Unknown Quantum States: The Quantum de Finetti Representation

By Carlton M. Caves, Christopher A. Fuchs, Ruediger Schack. 2001

432 citations recorded by the source corpus.

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...

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/8cedab8d7afa2debc07c6013fe85ecb3f68e0d6d

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Citations in the corpus

Quasi-probability representations of quantum theory with applications to quantum information science (2010, background): “Quantum Bayesianism (Caves et al 2002a, Fuchs 2002, Fuchs 2010, Fuchs and Schack 2009, Schack et al 2001) is an interpretation of quantum theory which sheds new light on not only the tradition ‘foundational’ problems (the ‘measurement problem’, for example) but also many concepts in quantum theory, ”
Quantum probabilities as Bayesian probabilities (2001, methodology): “The quantum de Finetti representation theorem [16,17] establishes that for any exchangeable sequence of density operators, ρ̂ can be written uniquely in the form”
Symmetric informationally complete quantum measurements (2003, background): “In quantum information theory such measurements are relevant to quantum state tomography [13], quantum cryptography [14], and to foundational studies [15] where they would make for a particularly interesting “standard quantum measurement”.”
Quantum Tomography under Prior Information (2011, methodology): “For illustrative purposes, let us use the above framework to confirm the well known fact that there exists an informationally complete POVM with d(2) outcomes and that d(2) is the minimal number for which informational completeness can be achieved for all of S(C) [11].”
Symmetry of large physical systems implies independence of subsystems (2007, background): “The connection between symmetric and i.i.d. states is of particular interest for foundational issues [Hud81, CFS02].”
The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement (2012, background): “We follow [40] in the construction of an informationally complete observable A.”
QBism, the Perimeter of Quantum Bayesianism (2010, background): “A case in point is the quantum de Finetti theorem [3,24].”
Geometry of Quantum States: Index (2007, background): “14 For an engaging account both backwards (with references to earlier literature) and forwards (to quantum information theory) see Caves, Fuchs and Schack [197].”
On Weyl-Heisenberg orbits of equiangular lines (2008, background): “Equiangular lines have several applications to quantum information such as quantum fingerprinting [15] , quantum tomography [5] and quantum cryptographic protocols [8].”