of informationally restricted correlations, namely the ones formed when the sender is allowed to prepare statistical mixtures of mixed states, showing that contrary to what happens in Bell nonlocality, mixed states can outperform pure ones. We then leverage these tools to derive device-independent witnesses of the information content of quantum communication, witnesses for different quantum information resources, and demonstrate that these methods can be used to develop a new avenue for semi-device independent random number generators.
Quantum communication leads to strong correlations, that can outperform classical ones. Complementary to previous works in this area, we investigate correlations in prepare-and-measure scenarios assuming a bound on the information content of the quantum communication, rather than on its Hilbert-space dimension. Specifically, we explore the extent of classical and quantum correlations given an upper bound on the one-shot accessible information. We provide a characterisation of the set of classical correlations and show that quantum correlations are stronger than classical ones. We also show tha…
In the present work, we suggest an approach for describing dynamics of finite-dimensional quantum systems in terms of pseudostochastic maps acting on probability distributions, which are obtained via minimal informationally complete quantum measurements. The suggested method for probability representation of quantum dynamics preserves the tensor product structure, which makes it favourable for the analysis of multi-qubit systems. A key advantage of the suggested approach is that minimal informationally complete positive operator-valued measures (MIC-POVMs) are easier to construct in comparison…
Although quantum entanglement has already been verified experimentally and applied in quantum computing, quantum sensing and quantum networks, most of the existing measures cannot characterize the entanglement faithfully. In this work, by exploiting the Schmidt decomposition of a bipartite state $|ψ\rangle_{AB}$, we first establish a one-to-one correspondence relation between the characteristic polynomial of the reduced state $ρ_A$ and the polynomials its trace. Then we introduce a family of entanglement measures which are given by the complete eigenvalues of the reduced density matrices of th…
It is well-known in classical frame theory that overcomplete representations of a given vector space provide robustness to additive noise on the frame coefficients of an unknown vector. We describe how the same robustness can be shown to exist in the context of quantum state estimation. A key element of the discussion is the application of classical frame theory to operator-valued vector spaces, or operator spaces, which arise naturally in quantum mechanics. Specifically, in the problem we describe the frame vectors are represented by the elements of an informationally complete or overcomplete…
The Generalized BEADS Representation: An Informationally Complete Visualization of Mixed Quantum States
Technical University of Munich, TUM School of Computation, Information and Technology（慕尼黑工业大学计算、信息与技术学院）；Technical University of Munich, TUM School of Natural Sciences（慕尼黑工业大学自然科学学院）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）
Recently introduced shadow tomography protocols use classical shadows of quantum states to predict many target functions of an unknown quantum state. Unlike full quantum state tomography, shadow tomography does not insist on accurate recovery of the density matrix for high rank mixed states. Yet, such a protocol makes multiple accurate predictions with high confidence, based on a moderate number of quantum measurements. One particular influential algorithm, proposed by Huang, Kueng, and Preskill arXiv:2002.08953, requires additional circuits for performing certain random unitary transformation…
Minimal Informationally Complete quantum measurements, or MICs, illuminate the structure of quantum theory and how it departs from the classical. Central to this capacity is their role as tomographically complete measurements with the fewest possible number of outcomes for a given finite dimension. Despite their advantages, little is known about them. We establish general properties of MICs, explore constructions of several classes of them, and make some developments to the theory of MIC Gram matrices. These Gram matrices turn out to be a rich subject of inquiry, relating linear algebra, numbe…
Informationally complete measurements are a dramatic discovery of quantum information science, and the symmetric IC measurements, known as SICs, are in many ways optimal among them. Close study of three of the "sporadic SICs" reveals an illuminating relation between different ways of quantifying the extent to which quantum theory deviates from classical expectations.
The rapid progress in quantum computing witnessed in recent years has sparked widespread interest in developing scalable quantum information theoretic methods to work with large quantum systems. For instance, several approaches have been proposed to bypass tomographic state reconstruction, and yet retain to a certain extent the capability to estimate multiple physical properties of a given state previously measured. In this paper, we introduce the Virtual Linear Map Algorithm (VILMA), a new method that enables not only to estimate multiple operator averages using classical post-processing of i…
Spekkens has introduced an epistemically restricted classical theory of discrete systems, based on discrete phase space. The theory manifests a number of quantum-like properties but cannot fully imitate quantum theory because it is noncontextual. In this paper we show how, for a certain class of quantum systems, the quantum description of an experiment can be decomposed into classical descriptions that are epistemically restricted, though in a different sense than in Spekkens' work. For each aspect of the experiment -- the preparation, the transformations, and the measurement -- the epistemic …
We devise a classical algorithm which efficiently computes the quantum expectation values arising in a class of continuous variable quantum circuits wherein the final quantum observable | after the Heisenberg evolution associated with the circuits | is at most second order in momentum. The classical computational algorithm exploits a specific epistemic restriction in classical phase space which directly captures the quantum uncertainty relation, to transform the quantum circuits in the complex Hilbert space into classical albeit unconventional stochastic processes in the phase space. The resul…
Given a classical channel---a stochastic map from inputs to outputs---the input can often be transformed to an intermediate variable that is informationally smaller than the input. The new channel accurately simulates the original but at a smaller transmission rate. Here, we examine this procedure when the intermediate variable is a quantum state. We determine when and how well quantum simulations of classical channels may improve upon the minimal rates of classical simulation. This inverts Holevo's original question of quantifying the capacity of quantum channels with classical resources. We …
We explore a particular way of reformulating quantum theory in classical terms, starting with phase space rather than Hilbert space, and with actual probability distributions rather than quasiprobabilities. The classical picture we start with is epistemically restricted, in the spirit of a model introduced by Spekkens. We obtain quantum theory only by combining a collection of restricted classical pictures. Our main challenge in this paper is to find a simple way of characterizing the allowed sets of classical pictures. We present one promising approach to this problem and show how it works ou…
We consider a communication scenario where classical information is encoded in an ensemble of quantum states that admit a power series expansion in a cost parameter and with the vanishing cost converge to a single zero-cost state. For a given measurement scheme, we derive an approximate expression for mutual information in the leading order of the cost parameter. The general results are applied to selected problems in optical communication, where coherent states of light are used as input symbols and the cost is quantified as the average number of photons per symbol. We show that for an arbitr…
We investigate the correlations that can arise between Alice and Bob in prepare-and-measure communication scenarios where the source (Alice) and the measurement device (Bob) can share prior entanglement. The paradigmatic example of such a scenario is the quantum dense coding protocol, where the communication capacity of a qudit can be doubled if a two-qudit entangled state is shared between Alice and Bob. We provide examples of correlations that actually require more general protocols based on higher-dimensional entangled states. This motivates us to investigate the set of correlations that ca…
A Restricted Boltzmann Machine with Quantum-State Visible Units
Institute for Advanced Study in Physics and School of Physics, Zhejiang University（浙江大学物理学院和高等研究院）；Wuhan Institute of Physics and Mathematics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences（中国科学院精密测量科学与技术创新研究院武汉物理与数学研究所）
Quantum processing units boost entanglement at the level of hardware and enable physical simulations of highly correlated electron states in molecules and intermolecular chemical bonds. The variational quantum eigensolver provides a hardware-efficient toolbox for ground state simulation; however, with limitations in precision. Even in the absence of noise, the algorithm may result into a biased energy estimation, particularly with some shallower ansatz types. Noise additionally degrades entanglement and hinders the ground state energy estimation (especially if the noise is not fully characteri…
Quantum computers solve ever more complex tasks using steadily growing system sizes. Characterizing these quantum systems is vital, yet becoming increasingly challenging. The gold-standard is quantum state tomography (QST), capable of fully reconstructing a quantum state without prior knowledge. Measurement and classical computing costs, however, increase exponentially in the system size - a bottleneck given the scale of existing and near-term quantum devices. Here, we demonstrate a scalable and practical QST approach that uses a single measurement setting, namely symmetric informationally com…
We consider a task in which classical information is encoded into a quantum system by an operation restricted by symmetry. The maximum amount of classical information that can be encoded under this restriction, namely the symmetry-restricted classical information capacity, depends on the initial state of the system. Our focus is on whether the capacity of an asymmetric state can be strictly larger than that of any symmetric states, whereas the latter is a strictly positive constant. That is, we ask whether an analog of superdense coding is implementable in the resource theory of asymmetry. We …
Operator representation of spatiotemporal quantum correlations
School of Mathematics and Statistics, Hainan University（海南大学数学与统计学院）；Hainan International Exchange Center for Theoretical Physics（海南理论物理交流中心）；Massachusetts Institute of Technology（麻省理工学院）
Expressive equivalence of classical and quantum restricted Boltzmann machines
Computation-Based Science and Technology Research Center, The Cyprus Institute（计算科学与技术研究中心、塞浦路斯研究所）；European Organisation for Nuclear Research (CERN)（欧洲核子研究中心）
Quantum-centric strong and dynamical electron correlation: A resource-efficient second-order $N$-electron valence perturbation theory formulation for near-term quantum devices
