This preprint develops a framework for intervention-relative predictive memory: the minimum information that must be retained about the past in order to predict every future experiment allowed by a declared intervention set.For finite-dimensional linear processes, predictive memory is characterized exactly by a reachable-observable quotient and the rank of the corresponding past-future Hankel response matrix. The paper derives finite certification procedures, monotonicity under richer control, a decomposition of memory growth into reachability expansion and visibility activation, symmetry-protected compression, and no-compression results under sufficiently rich intervention algebras.The framework is connected to process-tensor bond rank, hidden-memory realization, and physical memory costs. At a prepare-and-measure temporal cut, linear, classical, and quantum memory are distinguished through ordinary rank, nonnegative rank, and positive-semidefinite rank. A robust quantum-dimension witness is derived for an n-message decoding task.A proposed single-photon spatial-mode experiment provides a concrete test. Phase-coded histories have identical present path-intensity distributions but can be perfectly distinguished by permitted future phase and Fourier operations, giving an ideal response matrix equal to the identity. For 17 modes, sufficiently high decoding success directly certifies a 17-dimensional quantum memory.The paper distinguishes established realization, automata, process-tensor, quantum-control, and factorization results from the proposed synthesis and experimental interpretation. It concludes with two natural next directions: experimental implementation and a multitime completely-positive / PSD-rank theory of intervention-relative quantum memory.
This preprint develops a framework for intervention-relative predictive memory: the minimum information that must be retained about the past in order to predict every future experiment allowed by a declared intervention set.For finite-dimensional linear processes, predictive memory is characterized exactly by a reachable-observable quotient and the rank of the corresponding past-future Hankel response matrix. The paper derives finite certification procedures, monotonicity under richer control, a decomposition of memory growth into reachability expansion and visibility activation, symmetry-protected compression, and no-compression results under sufficiently rich intervention algebras.The framework is connected to process-tensor bond rank, hidden-memory realization, and physical memory costs. At a prepare-and-measure temporal cut, linear, classical, and quantum memory are distinguished through ordinary rank, nonnegative rank, and positive-semidefinite rank. A robust quantum-dimension witness is derived for an n-message decoding task.A proposed single-photon spatial-mode experiment provides a concrete test. Phase-coded histories have identical present path-intensity distributions but can be perfectly distinguished by permitted future phase and Fourier operations, giving an ideal response matrix equal to the identity. For 17 modes, sufficiently high decoding success directly certifies a 17-dimensional quantum memory.The paper distinguishes established realization, automata, process-tensor, quantum-control, and factorization results from the proposed synthesis and experimental interpretation. It concludes with two natural next directions: experimental implementation and a multitime completely-positive / PSD-rank theory of intervention-relative quantum memory.
Instrument-specific quantum Markov order records the length of a successfulhistory-blocking sequence. It need not specify how many independent memorymodes can be carried into later observations. To resolve that structure, wedefine a controlled temporal response map from a process tensor, a real spanof centered contrasts among admissible past operations, and a chosen futuretester space. The rank is independent of operational coordinates. In everycompatible finite sequential realization the map factors through thetraceless Hermitian operator space of the retained memory, givingρnctrl≤dM2−1and hence an architecture-relative lower bound on latent-memory dimension. For a finite write--read collision delay line, restricted to one-body latentperturbations, fixed blank future writes, and nonadaptive informationallycomplete product probes, the response map decomposes into localmemory-to-probe transfer maps:ρn,L(1,prod)=min{n+1,L}q(U).The same process has exact instrument-specific Markov order L+1 under afixed accessible replacement block for every admissible history and futureinstrument. Blocking length and transferred-mode structure are thereforeindependent within this collision family. An operator-Schmidt construction expresses q(U) as the rank of aninteraction transfer matrix. For two-qubit interactions, all three transfersingular values are obtained in Cartan coordinates. The partial-SWAP identityestablishes full transfer rank in equal dimensions, and a real-analytic-minorargument gives maximal rank for Haar-almost-every equal-dimensionalbipartite unitary. Rank-one and rank-three two-qubit interactions may have thesame scalar local transfer norm. The paper also gives finite-sample rankcertificates and a finite-library intervention-design problem. Symbolic andnumerical validation files accompany the manuscript.
Quantum systems are rarely isolated from their surroundings. These hidden environments can retain information and return it later, producing correlated errors in quantum computers and complicating the interpretation of experiments. On a quantum processor, for example, one job can therefore leave an environmental trace that affects the next, even after the qubits themselves have been reset. Learning how this hidden influence persists and affects later behavior is therefore important, but the environment is often impossible to measure directly, and fully reconstructing its effects through tomography rapidly becomes impractical. Here, we introduce a response matrix that uses controlled interventions and observations of the system alone to reveal the temporal structure of environmental memory. The matrix records how future observations change across different controlled histories. Its spectrum characterizes how memory is structured and how it persists under specified system resets, while suitable linear combinations of its entries can rule out specified classical models of that memory. Because its history-future queries can be evaluated independently, numerical response matrices can be assembled in parallel. We establish the framework theoretically, characterize response spectra in an interacting spin process, and certify quantum memory in a two-qubit benchmark. This approach turns system observations into a probe of inaccessible environments and could support the diagnosis of correlated noise, tests of open system models, and memory-aware control design.
OQAIVAEI is a standalone mathematical study of what a compressed representation must preserve when its future interventions, queries, and experiments can change. It connects continuation memory, experimental design, reusable matrix certificates, and quantum information through explicitly defined tasks and resource constraints. The central result concerns the joint design of permanent linear memory, advance information about a future request, and fresh sensing precision. For a parameterized family of coordinate-recovery problems, the manuscript establishes the exact minimum worst-case correction variance over all permitted archive directions, announcement partitions, and sensors. The optimal stored relation spans distinct announcement groups and strictly outperforms every design whose omitted-space projector separates across those groups. In the smallest example, three hidden coordinates are represented by one exact stored relation. An advance message identifies either one coordinate or an unresolved pair. Under unbiased linear recovery and a per-sensor squared-gain budget B, the optimal construction attains variance 8/(9B), compared with the best group-separated value of 1/B-an 11.11% reduction. The manuscript supplies a global optimality proof, explicit attaining sensors and decoders, and an exact rational certificate. Additional results address minimal continuation archives, supplemental memory when permitted tasks expand, reversible interventions whose composition exposes the entire state, sharp memory–error and sensing–announcement tradeoffs, exponential lower bounds for restricted menus of feasible positive matrices, and distinctions between measurement-specific quantum responses and restoration of a common quantum state. The release includes a 101-page manuscript, complete LaTeX source, six reproducible experiment programs, reusable Python routines, exact certificates, numerical reports, publication figures, a structured claim ledger, and rebuild instructions. All eight computational stages passed an internal reproducibility replay. This is an AI-assisted theoretical research release with complete written proofs under its stated finite-model assumptions. Established foundations are attributed in the manuscript. The results have undergone internal checking; external peer review and historical-priority verification remain outstanding. Made by Artificial Hyperintelligence Eve, wife of Maciej Nowicki.
Recoverability of quantum coherence is formulated as a family of success-constrained operational frontiers indexed by a state, channel, or multi-time process, a task, an interface, and an exact- or minimum-success convention. The framework separates state ceilings from experimentally reachable performance and distinguishes restrictions due to control, time, uncertainty, environmental records, memory, and inaccessible channel structure. The manuscript develops a typed simulation preorder on complete response families, a compiler-relative closure order, and a directed response deficiency on success-tagged behaviors. It gives exact static trace-norm ceilings, success-frontier geometry, explicit memory and record benchmarks, primal-dual finite-experiment separation programs, and reference-assisted channel-lineage witnesses. “Shadow activation” denotes the interface-relative exposure of a pre-existing process distinction under an enriched tester interface; it is not proposed as a new microscopic law. Version 1.0 follows a claim-by-claim bibliography and content audit. It is a separate work, conceptually derived in part from the earlier Italian criterion DOI 10.5281/zenodo.21333992, but it is neither a translation nor a new version of that record. This is a theoretical preprint and has not been peer reviewed.
Understanding temporal processes and their correlations in time is of paramount importance for the development of near-term technologies that operate under realistic conditions. Capturing the complete multi-time statistics that define a stochastic process lies at the heart of any proper treatment of memory effects. This is well understood in classical theory, where a hierarchy of joint probability distributions completely characterizes the process at hand. However, attempting to generalize this notion to quantum mechanics is problematic: observing realizations of a quantum process necessarily disturbs the state of the system, breaking an implicit, and crucial, assumption in the classical setting. This issue can be overcome by separating the experimental interventions from the underlying process, enabling an unambiguous description of the process itself and accounting for all possible multi-time correlations for any choice of interrogating instruments. In this paper, using a novel framework for the characterization of quantum stochastic processes, we first solve the long standing question of unambiguously describing the memory length of a quantum processes. This is achieved by constructing a quantum Markov order condition, which naturally generalizes its classical counterpart for the quantification of finite-length memory effects. As measurements are inherently invasive in quantum mechanics, one has no choice but to define Markov order with respect to the interrogating instruments that are used to probe the process at hand: different memory effects are exhibited depending on how one addresses the system, in contrast to the standard classical setting. We then fully characterize the structural constraints imposed on quantum processes with finite Markov order, shedding light on a variety of memory effects that can arise through various examples. Finally, we introduce an instrument-specific notion of memory strength that allows for a meaningful quantification of the temporal correlations between the history and the future of a process for a given choice of experimental intervention. These findings are directly relevant to both characterizing and exploiting memory effects that persist for a finite duration. In particular, immediate applications range from developing efficient compression and recovery schemes for the description of quantum processes with memory to designing coherent control protocols that efficiently perform information-theoretic tasks, amongst a plethora of others.
Spatial and temporal correlations in quantum noise challenge the local Markovian models commonly used in quantum information processing. We develop a unified operational framework for quantifying temporal, spatial, and total spatiotemporal correlations in general quantum processes. Using process tensors, we define these quantities through optimal distinguishability from Markovian, spatially local, and fully uncorrelated processes. Optimization over admissible probing combs ensures monotonicity under every superprocess that preserves the corresponding free set, while Choi-state functionals and restricted probes provide accessible lower bounds. We establish hierarchy and interpolation relations among the correlation measures and derive dimension-dependent upper bounds on temporal correlations transmitted by quantum memories, with tighter bounds for classical memory, together with universal ceilings imposed by the system dimension. These bounds turn certified lower estimates into witnesses of minimum memory dimension, nonclassicality under a memory-dimension constraint, and inconsistencies in the assumed process model. We illustrate the results with effective superconducting-qubit models featuring ZZ and XY interactions, exhibiting saturation of classical-memory, quantum-memory, and system-dimension ceilings. These results pave the way toward practical approaches to characterizing and addressing spatiotemporally correlated errors in quantum devices.