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Formalising an operational continuum limit of quantum combs

Clara Wassner, Jonáš Fuksa, Jens Eisert, Gregory A. L. White

Abstract

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 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 state becomes a field-theoretic state 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.

Formalising an operational continuum limit of quantum combs

Abstract

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 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 state becomes a field-theoretic state 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.
Paper Structure (34 sections, 2 theorems, 229 equations, 1 figure, 1 table)

This paper contains 34 sections, 2 theorems, 229 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Notation and conventions
  4. Process tensors and discrete quantum stochastic processes
  5. Continuous matrix product states
  6. The continuous process tensor framework
  7. The continuum limit of process tensors
  8. Getting back to finite process tensors
  9. Process tensor representations within the continuous matrix-product framework
  10. Continuous processes as quantum field theory states
  11. Instruments and the generalized Born rule
  12. Worked examples
  13. Infinite dimensional environment: Spin-boson model
  14. Control family
  15. Deterministic control
  16. ...and 19 more sections

Key Result

Proposition 1

Given any family of discrete processes tensors $\{{\Upsilon_{k:0}}\}_k$ defined on a fixed interval $[0,T]$ with respect to an underlying system-environment Hamiltonian $H^{\rm SE}(t)$ for $\|H^{\rm SE}\| < \infty$, the value of the limiting measure given in eq:dnm-inf is identically zero.

Figures (1)

  • Figure 1: A high-level depiction of the results presented in this work. (a) A discrete process tensor ${\Upsilon_{k:0}}$ is a multi-linear functional encoding the joint probabilities of all multi-time instruments $\mathbf{A}_{k:0}$ within the Choi state $\hat{{\Upsilon}}_{k:0}$. When viewing $\hat{{\Upsilon}}_{k:0}$ in a second-quantised form, these arbitrarily strong interventions can be interpreted as particles in a bosonic Fock space at higher and higher energy levels. (b) With this representation equipped, the continuum limit of this object may then be taken, arriving at a bosonic quantum field theory state vector $|\Upsilon_T \rangle$ with at most one particle excitation (the instrument generator) at each time.

Theorems & Definitions (9)

  • Definition 1: Process tensor
  • Definition 2: Continuous matrix product states
  • Definition 3: Process-canonical representation
  • Definition 4: Causality -- continuum
  • Definition 5: Positivity -- continuum
  • Definition 6: Continuous process tensor (cPT)
  • Proposition 1: Vanishing of discrete NM measures
  • Definition 7: $p$-number of cPT
  • Theorem 5.1: Recovering multi-time correlation functions