Characterization of the probabilistic models that can be embedded in quantum theory
Andrew J. P. Garner, Markus P. Mueller
备注：
6 pages, 0 figures. This paper has been superseded by arXiv:2112.09719 which has been published in Phys. Rev. X (DOI:10.1103/PhysRevX.13.041001). However, we decided to leave this preprint on the arXiv since other works refer to it
Quantum bits can be isolated to perform useful information-theoretic tasks, even though physical systems are fundamentally described by very high-dimensional operator algebras. This is because qubits can be consistently embedded into higher-dimensional Hilbert spaces. A similar embedding of classical probability distributions into quantum theory enables the emergence of classical physics via decoherence. Here, we ask which other probabilistic models can similarly be embedded into finite-dimensional quantum theory. We show that the embeddable models are exactly those that correspond to the Euclidean special Jordan algebras: quantum theory over the reals, the complex numbers, or the quaternions, and "spin factors" (qubits with more than three degrees of freedom), and direct sums thereof. Among those, only classical and standard quantum theory with superselection rules can arise from a physical decoherence map. Our results have significant consequences for some experimental tests of quantum theory, by clarifying how they could (or could not) falsify it. Furthermore, they imply that all unrestricted non-classical models must be contextual.
In quantum mechanics, often it is important for the representation of quantum system to study the structure-preserving bijective maps of the quantum system. Such maps are also called isomorphisms or automorphisms. In this note, using the Uhlhorn-type of Wigner's theorem, we characterize all affine automorphisms and ortho-order automorphisms of quantum probabilistic maps.
Probabilistically creating n perfect clones from m copies for one of N priori known quantum states with minimum failure probability is a long-standing problem. We provide a rigorous proof for the geometric approach to this probabilistic quantum cloning problem when N = 2. Then, we give the general geometric form of the sufficient and necessary condition of probabilistic cloning for N known quantum states. By this general geometric approach, we realize the optimal probabilistic quantum cloning of N known quantum states with priori probabilities. The results are also applicable to the identifica…
As a modern approach for the foundation of quantum theory, existing studies of General Probabilistic Theories gave various models of states and measurements that are quite different from quantum theory. In this paper, to seek a more realistic situation, we investigate models approximately close to quantum theory. We define larger measurement classes that are smoothly connected with the class of POVMs via a parameter, and investigate the performance of perfect discrimination. As a result, we give a sufficient condition of perfect discrimination, which shows a significant improvement beyond the …
General probabilistic theories are designed to provide operationally the most general probabilistic models including both classical and quantum theories. In this letter, we introduce a systematic method to construct a series of entropies, all of which generalize Shannon entropy in classical system and von Neumann entropy in quantum system. Using these entropies, the Holevo bound, an upper bound of the accessible information from a quantum system, is generalized to hold in any general probabilistic theory.
Quantum Artificial Intelligence Laboratory, NASA Ames Research Center（美国国家航空航天局艾姆斯研究中心量子人工智能实验室）；Department of Computer Science, University College London（伦敦大学学院计算机科学系）；Cambridge Quantum Computing Limited（剑桥量子计算有限公司）；SGT Inc.（SGT公司）；Instituto de Matemáticas Aplicadas, Universidad de Cartagena（卡塔赫纳大学应用数学系）；Exploration Technology Directorate, NASA Ames Research Center（美国国家航空航天局艾姆斯研究中心探索技术部）；USRA Research Institute for Advanced Computer Science（美国航天研究所高级计算机科学研究所）
Mainstream machine-learning techniques such as deep learning and probabilistic programming rely heavily on sampling from generally intractable probability distributions. There is increasing interest in the potential advantages of using quantum computing technologies as sampling engines to speed up these tasks or to make them more effective. However, some pressing challenges in state-of-the-art quantum annealers have to be overcome before we can assess their actual performance. The sparse connectivity, resulting from the local interaction between quantum bits in physical hardware implementation…
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Probabilistic metrology attempts to improve parameter estimation by occasionally reporting an excellent estimate and the rest of the time either guessing or doing nothing at all. Here we show that probabilistic metrology can never improve quantum limits on estimation of a single parameter, both on average and asymptotically in number of trials, if performance is judged relative to mean-square estimation error. We extend the result by showing that for a finite number of trials, the probability of obtaining better estimates using probabilistic metrology, as measured by mean-square error, decreas…
