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AIP conference proceedings

Exchangeability and de Finetti’s theorem: from probabilities to quantum-mechanical states

Rüdiger Schack

Published Jan 1, 2004• 0 citations
Publisher

Abstract

This paper gives a simple introduction to the quantum mechanical concept of a probability operator valued measurement (POVM) and shows how it can be used to derive a quantum version of de Finetti’s representation theorem.
DOI
10.1063/1.1835257

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Il teorema di de Finetti classico mostra come, se l'ordine di un numero arbitrario di misure e irrilevante, sia sempre possibile rappresentare le probabilita bayesiane in termini di una media sulla probabilita oggettiva di ottenere i singoli risultati. La tesi, dopo aver introdotto il teorema nell'ambito della Fisica Classica, intende illustrarne l'estensione al caso di stati quantistici e le sue principali conseguenze nell'ambito dell'interpretazione bayesiana della Meccanica Quantistica, nella quale gli stati codificano il grado soggettivo di conoscenza sul sistema.
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.
In quantum-process tomography it is possible to express the experimenter's prior information as a sequence of quantum operations, i.e., trace-preserving completely positive maps. In analogy to de Finetti's concept of exchangeability for probability distributions, we give a definition of exchangeability for sequences of quantum operations. We then state and prove a representation theorem for such exchangeable sequences. The theorem leads to a simple characterization of admissible priors for quantum process tomography and solves to a Bayesian's satisfaction the problem of an unknown quantum operation.