Advances Quantum Combs: Formalising a Continuum Limit for Multi-Time Processes

Quantum combs represent a powerful, yet often abstract, framework for understanding multi-time processes central to information theory. Clara Wassner, Jonáš Fuksa, and Jens Eisert, alongside Gregory A. L. White , all from the Dahlem Center for Complex Quantum Systems at Freie Universität Berlin, and in Eisert’s case also the Helmholtz-Zentrum Berlin , now address a fundamental limitation within this field: the lack of a clear connection between discrete comb structures and continuous physical time. Their research formalises an operational continuum limit of quantum combs, bridging a conceptual gap by demonstrating how discrete comb states evolve into field-theoretic states within bosonic Fock space. This breakthrough not only closes a significant hole in the information literature, but also promises to unlock new avenues for applying insights from physics to the study of stochastic processes occurring continuously in time.

Continuous Process Tensors Resolve Quantum Time Incompatibility

This breakthrough addresses a long-standing conceptual problem: the incompatibility of quantum combs, powerful tools for capturing multi-time processes, with a meaningful physical connection to time. The study reveals that traditional process tensors, while effective for studying non-Markovian Open quantum systems, lack a clear mathematical connection to Continuous dynamics, creating an uncomfortable conceptual gap. This innovative approach allows for a natural continuum limit, where only the lowest energy levels survive, effectively smoothing out the discrete steps and establishing a continuous dynamic. By mapping Temporal correlations onto spatial ones using the generalised Choi-Jamiołkowski isomorphism, the research provides an operational understanding of quantum processes, allowing scientists to analyse multi-time dynamics using established quantum information tools.
The study unveils that the resulting object is not a spin system, but a quantum field theory state, offering a more physically realistic representation of continuous processes. Furthermore, the research establishes a connection between seemingly disparate frameworks, quantum combs, operator tensors, and process matrices, all of which share a common mathematical structure as multi-linear functionals encoding joint probabilities across space and time. This translation allows for a deeper understanding of the underlying dynamics, moving away from a purely discrete picture towards a more nuanced and continuous representation of quantum evolution.,.

Continuous Process Tensors via Continuous Matrix Product States

Initially, the team constructed a k-step process tensor Υk:0 using a series of system-environment unitaries {USE j:j−1}k, each associated with a Hamiltonian generator {HSE j} and evolution time tε. Crucially, the researchers recognised that naively shrinking time steps (∆t →0) would lead to an ill-defined limit, creating an infinite number of Bell pairs, instead, they framed instruments as transformations acting over finite time windows. This approach ensured that the total energy, or excitations in the field theory, remained finite, rendering the continuous process tensor “UV finite”. The vectorised process tensor |Υtε k:0⟩⟩ was then expressed as a sum over orthonormal basis states, revealing its structure as a matrix product state defined by Eq. (19). This formulation distinguishes between a virtual space, encoding uncontrollable system-environment evolution, and physical indices representing controllable experimental operations.

Continuous Process Tensors and Bosonic Fock Space

Experiments revealed that, upon taking this limit, only the zeroth and first energy levels persist, signifying that the process tensor becomes a multi-linear functional on instrument generators rather than their propagators. Measurements confirm that introducing a ‘wall clock’ into process tensor theory is a necessary consequence of this transformation, aligning the framework with a more physically realistic depiction of time evolution. The work demonstrates that quantum combs, specifically process tensors, can be understood as multi-linear functionals mapping control operations across spacetime coordinates to output states, encoding all non-commutative joint probability distributions of a quantum system. Data shows that this approach allows for the operational understanding of temporal structure using quantum information tools, mapping temporal correlations onto spatial ones via the generalised Choi-Jamio lkowski isomorphism. Tests prove that by casting process tensors in a second-quantised form, the Hilbert spaces representing system inputs and outputs are mapped onto a Fock space, with each instrument order corresponding to particle creation at higher energy levels. Furthermore, the research anticipates applications in tomography of non-Markovian noise, enhancing the efficiency of simulation in non-Markovian open quantum systems, and providing a more rigorous treatment of non-Markovianity measures.

Continuous Combs and Field-Theoretic States offer a novel

This work establishes a framework for treating information-theoretic problems in the continuum, potentially enabling the application of physics-based understandings to stochastic processes. The study highlights the importance of finite energy levels for instruments probing the system, ensuring the framework remains “UV finite”, and acknowledges that assuming an underlying generator for the dynamics limits the full generality of the classification of quantum stochastic processes. The authors note a limitation in that their framework prioritises an operational approach to continuous process tensors, rather than a fully general classification, though they suggest a fully general approach may lack clear physical meaning. Future research could explore extending this framework to processes not governed by stochastic differential equations, potentially broadening its applicability. This advancement offers a new perspective on multi-time correlations and opens avenues for integrating information theory with concepts from field theory and quantum stochastic processes.