Formalism and Interpretation in Quantum Theory

Wilce, Alexander

doi:10.1007/s10701-010-9410-x

Cited by 15 publications

(19 citation statements)

References 40 publications

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“…This notation is supposed to suggest the reading 'general probabilistic' in the sense of general probabilistic theories [12]. In the terminology of test spaces [99], G(H) is the set of states over H; unfortunately, the term 'probabilistic model' also exists in the test space formalism, but refers to a different concept.…”

Section: Probabilistic Modelsmentioning

confidence: 99%

A Combinatorial Approach to Nonlocality and Contextuality

Acín

Fritz

Leverrier

et al. 2015

Commun. Math. Phys.

174

416

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Abstract. So far, most of the literature on (quantum) contextuality and the Kochen-Specker theorem seems either to concern particular examples of contextuality, or be considered as quantum logic. Here, we develop a general formalism for contextuality scenarios based on the combinatorics of hypergraphs which significantly refines a similar recent approach by Cabello, Severini and Winter (CSW). In contrast to CSW, we explicitly include the normalization of probabilities, which gives us a much finer control over the various sets of probabilistic models like classical, quantum and generalized probabilistic. In particular, our framework specializes to (quantum) nonlocality in the case of Bell scenarios, which arise very naturally from a certain product of contextuality scenarios due to Foulis and Randall. In the spirit of CSW, we find close relationships to several graph invariants. The recently proposed Local Orthogonality principle turns out to be a special case of a general principle for contextuality scenarios related to the Shannon capacity of graphs. Our results imply that it is strictly dominated by a low level of the Navascués-Pironio-Acín hierarchy of semidefinite programs, which we also apply to contextuality scenarios.We derive a wealth of results in our framework, many of these relating to quantum and supraquantum contextuality and nonlocality, and state numerous open problems. For example, we show that the set of quantum models on a contextuality scenario can in general not be characterized in terms of a graph invariant.In terms of graph theory, our main result is this: there exist two graphs G 1 and G 2

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Section: Probabilistic Modelsmentioning

confidence: 99%

A Combinatorial Approach to Nonlocality and Contextuality

Acín

Fritz

Leverrier

et al. 2015

Commun. Math. Phys.

174

416

View full text Add to dashboard Cite

show abstract

“…Specifically, this paper adopts the framework known as operational-probabilistic theories (OPTs) [57-59, 61, 68-71]. The OPT framework differs from other frameworks for general probabilistic theories, such as the convex set framework [55,[72][73][74], in the particular way it treats the composition of systems. While in the convex set framework one generally starts from convex sets associated with individual systems, and builds composites from them, the OPT framework takes the composition of physical processes as primitive.…”

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confidence: 99%

Microcanonical thermodynamics in general physical theories

Chiribella

Scandolo

2017

New J. Phys.

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Microcanonical thermodynamics studies the operations that can be performed on systems with welldefined energy. So far, this approach has been applied to classical and quantum systems. Here we extend it to arbitrary physical theories, proposing two requirements for the development of a general microcanonical framework. We then formulate three resource theories, corresponding to three different sets of basic operations: (i) random reversible operations, resulting from reversible dynamics with fluctuating parameters, (ii) noisy operations, generated by the interaction with ancillas in the microcanonical state, and (iii) unital operations, defined as the operations that preserve the microcanonical state. We focus our attention on a class of physical theories, called sharp theories with purification, where these three sets of operations exhibit remarkable properties. Firstly, each set is contained into the next. Secondly, the convertibility of states by unital operations is completely characterised by a majorisation criterion. Thirdly, the three sets are equivalent in terms of state convertibility if and only if the dynamics allowed by theory satisfy a suitable condition, which we call unrestricted reversibility. Under this condition, we derive a duality between the resource theories of microcanonical thermodynamics and the resource theory of pure bipartite entanglement.1. random unitary channels [43][44][45], arising from unitary dynamics with randomly fluctuating parameters;2. noisy operations [46][47][48], generated by preparing ancillas in the microcanonical state, turning on a unitary dynamics, and discarding the ancillas;3. unital channels [49,50], defined as the quantum processes that preserve the microcanonical state.These three sets are strictly different: the set of random unitary channels is strictly contained in the set of noisy operations [51], and the latter is strictly contained in the set of unital channels [52]. In spite of this, the three sets are equivalent in terms of state convertibility [48]. This means that all the natural candidates for the sets of free operations induce the same notion of resource. This resource is generally called purity, and plays a fundamental role in many quantum protocols [53].In this paper we extend the paradigm of microcanonical thermodynamics from quantum theory to arbitrary physical theories [54][55][56][57][58][59][60][61]. We propose two minimal requirements a probabilistic theory must satisfy in order to support a microcanonical description, and, when these requirements are satisfied, we provide a general operational definition of random reversible, noisy, and unital operations. We then focus on a special class of theories, called sharp theories with purification, which are appealing for the foundations of thermodynamics [62], and have also been studied for their computational power [63, 64] and interference properties [65]. In sharp theories with purification, we show that the three sets of operations satisfy the same inclusion relations as in quantum theory, with...

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“…Following A. Wilce [23], we shall term the latter classical test spaces and it is worth commenting on the motivation for the generalization. 4 Classical test spaces are equivalent to manuals with only a single experiment and, intuitively, one would suppose that, for any manual with n experiments, one can merge the experiments of M by taking the set of all sets of n members where the first member is a member of the first experiment, the second member is a member of the second experiment and the nth member is a member of the nth experiment and none of the members are orthogonal to each other.…”

Section: Manuals and Systemsmentioning

confidence: 99%

“…Using Wilce's terminology [23], let us term 'classical' any manual that has only one experiment and 'semi-classical' any manual that has multiple disjointed experiments. Let us also term 'non-classical' any manual that has multiple experiments that are not disjointed.…”

Section: Manuals and Systemsmentioning

confidence: 99%

Does Science Influence the Logic we Ought to Use: A Reflection on the Quantum Logic Controversy

Ashcroft

2010

Stud Logica

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In this article I argue that there is a sense in which logic is empirical, and hence open to influence from science. One of the roles of logic is the modelling and extending of natural language reasoning. It does so by providing a formal system which succeeds in modelling the structure of a paradigmatic set of our natural language inferences and which then permits us to extend this structure to novel cases with relative ease. In choosing the best system of those that succeed in this, we seek certain virtues of such structures such as simplicity and naturalness (which will be explained).Science can influence logic by bringing us, as in the case of quantum mechanics, to make natural language inferences about new kinds of systems and thereby extend the set of paradigmatic cases that our formal logic ought to model as simply and naturally as possible. This can alter which structures ought to be used to provide semantics for such models. I show why such a revolution could have led us to reject one logic for another through explaining why complex claims about quantum mechanical systems failed to lead us to reject classical logic for quantum logic.

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Formalism and Interpretation in Quantum Theory

Cited by 15 publications

References 40 publications

A Combinatorial Approach to Nonlocality and Contextuality

A Combinatorial Approach to Nonlocality and Contextuality

Microcanonical thermodynamics in general physical theories

Does Science Influence the Logic we Ought to Use: A Reflection on the Quantum Logic Controversy

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