Which properties of a quantum many-body system are operationally accessible is a central question underlying spectroscopy, thermodynamics, and quantum information science. Conventional response theory answers this question within a system-only paradigm: one perturbs and measures the matter itself, obtaining susceptibility built from causally ordered nested commutators. Here we show that coherently controlled quantum probes, when measured at the end, define a strictly larger operational learning framework beyond that accessible from response theory. We establish this through a quantum-circuit description that unifies spectroscopy, probe microscopy, and probe-based quantum technologies within a common operational framework, from which we develop quantum protocols for learning many-body properties from probe readout only. This advantage arises because the reduced dynamics of quantum probes generically encode anti-commutator and mixed-order correlators of the target; therefore, measurements on the probe provide access to fluctuations, non-equilibrium structure, and entanglement entropy that are in general not accessible through response functions or a single probe alone. Moreover, we demonstrate that entangled probes can access many-body properties such as von Neumann entropy. We prove that the required probe resources scale with the complexity of the target correlations rather than with the size of the many-body system. Quantum probes are therefore not merely more sensitive sensors but provide a new way to learn many-body properties distinct from those of tomography or quantum simulation.
Which properties of a quantum many-body system are operationally accessible is a central question underlying spectroscopy, thermodynamics, and quantum information science. Conventional response theory answers this question within a system-only paradigm: one perturbs and measures the matter itself, obtaining susceptibility built from causally ordered nested commutators. Here we show that coherently controlled quantum probes, when measured at the end, define a strictly larger operational learning framework beyond that accessible from response theory. We establish this through a quantum-circuit description that unifies spectroscopy, probe microscopy, and probe-based quantum technologies within a common operational framework, from which we develop quantum protocols for learning many-body properties from probe readout only. This advantage arises because the reduced dynamics of quantum probes generically encode anti-commutator and mixed-order correlators of the target; therefore, measurements on the probe provide access to fluctuations, non-equilibrium structure, and entanglement entropy that are in general not accessible through response functions or a single probe alone. Moreover, we demonstrate that entangled probes can access many-body properties such as von Neumann entropy. We prove that the required probe resources scale with the complexity of the target correlations rather than with the size of the many-body system. Quantum probes are therefore not merely more sensitive sensors but provide a new way to learn many-body properties distinct from those of tomography or quantum simulation.
Learning the properties of dynamical quantum systems underlies applications ranging from nuclear magnetic resonance spectroscopy to quantum device characterization. A central challenge in this pursuit is the learning of strongly interacting systems, where conventional observables decay quickly in time and space, limiting the information that can be learned from their measurement. In this work, we introduce a new class of observables into the context of quantum learning---the out-of-time-order correlator---which we show can substantially improve the learnability of strongly interacting systems by virtue of displaying informative physics at large times and distances. We identify two general scenarios in which out-of-time-order correlators provide a significant learning advantage: (i) when experimental access to the system is spatially restricted, for example, via a single ``probe'' degree of freedom, and (ii) when one desires to characterize weak interactions whose strength is much less than the typical interaction strength. We numerically characterize these advantages across a variety of learning problems, and find that they are robust to both read-out error and decoherence. Motivated by these physical scenarios, we introduce several learning tasks---including Clifford tomography, and learning the connectivity of an unknown unitary---in which out-of-time-order experiments have a provable exponential advantage over any learning protocol involving only time-ordered operations.
Characterizing quantum many-body systems is a fundamental problem across physics, chemistry, and materials science. While significant progress has been made, many existing Hamiltonian learning protocols demand digital quantum control over the entire system, creating a disconnect from many real-world settings that provide access only through small, local probes. Motivated by this, we introduce and formalize the problem of quantum probe tomography, where one seeks to learn the parameters of a many-body Hamiltonian using a single local probe access to a small subsystem of a many-body thermal state undergoing time evolution. We address the identifiability problem of determining which Hamiltonians can be distinguished from probe data through a new combination of tools from algebraic geometry and smoothed analysis. Using this approach, we prove that generic Hamiltonians in various physically natural families are identifiable up to simple, unavoidable structural symmetries. Building on these insights, we design the first efficient end-to-end algorithm for probe tomography that learns Hamiltonian parameters to accuracy ε, with query complexity scaling polynomially in 1/ε and classical post-processing time scaling polylogarithmically in 1/ε. In particular, we demonstrate that translation- and rotation-invariant nearest-neighbor Hamiltonians on square lattices in one, two, and three dimensions can be efficiently reconstructed from single-site probes of the Gibbs state, up to inversion symmetry about the probed site. Our results demonstrate that robust Hamiltonian learning remains achievable even under severely constrained experimental access.
Dynamical quantum simulations are commonly designed to access a particular response function or correlation function. Yet the underlying protocol prepares a family of many-body states that contains information well beyond the observable for which the simulation was originally designed. In this work, we explore the idea of using these states as laboratories for quantum information. As a case study, we consider the local-pulse protocols used to compute Green functions in finite XX and XXX spin chains, and we interpret opposite source rotations as encoding a classical bit into the many-body system. This allows us to follow how the resulting record propagates, how it is encoded across local subsystems, and how readily it can be accessed with different measurements. We also find a clear separation between representation and readout. While fermionic covariance data exactly reconstruct the signed response in the XX chain, simple spin measurements recover a larger fraction of the locally available information in the XXX chain. This reversed ordering persists when we move from finite-pulse discrimination to local source-angle estimation. Going beyond the source record, the same reduced states also allow us to reinterpret results in terms of Bell state recovery and quantum communication rates, as well as providing direct access to the evolving bipartite and genuine multipartite entanglement of the system. Our results show that dynamical quantum simulations developed for response calculations can naturally be used to ask a much broader set of quantum-information questions about the many-body states they already prepare.
Quantum resources can dramatically reduce the data required to learn physical systems, with exponential improvements in sample complexity demonstrated in several quantum learning tasks. However, these advantages have typically relied on quantum resources that scale with problem complexity, most notably increasing system dimension or entanglement. This raises a fundamental question: can exponential quantum learning advantages arise within a fixed, unentangled quantum system? In this paper, we address this question by considering the learning of an unknown random-displacement distribution whose complexity is determined not by the dimensionality of the physical system, but by the Fourier resolution of its features. We show that the quantum-limited noise of vacuum probes progressively obscures high-frequency features, leading to an exponential growth in sample complexity. We establish an information-theoretic lower bound for arbitrary classical-state probes and show that squeezing overcomes this classical limit by extending the accessible Fourier bandwidth. Experimentally, we demonstrate an exponential reduction in sample complexity using squeezed vacuum probes for both binary hypothesis testing and characteristic-function reconstruction. Our results show that a single bosonic mode can exhibit exponential quantum learning advantages without entanglement or an increase in system size, identifying accessible Fourier bandwidth as a fundamentally distinct resource for quantum-enhanced learning.
Out-of-equilibrium quantum many-body systems stand at the forefront of modern theoretical physics, addressing fundamental questions on thermalization, transport, and universal phenomena. These fields, already well established in condensed matter, statistical physics, quantum optics, and quantum information theory, are progressively gaining even greater relevance with the rapid development of quantum technologies and simulation, which inherently operate in dynamical regimes. In recent years, the traditional paradigm of unitary quantum evolution has been expanded to include measurements, opening new directions in out-of-equilibrium physics. At the core of these advances lie measurement-induced phase transitions (MIPTs), which have emerged as a new class of dynamical critical phenomena characterizing the general behavior of monitored quantum dynamics. When external monitoring intertwines with unitary evolution, many-body quantum correlations change their structure, giving rise to distinct entanglement phases of matter. This discovery has sparked enormous interest in MIPTs, leading to significant advances in open quantum systems, entanglement theory, and more broadly quantum complexity. Despite much progress, a full understanding of monitored many-body dynamics is far from complete, leaving several open questions on the nature of MIPTs, their experimental observability, and the possibilities offered by measurements to enhance control over synthetic quantum matter. These issues persist due to the intrinsic complexity of the problem and the lack of efficient tools to study it, mainly caused by the stochastic character of monitored evolution. This thesis addresses these challenges by expanding the investigation of measurement-induced phenomena in new settings and introducing innovative probes of entanglement and many-body quantum complexity for MIPTs. A core question we investigate is the role of symmetries, non-ergodicity, and especially integrability in measurement-induced criticality, which dramatically affect the non-equilibrium phases. We further explore how these phenomena extend beyond bipartite quantum correlations to multipartite entanglement and quantum non-stabilizerness, highlighting the non-trivial interplay between measurements and complexity notions rooted in quantum information theory. Finally, we focus on the compelling problem of decoherence, modeling how noise spoils entanglement structures. These findings, supported by advanced numerical simulations and theoretical analysis, deepen the current understanding of entanglement, complexity, and integrability in monitored quantum many-body systems, offering new perspectives on their rich behavior. In parallel, we address the experimental problem of dissipation in MIPTs, which is of key relevance for practical implementations. We anticipate the present investigation to foster future research on the nature of monitored dynamical critical phenomena and, more broadly, the applications of measurements in quantum state evolution.
Understanding and harnessing the power of quantum systems has the potential to transform many domains in science and technology. However, before we can achieve these aspirations, we must first build a better understanding of how quantum systems fundamentally behave. In this thesis, we approach this question through the lens of learning theory to develop new paradigms for learning about quantum systems and understanding their structural properties. We deliver several surprising results, upending previous beliefs about even fundamental laws and giving provably efficient algorithms for learning about quantum systems in settings previously conjectured to be intractable. Typically in quantum many-body systems, the particles in the system interact locally with respect to some geometry as described by a local Hamiltonian. Two key questions are first, understanding equilibrium properties of a system with a given Hamiltonian and second, recovering the Hamiltonian from measurements of the properties of the system. For the first, we prove a universal law that there is a sudden death of entanglement, at a critical temperature depending only on the geometry but not on the system size. For the second, we give the first efficient algorithm for recovering the Hamiltonian at any temperature, breaking a conjectured barrier at low temperatures. Beyond systems with local interactions, we also consider learning and testing properties of general quantum states, focusing on the interplay between statistical complexity and near-term quantum device constraints, only allowing for entangled measurements over a limited number of copies of the state. We characterize the optimal rates for learning and testing with single-copy measurements and for multi-copy measurements in many relevant near-term regimes.