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Paul Busch

University of York, Mathematics, Faculty Member

University of Turku, Department of Physics and Astronomy, Adjunct

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Mathematical physicist with extensive research experience in
conceptual and mathematical foundations of quantum physics. Current research interests in the quantum-classical contrast, quantum limitations of measurements and measurement uncertainty relations, with implications for quantum information processing and metrology, and issues between quantum theory and relativity. Coauthor of two monographs, "The Quantum Theory of Measurement" (with P Lahti and P Mittelstaedt), and "Operational Quantum Physics" (with M Grabowski and P Lahti).
Phone: +441904323082
Address: Department of Mathematics
University of York
Heslington
York YO10 5DD
UK

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University of Zagreb

Oleg Yu Vorobyev

Siberian Federal University

Serpil Oppermann

Kapadokya Üniversitesi

Kansas State University

Armando Marques-Guedes

UNL - New University of Lisbon

Gordon N Fleming

Pennsylvania State University

Viacheslav Kuleshov

Stockholm University

Francisco Caruso

Centro Brasileiro de Pesquisas Físicas

The University Of Jordan

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Papers by Paul Busch

Approximating relational observables by absolute quantities: a quantum accuracy-size trade-off

by Leon Loveridge and Paul Busch

The notion that any physical quantity is defined and measured relative to a reference frame is tr... more The notion that any physical quantity is defined and measured relative to a reference frame is traditionally not explicitly reflected in the theoretical description of physical experiments where, instead, the relevant observables are typically represented as " absolute " quantities. However, the emergence of the resource theory of quantum reference frames as a new branch of quantum information science in recent years has highlighted the need to identify the physical conditions under which a quantum system can serve as a good reference. Here we investigate the conditions under which, in quantum theory, an account in terms of absolute quantities can provide a good approximation of relative quantities. We find that this requires the reference system to be large in a suitable sense.

Symmetry and the Relativity of States and Observables in Quantum Mechanics

by Leon Loveridge and Paul Busch

Motivated by symmetry, we construct a relational, or relative description of quantum states and o... more Motivated by symmetry, we construct a relational, or relative description of quantum states and observables. From the premise that the notion of an observable quantity includes the specification of a reference frame, and using a relativization map which makes the frame-dependence explicit, we show that the usual quantum description approximates the relative one precisely when the reference system admits an appropriate localizable quantity and a localized state. From this follows a new perspective on the nature and reality of quantum superpositions and optical coherence.

Direct tests of measurement uncertainty relations: what it takes (Title of earlier version: Determining measurement errors: where the good meet the bad)

The uncertainty principle being a cornerstone of quantum mechanics, it is surprising that in near... more The uncertainty principle being a cornerstone of quantum mechanics, it is surprising that in nearly 90 years there have been no direct tests of measurement uncertainty relations. This lacuna was due to the absence of two essential ingredients: appropriate measures of measurement error (and disturbance), and precise formulations of such relations that are {\em universally valid}and {\em directly testable}. We formulate two distinct forms of direct tests, based on different measures of error. We present a prototype protocol for a direct test of measurement uncertainty relations in terms of {\em value deviation errors} (hitherto considered nonfeasible), highlighting the lack of universality of these relations. This shows that the formulation of universal, directly testable measurement uncertainty relations for {\em state-dependent} error measures remains an important open problem. Recent experiments that were claimed to constitute invalidations of Heisenberg's error-disturbance relation are shown to conform with the spirit of Heisenberg's principle if interpreted as direct tests of measurement uncertainty relations for error measures that quantify {\em distances between observables}.

Note on non-entangling measurements

Noise operators and measures of rms error and disturbance in quantum mechanics

Peter Mittelstaedt: List of Publications until 2010

Foundations of Physics, 2010

In a period of over 50 years, Peter Mittelstaedt has made substantial and lasting contributions t... more In a period of over 50 years, Peter Mittelstaedt has made substantial and lasting contributions to several fields in theoretical physics as well as the foundations and philosophy of physics. Here we present an overview of his achievements in physics and its foundations which may serve as a guide to the bibliography (printed in this Festschrift) of his publications. An appraisal of Peter Mittelstaedt's work in the philosophy of physics is given in a separate contribution by B. Falkenburg.

Symposium on the Foundations of Modern Physics 1993

Foundations of Physics, 1993

Unbestimmtheitsrelation und simultane messungen in der quantent

SYMPOSIUM ON THE FOUNDATIONS OF MODERN PHYSICS, pp. 343-357 edited by P. Lahti & P. Mittelstaedt© 1985 by World Scientific Publishing Co

Symposium on the Foundations of Modern Physics: 50 Years of the Einstein-Podolsky-Rosen Gedankenexperiment, Joensuu, Finland, 16-20 June 1985, 1985

Feasibility study of spin correlations in {sup 2} He ({sup 1} S {sub 0})

Verhandlungen der Deutschen Physikalischen Gesellschaft, Jul 1, 2002

There may be additional, and more current, records on this topic in the Energy Technology Data Ex... more

Complementarity and uncertainty--entangled in joint path-interference measurements

Phys. Rev. A, Jul 15, 2002

‘Resurrection of von Neumann’s no-hidden-variables theorem,’’

arXiv preprint quant-ph/9909073, Sep 24, 1999

Von Neumann's theorem on the impossibility of non-contextual hidden variable theories for qu... more Von Neumann's theorem on the impossibility of non-contextual hidden variable theories for quantum mechanics has been criticized on account of its unfounded assumption of the additivity of the valuation function for noncommuting observables. Starting from the fact that the additivity is operationally meaningful for noncommuting sets of effects whose sum is an effect, we show that any effect valuation extends to a unique linear functional on the space of bounded self-adjoint operators and hence must be a quantum state. It follows ...

On the sharpness of an effect

arXiv preprint math-ph/0706.3532, Jun 26, 2007

I present a measure of the sharpness of an effect $ A $ as a function $ A\ mapsto\ S (A) $ that a... more I present a measure of the sharpness of an effect $ A $ as a function $ A\ mapsto\ S (A) $ that assumes values between 0 and 1 such that $\ S (A)= 0$ exactly when $ A $ is trivial and $\ S (A)= 1$ exactly when $ A $ is a nontrivial projection. This is followed by a list of open questions arising from this definition.

Quantum states as effect valuations: giving operational content to von Neumann’s no-hidden-variables theorem

arXiv preprint quant-ph/9909073, 2003

A quantum state can be understood in a loose sense as a map that assigns a value to every observa... more A quantum state can be understood in a loose sense as a map that assigns a value to every observable. Formalizing this characterization of states in terms of effect valuations, we employ a reinterpretation of a theorem due to von Neumann to obtain a simple proof of the result that any quantum state is given by a density operator.

promoting access to White Rose research papers

Direct tests of measurement uncertainty relations: what it takes

The uncertainty principle being a cornerstone of quantum mechanics, it is surprising that in near... more The uncertainty principle being a cornerstone of quantum mechanics, it is surprising that in nearly 90 years there have been no direct tests of measurement uncertainty relations. This lacuna was due to the absence of two essential ingredients: appropriate measures of measurement error (and disturbance), and precise formulations of such relations that are universally valid and directly testable. We formulate two distinct forms of direct tests, based on different measures of error. We present a prototype protocol for a direct test of measurement uncertainty relations in terms of value deviation errors (hitherto considered nonfeasible), highlighting the lack of universality of these relations. This shows that the formulation of universal, directly testable measurement uncertainty relations for state-dependent error measures remains an important open problem. Recent experiments that were claimed to constitute invalidations of Heisenberg’s error-disturbance relation, are shown to conform with the spirit of Heisenberg’s principle if interpreted as direct tests of measurement uncertainty relations for error measures that quantify distances between observables.

Colloquium: Quantum root-mean-square error and measurement uncertainty relations

Recent years have witnessed a controversy over Heisenberg’s famous error-disturbance relation. He... more Recent years have witnessed a controversy over Heisenberg’s famous error-disturbance relation. Here we resolve the conflict by way of an analysis of the possible conceptualizations of measurement error and disturbance in quantum mechanics. We discuss two approaches to adapting the classic notion of root-mean-square error to quantum measurements. One is based on the concept of noise operator; its natural operational content is that of a mean deviation of the values of two observables measured jointly, and thus its applicability is limited to cases where such joint measurements are available. The second error measure quantifies the differences between two probability distributions obtained in separate runs of measurements and is of unrestricted applicability. We show that there are no nontrivial unconditional joint-measurement bounds for state-dependent errors in the conceptual framework discussed here, while Heisenberg-type measurement uncertainty relations for state-independent errors have been proven.

Focusing in Arthurs-Kelly-Type Joint Measurements with Correlated Probes

Joint approximate measurement schemes of position and momentum provide us with a means of inferri... more Joint approximate measurement schemes of position and momentum provide us with a means of inferring pieces of complementary information if we allow for the irreducible noise required by quantum theory. One such scheme is given by the Arthurs-Kelly model, where information about a system is extracted via indirect probe measurements, assuming separable uncorrelated probes. Here, following Di Lorenzo [Phys. Rev. Lett. 110, 120403 (2013)], we extend this model to both entangled and classically correlated probes, achieving full generality. We show that correlated probes can produce more precise joint measurement outcomes than the same probes can achieve if applied alone to realize a position or momentum measurement. This phenomenon of focusing may be useful where one tries to optimize measurements with limited physical resources. Contrary to Di Lorenzo’s claim, we find that there are no violations of Heisenberg’s error-disturbance relation in these generalized Arthurs-Kelly models. This is simply due to the fact that, as we show, the measured observable of the system under consideration is covariant under phase space translations and as such is known to obey a tight joint measurement error relation.

Uncertainty in the context of multislit interferometry

A pair of uncertainty relations relevant for quantum states of multislit interferometry is derive... more A pair of uncertainty relations relevant for quantum states of multislit interferometry is derived, based on the mutually commuting “modular” position and momentum operators and their complementary counterparts, originally introduced by Aharonov and co-workers. We provide a precise argument as to why these relations are superior to the standard Heisenberg uncertainty relation at expressing the complementarity between spatial localization and the appearance of fringes. We further support the argument with explicit computations involving wave functions specifically tailored to the interference setup. Conceptually developing the idea of Aharonov and co-workers, we show how the modular momentum should reflect the given experimental setup, yielding a refined observable that accurately captures the fine structure of the interference pattern.

Measurement Uncertainty Relations

Measurement uncertainty relations are quantitative bounds on the errors in an approximate joint m... more Measurement uncertainty relations are quantitative bounds on the errors in an approximate joint measurement of two observables. They can be seen as a generalization of the error/disturbance tradeoff first discussed heuristically by Heisenberg. Here we prove such relations for the case of two canonically conjugate observables like position and momentum, and establish a close connection with the more familiar preparation uncertainty relations constraining the sharpness of the distributions of the two observables in the same state. Both sets of relations are generalized to means of order α rather than the usual quadratic means, and we show that the optimal constants are the same for preparation and for measurement uncertainty. The constants are determined numerically and compared with some bounds in the literature. In both cases the near-saturation of the inequalities entails that the state (resp. observable) is uniformly close to a minimizing one.

Approximating relational observables by absolute quantities: a quantum accuracy-size trade-off

by Leon Loveridge and Paul Busch

The notion that any physical quantity is defined and measured relative to a reference frame is tr... more The notion that any physical quantity is defined and measured relative to a reference frame is traditionally not explicitly reflected in the theoretical description of physical experiments where, instead, the relevant observables are typically represented as " absolute " quantities. However, the emergence of the resource theory of quantum reference frames as a new branch of quantum information science in recent years has highlighted the need to identify the physical conditions under which a quantum system can serve as a good reference. Here we investigate the conditions under which, in quantum theory, an account in terms of absolute quantities can provide a good approximation of relative quantities. We find that this requires the reference system to be large in a suitable sense.

Symmetry and the Relativity of States and Observables in Quantum Mechanics

by Leon Loveridge and Paul Busch

Motivated by symmetry, we construct a relational, or relative description of quantum states and o... more Motivated by symmetry, we construct a relational, or relative description of quantum states and observables. From the premise that the notion of an observable quantity includes the specification of a reference frame, and using a relativization map which makes the frame-dependence explicit, we show that the usual quantum description approximates the relative one precisely when the reference system admits an appropriate localizable quantity and a localized state. From this follows a new perspective on the nature and reality of quantum superpositions and optical coherence.

Direct tests of measurement uncertainty relations: what it takes (Title of earlier version: Determining measurement errors: where the good meet the bad)

The uncertainty principle being a cornerstone of quantum mechanics, it is surprising that in near... more The uncertainty principle being a cornerstone of quantum mechanics, it is surprising that in nearly 90 years there have been no direct tests of measurement uncertainty relations. This lacuna was due to the absence of two essential ingredients: appropriate measures of measurement error (and disturbance), and precise formulations of such relations that are {\em universally valid}and {\em directly testable}. We formulate two distinct forms of direct tests, based on different measures of error. We present a prototype protocol for a direct test of measurement uncertainty relations in terms of {\em value deviation errors} (hitherto considered nonfeasible), highlighting the lack of universality of these relations. This shows that the formulation of universal, directly testable measurement uncertainty relations for {\em state-dependent} error measures remains an important open problem. Recent experiments that were claimed to constitute invalidations of Heisenberg's error-disturbance relation are shown to conform with the spirit of Heisenberg's principle if interpreted as direct tests of measurement uncertainty relations for error measures that quantify {\em distances between observables}.

Note on non-entangling measurements

Noise operators and measures of rms error and disturbance in quantum mechanics

Peter Mittelstaedt: List of Publications until 2010

Foundations of Physics, 2010

In a period of over 50 years, Peter Mittelstaedt has made substantial and lasting contributions t... more In a period of over 50 years, Peter Mittelstaedt has made substantial and lasting contributions to several fields in theoretical physics as well as the foundations and philosophy of physics. Here we present an overview of his achievements in physics and its foundations which may serve as a guide to the bibliography (printed in this Festschrift) of his publications. An appraisal of Peter Mittelstaedt's work in the philosophy of physics is given in a separate contribution by B. Falkenburg.

Symposium on the Foundations of Modern Physics 1993

Foundations of Physics, 1993

Unbestimmtheitsrelation und simultane messungen in der quantent

SYMPOSIUM ON THE FOUNDATIONS OF MODERN PHYSICS, pp. 343-357 edited by P. Lahti & P. Mittelstaedt© 1985 by World Scientific Publishing Co

Symposium on the Foundations of Modern Physics: 50 Years of the Einstein-Podolsky-Rosen Gedankenexperiment, Joensuu, Finland, 16-20 June 1985, 1985

Feasibility study of spin correlations in {sup 2} He ({sup 1} S {sub 0})

Verhandlungen der Deutschen Physikalischen Gesellschaft, Jul 1, 2002

There may be additional, and more current, records on this topic in the Energy Technology Data Ex... more

Complementarity and uncertainty--entangled in joint path-interference measurements

Phys. Rev. A, Jul 15, 2002

‘Resurrection of von Neumann’s no-hidden-variables theorem,’’

arXiv preprint quant-ph/9909073, Sep 24, 1999

Von Neumann's theorem on the impossibility of non-contextual hidden variable theories for qu... more Von Neumann's theorem on the impossibility of non-contextual hidden variable theories for quantum mechanics has been criticized on account of its unfounded assumption of the additivity of the valuation function for noncommuting observables. Starting from the fact that the additivity is operationally meaningful for noncommuting sets of effects whose sum is an effect, we show that any effect valuation extends to a unique linear functional on the space of bounded self-adjoint operators and hence must be a quantum state. It follows ...

On the sharpness of an effect

arXiv preprint math-ph/0706.3532, Jun 26, 2007

I present a measure of the sharpness of an effect $ A $ as a function $ A\ mapsto\ S (A) $ that a... more I present a measure of the sharpness of an effect $ A $ as a function $ A\ mapsto\ S (A) $ that assumes values between 0 and 1 such that $\ S (A)= 0$ exactly when $ A $ is trivial and $\ S (A)= 1$ exactly when $ A $ is a nontrivial projection. This is followed by a list of open questions arising from this definition.

Quantum states as effect valuations: giving operational content to von Neumann’s no-hidden-variables theorem

arXiv preprint quant-ph/9909073, 2003

A quantum state can be understood in a loose sense as a map that assigns a value to every observa... more A quantum state can be understood in a loose sense as a map that assigns a value to every observable. Formalizing this characterization of states in terms of effect valuations, we employ a reinterpretation of a theorem due to von Neumann to obtain a simple proof of the result that any quantum state is given by a density operator.

promoting access to White Rose research papers

Direct tests of measurement uncertainty relations: what it takes

The uncertainty principle being a cornerstone of quantum mechanics, it is surprising that in near... more The uncertainty principle being a cornerstone of quantum mechanics, it is surprising that in nearly 90 years there have been no direct tests of measurement uncertainty relations. This lacuna was due to the absence of two essential ingredients: appropriate measures of measurement error (and disturbance), and precise formulations of such relations that are universally valid and directly testable. We formulate two distinct forms of direct tests, based on different measures of error. We present a prototype protocol for a direct test of measurement uncertainty relations in terms of value deviation errors (hitherto considered nonfeasible), highlighting the lack of universality of these relations. This shows that the formulation of universal, directly testable measurement uncertainty relations for state-dependent error measures remains an important open problem. Recent experiments that were claimed to constitute invalidations of Heisenberg’s error-disturbance relation, are shown to conform with the spirit of Heisenberg’s principle if interpreted as direct tests of measurement uncertainty relations for error measures that quantify distances between observables.

Colloquium: Quantum root-mean-square error and measurement uncertainty relations

Recent years have witnessed a controversy over Heisenberg’s famous error-disturbance relation. He... more Recent years have witnessed a controversy over Heisenberg’s famous error-disturbance relation. Here we resolve the conflict by way of an analysis of the possible conceptualizations of measurement error and disturbance in quantum mechanics. We discuss two approaches to adapting the classic notion of root-mean-square error to quantum measurements. One is based on the concept of noise operator; its natural operational content is that of a mean deviation of the values of two observables measured jointly, and thus its applicability is limited to cases where such joint measurements are available. The second error measure quantifies the differences between two probability distributions obtained in separate runs of measurements and is of unrestricted applicability. We show that there are no nontrivial unconditional joint-measurement bounds for state-dependent errors in the conceptual framework discussed here, while Heisenberg-type measurement uncertainty relations for state-independent errors have been proven.

Focusing in Arthurs-Kelly-Type Joint Measurements with Correlated Probes

Joint approximate measurement schemes of position and momentum provide us with a means of inferri... more Joint approximate measurement schemes of position and momentum provide us with a means of inferring pieces of complementary information if we allow for the irreducible noise required by quantum theory. One such scheme is given by the Arthurs-Kelly model, where information about a system is extracted via indirect probe measurements, assuming separable uncorrelated probes. Here, following Di Lorenzo [Phys. Rev. Lett. 110, 120403 (2013)], we extend this model to both entangled and classically correlated probes, achieving full generality. We show that correlated probes can produce more precise joint measurement outcomes than the same probes can achieve if applied alone to realize a position or momentum measurement. This phenomenon of focusing may be useful where one tries to optimize measurements with limited physical resources. Contrary to Di Lorenzo’s claim, we find that there are no violations of Heisenberg’s error-disturbance relation in these generalized Arthurs-Kelly models. This is simply due to the fact that, as we show, the measured observable of the system under consideration is covariant under phase space translations and as such is known to obey a tight joint measurement error relation.

Uncertainty in the context of multislit interferometry

A pair of uncertainty relations relevant for quantum states of multislit interferometry is derive... more A pair of uncertainty relations relevant for quantum states of multislit interferometry is derived, based on the mutually commuting “modular” position and momentum operators and their complementary counterparts, originally introduced by Aharonov and co-workers. We provide a precise argument as to why these relations are superior to the standard Heisenberg uncertainty relation at expressing the complementarity between spatial localization and the appearance of fringes. We further support the argument with explicit computations involving wave functions specifically tailored to the interference setup. Conceptually developing the idea of Aharonov and co-workers, we show how the modular momentum should reflect the given experimental setup, yielding a refined observable that accurately captures the fine structure of the interference pattern.

Measurement Uncertainty Relations

Measurement uncertainty relations are quantitative bounds on the errors in an approximate joint m... more Measurement uncertainty relations are quantitative bounds on the errors in an approximate joint measurement of two observables. They can be seen as a generalization of the error/disturbance tradeoff first discussed heuristically by Heisenberg. Here we prove such relations for the case of two canonically conjugate observables like position and momentum, and establish a close connection with the more familiar preparation uncertainty relations constraining the sharpness of the distributions of the two observables in the same state. Both sets of relations are generalized to means of order α rather than the usual quadratic means, and we show that the optimal constants are the same for preparation and for measurement uncertainty. The constants are determined numerically and compared with some bounds in the literature. In both cases the near-saturation of the inequalities entails that the state (resp. observable) is uniformly close to a minimizing one.

Quantum Measurement

by Paul Busch and Kari Ylinen

This is a book about the Hilbert space formulation of quantum mechanics and its measurement theor... more This is a book about the Hilbert space formulation of quantum mechanics and its measurement theory. It contains a synopsis of what became of the Mathematical Foundations of Quantum Mechanics since von Neumann’s classic treatise with this title. Fundamental non-classical features of quantum mechanics—indeterminacy and incompatibility of observables, unavoidable measurement disturbance, entanglement, nonlocality—are explicated and analysed using the tools of operational quantum theory.
The book is divided into four parts: 1. Mathematics provides a systematic exposition of the Hilbert space and operator theoretic tools and relevant measure and integration theory leading to the Naimark and Stinespring dilation theorems; 2. Elements develops the basic concepts of quantum mechanics and measurement theory with a focus on the notion of approximate joint measurability; 3. Realisations offers in-depth studies of the fundamental observables of quantum mechanics and some of their measurement implementations; and 4. Foundations discusses a selection of foundational topics (quantum-classical contrast, Bell nonlocality, measurement limitations, measurement problem, operational axioms) from a measurement theoretic perspective.
The book is addressed to physicists, mathematicians and philosophers of physics with an interest in the mathematical and conceptual foundations of quantum physics, specifically from the perspective of measurement theory.

Operational Quantum Physics

The Editorial Policy for Monographs The series Lecture Notes in Physics reports new developments ... more

Symposium on the Foundations of Modern Physics 1993 - Quantum Measurement, Irreversibility, and the Physics of Information

The quantum theory of measurement

The Editorial Policy for Monographs The series Lecture Notes in Physics reports new developments ... more

Classical versus Quantum Ontology-Book Review Essay: D. Home, Conceptual Foundations of Quantum Mechanics

Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics, 2002

Book Review: The mathematical language of quantum theory (by T Heinosaari and M Ziman)

Book review: Physical Theory and its Interpretation-Essays in Honor of Jeffrey Bub

This book is an hommage to Jeffrey Bub, with twelve contributions from colleagues and friends – p... more This book is an hommage to Jeffrey Bub, with twelve contributions from colleagues and friends – philosophers, physicists and mathematicians – working in the foundations and philosophy of modern physics. The range of topics covered reflects the scope of Bub's broad interests and life work in this field, and each article is a tribute to his commitment to conceptual and mathematical rigor. Some of the contributions are more didactic in nature, giving an accessible introduction to or survey of their subject.

Book Review: Foundations of quantum mechanics, an empiricist approach (by W de Muynck)

Essay Book Review: Classical versus quantum ontology

Quantum mechanics faces a strange dilemma. On the one hand it has long been claimed to be an irre... more Quantum mechanics faces a strange dilemma. On the one hand it has long been claimed to be an irreducibly statistical theory, allowing the calculation of measurement outcome statistics while being unable to predict the behaviour of individual microphysical processes. On the other hand, quantum mechanics has been increasingly used, with stunning success in the past few decades, to gain experimental control over individual objects on an atomic scale.

Book Review: Interpreting the quantum world (by J Bub)

Book Review: Black-body theory and the quantum discontinuity (by TS Kuhn)

Quantum rms error and Heisenberg’s error-disturbance relation

Wigner 111 – Colourful & Deep, Scientific Symposium, Budapest, Hungary, November 11-13, 2013

Reports on experiments recently performed in Vienna [Erhard et al, Nature Phys. 8, 185 (2012)] an... more Reports on experiments recently performed in Vienna [Erhard et al, Nature Phys. 8, 185 (2012)] and Toronto [Rozema et al, Phys. Rev. Lett. 109, 100404 (2012)] include claims of a violation of Heisenberg’s error-disturbance relation. In contrast, a Heisenberg-type tradeoff relation for joint measurements of position and momentum has been formulated and proven in [Phys. Rev. Lett. 111, 160405 (2013)]. Here I show how the apparent conflict is resolved by a careful consideration of the quantum generalisation of the notion of root-mean-square error. The claim of a violation of Heisenberg’s principle is untenable as it is based on a historically wrong attribution of an incorrect relation to Heisenberg, which is in fact trivially violated. We review a new general trade-off relation for the necessary errors in approximate joint measurements of incompatible qubit observables that is in the spirit of Heisenberg’s intuitions. The experiments mentioned may directly be used to test this new error inequality.

Quantum Measurements Constrained by Symmetries

by Paul Busch and Leon Loveridge

Symmetries and Groups in Contemporary Physics, Proceedings, The XXIXth International Colloquium on Group-Theoretical Methods in Physics

We revisit the theorem of Wigner, Araki and Yanase (WAY) describing limitations to repeatable qua... more We revisit the theorem of Wigner, Araki and Yanase (WAY) describing limitations to repeatable quantum measurements that arise from the presence of conservation laws. We will review a strengthening of this theorem by exhibiting and discussing a condition that has hitherto not been identified as a relevant factor. We will also show that an extension of the theorem to continuous variables such as position and momentum can be obtained if the degree of repeatability is suitably quantified.

Quantum mechanics as a framework for dealing with uncertainty

Quantum uncertainty is described here in two guises: indeterminacy with its concomitant indetermi... more Quantum uncertainty is described here in two guises: indeterminacy with its concomitant indeterminism of measurement outcomes, and fuzziness, or unsharpness. Both features were long seen as obstructions of experimental possibilities that were available in the realm of classical physics. The birth of quantum information science was due to the realization that such obstructions can be turned into powerful resources.

“No Information Without Disturbance”: Quantum Limitations of Measurement

Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle (eds. W.C. Myrvold, J. Christian), pp. 229-256, 2009

In this contribution I review rigorous formulations of a variety of limitations of measurability ... more In this contribution I review rigorous formulations of a variety of limitations of measurability in quantum mechanics. To this end I begin with a brief presentation of the conceptual tools of modern measurement theory. I will make precise the notion that quantum measurements necessarily alter the system under investigation and elucidate its connection with the complementarity and uncertainty principles.

Less (precision) is more (information): quantum information in fuzzy probability theory

Proceedings, International Conference in Quantum Theory: Reconsideration of Foundations - 2, Växjö, Sweden, June 1-6, 2003 (eds. H. Barnum, C.A. Fuchs, A. Khrennikov), pp. 113-128, 2004

A comparison of structural features of quantum and classical physical theories, such as the infor... more A comparison of structural features of quantum and classical physical theories, such as the information capacity of systems subject to these theories, requires a common formal framework for the presentation of corresponding concepts (such as states, observables, probability, entropy). Such a framework is provided by the notion of statistical model developed in the convexity approach to statistical physical theories.

EPR-Bell Tests with Unsharp Observables and Relativistic Quantum Measurement

Non-Locality and Modality (eds. T. Placek, J. Butterfield), 2002

Abstract. The Einstein-Podolsky-Rosen argument will be revisited and its validity checked for the... more Abstract. The Einstein-Podolsky-Rosen argument will be revisited and its validity checked for the case of realistic, imperfect measurements described in terms of positive operator valued measures. It is shown how Bell's inequalities can be satisfied if the degree of unsharpness in the observables involved is sufficiently large. The EPR contradiction between the assumptions of (usharp) reality, locality and the validity of quantum mechanics is resolved by maintaining realism and quantum mechanics and weakening locality.

Remarks on unsharp quantum observables, objectification, and modal interpretations

The Modal Interpretation of Quantum Mechanics, Feb 1998

In this contribution I wish to pose a question that emerged from discussions with Dennis Dieks re... more In this contribution I wish to pose a question that emerged from discussions with Dennis Dieks regarding my talk at the Utrecht Workshop. The question is whether a variant of a modal interpretation is conceivable that could accommodate property ascriptions associated with nonorthogonal resolutions of the unity and nonorthogonal families of relative states as they occur in imperfect or genuinely unsharp measurements.

The Status of Quantum Mechanics in the Light of the Objectification Problem

The Interpretation of Quantum Mechanics - Where do we stand?, 1994

Weakly Disturbing Phase Space Measurements in Quantum Mechanics

Quantum Communications and Measurement (1995) 155-163

The existence of weakly disturbing quantum measurements is demonstrated by means of a phase space... more

Quantum-to-Classical transition for Galilei-Covariant Observables

Symposium on the Foundations of Modern Physics 1993: Quantum Measurement, Irreversibility and the Physics of Information

The Quantum Theory of Unsharp Measurements

Classical and Quantum Systems - Foundations and Symmetries. Proceedings of the 2nd International Wigner Symposium, Goslar, Germany, 16-20 July 1991, 1993

Macroscopic Quantum Systems and the Objectification Problem

Symposium on the Foundations of Modern Physics 1990: Quantum Measurement Theory and its Philosophical Implications, 1991

A `Natural' Explanation of the Aharonov et al `Surprising Quantum Effect'

Proceedings of the 3rd International Symposium on Foundations of Quantum Mechanics in the Light of New Technology

Unsharp reality and the question of quantum systems

Symposium on the Foundations of Modern Physics 1987, 1987

Elements of Unsharp Reality in the EPR Experiment

Symposium on the Foundations of Modern Physics 1985: 50 Years of the Einstein-Podolsky-Rosen Gedanken Experiment

Quantum physical quantities are operationally unsharp observables. Therefore, a re-examination of... more Quantum physical quantities are operationally unsharp observables. Therefore, a re-examination of the EPR argument - which has been formulated in terms of discrete, sharp spin properties - is necessary and will be attempted in this note. It turns out that under certain conditions the well-known contradiction between the EPR assumptions and quantum mechanics disappears; otherwise it can be resolved along the lines proposed recently by Mittelstaedt and Stachow.

Can quantum theoretical reality be considered sharp?

Recent Developments in Quantum Logic (Bibliographisches Institut), 1985

Compendium of Quantum Physics, 2009

The term effect was introduced by G. Ludwig [1] as a technical term in his axiomatic reconstructi... more The term effect was introduced by G. Ludwig [1] as a technical term in his axiomatic reconstruction of quantum mechanics. Intuitively, this term refers to the “effect” of a physical object on a measuring device. Every experiment is understood to be carried out on a particular ensemble (“Gesamtheit”) of objects (► ensembles in quantum mechanics), all of which are subjected to the same preparation procedure; each object interacting with the measuring device triggers one of the different possible measurement outcomes. ...

Compendium of Quantum Physics, 2009

The term observable has become the standard name in quantum mechanics for what used to be called ... more The term observable has become the standard name in quantum mechanics for what used to be called physical quantity or measurable quantity in classical physics. This term derives from observable quantity (“beobachtbare Grösse”), which was used by Werner Heisenberg in his groundbreaking work on► matrix mechanics [1] to emphasize that the meaning of a physical quantity must be specified by means of an operational definition. Together with a state (► states in quantum mechanics), an observable determines the probabilities of the ...

Compendium of Quantum Physics, 2009

Lüders measurements offer an important characterization of the compatibility of► observables A, B... more Lüders measurements offer an important characterization of the compatibility of► observables A, B with discrete spectra: A and B commute if and only if the expectation value of B is not changed by a nonselective Lüders operation of A in any state T [1]. This result is the basis for the axiom of local commutativity in relativistic quantum field theory: the mutual commutativity of observables from local algebras associated with two spacelike separated regions of spacetime ensures, and is necessitated by, the impossibility of influencing the ...

Heisenberg Uncertainty Relation (Indeterminacy Relations)

by Paul Busch and Brigitte Falkenburg

The term Heisenberg uncertainty relation is a name for not one but three distinct trade-off relat... more The term Heisenberg uncertainty relation is a name for not one but three distinct trade-off relations which are all formulated in a more or less intuitive and vague way in Heisenberg's seminal paper of 1927 [1]. These relations are expressions and quantifications of three fundamental limitations of the operational possibilities of preparing and measuring quantum mechanical systems which are stated here informally with reference to position and momentum as a paradigmatic example of canonically conjugate pairs of quantities:

Which-Way or Welcher-Weg-Experiments

Heisenberg's Uncertainty Relation (Compendium entry)

The term Heisenberg uncertainty relation is a name for not one but three distinct trade-off relat... more The term Heisenberg uncertainty relation is a name for not one but three distinct trade-off relations which are all formulated in a more or less intuitive and vague way in Heisenberg's seminal paper of 1927 [1]. These relations are expressions and quantifications of three fundamental limitations of the operational possibilities of preparing and measuring quantum mechanical systems which are stated here informally with reference to position and momentum as a paradigmatic example of canonically conjugate pairs of quantities: ... (A) It is impossible to prepare states in ...

Heisenberg's Uncertainty Relation (Compendium entry)

by Paul Busch and Brigitte Falkenburg

The term Heisenberg uncertainty relation is a name for not one but three distinct trade-off relat... more The term Heisenberg uncertainty relation is a name for not one but three distinct trade-off relations which are all formulated in a more or less intuitive and vague way in Heisenberg's seminal paper of 1927 [1]. These relations are expressions and quantifications of three fundamental limitations of the operational possibilities of preparing and measuring quantum mechanical systems which are stated here informally with reference to position and momentum as a paradigmatic example of canonically conjugate pairs of quantities: ... (A) It is impossible to prepare states in ...

Zur Quantentheorie der Messung

Quanten, Chaos und Dämonen (eds. K. Mainzer, W. Schirmacher), 1993

Effect (Compendium entry)

Compendium of Quantum Physics (eds. D Greenberger, K Hentschel, F Weinert), pp. 179-180

The term effect was introduced by G. Ludwig [1] as a technical term in his axiomatic reconstructi... more The term effect was introduced by G. Ludwig [1] as a technical term in his axiomatic reconstruction of quantum mechanics. Intuitively, this term refers to the “effect” of a physical object on a measuring device. Every experiment is understood to be carried out on a particular ensemble (“Gesamtheit”) of objects (► ensembles in quantum mechanics), all of which are subjected to the same preparation procedure; each object interacting with the measuring device triggers one of the different possible measurement outcomes. Technically, preparation procedures and effects are used as primitive concepts to postulate the existence of probability assignments: each measurement outcome, identified by its effect, and each preparation procedure are assumed to determine a unique probability which represents the probability of the occurrence of that particular outcome. Thus, an effect can be taken to be the probability assignment, associated with a given outcome, to an ensemble of objects, or the preparation procedure applied to this ensemble [3]. In Hilbert space quantum mechanics, an effect is defined as an affine map from the set of states to the interval [0,1], or equivalently, as a linear operator E whose expectation value tr[ρE] for any state (► density operator) ρ lies within [0,1]. From this it follows that E is a positive bounded, hence selfadjoint, ► operator.

Measurement Theory (Compendium entry)

Compendium of Quantum Physics (eds. D Greenberger, K Hentschel, F Weinert), pp. 374-379

The term measurement theory refers to that part of a physical theory in which the empirical and o... more The term measurement theory refers to that part of a physical theory in which the empirical and operational content of the concepts of the theory is determined. Measurements are analyzed both as operational procedures defining the → observables of the theory and as physical processes which are themselves subject to the laws of physics.

Lüders Rule (Compendium Entry)

Compendium of Quantum Physics (eds. D Grenberger, K Hentschel, F Weinert), pp. 356-358

Observable (Compendium entry)

Compendium of Quantum Physics, Concepts, Experiments, History and Philosophy (eds. D. Greenberger, K. Hentschel, F. Weinert), pp. 425-428, 2009

The generalized representation of observables as positive operator measures was discovered by sev... more The generalized representation of observables as positive operator measures was discovered by several authors in the 1960s (eg, [6, 7, 10, 11, 12, 13]) and has by now become a standard element of quantum mechanics. It has greatly advanced the mathematical coherence and conceptual clarity of the theory. For instance, the problem of the (approximate) joint measurability of noncommuting observables such as position and momentum and the relevance of the → Heisenberg uncertainty relations to this question is now fully understood.

Which-Way or Welcher-Weg Experiment (Compendium entry)

Compendium of Quantum Physics, Concepts, Experiments, History and Philosophy (eds. D. Greenberger, K. Hentschel, F. Weinert), pp. 845-851, 2008

The issue of the → wave-particle duality of light and matter is commonly illustrated by the → dou... more The issue of the → wave-particle duality of light and matter is commonly illustrated by the → double-slit experiment, in which a quantum object of relatively well defined momentum (such as a photon, electron, neutron, atom, or molecule) is sent through a diaphragm containing two slits, after which it is detected at a capture screen. It is found that an interference pattern characteristic of wave behaviour emerges as a large number of similarly prepared quantum objects is detected on the screen.

Heisenberg's Uncertainty Relation (Indeterminacy Relations) (Compendium entry)

Compendium of Quantum Physics, Concepts, Experiments, History and Philosophy (eds. D. Greenberger, K. Hentschel, F. Weinert), pp. 281-283, 2009

The term Heisenberg uncertainty relation is a name for not one but three distinct trade-off relat... more The term Heisenberg uncertainty relation is a name for not one but three distinct trade-off relations which are all formulated in a more or less intuitive and vague way in Heisenberg's seminal paper of 1927 [1]. These relations are expressions and quantifications of three fundamental limitations of the operational possibilities of preparing and measuring quantum mechanical systems which are stated here informally with reference to position and momentum as a paradigmatic example of canonically conjugate pairs of quantities: It is impossible to prepare states in which position and momentum are simultaneously arbitrarily well localized. In every state, the probability distributions of these ► observables have widths that obey an uncertainty relation. It is impossible to make joint measurements of position and momentum. But it is possible to make approximate joint measurements of these observables, with inaccuracies that obey an uncertainty relation. It is impossible to measure position without disturbing momentum, and vice versa. The inaccuracy of the position measurement and the disturbance of the momentum distribution obey an uncertainty relation.

The time-energy uncertainty relation

Time in Quantum Mechanics, 2002

The time-energy uncertainty relation ΔT ΔE≥ 1/2ħ (3.1) has been a controversial issue since the a... more The time-energy uncertainty relation ΔT ΔE≥ 1/2ħ (3.1) has been a controversial issue since the advent of quantum theory, with respect to appropriate formalisation, validity and possible meanings. Already the first formulations due to Bohr, Heisenberg, Pauli and Schrödinger are very different, as are the interpretations of the terms used. A comprehensive account of the development of this subject up to the 1980s is provided by a combination of the reviews of Jammer [1], Bauer and Mello [2], and Busch [3, 4].

Is the quantum state (an) observable?

Potentiality, Entanglement and Passion-at-a-Distance: Quantum Mechanical Studies for Abner Shimony (eds. R.S. Cohen, M.A. Horne, J. Stachel), 1997

We explore the sense in which the state of a physical system may or may not be regarded (an) obse... more We explore the sense in which the state of a physical system may or may not be regarded (an) observable in quantum mechanics. Simple and general arguments from various lines of approach are reviewed which demonstrate the following no-go claims: (1) the structure of quantum mechanics precludes the determination of the state of a single system by means of measurements performed on that system only; (2) there is no way of using entangled two-particle states to transmit superluminal signals.

Quantum Groups, Quantum Foundations and Quantum Information: a Festschrift for Tony Sudbery

by Paul Busch and Stefan Weigert

Journal of Physics: Conference Series, 2010

... There is no doubt that he will utilise this newly-gained freedom to continue to inspire and c... more

Philosophical Problems of Modern Physics — Peter Mittelstaedt 1929–2014

The University of Cologne and the international community of researchers in foundations of physic... more The University of Cologne and the international community of researchers in foundations of physics mourn the loss of Peter Mittelstaedt, who passed away on November 21, 2014, after a short period of illness. Peter Mittelstaedt held a chair in theoretical physics at the University of Cologne from 1965 until his retirement in 1995. In addition to his engagement as a scientist and academic teacher he was elected first as Dean of the Faculty of Science (1968–1969) and then Rector of the University of Cologne (1970–1971). Subsequently he served as Prorector (1971–1973) and Prorector for Research (1991–1994). He was an elected member of l’Académie Internationale de Philosophie des Sciences and founding member and president of the International Quantum Structures Association (1994–1996).Peter Mittelstaedt was born in Leipzig on November 24, 1929. In his childhood home he may already have witnessed the spirit of philosophical discourse about the world-picture of modern physics. For Werner Heis ...

Editorial: Quantum Groups, Quantum Foundations and Quantum Information: a Festschrift for Tony Sudbery,

This Festschrift, also made possible by the generosity of the IOP and the LMS, reproduces the maj... more

Editorial: Between Physics and Philosophy—Festschrift for Peter Mittelstaedt on His 80th Birthday

Editorial: Pekka Johannes Lahti – 60th Birthday

by Paul Busch and Dennis Dieks

On 26 December 2008, Pekka Johannes Lahti celebrated his 60th birthday. To mark this occasion, tw... more

Obituary: Sławomir Bugajski 1941-2003

Peter Mittelstaedt: Philosopher-physicist

Error and unsharpness in approximate joint measurements of position and momentum

In recent years, novel quantifications of measurement error in quantum mechanics have for the fir... more In recent years, novel quantifications of measurement error in quantum mechanics have for the first time enabled precise formulations of Heisenberg’s famous but often challenged measurement uncertainty relation. This relation takes the form of a trade-off for the necessary errors in joint approximate measurements of position and momentum and other incompatible pairs of observables. Much work remains to be done to obtain a better understanding of the new error measures and their suitability. To this end we review here some of these error measures and associated measurement uncertainty relations. We investigate the properties and suitability of these measures, give examples to show how they can be computed in specific cases, and compare their relative strengths as criteria for “good” approximations.

Comment on "Experimental Test of Error-Disturbance Uncertainty Relations by Weak Measurement"

In this comment on the paper by F. Kaneda, S.-Y. Baek, M. Ozawa and K. Edamatsu [Phys. Rev. Lett.... more In this comment on the paper by F. Kaneda, S.-Y. Baek, M. Ozawa and K. Edamatsu [Phys. Rev. Lett. 112, 020402, 2014, arXiv:1308.5868], we point out that the claim of having refuted Heisenberg's error-disturbance relation is unfounded since it is based on the choice of unsuitable and operationally problematical quantifications of measurement error and disturbance. As we have shown elsewhere [PRL 111, 160405, 2013], for appropriate choices of operational measures of error and disturbance, Heisenberg's heuristic relation can be turned into a precise inequality which is a rigorous consequence of quantum mechanics.

Comment on "Noise and Disturbance in Quantum Measurements: An Information-Theoretic Approach"

In this comment on the work of F. Buscemi, M.J.W. Hall, M. Ozawa and M.M. Wilde [PRL 112, 050401,... more In this comment on the work of F. Buscemi, M.J.W. Hall, M. Ozawa and M.M. Wilde [PRL 112, 050401, 2014, arXiv:1310.6603], we point out a misrepresentation of measures of error and disturbance introduced in our recent work [PRL 111, 160405, 2013, arXiv:1306.1565] as being "purely formal, with no operational counterparts". We also exhibit an tension in the authors' message, in that their main result is an error-disturbance relation for state-independent measures, but its importance is declared to be limited to discrete variables. In contrast, we point out the separate roles played by such relations for either state-dependent or state-independent measures of error and disturbance.

Measurement Uncertainty: Reply to Critics

In a recent publication [PRL 111, 160405 (2013)] we proved a version of Heisenberg's error-distur... more In a recent publication [PRL 111, 160405 (2013)] we proved a version of Heisenberg's error-disturbance tradeoff. This result was in apparent contradiction to claims by Ozawa of having refuted these ideas of Heisenberg. In a direct reaction [arXiv:1308.3540] Ozawa has called our work groundless, and has claimed to have found both a counterexample and an error in our proof. Here we answer to these allegations. We also comment on the submission [arXiv:1307.3604] by Rozema et al, in which our approach is unfavourably compared to that of Ozawa.

Quantum, Matter, Spacetime: Peter Mittelstaedt’s Contributions to Physics and Its Foundations

by Paul Busch and EW Stachow

In a period of over 50 years, Peter Mittelstaedt has made substantial and lasting contributions t... more In a period of over 50 years, Peter Mittelstaedt has made substantial and lasting contributions to several fields in theoretical physics as well as the foundations and philosophy of physics. Here we present an overview of his achievements in physics and its foundations which may serve as a guide to the bibliography (printed in this Festschrift) of his publications. An appraisal of Peter Mittelstaedt’s work in the philosophy of physics is given in a separate contribution by B. Falkenburg.

Measurement of quantum mechanical Operators - Translation of "Die Messung quantenmechanischer Operatoren" by E Wigner

This is a'facsimile-style'translation of Wigner's seminal paper on measurement limitations in the... more This is a'facsimile-style'translation of Wigner's seminal paper on measurement limitations in the presence of additive conservation laws. A critical survey of the history of subsequent extensions and variations of what is now known as the Wigner-Araki-Yanase (WAY) Theorem is provided in a paper published concurrently.