Paul Busch

• Dr. rer. nat.
• Professor at University of York

We propose that observables in quantum theory are properly understood as representatives of symmetry-invariant quantities relating one system to another, the latter to be called a reference system. We provide a rigorous mathematical language to introduce and study quantum reference systems, showing that the orthodox "absolute" quantities are good r...

The uncertainty relations, pioneered by Werner Heisenberg nearly 90 years ago, set a fundamental limitation on the joint measurability of complementary observables. This limitation has long been a subject of debate, which has been reignited recently due to new proposed forms of measurement uncertainty relations. The present work is associated with...

The uncertainty relations, pioneered by Werner Heisenberg nearly 90 years ago, set a fundamental limitation on the joint measurability of complementary observables. This limitation has long been a subject of debate, which has been reignited recently due to new proposed forms of measurement uncertainty relations. The present work is associated with...

Under the principle that quantum mechanical observables are invariant under relevant symmetry transformations, we explore how the usual, non-invariant quantities may capture measurement statistics. Using a relativisation mapping, viewed as the incorporation of a quantum reference frame, we show that the usual quantum description approximates the re...

We propose that observables in quantum theory are properly understood as representatives of symmetry-invariant quantities relating one system to another, the latter to be called a reference system. We provide a rigorous mathematical language to introduce and study quantum reference systems, showing that the orthodox "absolute" quantities are good r...

One purpose of Chapter 4 is to present the spectral theory for arbitrary bounded selfadjoint operators. We begin with auxiliary techniques from the (scalar) theory of measure and integration. Then the notion of positive operator measure is introduced and studied. This is a key concept for the rest of the book, but in this chapter its special case,...

It is one of the key features of quantum mechanics that not all of the observables of this theory can be measured jointly; in other words, many pairs of larger families of observables are incompatible. In order to study this basic feature of quantum mechanics, this chapter will make precise minimal requirements of what constitutes a joint measureme...

In Chap. 11 we have seen that noncommutativity prohibits the joint measurability of two sharp observables but not necessarily that of two unsharp observables. We have also observed that adding noise to two incompatible observables can turn them into jointly measurable observables. This leads to the idea of realising an approximate joint measurement...

The term Bell inequalities is a collective name for a class of inequalities involving probabilities that are used to demonstrate the nonlocal nature of correlations between quantum systems in certain entangled states. A demonstration of violations of Bell inequalities in quantum mechanics typically involves the use of incompatible observables. Ther...

In Chap. 22 make precise the so-called quantum measurement problem–the question whether an account can be provided, in terms of quantum mechanics, of the occurrence of a definite value of the pointer observable at the end of a measurement, and what inference, if any, may be possible on the value of the measured object observable E after the measure...

In this chapter we investigate the joint measurability of the observables of a qubit system, that is, a quantum system whose relevant degrees of freedom are represented by a two-dimensional complex Hilbert space. We will give a full characterisation of the pairs of simple qubit observables that are jointly measurable. We also develop the theory of...

In this chapter we discuss some common examples of semispectral measures and their natural dilations.

Following the introduction and study of a range of concrete operator measures representing quantum mechanical observables in earlier chapters (notably Chaps. 8, 14–17), we now apply the tools of measurement theory developed in Chap. 10 to illustrate the implementation of more or less realistic measurement schemes for typical observables.

In classical physics the phase of an electromagnetic field is well defined both theoretically and by interference experiments. Diffraction of light, holography, and many other phase dependent phenomena are well understood. As we will see in Sect. 19. 3, it is also easy to describe classically eight-port homodyne detection and other direct measureme...

The probabilistic structure of quantum mechanics is a reflection of the fact that observations on quantum physical objects typically yield uncertain outcomes. Formally this uncertainty is encoded in the probability distribution of an observable in the state of the physical system. It is a fundamental feature of quantum mechanics that there are pair...

In this final chapter we address the question of justifying the Hilbert space formulation of quantum mechanics. We show that this theory can essentially be derived from physically plausible assumptions using the general frame of statistical dualities sketched briefly already in the Introduction.

Naimark’s work in the 1940s on the representation of positive operator measures in terms of projection valued measures acting in a larger Hilbert space may be seen as the starting point for the discovery of a vast variety of dilation theorems, an early highlight being Stinespring’s dilation theorem from the mid 1950s. This chapter contains a unifie...

This chapter is devoted to a study of informational completeness of a measurement outcome statistics and the related problem of state reconstruction. Special attention is given to the continuous variable case. After introducing the key concepts and the basic results, including a short discussion of the qubit case, the Pauli problem and the two basi...

The realisability of measurements of an observable depends on the availability of appropriate measurement couplings between object system and apparatus or probe system. It is a well-known fact that the fundamental interactions between physical systems are subject to symmetries. Limitations of the measurability of quantum observables in the presence...

In this chapter we review the basic elements and structures of Hilbert space quantum mechanics. We build on the idea of a statistical duality arising from the analysis of an experiment as a preparation-measurement-registration scheme, as sketched in Sect. 1. 2. The description of a physical system \(\mathcal {S}\) is thus based on the notions of st...

Space and time have a double role in quantum mechanics. On the one hand, they form the arena for physical events; physical systems occur in spacetime. On the other hand, the symmetries of the underlying spacetime specify the description of the system and dictate its basic properties. In the relativistic case spacetime symmetries are expressed with...

As the present work is about Hilbert space quantum mechanics, it is mandatory that the reader has sufficient grounding in Hilbert space theory. This short chapter is designed to indicate what sort of basic equipment one needs in the ensuing more sophisticated chapters. At the same time it can be used as an introduction to elementary Hilbert space t...

In the traditional Hilbert space quantum mechanics, (bounded or unbounded) selfadjoint operators are taken as physical observables. In the present work a more comprehensive approach is used: observables are represented as normalised positive operator measures. We develop the spectral representation theory of generally unbounded selfadjoint operator...

In traditional Hilbert space quantum mechanics, the spectral representation of a (generally unbounded) selfadjoint operator plays a key role. In this chapter the main focus is on the class of compact operators and its two subsets, the trace class and the Hilbert-Schmidt class. This theory is not just an elementary introduction to the general case b...

The purpose of a measurement is the determination of properties of the physical system under investigation. In this sense the general conception of measurement is that of an unambiguous comparison: the object system prepared in a state is brought into a suitable contact—a measurement coupling—with another, independently prepared system, the measuri...

Position and momentum are the most basic kinematic and dynamic variables of a quantum object. They reflect the structure of space and time where the object resides. In the nonrelativistic quantum theory, which is followed here, position and momentum are intertwined through the projective nature of the Weyl representation of space translations and v...

The level of mathematical sophistication assumed of the reader cannot be perfectly uniform throughout the work, although the present chapter has been designed to alleviate this situation. In the sequel we at times need to refer to outside literature for some rather deep tools. This chapter contains some such results, though many statements are give...

Under the principle that quantum mechanical observables are invariant under relevant symmetry transformations, we explore how the usual, non-invariant quantities may capture measurement statistics. Using a relativisation mapping, viewed as the incorporation of a quantum reference frame, we show that the usual quantum description approximates the re...

We study uncertainty relations for pairs of conjugate variables like number and angle, of which one takes integer values and the other takes values on the unit circle. The translation symmetry of the problem in either variable implies that measurement uncertainty and preparation uncertainty coincide quantitatively, and the bounds depend only on the...

We study uncertainty relations for pairs of conjugate variables like number and angle, of which one takes integer values and the other takes values on the unit circle. The translation symmetry of the problem in either variable implies that measurement uncertainty and preparation uncertainty coincide quantitatively, and the bounds depend only on the...

The fact that any physical quantity is defined and measured relative to a reference frame is generally not explicitly reflected in the theoretical description of physical experiments. Instead, the relevant observables are typically represented as "absolute" quantities. Here we investigate the conditions under which, in quantum theory, an account in...

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,...

We consider the task of characterizing optimal protocols for approximating incompatible observables via jointly measurable observables. This amounts to jointly minimizing the approximation errors (suitably quantified), subject to the compatibility constraint. As a case study we consider the approximation of two-valued qubit observables and scrutini...

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...

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 th...

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. On...

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...

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 et al.. We provide a precise argument as to why these relations are superior to the standard Heisenberg...

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. These relations take the form of a trade-off for the necessary errors in joint approximate measurements of position and momentum and other...

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. So far, only sep...

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 a...

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 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 an...

Measurement uncertainty relations are quantitative bounds on the errors in an approximate joint measurement of two complementary observables like position and momentum. They can be seen as a generalization of the error/disturbance tradeoff first discussed heuristically by Heisenberg. Here we prove such relations and establish a close connection wit...

We show that connections between a degree of incompatibility of pairs of observables and the strength of violations of Bell's inequality found in recent investigations can be extended to a general class of probabilistic physical models. It turns out that the property of universal uniform steering is sufficient for the saturation of a generalised Ts...

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, we have presented and proven a Heisenberg-type relation for joint measurements of position and momentum...

We revisit the theorem of Wigner, Araki and Yanase (WAY) describ-ing limitations to repeatable quantum measurements that arise from the presence of conservation laws. We will review a strengthening of this the-orem by exhibiting and discussing a condition that has hitherto not been identified as a relevant factor. We will also show that an extensio...

We introduce a new way of quantifying the degrees of incompatibility of two ob- servables in a probabilistic physical theory and, based on this, a global measure of the degree of incompatibility inherent in such theories, across all observable pairs. This opens up a novel and flexible way of comparing probabilistic theories with respect to the nonc...

We exhibit an operational connection between mutually unbiased bases and symmetric infomationally complete positive operator-valued measures. Assuming that the latter exists, we show that there is a strong link between these two structures in all prime power dimensions. We also demonstrate that a similar link cannot exists in dimension 6.

We exhibit an operational connection between mutually unbiased bases and symmetric informationally complete positive operator-valued measures. Assuming that the latter exists, we show that there is a strong link between these two structures in all prime power dimensions. We also demonstrate that a similar link cannot exist in dimension 6.

In a recent, modified double-pinhole diffraction experiment the existence of an interference pattern was established indirectly along with a near-perfect imaging of the double pinhole. Our theoretical analysis shows that the experiment constitutes a preparation of a quantum state that is, to a good approximation, a joint eigenstate of commuting fun...

We consider the properties of the Shannon entropy for two probability distributions which stand in the relationship of majorization. Then we give a generalization of a theorem due to Uhlmann, extending it to infinite dimensional Hilbert spaces. Finally we show that for any quantum channel $\Phi$, one has $S(\Phi(\rho))=S(\rho)$ for all quantum stat...

While the slogan "no measurement without disturbance" has established itself under the name Heisenberg effect in the consciousness of the scientifically interested public, a precise statement of this fundamental feature of the quantum world has remained elusive, and serious attempts at rigorous formulations of it as a consequence of quantum theory...

We present a hitherto unknown fundamental limitation to a basic measurement: that of the position of a quantum object when the total momentum of the object and apparatus is conserved. This result extends the famous Wigner-Araki-Yanase theorem, and shows that accurate position measurements are only practically feasible if there is a large momentum u...

Working within the framework of stochastic quantum mechanics, we prove that a mild reality constraint on the radial part of the resolution generator renders the (probability) current, for a single particle with only positive (or only negative) energies and having arbitrary integer spin, conserved. The result is proven both nonrelativistically and r...

The Wigner-Araki-Yanase (WAY) theorem states a remarkable limitation to quantum mechanical measurements in the presence of additive conserved quantities. Discovered by Wigner in 1952, this limitation is known to induce constraints on the control of individual quantum systems in the context of information processing. It is therefore important to und...

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.

The notion of coexistence of quantum observables was introduced to describe the possibility of measuring two or more observables together. Here we survey the various different formalisations of this notion and their connections. We review examples illustrating the necessary degrees of unsharpness for two noncommuting observables to be jointly measu...

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...

This bibliography has been compiled by P. Busch using the list of publications on a webpage dedicated to Peter Mittelstaedt and maintained by H. Fink at http://www.peter-mittelstaedt.de. A list of Peter Mittelstaedt’s doctoral students is included, with dissertation titles given in English translation.

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...

The positive operator (valued) measures (POMs) allow one to generalize the notion of observable beyond the traditional one based on projection valued measures (PVMs). Here, we argue that this generalized conception of observable enables a consistent notion of unsharp reality and with it an adequate concept of joint properties. A sharp or unsharp pr...

We present a hitherto unknown fundamental limitation to a basic measurement: that of the position of a quantum object when the total momentum of the object and apparatus is conserved. This result extends the famous Wigner-Araki-Yanase (WAY) theorem, and shows that accurate position measurements are only practically feasible if there is a large mome...

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 realiz...

Two quantum events, represented by positive operators (effects), are coexistent if they can occur as possible outcomes in a single measurement scheme. Equivalently, the corresponding effects are coexistent if and only if they are contained in the ranges of a single (joint) observable. Here we give several equivalent characterizations of coexistent...

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.

Two quantum events, represented by positive operators (effects), are coexistent if they can occur as possible outcomes in a single measurement scheme. Equivalently, the corresponding effects are coexistent if and only if they are contained in the ranges of a single (joint) observable. Here we give several equivalent characterizations of coexistent...

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 mea...

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 theor...

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 tha...

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), al...

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...

The question of quantifying the sharpness (or unsharpness) of a quantum mechanical effect is investigated. Apart from sharpness, another property, bias, is found to be relevant for the joint measurability or coexistence of two effects. Measures of bias will be defined and examples given. Dedication The impossibility of measuring jointly certain pai...

Pekka’s research interest has primarily been in conceptual and mathematical foundations of quantum mechanics. He has written well over 100 publications and a number of books on this subject. In a career spanning until now three decades, he has been influential in many ways.

The notion coexistence of quantum observables was introduced to describe the possibility of measuring two or more observables together. Here we survey the various different formalisations of this notion and their connections. We review examples illustrating the necessary degrees of unsharpness for two noncommuting observables to be jointly measurab...

Two quantum events, represented by positive operators (effects), are coexistent if they can occur as possible outcomes in a single measurement scheme. Equivalently, the corresponding effects are coexistent if and only if they are contained in the ranges of a single (joint) observable. Here we give several equivalent characterizations of coexistent...

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 connect...

This is a book review. To be published in Studies in History and Philosophy of Modern Physics.

This is an entry to the Compendium of Quantum Physics, edited by F Weinert, K Hentschel and D Greenberger, to be published by Springer-Verlag.

This is an entry to the Compendium of Quantum Physics, edited by F Weinert, K Hentschel and D Greenberg, to be published by Springer-Verlag.

This is an entry to the Compendium of Quantum Physics, edited by F Weinert, K Hentschel and D Greenberger, to be published by Springer-Verlag.

This is an entry to the Compendium of Quantum Physics, edited by F Weinert, K Hentschel and D Greenberger, to be published by Springer-Verlag.

This is an entry to the Compendium of Quantum Physics, edited by F Weinert, K Hentschel and D Greenberger, to be published by Springer-Verlag.

We describe an operational scheme for determining both the position and momentum distributions in a large class of quantum states, together with an experimental implementation.

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 compre...

Heisenberg's uncertainty principle is usually taken to express a limitation of operational possibilities imposed by quantum mechanics. Here we demonstrate that the full content of this principle also includes its positive role as a condition ensuring that mutually exclusive experimental options can be reconciled if an appropriate trade-off is accep...

On the basis of a suggestive definition of a classical extension of quantum mechanics in terms of statistical models, we prove that every such classical extension is essentially given by the so-called Misra-Bugajski reduction map. We consider how this map enables one to understand quantum mechanics as a reduced classical statistical theory on the p...

Joint measurements of qubit observables have recently been studied in conjunction with quantum information processing tasks such as cloning. Considerations of such joint measurements have until now been restricted to a certain class of observables that can be characterized by a form of covariance. Here we investigate conditions for the joint measur...

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 connect...

We formulate and prove a new, universally valid uncertainty relation for the necessary errors bar widths in any approximate joint measurement of position and momentum.

Heisenberg's uncertainty principle is usually taken to express a limitation of operational possibilities imposed by quantum mechanics. Here we demonstrate that the full content of this principle also includes its positive role as a condition ensuring that mutually exclusive experimental options can be reconciled if an appropriate trade-off is accep...

A coherent account of the connections and contrasts between the principles of complementarity and uncertainty is developed starting from a survey of the various formalizations of these principles. The conceptual analysis is illustrated by means of a set of experimental schemes based on Mach–Zehnder interferometry. In particular, path detection via...

Heisenberg's uncertainty principle is usually taken to express a limitation of operational possibilities imposed by quantum mechanics. Here we demonstrate that the full content of this principle also includes its positive role as a condition ensuring that mutually exclusive experimental options can be reconciled if an appropriate trade-off is accep...

A coherent account of the connections and contrasts between the principles of com- plementarity and uncertainty is developed starting from a survey of the various formalizations of these principles. The conceptual analysis is illustrated by means of a set of experimental schemes based on Mach-Zehnder interferometry. In particu- lar, path detection...

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...

The operational meaning of some measures of noise and disturbance in measurements is analyzed and their limitations are pointed out. The cases of minimal noise and least disturbance are characterized.