Unknown Quantum States and Operations, a Bayesian View

TL;DR

This paper extends the classical de Finetti theorem to quantum mechanics, providing operational definitions for unknown quantum states and operations within a Bayesian framework, crucial for quantum-state and process tomography.

Contribution

It introduces quantum de Finetti theorems for states and operations, advancing the Bayesian interpretation of quantum mechanics by defining unknowns as states of belief.

Findings

Quantum de Finetti theorem for states established

Quantum de Finetti theorem for operations established

Supports Bayesian approach to quantum mechanics

Abstract

The classical de Finetti theorem provides an operational definition of the concept of an unknown probability in Bayesian probability theory, where probabilities are taken to be degrees of belief instead of objective states of nature. In this paper, we motivate and review two results that generalize de Finetti's theorem to the quantum mechanical setting: Namely a de Finetti theorem for quantum states and a de Finetti theorem for quantum operations. The quantum-state theorem, in a closely analogous fashion to the original de Finetti theorem, deals with exchangeable density-operator assignments and provides an operational definition of the concept of an "unknown quantum state" in quantum-state tomography. Similarly, the quantum-operation theorem gives an operational definition of an "unknown quantum operation" in quantum-process tomography. These results are especially important for a Bayesian interpretation of quantum mechanics, where quantum states and (at least some) quantum operations are taken to be states of belief rather than states of nature.