Quantum Theory: Concepts and Methods

Peres, Asher; Ballentine, L. E.

doi:10.1119/1.17946

Cited by 1,387 publications

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“…In quantum probability theory, the law of reciprocity (Peres, 1998) states that if two events X and E are represented by a single dimensional subspace (i.e., a ray), then p(X|E) is exactly the same as p(E|X). In our hierarchy of representations, reciprocity is predicted by the 2D model because events are represented by rays.…”

Section: Reciprocitymentioning

confidence: 99%

A quantum probability framework for human probabilistic inference.

Trueblood

Yearsley

Pothos

2017

Journal of Experimental Psychology: General

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This is the accepted version of the paper.This version of the publication may differ from the final published version. Permanent repository link AbstractThere is considerable variety in human inference (e.g., a doctor inferring the presence of a disease, a juror inferring the guilt of a defendant, or someone inferring future weight loss based on diet and exercise). As such, people display a wide range of behaviors when making inference judgments.Sometimes, people's judgments appear Bayesian (i.e., normative), but in other cases, judgments deviate from the normative prescription of classical probability theory. How can we combine bothBayesian and non-Bayesian influences in a principled way? We propose a unified explanation of human inference using quantum probability theory. In our approach, we postulate a hierarchy of mental representations, from 'fully' quantum to 'fully' classical, which could be adopted in different situations. In our hierarchy of models, moving from the lowest level to the highest involves changing assumptions about compatibility (i.e., how joint events are represented). Using results from three experiments, we show that our modeling approach explains five key phenomena in human inference including order effects, reciprocity (i.e., the inverse fallacy), memorylessness, violations of the Markov condition, and anti-discounting. As far as we are aware, no existing theory or model can explain all five phenomena. We also explore transitions in our hierarchy, examining how representations change from more quantum to more classical. We show that classical representations provide a better account of data as individuals gain familiarity with a task.We also show that representations vary between individuals, in a way that relates to a simple measure of cognitive style, the Cognitive Reflection Test.Keywords: Human judgment, quantum probability theory, Bayes' rule, order effects, Markov condition A Quantum Framework for Probabilistic Inference 3 A Quantum Probability Framework for Human Probabilistic InferenceEveryday we face situations where we must make inferences about the world around us. For example, a doctor must determine the likelihood that a patient has a disease based on a set of symptoms. A juror must decide the probability that a defendant is guilty after hearing the cases made by the prosecution and defense. Or, maybe you want to judge the likelihood that you will weigh less next month if you start exercising more regularly and you improve your diet. In general, the inference problem involves judging the likelihood of some hypothesis (e.g., presence of a disease, guilt of a defendant, future weight loss) based on a series of evidence (e.g., medicalsymptoms, prosecution and defense cases, changes in your diet and exercise).Bayesian inference is widely accepted as the normative approach to inference. However, decades of research in human judgment and decision-making have suggested that people's judgments often violate the rules of Bayesian inference and classical probability theory (Tversky...

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Section: Reciprocitymentioning

confidence: 99%

A quantum probability framework for human probabilistic inference.

Trueblood

Yearsley

Pothos

2017

Journal of Experimental Psychology: General

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“…For a selection of more recent statements see, for example, Fuchs and Peres (2000); Mermin (2001bMermin ( , 2003; Peierls (1986Peierls ( , 1991Peres (1995);Wheeler (1986Wheeler ( , 1990Zeilinger (1999b).…”

Section: Moralsmentioning

confidence: 99%

“…Any (projective) measurement that does not project onto the eigenbasis of ρ will result in a post-measurement ensemble that is more mixed than ρ (see e.g. Nielsen (2001); Peres (1995) and below) and hence will have a greater uncertainty, thus a greater Shannon information, or any other measure of information gain. .1: The probability distribution p = { 1 2 , 1 3 , 1 6 } can be considered as given for the three outcomes directly, or we could consider first a choice of two equiprobable events, followed by a second choice of two events with probabilities 2 3 , 1 3 , conditional on the second, say, of the first two events occurring, a 'decomposition' of a single choice into two successive choices, the latter of which will only be made half the time.…”

Section: Introductionmentioning

confidence: 99%

Quantum Information Theory and the Foundations of Quantum Mechanics

Timpson¹

2013

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of Thesis Submitted for the Degree of Doctor of PhilosophyThis thesis is a contribution to the debate on the implications of quantum information theory for the foundational problems of quantum mechanics. In Part I an attempt is made to shed some light on the nature of information and quantum information theory. It is emphasized that the everyday notion of information is to be firmly distinguished from the technical notions arising in information theory;however it is maintained that in both settings 'information' functions as an abstract noun, hence does not refer to a particular or substance. The popular claim 'Information is Physical' is assessed and it is argued that this proposition faces a destructive dilemma. Accordingly, the slogan may not be understood as an ontological claim, but at best, as a methodological one. A novel argument is provided against Dretske's (1981) attempt to base a semantic notion of information on ideas from information theory.The function of various measures of information content for quantum systems is explored and the applicability of the Shannon information in the quantum context maintained against the challenge of Brukner and Zeilinger (2001). The phenomenon of quantum teleportation is then explored as a case study serving to emphasize the value of recognising the logical status of 'information' as an abstract noun: it is argued that the conceptual puzzles often associated with this phenomenon result from the familiar error of hypostatizing an abstract noun.The approach of Deutsch and Hayden (2000) to the questions of locality and information flow in entangled quantum systems is assessed. It is suggested that the approach suffers from an equivocation between a conservative and an ontological reading; and the differing implications of each is examined. Some results are presented on the characterization of entanglement in the Deutsch-Hayden formalism.Part I closes with a discussion of some philosophical aspects of quantum computation.In particular, it is argued against Deutsch that the Church-Turing hypothesis is not underwritten by a physical principle, the Turing Principle. Some general morals are drawn concerning the nature of quantum information theory.In Part II, attention turns to the question of the implications of quantum information theory for our understanding of the meaning of the quantum formalism. Following some preliminary remarks, two particular information-theoretic approaches to the foundations of quantum mechanics are assessed in detail. It is argued that Zeilinger's (1999) Foundational Principle is unsuccessful as a foundational principle for quantum mechanics. The information-theoretic characterization theorem of Clifton, is assessed more favourably, but the generality of the approach is questioned and it is argued that the implications of the theorem for the traditional foundational problems in quantum mechanics remains obscure.

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“…After their interaction, which we describe as an arbitrary unitary evolution of the composite system, the initial density matrix ρ L ⊗ ρ R has evolved to ρ ′ LR . As a consequence of subadditivity of the entropy, it is easy to see that the partial traces ρ ′ L and [8,9]. Of course, if the initial state is a product of pure states, S L = S R = 0.…”

Section: Entanglement Entropy Of a Bipartite Systemmentioning

confidence: 99%

“…Entanglement or non-separability refers to the existence of quantum correlations between two sets of degrees of freedom of a physical system that can be considered as subsystems [8,9]. Two (sub)systems in interaction are entangled and their entanglement continues after their interaction has ceased.…”

Section: Entanglement Entropy and Quantum Informationmentioning

confidence: 99%

Renormalization group and quantum information

Gaite¹

2006

J. Phys. A: Math. Gen.

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The renormalization group is a tool that allows one to obtain a reduced description of systems with many degrees of freedom while preserving the relevant features. In the case of quantum systems, in particular, one-dimensional systems defined on a chain, an optimal formulation is given by White's "density matrix renormalization group". This formulation can be shown to rely on concepts of the developing theory of quantum information. Furthermore, White's algorithm can be connected with a peculiar type of quantization, namely, angular quantization. This type of quantization arose in connection with quantum gravity problems, in particular, the Unruh effect in the problem of black-hole entropy and Hawking radiation. This connection highlights the importance of quantum system boundaries, regarding the concentration of quantum states on them, and helps us to understand the optimal nature of White's algorithm.

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Quantum Theory: Concepts and Methods

Abstract: Peres has given us a clear and fully elaborated statement of the epistemology of quantum mechanics, and a rich source of examples of how ordinary questions can be posed in the theory,

Cited by 1,387 publications

References 0 publications

A quantum probability framework for human probabilistic inference.

A quantum probability framework for human probabilistic inference.

Quantum Information Theory and the Foundations of Quantum Mechanics

Renormalization group and quantum information

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