Quantum probabilities as Bayesian probabilities
TL;DR: In this paper, it was shown that the distinction between classical and quantum probabilities lies not in their definition, but in the nature of the information they encode, and that any Bayesian probability assignment in quantum mechanics must have the form of the quantum probability rule.
Abstract: In the Bayesian approach to probability theory, probability quantifies a degree of belief for a single trial, without any a priori connection to limiting frequencies. In this paper, we show that, despite being prescribed by a fundamental law, probabilities for individual quantum systems can be understood within the Bayesian approach. We argue that the distinction between classical and quantum probabilities lies not in their definition, but in the nature of the information they encode. In the classical world, maximal information about a physical system is complete in the sense of providing definite answers for all possible questions that can be asked of the system. In the quantum world, maximal information is not complete and cannot be completed. Using this distinction, we show that any Bayesian probability assignment in quantum mechanics must have the form of the quantum probability rule, that maximal information about a quantum system leads to a unique quantum-state assignment, and that quantum theory provides a stronger connection between probability and measured frequency than can be justified classically. Finally, we give a Bayesian formulation of quantum-state tomography.
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TL;DR: In this paper, a no-go theorem on the reality of the quantum state is demonstrated, which states that if the quantum states merely represent information about the physical state of a system, then predictions that contradict those of quantum theory are obtained.
Abstract: A no-go theorem on the reality of the quantum state is demonstrated. If the quantum state merely represents information about the physical state of a system, then predictions that contradict those of quantum theory are obtained.
845 citations
TL;DR: A framework in which a variety of probabilistic theories can be defined, including classical and quantum theories, and many others, is introduced, and a tensor product rule for combining separate systems can be derived.
Abstract: I introduce a framework in which a variety of probabilistic theories can be defined, including classical and quantum theories, and many others. From two simple assumptions, a tensor product rule for combining separate systems can be derived. Certain features, usually thought of as specifically quantum, turn out to be generic in this framework, meaning that they are present in all except classical theories. These include the nonunique decomposition of a mixed state into pure states, a theorem involving disturbance of a system on measurement (suggesting that the possibility of secure key distribution is generic), and a no-cloning theorem. Two particular theories are then investigated in detail, for the sake of comparison with the classical and quantum cases. One of these includes states that can give rise to arbitrary nonsignaling correlations, including the superquantum correlations that have become known in the literature as nonlocal machines or Popescu-Rohrlich boxes. By investigating these correlations in the context of a theory with well-defined dynamics, I hope to make further progress with a question raised by Popescu and Rohrlich, which is why does quantum theory not allow these strongly nonlocal correlations? The existence of such correlations forces much of the dynamics in this theory to be, in a certain sense, classical, with consequences for teleportation, cryptography, and computation. I also investigate another theory in which all states are local. Finally, I raise the question of what further axiom(s) could be added to the framework in order to identify quantum theory uniquely, and hypothesize that quantum theory is optimal for computation.
756 citations
Book•
26 Jul 2012TL;DR: The foundations for modelling probabilistic-dynamic systems using two aspects of quantum theory, 'contextuality' and 'quantum entanglement', are introduced, which allow cognitive phenomena to be modeled in non-reductionist ways.
Abstract: Much of our understanding of human thinking is based on probabilistic models. This innovative book by Jerome R. Busemeyer and Peter D. Bruza argues that, actually, the underlying mathematical structures from quantum theory provide a much better account of human thinking than traditional models. They introduce the foundations for modelling probabilistic-dynamic systems using two aspects of quantum theory. The first, 'contextuality', is a way to understand interference effects found with inferences and decisions under conditions of uncertainty. The second, 'quantum entanglement', allows cognitive phenomena to be modeled in non-reductionist ways. Employing these principles drawn from quantum theory allows us to view human cognition and decision in a totally new light. Introducing the basic principles in an easy-to-follow way, this book does not assume a physics background or a quantum brain and comes complete with a tutorial and fully worked-out applications in important areas of cognition and decision.
745 citations
TL;DR: The toy theory of as discussed by the authors states that the number of questions about the physical state of a system that are answered must always be equal to the number that are unanswered in a state of maximal knowledge.
Abstract: We present a toy theory that is based on a simple principle: the number of questions about the physical state of a system that are answered must always be equal to the number that are unanswered in a state of maximal knowledge. Many quantum phenomena are found to have analogues within this toy theory. These include the noncommutativity of measurements, interference, the multiplicity of convex decompositions of a mixed state, the impossibility of discriminating nonorthogonal states, the impossibility of a universal state inverter, the distinction between bipartite and tripartite entanglement, the monogamy of pure entanglement, no cloning, no broadcasting, remote steering, teleportation, entanglement swapping, dense coding, mutually unbiased bases, and many others. The diversity and quality of these analogies is taken as evidence for the view that quantum states are states of incomplete knowledge rather than states of reality. A consideration of the phenomena that the toy theory fails to reproduce, notably, violations of Bell inequalities and the existence of a Kochen-Specker theorem, provides clues for how to proceed with this research program.
726 citations
Posted Content•
TL;DR: In this regard, no tool appears better calibrated for a direct assault than quantum information theory as discussed by the authors, and this method holds promise precisely because a large part of the structure of quantum theory has always concerned information.
Abstract: In this paper, I try once again to cause some good-natured trouble The issue remains, when will we ever stop burdening the taxpayer with conferences devoted to the quantum foundations? The suspicion is expressed that no end will be in sight until a means is found to reduce quantum theory to two or three statements of crisp physical (rather than abstract, axiomatic) significance In this regard, no tool appears better calibrated for a direct assault than quantum information theory Far from a strained application of the latest fad to a time-honored problem, this method holds promise precisely because a large part--but not all--of the structure of quantum theory has always concerned information It is just that the physics community needs reminding
This paper, though taking quant-ph/0106166 as its core, corrects one mistake and offers several observations beyond the previous version In particular, I identify one element of quantum mechanics that I would not label a subjective term in the theory--it is the integer parameter D traditionally ascribed to a quantum system via its Hilbert-space dimension
460 citations
References
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Book•
01 Jan 1939
TL;DR: In this paper, the authors introduce the concept of direct probabilities, approximate methods and simplifications, and significant importance tests for various complications, including one new parameter, and various complications for frequency definitions and direct methods.
Abstract: 1. Fundamental notions 2. Direct probabilities 3. Estimation problems 4. Approximate methods and simplifications 5. Significance tests: one new parameter 6. Significance tests: various complications 7. Frequency definitions and direct methods 8. General questions
7,086 citations
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TL;DR: The author revealed that quantum teleportation as “Quantum one-time-pad” had changed from a “classical teleportation” to an “optical amplification, privacy amplification and quantum secret growing” situation.
Abstract: Quantum cryptography could well be the first application of quantum mechanics at the individual quanta level. The very fast progress in both theory and experiments over the recent years are reviewed, with emphasis on open questions and technological issues.
6,949 citations
"Quantum probabilities as Bayesian p..." refers background in this paper
...e of the states must be a mixed state—and hence they will assign different probabilities to the outcomes of some measurements. This situation is commonly encountered in quantum cryptographic protocols [3], where the different players, possibly including an eavesdropper, have different information about the quantum systems being sent. In a Bayesian framework, the probabilities assigned by the different pl...
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IBM1
TL;DR: In information processing, as in physics, the classical world view provides an incomplete approximation to an underlying quantum reality that can be harnessed to break codes, create unbreakable codes, and speed up otherwise intractable computations.
Abstract: In information processing, as in physics, our classical world view provides an incomplete approximation to an underlying quantum reality. Quantum effects like interference and entanglement play no direct role in conventional information processing, but they can--in principle now, but probably eventually in practice--be harnessed to break codes, create unbreakable codes, and speed up otherwise intractable computations.
3,080 citations
"Quantum probabilities as Bayesian p..." refers background in this paper
...m state,” irreconcilable with the idea of quantum states as states of knowledge, can be banished from quantum-state tomography using the quantum de Finetti representation. Quantum information science [21] is an emerging field that uses quantum states to escape the constraints imposed on information processing in a realistic/deterministic world. The rewards in quantum information science are great: tele...
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TL;DR: In this paper, a measure on the closed subspaces of a Hilbert space is defined, which assigns to every closed subspace a non-negative real number such that if the subspace is a countable collection of mutually orthogonal sub-spaces having closed linear span B, then
Abstract: In his investigations of the mathematical foundations of quantum mechanics, Mackey1 has proposed the following problem: Determine all measures on the closed subspaces of a Hilbert space. A measure on the closed subspaces means a function μ which assigns to every closed subspace a non-negative real number such that if {Ai} is a countable collection of mutually orthogonal subspaces having closed linear span B, then
$$ \mu (B) = \sum {\mu \left( {{A_i}} \right)} $$
.
1,322 citations
"Quantum probabilities as Bayesian p..." refers background or methods in this paper
...ed limiting frequencies. The keys to these results are Gleason’s theorem and a quantum variant of the Dutch-book argument of the previous paragraph. Gleason’s theorem and the quantum probability rule [12]. In order to derive the quantum probability rule, we make the following assumptions about a quantum system that is described by a D-dimensional Hilbert space: (i) Each set of orthogonal one-dimension...
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... system? In this paper we give answers to these questions. These answers turn out to be simple and straightforward. After a brief introduction to Bayesian probability theory, we use Gleason’s theorem [12] to show that any subjective quantum probability assignment must have the form of the standard quantum probability rule. We then use a version of the so-called Dutch-book argument [13, 14] to show tha...
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