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arXiv:2610.22489

Multi-Time States in Quantum and Operational Probabilistic Theories: An Axiomatic Framework

Kaumudibikash Goswami, Archana Barvar, Manabendra Nath Bera

quant-phgr-qcmath-phSubmitted Oct 15, 2026• Announced Oct 19, 2026• 0 citations
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Abstract

Quantum states over time were introduced, in part, to extend Bayesian probability to quantum theory and have since found applications across diverse areas, ranging from quantum causality and open-system dynamics to tests of macroscopic realism. While axiomatic characterizations of these states exist for Markovian dynamics, a general framework accommodating non-Markovian dynamics is still lacking. Moreover, extending this framework beyond quantum mechanics to general physical theories also remains elusive. In this work, we develop an axiomatic framework for multi-time states within a class of operational probabilistic theories (OPTs) that includes quantum mechanics as a special case. Based on three assumptions---compositionality, reconstructibility, and spatio-temporal marginal consistency---the framework accommodates both Markovian and non-Markovian dynamics, and provides sufficient conditions to certify signaling and temporal memory within OPT. As a concrete application of our OPT framework, we fully characterize the unique multi-time state in boxworld and develop a method for Bayesian retrodiction of physical transformations there. Our general framework is consistent with quantum theory, reproducing several existing results. In the non-Markovian quantum dynamics, we establish a correspondence between multi-time states and the process-matrix framework, and fully characterize the multi-time states compatible with general dynamics. This correspondence also lets us infer quantum causal models from multi-time states, and as a proof of principle, we propose a semidefinite program to infer causal order from quantum multi-time states. These results provide a unified framework for studying spatio-temporal states, causal relations, and Bayesian reasoning within OPTs, with implications for both fundamental and applied research.

Comments
22+19 pages, 7+2 figures, comments welcome

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Quantum states over time were introduced, in part, to extend Bayesian probability to quantum theory and have since found applications across diverse areas, ranging from quantum causality and open-system dynamics to tests of macroscopic realism. While axiomatic characterizations of these states exist for Markovian dynamics, a general framework accommodating non-Markovian dynamics is still lacking. Moreover, extending this framework beyond quantum mechanics to general physical theories also remains elusive. In this work, we develop an axiomatic framework for multi-time states within a class of operational probabilistic theories (OPTs) that includes quantum mechanics as a special case. Based on three assumptions---compositionality, reconstructibility, and spatio-temporal marginal consistency---the framework accommodates both Markovian and non-Markovian dynamics, and provides sufficient conditions to certify signaling and temporal memory within OPT. As a concrete application of our OPT framework, we fully characterize the unique multi-time state in boxworld and develop a method for Bayesian retrodiction of physical transformations there. Our general framework is consistent with quantum theory, reproducing several existing results. In the non-Markovian quantum dynamics, we establish a correspondence between multi-time states and the process-matrix framework, and fully characterize the multi-time states compatible with general dynamics. This correspondence also lets us infer quantum causal models from multi-time states, and as a proof of principle, we propose a semidefinite program to infer causal order from quantum multi-time states. These results provide a unified framework for studying spatio-temporal states, causal relations, and Bayesian reasoning within OPTs, with implications for both fundamental and applied research.
Currently, there is no systematic way to describe a quantum process with memory solely in terms of experimentally accessible quantities. However, recent technological advances mean we have control over systems at scales where memory effects are non-negligible. The lack of such an operational description has hindered advances in understanding physical, chemical, and biological processes, where often unjustified theoretical assumptions are made to render a dynamical description tractable. This has led to theories plagued with unphysical results and no consensus on what a quantum Markov (memoryless) process is. Here, we develop a universal framework to characterize arbitrary non-Markovian quantum processes. We show how a multitime non-Markovian process can be reconstructed experimentally, and that it has a natural representation as a many-body quantum state, where temporal correlations are mapped to spatial ones. Moreover, this state is expected to have an efficient matrix-product-operator form in many cases. Our framework constitutes a systematic tool for the effective description of memory-bearing open-system evolutions.
Continuous measurements are central to quantum control and sensing, yet lack a model-independent operational description that can be applied to arbitrary non-Markovian processes without specifying a microscopic measurement model. Existing multi-time frameworks, such as process matrices, allow for an arbitrary sequence of operations to be applied on a general process, but are restricted to interventions at discrete times and cannot represent measurements of finite duration. We introduce a continuous-time extension of multi-time quantum processes based on process and operation functionals, which generalise the Feynman-Vernon influence functional and yield a continuous Born rule that cleanly separates processes from operations. This framework provides a consistent representation of non-Markovian dynamics under continuous monitoring and leads to natural definitions of causality and Markovianity in continuous time, opening the way to formalising quantum causal structures in the continuum. We illustrate the formalism by analysing continuous measurements in a generalised Caldeira-Leggett model, demonstrating its applicability to realistic non-Markovian scenarios.
This is a book about operational probabilistic theories. The standard approach in such theories is from a time forward perspective. In this book we mostly take a time symmetric perspective. This presents a branding problem. Is this a niche book merely about time symmetry? No. This is a comprehensive book about operational probabilistic theories, but mostly from a time symmetric perspective. In fact, this book consists of (1) a simple book about simple operations having simple causal structure (where all the inputs are before all the outputs), and (2) a complex book about complex operations that can have complicated causal structure (a complex operation is equipped with a causal diagram). For the simple case we are able to show that the time symmetric perspective is equivalent to the time forward perspective. In each book we set up (A) operational probabilistic theories (OPTs) in terms of operations, (B) Operational Quantum Theory (OQT) in terms of operator tensors which correspond to operations, and (C) the theory of Hilbert objects which can be doubled up to give operator tensors. Operations are required to be physical which guarantees that circuits built out of operations have probabilities between 0 and 1 and that certain causality conditions are met. We prove that when we wire together operations the resulting networks are also physical. We model Sorkin's impossible measurements with complex operations and show that physicality prevents anomalous signalling. We develop diagrammatic notation for Hilbert objects. This includes mirrors for doubling up and mirror theorems. We use this framework to prove time symmetric causal dilation theorems for various causal diagrams.
Quantum combs are powerful conceptual tools for capturing multi-time processes in quantum information theory, constituting the most general quantum mechanical process. But, despite their causal nature, they lack a meaningful physical connection to time -- and are, by and large, arguably incompatible with it without extra structure. The subclass of quantum combs which assumes an underlying process is described by the so-called process tensor framework, which has been successfully used to study and characterise non-Markovian open quantum systems. But, although process tensors are motivated by an underlying dynamics, it is not a priori clear how to connect them to a continuous process tensor object mathematically -- leaving an uncomfortable conceptual gap. In this work, we take a decisive step toward remedying this situation. We introduce a fully continuous process tensor framework by showing how the discrete multi-partite Choi matrix becomes a vector in bosonic Fock space, which is intrinsically and rigorously defined in the continuum. With this equipped, we lay out the core structural elements of this framework and its properties. This translation allows for an information-theoretic treatment of multi-time correlations in the continuum via the analysis of their continuous matrix product state representatives. Our work closes a gap in the quantum information literature, and opens up the opportunity for the application of many-body physics insights to our understanding of quantum stochastic processes in the continuum.
Characterization of non-Markovian open quantum dynamics is both of theoretical and practical relevance. In a seminal work [Phys. Rev. Lett. 120, 040405 (2018)], a necessary and sufficient quantum Markov condition is proposed, with a clear operational interpretation and correspondence with the classical limit. Here we propose two non-Markovianity measures for general open quantum dynamics, which are fully reconciled with the Markovian limit and can be efficiently calculated based on the multi-time quantum measurements of the system. A heuristic algorithm for reconstructing the underlying open quantum dynamics is proposed, whose complexity is directly related to the proposed non-Markovianity measures. The non-Markovianity measures and the reconstruction algorithm are demonstrated with numerical examples, together with a careful reexamination of the non-Markovianity in quantum dephasing dynamics.
Thermodynamics imposes fundamental constraints on the evolution of quantum systems. These constraints and their dynamical consequences have been formulated within distinct paradigms, including axiomatic approaches based on quantum master equations, microscopic descriptions of open-system dynamics, and operational formulations rooted in resource theories. While each perspective has yielded important insights into thermodynamically consistent quantum dynamics, their precise relationship has remained unresolved. Here we establish the exact equivalence of these three paradigms in the Markovian regime. We prove that thermal Lindbladians satisfying Markovianity, time-translation symmetry, and quantum detailed balance are precisely those admitting a microscopic realisation as an energy-conserving thermal collision model and, equivalently, those generating Markovian thermal operations. This unifies the existing approaches to Markovian quantum thermodynamics and identifies its dynamical underpinnings. We further provide an explicit microscopic protocol for simulating thermal Markovian processes with controlled finite-time simulation errors. We illustrate its applicability by providing faithful thermal collision-model implementations of a qubit thermalising in a bosonic environment and of a three-level autonomous thermal machine. In the latter case, the protocol gives rise to a finite-stroke thermal engine that not only reproduces the continuous-time dynamics but also its steady-state thermodynamic performance. As a whole, these results establish a unified foundation for Markovian quantum thermodynamics, showing that its axiomatic, microscopic, and operational formulations are exactly equivalent and providing a universal protocol for implementing thermal processes and machines.