This article serves as a brief report on the recent affairs in quantum compressive tomography, which is a class of techniques that significantly reduce the physical resources needed to uniquely reconstruct any rank-deficient quantum state, channel or measurement without any assumption about the unknown quantum object (be it the rank, sparsity, eigenbasis, etc.) of a given Hilbert-space dimension.
This article serves as a brief report on the recent affairs in quantum compressive tomography, which is a class of techniques that significantly reduce the physical resources needed to uniquely reconstruct any rank-deficient quantum state, channel or measurement without any assumption about the unknown quantum object (be it the rank, sparsity, eigenbasis, etc.) of a given Hilbert-space dimension.
This review serves as a concise introductory survey of modern compressive tomography developed since 2019. These are schemes meant for characterizing arbitrary low-rank quantum objects, be it an unknown state, a process or detector, using minimal measuring resources (hence compressive) without any a priori assumptions (rank, sparsity, eigenbasis, etc.) about the quantum object. This paper contains a reasonable amount of technical details for the quantum-information community to start applying the methods discussed here. To facilitate the understanding of formulation logic and physics of compressive tomography, the theoretical concepts and important numerical results (both new and cross-referenced) shall be presented in a pedagogical manner.
Quantum state tomography is both a crucial component in the field of quantum information and computation and a formidable task that requires an incogitable number of measurement configurations as the system dimension grows. We propose and experimentally carry out an intuitive adaptive compressive tomography scheme, inspired by the traditional compressed-sensing protocol in signal recovery, that tremendously reduces the number of configurations needed to uniquely reconstruct any given quantum state without any additional a priori assumption whatsoever (such as rank information, purity, etc.) about the state, apart from its dimension.
We perform several numerical studies for our recently published adaptive compressive tomography scheme [D. Ahn et al., Phys. Rev. Lett. 122, 100404 (2019)], which significantly reduces the number of measurement settings to unambiguously reconstruct any rank-deficient state without any a priori knowledge besides its dimension. We show that both entangled and product bases chosen by our adaptive scheme perform comparably well with recently known compressed-sensing element-probing measurements, and also beat random measurement bases for low-rank quantum states. We also numerically conjecture asymptotic scaling behaviors for this number as a function of the state rank for our adaptive schemes. These scaling formulas appear to be independent of the Hilbert-space dimension. As a natural development, we establish a faster hybrid compressive scheme that first chooses random bases, and later adaptive bases as the scheme progresses. As an epilogue, we reiterate important elements of informational completeness for our adaptive scheme.
Quantum state tomography (QST) is a fundamental task in quantum information science that aims to reconstruct unknown quantum states from measurement data. However, the exponential growth of Hilbert-space dimension with system size makes full tomography of general quantum states statistically and computationally prohibitive. This challenge has motivated extensive research on structured quantum state tomography, where prior structure, such as low-rankness, tensor-network representations, shallow quantum circuits, and neural quantum states, can substantially reduce the effective degrees of freedom and enable scalable recovery. In this review, we provide a unified perspective on QST for structured quantum states through three closely related themes: compact state representations, measurement design, and computational algorithms. After reviewing common models for structured quantum states, we survey existing work on geometric preservation properties of measurement frameworks, ranging from informationally complete POVMs to randomized measurements, and their implications for sample complexity. On the algorithmic side, we review optimization methods for reconstructing structured quantum states from empirical measurements. By connecting QST with broader principles from compressive sensing, matrix sensing, and structured inverse problems, this survey highlights common theoretical foundations underlying sample complexity, measurement efficiency, and scalable recovery.
We establish methods for quantum state tomography based on compressed sensing. These methods are specialized for quantum states that are fairly pure, and they offer a significant performance improvement on large quantum systems. In particular, they are able to reconstruct an unknown density matrix of dimension d and rank r using O(rdlog²d) measurement settings, compared to standard methods that require d² settings. Our methods have several features that make them amenable to experimental implementation: they require only simple Pauli measurements, use fast convex optimization, are stable against noise, and can be applied to states that are only approximately low rank. The acquired data can be used to certify that the state is indeed close to pure, so no a priori assumptions are needed.
Quantum state tomography (QST) is a crucial technique for characterizing the state of quantum systems. However, traditional QST requires a large number of measurements, leading to significant experimental overhead in terms of time, resources, and data acquisition. This work proposes a novel approach to QST by leveraging compressed sensing (CS) techniques. CS allows for accurate reconstruction of the quantum state from a significantly reduced set of measurements, thereby mitigating the experimental burden. We employ sparsity regularization, a key component of CS, to minimize the number of measurements needed. The theoretical framework and the underlying principles are presented, focusing on the mathematical formulation of the problem and the optimization strategies involved. This method offers a pathway to more efficient and cost-effective QST, particularly for complex quantum states or when limited experimental resources are available. We demonstrate the potential of this approach through a detailed mathematical analysis.