Unknown quantum states: The quantum de Finetti representation

TLDR

Must summarize each label content. Let's gather content. Purpose, Mechanism sentence: "We present an elementary proof of the quantum de Finetti representation theorem, a quantum analog of de Finetti’s classical theorem on exchangeable probability assignments." Background sentences: multiple. Summarize field context: classical de Finetti theorem gives operational definition of unknown probability; quantum version deals with exchangeable density operators and defines unknown quantum state; important for information-based interpretations; contrasts with original proof by Hudson and Moody which relies on advanced math and not generalizable. Also mention that theorem fails for real Hilbert spaces?

Abstract

We present an elementary proof of the quantum de Finetti representation theorem, a quantum analog of de Finetti’s classical theorem on exchangeable probability assignments. This contrasts with the original proof of Hudson and Moody [Z. Wahrschein. verw. Geb. 33, 343 (1976)], which relies on advanced mathematics and does not share the same potential for generalization. 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. The quantum de Finetti theorem, in a closely analogous fashion, deals with exchangeable density-operator assignments and provides an operational definition of the concept of an “unknown quantum state” in quantum-state tomography. This result is especially important for information-based interpretations of quantum mechanics, where quantum states, like probabilities, are taken to be states of knowledge rather than states of nature. We further demonstrate that the theorem fails for real Hilbert spaces and discuss the significance of this point.