Versuch einer axiomatischen Grundlegung der Quantenmechanik und allgemeinerer physikalischer Theorien

Ludwig, Günther

doi:10.1007/bf01418533

Cited by 93 publications

(83 citation statements)

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“…This reduces to the joint measurability condition η ≤ η opt for Alice's local unsharp measurements, as one can identify a striking agreement between the degree of incompatibility η opt (given by (12) and listed in Table. II)) and the upper bound f (N ) (given in (36)) of the finite setting linear steering inequality (35). This is a clear example of the intrinsic connection (established in Refs.…”

Section: Joint Measurability Condition From Linear Steering Inequamentioning

confidence: 73%

“…However, non-commuting observables are declared as incompatible under the regime of PV measurements. Introduction of POVMs in 1960's by Ludwig [35] and subsequent investigations on their applicability [36], led to a mathematically rigorous generalization of measurement theory. It is the notion of compatibility (the notion of compatibility of a set of POVMs will be defined in the following) -rather than commutativity -which gains importance so as to recognize if a given set of POVMs are jointly measurable or not [37].…”

Section: Joint Measurability Of Povmsmentioning

confidence: 99%

“…This bound is not obeyed, in general, if the expectation value of the observable σ θ is replaced by its conditional expectation value σ θ a θ |θ , evaluated in a sequential measurement, with the first measurement resulting in an outcome a θ . In particular, in the setting where finite number of pairwise sequential measurements of the equatorial qubit observable σ θ k , with same angle θ k , are carried out [53], the analogue of the steering inequality (35) …”

Section: Local Analogue Of the Linear Steering Inequalitymentioning

confidence: 99%

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N-term pairwise-correlation inequalities, steering, and joint measurability

et al. 2017

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Chained inequalities involving pairwise correlations of qubit observables in the equatorial plane are constructed based on the positivity of a sequence of moment matrices. When a jointly measurable set of fuzzy POVMs is employed in first measurement of every pair of sequential measurements, the chained pairwise correlations do not violate the classical bound imposed by the moment matrix positivity. We identify that incompatibility of the set of POVMs employed in first measurements is only necessary, but not sufficient, in general, for the violation of the inequality. On the other hand, there exists a one-to-one equivalence between the degree of incompatibility (which quantifies the joint measurability) of the equatorial qubit POVMs and the optimal violation of a non-local steering inequality, proposed by Jones and Wiseman (Phys. Rev. A, 84, 012110 (2011)). To this end, we construct a local analogue of this steering inequality in a single qubit system and show that its violation is a mere reflection of measurement incompatibility of equatorial qubit POVMs, employed in first measurements in the sequential unsharp-sharp scheme.

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Section: Joint Measurability Condition From Linear Steering Inequamentioning

confidence: 73%

Section: Joint Measurability Of Povmsmentioning

confidence: 99%

Section: Local Analogue Of the Linear Steering Inequalitymentioning

confidence: 99%

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N-term pairwise-correlation inequalities, steering, and joint measurability

et al. 2017

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“…Thus, in the later development of quantum theory, several authors suggested the generalized representation of observables as positive operator measures [9][10][11]). This concept advanced the mathematical coherence and conceptual clarity at first of quantum mechanics and then of quantum field theory.…”

Section: The Ubiquity Of Natural Operatorsmentioning

confidence: 99%

Operators in Nature, Science, Technology, and Society: Mathematical, Logical, and Philosophical Issues

Burgin

Brenner²

2017

Philosophies

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Abstract:The concept of an operator is used in a variety of practical and theoretical areas. Operators, as both conceptual and physical entities, are found throughout the world as subsystems in nature, the human mind, and the manmade world. Operators, and what they operate, i.e., their substrates, targets, or operands, have a wide variety of forms, functions, and properties. Operators have explicit philosophical significance. On the one hand, they represent important ontological issues of reality. On the other hand, epistemological operators form the basic mechanism of cognition. At the same time, there is no unified theory of the nature and functions of operators. In this work, we elaborate a detailed analysis of operators, which range from the most abstract formal structures and symbols in mathematics and logic to real entities, human and machine, and are responsible for effecting changes at both the individual and collective human levels. Our goal is to find what is common in physical objects called operators and abstract mathematical structures, with the name operator providing foundations for building a unified but flexible theory of operators. The paper concludes with some reflections on functionalism and other philosophical aspects of the 'operation' of operators.

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“…More specifically, joint measurability or compatibility of observables is entirely linked with their commutativity, if sharp PV measurements are employed. However, with the introduction of generalized measurements -formulated in terms of positive operator valued measures (POVMs) -it has been possible to refine the notion of joint measurability of non-commuting observables [1][2][3][4] . While sharp values for non-commuting observables can not be assigned jointly via their PV measurements, their unsharp joint values could be perceived when compatible POVMs are employed.…”

Section: Introductionmentioning

confidence: 99%

Unsharp Measurements and Joint Measurability

2015

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We give an overview of joint unsharp measurements of non-commuting observables using positive operator valued measures (POVMs). We exemplify the role played by joint measurability of POVMs in entropic uncertainty relation for Alice's pair of non-commuting observables in the presence of Bob's entangled quantum memory. We show that Bob should record the outcomes of incompatible (non-jointly measurable) POVMs in his quantum memory so as to beat the entropic uncertainty bound. In other words, in addition to the presence of entangled Alice-Bob state, implementing incompatible POVMs at Bob's end is necessary to beat the uncertainty bound and hence, predict the outcomes of non-commuting observables with improved precision. We also explore the implications of joint measurability to validate a moment matrix constructed from average pairwise correlations of three dichotomic non-commuting qubit observables. We prove that a classically acceptable moment matrix -which ascertains the existence of a legitimate joint probability distribution for the outcomes of all the three dichotomic observables -could be realized if and only if compatible POVMs are employed.

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Versuch einer axiomatischen Grundlegung der Quantenmechanik und allgemeinerer physikalischer Theorien

Cited by 93 publications

References 0 publications

N-term pairwise-correlation inequalities, steering, and joint measurability

N-term pairwise-correlation inequalities, steering, and joint measurability

Operators in Nature, Science, Technology, and Society: Mathematical, Logical, and Philosophical Issues

Unsharp Measurements and Joint Measurability

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