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Many-Body Quantum Shadow-State Tomography: Unveiling Hidden Layer Structures via Higher-Order Quantum Correlations

Jincheng Zhang

Published Aug 11, 2026• 0 citations
Publisher

Abstract

Quantum many-body systems often harbor structural secrets that elude standard tomographic techniques, which typically focus on low-order marginals or direct system observables. In this paper, we introduce the framework of Many-Body Quantum Shadow-State Tomography (MQSST) to address the fundamental question of whether quantum states possess hidden layers of structure that do not directly participate in ordinary measurements yet fundamentally dictate observable behavior through higher-order correlations. We formalize the concept of a shadow-state layer, proposing that the total density matrix of a complex quantum system can be mapped onto an observable sector via a partial trace or projection over a shadow subspace. To reconstruct this hidden layer without direct physical access, we develop an advanced inversion theory driven by multi-point correlation functions up to fourth order, denoted as C^(2), C^(3), and C^(4), combined with a specialized inverse transformation operator T^-1. Our formulation establishes a rigorous mathematical bridge between visible low-energy sectors and invisible shadow spaces, offering profound implications for understanding macroscopic quantum phenomena, entanglement structures, and complex phase transitions.

DOI
10.5281/zenodo.21885418

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Quantum many-body systems often harbor structural secrets that elude standard tomographic techniques, which typically focus on low-order marginals or direct system observables. In this paper, we introduce the framework of Many-Body Quantum Shadow-State Tomography (MQSST) to address the fundamental question of whether quantum states possess hidden layers of structure that do not directly participate in ordinary measurements yet fundamentally dictate observable behavior through higher-order correlations. We formalize the concept of a shadow-state layer, proposing that the total density matrix of a complex quantum system can be mapped onto an observable sector via a partial trace or projection over a shadow subspace. To reconstruct this hidden layer without direct physical access, we develop an advanced inversion theory driven by multi-point correlation functions up to fourth order, denoted as C^(2), C^(3), and C^(4), combined with a specialized inverse transformation operator T^-1. Our formulation establishes a rigorous mathematical bridge between visible low-energy sectors and invisible shadow spaces, offering profound implications for understanding macroscopic quantum phenomena, entanglement structures, and complex phase transitions.
Quantum shadow tomography based on the classical shadow representation provides an efficient way to estimate properties of an unknown quantum state without performing a full quantum state tomography. In scenarios where estimating the expectation values for only certain classes of observables is required, obtaining information about the entire density matrix is unnecessary. We propose a partial quantum shadow tomography protocol that estimates a subset of density matrix elements relevant to the expectation values of structured observables. Specifically, we identify specific subsets of the tomographically complete set Cl(2)⊗n{\mathrm{Cl}}(2)^{\otimes n}Cl(2)⊗n and a simple pseudo-inverse of the associated channel, which can be used to estimate all elements of the density matrix with the same active order. By restricting the protocol to smaller subsets of single-qubit Pauli measurements, it becomes experimentally more efficient. We demonstrate the advantage over unitary designs, such as the Clifford, full Pauli basis, and methods utilizing mutually unbiased bases, by analytically deriving error bounds and numerically evaluating the protocol on structured operators. We experimentally demonstrate the partial shadow estimation scheme for a wide class of two-qubit states (pure, entangled, and mixed) in the nuclear magnetic resonance (NMR) platform. The full density matrix, reconstructed experimentally by combining different partial estimators, achieves fidelities around 99%.
Large-scale quantum systems have advanced rapidly via the exploration of more particles and higher dimensions, offering great potential for developing quantum technologies. However, their characterization becomes prohibitive with increasing local dimensionality and particle number. Here we propose high-dimensional quantum overlapping tomography based on a graph-theoretic formulation, which allows one to efficiently reconstruct few-body marginals of multipartite high-dimensional quantum systems. We experimentally realize it on a photonic four-party entangled state in a 4×4×2×24 \times 4 \times 2 \times 24×4×2×2 system. Using measurements in mutually unbiased bases, we reconstruct all six two-body marginals with only 25 projective measurement settings, compared with 94 and 225 settings for independent tomography of all two-body reduced states and full state tomography, respectively. The reconstructed marginals reveal a layered entanglement structure vital for high-dimensional quantum networks. We further show that these marginals enable more noise-resilient certification of multipartite high-dimensional entanglement than the fidelity-based criterion. Our work thus offers a scalable route for learning multidimensional quantum systems.
Quantum state tomography provides a powerful and versatile framework for characterizing quantum states across diverse quantum systems. However, tomography is still largely restricted to systems with a few degrees of freedom, while for continuous systems, it is typically tailored to quantum-optical platforms. For other many-body systems described by continuous quantum fields such as ultracold atomic gases, tomographic technique is not as well-established. The present thesis addresses this gap by providing new approaches to measurement and tomography of ultracold Bose gases trapped in one (1D) and two dimensions (2D). These systems serve as analog quantum field theory (QFT) simulators and exhibit novel physics absent in higher dimensions. We begin by systematically assessing state-of-the-art method for extracting the relative phase field in parallel 1D Bose gases with matter-wave interference. In particular, we derive a formula for the systematic phase shift due to longitudinal dynamics, a factor that is often ignored in previous modeling of the measurement. We further examine how the single-shot readout error propagates to correlation and distribution functions of the relative phase. Then, we broaden the scope of experimentally accessible quantities by developing a new methodology to extract the common phase field---the sum of the phase fields of the two parallel gases---from the already available density ripple data. We rigorously benchmark this new readout with numerical simulations and experimental data in 1D systems and further generalize it to 2D systems. By combining the relative and common phases information, we improve Gaussian state tomography in parallel 1D Bose gases and apply it to probe new physical phenomena in quantum field simulators. Specifically, we focus on two applications: (i) field-space entanglement detection at finite temperatures and out of equilibrium, and (ii) probing non-linear relaxation in near-integrable quantum systems. This thesis bridges theory and experiment in the simulation of QFTs and reveals new opportunities for exploring many-body quantum effects in ultracold bosonic gases.
While classical shadows can efficiently predict key quantum state properties, their suitability for certified quantum state tomography remains uncertain. In this paper, we address this challenge by introducing a projected classical shadow (PCS) that extends the standard classical shadow by incorporating a projection step onto the target subspace. For a general quantum state consisting of n qubits, our method requires a minimum of O (4 n ) total state copies to achieve a bounded recovery error in the Frobenius norm between the reconstructed and true density matrices, reducing to O (2 n r ) for states of rank r < 2 n —meeting information-theoretic optimal bounds in both cases. For matrix product operator states, we demonstrate that the PCS can recover the ground-truth state with O ( n 2 ) total state copies, improving upon the previously established Haar-random bound of O ( n 3 ). Numerical simulations validate our scaling results and demonstrate the practical accuracy of the proposed PCS method.
Shadow tomography aims to build a classical description of a quantum state from a sequence of simple random measurements. Physical observables are then reconstructed from the resulting classical shadow. Shadow protocols which use single-body random measurements are simple to implement and capture few-body observables efficiently, but do not apply to systems with fundamental number conservation laws, such as ultracold atoms. We address this shortcoming by proposing and analyzing a new local shadow protocol adapted to such systems. The All-Pairs protocol requires one layer of two-body gates and only poly(V) samples to reconstruct arbitrary few body observables. Moreover, by exploiting the permutation symmetry of the protocol, we derive a linear time postprocessing algorithm which applies to both hardcore bosons and spinless fermions in any spatial dimension. We provide a proof-of-principle reference implementation and demonstrate the reconstruction of two- and four-point functions in a paired Luttinger liquid of hardcore bosons.
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 method for this task is quantum state tomography (QST), capable of fully reconstructing a quantum state without prior knowledge.The measurement and classical computing costs, however, increase exponentially with the number of constituents (e.g., qubits)-a daunting bottleneck given the scale of existing and nearterm quantum devices.Here, we demonstrate a scalable and practical QST approach that only uses a single measurement setting, namely symmetric informationally complete (SIC) positive operator-valued measures (POVMs).We implement these nonorthogonal measurements on an ion trap quantum processor by utilizing additional energy levels within each ion-without requiring ancillary ions to assist in measurements.More precisely, we locally map the SIC POVM to orthogonal states embedded in a higher-dimensional system, which we read out using repeated in-sequence detections, thereby providing full tomographic information in every shot.Combining this SIC tomography with the recently developed randomized measurement toolbox ("classical shadows") proves to be a powerful combination.SIC tomography alleviates the need for choosing measurement settings at random ("derandomization"), while classical shadows enable the estimation of arbitrary polynomial functions of the density matrix orders of magnitudes faster than standard methods.The latter enables in-depth entanglement characterization, which we experimentally showcase on a five-qubit absolutely maximally entangled state.Moreover, the fact that the full tomography information is available in every shot enables online QST in real time (i.e., while the experiment is running).We demonstrate this on an eight-qubit entangled state (which has 2 8 • 2 8 -1 = 65 535 degrees of freedom), as well as for fast state identification.All in all, these features single out SIC-based classical shadow estimation as a highly scalable and convenient tool for quantum state characterization.