Unknown Quantum States and Operations, a Bayesian View

By Christopher A. Fuchs, Ruediger Schack. 2004

1,081 citations recorded by the source corpus.

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.

Subfields: Atomic and Molecular Physics, and Optics, Electrical and Electronic Engineering, Spectroscopy

View Paper: https://www.semanticscholar.org/paper/ad214f91aeb3c1f1393b95f15c8c71935426385c

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Citations in the corpus

Quantum Generative Adversarial Networks for learning and loading random distributions (2019, background): “Some focus on quantum state estimation [13], i.”
Validating quantum computers using randomized model circuits (2018, background): “Methods such as randomized benchmarking [5], state and process tomography [6], and gateset tomography [7] are valued for measuring the performance of operations on a few qubits, yet they fail to account for errors arising from interactions with spectator qubits [8, 9].”
Robust, self-consistent, closed-form tomography of quantum logic gates on a trapped ion qubit (2013, background): “Quantum process tomography [13, 14] is very similar, but in addition to an informationally complete set of measurement outcomes, we must also prepare a set of test states ρj that span B(H), and apply the unknown process G to them.”
Experimental one-way quantum computing (2005, methodology): “Instead of a direct linear combination of measurement results which can lead to unphysical density matrices due to experimental noise, we employ a maximum-likelihood reconstruction technique 30-32 .”
Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators (2012, methodology): “They include: constrained trace-minimization (a.k.a. the matrix Dantzig selector), least squares with trace-norm regularization (a.k.a. the matrix Lasso), as well as a standard maximum likelihood estimation (MLE) [35–37] for comparison.”
Heralded generation of entangled photon pairs (2010, methodology): “The most likely physical density matrix for our two-qubit state was extracted using a maximum-likelihood reconstructio”
Quantum computing with trapped ions (2008, neutral): “the experimentally recorded set of measurement results. Even though maximum likelihood estimation has been criticized (Blume-Kohout, 2006) for being less accurate than Bayesian estimation techniques (Paris and Rehacek, 2004), it has the practical merit of being easily implemented in experiments. Sta”