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Quantum theory in finite dimension cannot explain every general process with finite memory

Marco Fanizza, Josep Lumbreras, Andreas Winter

Abstract

Arguably, the largest class of stochastic processes generated by means of a finite memory consists of those that are sequences of observations produced by sequential measurements in a suitable generalized probabilistic theory (GPT). These are constructed from a finite-dimensional memory evolving under a set of possible linear maps, and with probabilities of outcomes determined by linear functions of the memory state. Examples of such models are given by classical hidden Markov processes, where the memory state is a probability distribution, and at each step it evolves according to a non-negative matrix, and hidden quantum Markov processes, where the memory state is a finite dimensional quantum state, and at each step it evolves according to a completely positive map. Here we show that the set of processes admitting a finite-dimensional explanation do not need to be explainable in terms of either classical probability or quantum mechanics. To wit, we exhibit families of processes that have a finite-dimensional explanation, defined manifestly by the dynamics of explicitly given GPT, but that do not admit a quantum, and therefore not even classical, explanation in finite dimension. Furthermore, we present a family of quantum processes on qubits and qutrits that do not admit a classical finite-dimensional realization, which includes examples introduced earlier by Fox, Rubin, Dharmadikari and Nadkarni as functions of infinite dimensional Markov chains, and lower bound the size of the memory of a classical model realizing a noisy version of the qubit processes.

Quantum theory in finite dimension cannot explain every general process with finite memory

Abstract

Arguably, the largest class of stochastic processes generated by means of a finite memory consists of those that are sequences of observations produced by sequential measurements in a suitable generalized probabilistic theory (GPT). These are constructed from a finite-dimensional memory evolving under a set of possible linear maps, and with probabilities of outcomes determined by linear functions of the memory state. Examples of such models are given by classical hidden Markov processes, where the memory state is a probability distribution, and at each step it evolves according to a non-negative matrix, and hidden quantum Markov processes, where the memory state is a finite dimensional quantum state, and at each step it evolves according to a completely positive map. Here we show that the set of processes admitting a finite-dimensional explanation do not need to be explainable in terms of either classical probability or quantum mechanics. To wit, we exhibit families of processes that have a finite-dimensional explanation, defined manifestly by the dynamics of explicitly given GPT, but that do not admit a quantum, and therefore not even classical, explanation in finite dimension. Furthermore, we present a family of quantum processes on qubits and qutrits that do not admit a classical finite-dimensional realization, which includes examples introduced earlier by Fox, Rubin, Dharmadikari and Nadkarni as functions of infinite dimensional Markov chains, and lower bound the size of the memory of a classical model realizing a noisy version of the qubit processes.
Paper Structure (14 sections, 6 theorems, 65 equations, 2 figures)

This paper contains 14 sections, 6 theorems, 65 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Stationary stochastic processes and quasi-realizations
  4. Classical and quantum processes
  5. HMM vs HQMM
  6. Processes without quantum realization
  7. Discussion
  8. Quantum realizations of the FRDN processes: proof of Theorem \ref{['qrel']}
  9. Quasi-realization
  10. Quantum (completely positive) realization
  11. Noise robustness of the size of classical memory: proof of Theorem \ref{['theonoise']}
  12. Processes without a quantum realization
  13. Proof of Theorem \ref{['theoexp']}: Exponential cone
  14. Proof of Theorem \ref{['theopow']}: Power cone

Key Result

Proposition 2

A quasi-realization defines a non-negative measure if and only if there is a convex cone $\mathcal{C}\subset\mathcal{V}$ such that $\tau\in\mathcal{C}$, $D^{({\bf u})}(\mathcal{C})\subseteq\mathcal{C}$, $\pi\in\mathcal{C}^* := \{ f\in\mathcal{V}^* : f(x) \geq 0 \ \forall x\in\mathcal{C} \}$, the dua

Figures (2)

  • Figure 1: A depiction of a general stationary process with finite memory. The probability of a sequence $u_{-k,..,u_0,...,u_l}$ can be computed as the inner product between a right stationary state $\pi$, evolved through a sequence of linear maps $D_{u_{-k}},..,D_{u_{l}}$ acting from the right, and a right stationary state $\tau$. The hidden vector space in which $\pi D_{u_{-k}},...D_{u_{l}}$ lives represents the memory of the process. For quantum hidden Markov models, $\pi$ is a state and $\tau$ is the trace functional in the dual of the state space, while $D_{u}$ are CP maps such that $\sum_{u\in \mathbb M} D_u$ is unital.
  • Figure 2: Exponential cone \ref{['eq:exponentialcone']} (left) and power cone \ref{['eq:powercone']} (right) for $\alpha = \frac{1}{\sqrt{2}}$.

Theorems & Definitions (8)

  • Definition 1
  • Proposition 2
  • Proposition 3
  • Definition 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8