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, giving\(\rho_n^{\mathrm{ctrl}}\leq d_M^2-1\)and 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:\[ \rho_{n,L}^{(1,\mathrm{prod})} =\min\{n+1,L\}\,q(U).\]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.
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, giving(\rho_n^{\mathrm{ctrl}}\leq d_M^2-1)and 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:[ \rho_{n,L}^{(1,\mathrm{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.
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.
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.
How much internal quantum memory is needed to reproduce a stationary stochastic process, and how does reducing that memory change the environmental resources required when output records are retained? This paper establishes exact quantum-source constructions and resource bounds under explicitly specified readout and retained-output conditions. A general completion theorem identifies the minimum Hilbert-space dimension for stationary triple data in the orthogonal-current-symbol source class: it equals the sum of the complex positive-semidefinite ranks of the middle-symbol slices. An attaining construction reproduces the entire canonical stationary order-two process. For the delayed trine process, the minimum classical hidden Markov model requires nine states at every non-independent parameter value. Quantum instruments reproduce the same complete output law using four-dimensional memory throughout the parameter range, with four dimensions proved necessary in a certified low-noise regime and three dimensions both sufficient and necessary on an explicit high-noise interval. The paper then separates reusable memory dimension from environmental dilation cost. On the certified high-noise interval, the exact one-step Pareto frontier consists of a three-dimensional source with recorded-channel Choi rank nine and a four-dimensional source with rank three. Here, Choi rank measures the minimum initially pure environmental remainder dimension for the channel retaining both the emitted classical symbol and final quantum memory. A separate mixed-support qutrit construction at noise parameter 0.99 improves complete-history costs and gives upper bounds close to universal lower bounds. These results show why optimizing a single step need not optimize a retained multi-step history. The analysis also distinguishes complete channels, fixed-input states and archive-only outputs, and demonstrates how access to a retained reference can distinguish generators with identical passive output statistics. Full-word proofs, algebraic complete-positivity certificates, a technical companion and reproducibility materials accompany the results. The exact intermediate memory boundary and complete-history optimum remain open. All results use standard quantum probabilities and apply to the stated operational resource models; they do not establish total hardware, energy or thermodynamic minima.
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.
Relational quantum causal processes formulate finite operational contexts as normal positive functionals on local completely positive maps. Response differences generate an influence algebra, and its central projections define jointly readable Boolean events. We develop this starting point through a sequence of exact and controlled models. Fresh-environment unitary collision circuits produce dephasing-exchange kinetics with an exact charge-center fixed algebra, a uniform finite-step limit at fixed response order, and graph-controlled metastable Markov dynamics. An absorbing-state model exhibits a sharp transition between non-Abelian quantum memory and Boolean records. A reversal-covariant defect dynamics generates a locally finite partial order on a restricted graph family without assuming a Lyapunov time. Conditional on a certified order, a positive additive record measure, compactness, and identifiability, we prove subsequential convergence to a Lorentzian metric-measure space, finite reconstruction bounds, and uniqueness of admissible smooth limits. Complementary finite regulators provide controlled tests of modular-to-boost response, null tomography, same-update variational identities, induced quadratic gravity, and compatible common-refinement limits. These results are exact or controlled within their stated models, but they do not yet constitute a single background-independent microscopic law that jointly generates adjacency, time, volume normalization, dimension, signature, nonlinear Einstein constraints, and quantum matter. We therefore present RQCP-QG as a theorem-indexed framework that separates established mechanisms, conditional compositions, and open assumptions.
Non-Markovian quantum processes exhibit different memory effects when measured in different ways; an unambiguous characterization of memory length requires accounting for the sequence of instruments applied to probe the system dynamics. This instrument-specific notion of quantum Markov order displays stark differences to its classical counterpart. Here, we explore the structure of quantum stochastic processes with finite memory length in detail. We begin by examining a generalized collision model with memory, before framing this instance within the general theory. We detail the constraints that are placed on the underlying system-environment dynamics for a process to exhibit finite Markov order with respect to natural classes of probing instruments, including deterministic (unitary) operations and sequences of generalized quantum measurements with informationally complete repreparations. Lastly, we show how processes with vanishing quantum conditional mutual information form a special case of the theory. Throughout, we provide a number of representative, pedagogical examples to display the salient features of memory effects in quantum processes.