Special Issue "Mathematics of Quantum Uncertainty"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 March 2016).

Special Issue Editors

Prof. Dr. Paul Busch
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Guest Editor
Department of Mathematics and York Centre for Quantum Technologies, University of York, York, YO10 5DD, UK
Interests: conceptual and mathematical foundations of quantum theory; quantum measurement and information; quantum theory and relativity
Assoc. Prof. Takayuki Miyadera
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Guest Editor
Department of Nuclear Engineering, Kyoto University, 6158540 Kyoto, Japan
Interests: foundations of quantum theory; quantum information; mathematical physics
Special Issues and Collections in MDPI journals
Dr. Teiko Heinosaari
Website
Guest Editor
Turku Centre for Quantum Physics, Department of Physics and Astronomy, University of Turku, Finland
Interests: foundations of quantum theory; quantum information; mathematical physics

Special Issue Information

Dear Colleagues,

The uncertainty principle is a cornerstone of one of the fundamental theories of modern physics, quantum mechanics. Somewhat surprisingly, an important aspect of the principle—Heisenberg’s famous error-disturbance relation—has, until recent years, led a “life in the shadows”. It has taken until the late 1990s before first attempts were made at finding precise formulations of trade-off relations for the errors in joint measurements of incompatible quantities, and at understanding the role of measurement disturbance. The underlying issues that prevented an earlier development are at once of a mathematical and conceptual nature. Firstly, it has taken several decades before mathematical tools for the representation of unsharp and approximate measurements were developed; these tools include the theory of operator-valued measures and their (measurement) dilations. Secondly, the concept of joint measurability had to be extended to encompass sets of noncommuting observables. Finally, a notion of approximation of one observable by another had to be developed along with appropriate measures of error. Meanwhile there are a variety of approaches for such formalizations, yielding new insights such as a deep connection between measurement uncertainty and preparation uncertainty, but also resulting in an ongoing controversy over what constitutes a good quantum generalization of the time-honored Gaussian root-mean-square deviation. The framework of operational quantum theory, now routinely used in quantum information theory, provides the basis for the ongoing search of novel forms of tight preparation and measurement uncertainty relations that are open to experimental testing and may be expected to inform ultimate quantum bounds for high-precision measurement protocols and support the inception of new quantum information tasks, such as cryptographic protocols that utilize entropic uncertainty relations adapted to the presence of quantum memories.

Needless to say, the investigation of quantum uncertainty is not restricted to uncertainty relations, but also addresses related structural aspects of quantum mechanics including, inter alia, the study of the notion of incompatibility of observables and of theories, limitations of measurements due to symmetry, connections between incompatibility and nonlocality, and the relationship between joint measurability and approximate cloning.

The purpose of this Special Issue is to establish a collection of articles that reflect the latest mathematical and conceptual developments in the field of quantum uncertainty and explore the scope for applications in areas such as quantum cryptography, quantum control, and quantum metrology.

Prof. Dr. Paul Busch
Dr. Takayuki Miyadera
Dr. Teiko Heinosaari
Guest Editors

Manuscript Submission Information

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Keywords

  • uncertainty principle
  • uncertainty relations
  • quantum measurement
  • incompatibility
  • error-disturbance relation
  • Heisenberg effect
  • measurement disturbance
  • joint measurability
  • unsharp observable

Published Papers (12 papers)

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Research

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Open AccessArticle
Quantum Measurements, Stochastic Networks, the Uncertainty Principle, and the Not So Strange “Weak Values”
Mathematics 2016, 4(3), 56; https://doi.org/10.3390/math4030056 - 15 Sep 2016
Cited by 7
Abstract
Suppose we make a series of measurements on a chosen quantum system. The outcomes of the measurements form a sequence of random events, which occur in a particular order. The system, together with a meter or meters, can be seen as following the [...] Read more.
Suppose we make a series of measurements on a chosen quantum system. The outcomes of the measurements form a sequence of random events, which occur in a particular order. The system, together with a meter or meters, can be seen as following the paths of a stochastic network connecting all possible outcomes. The paths are shaped from the virtual paths of the system, and the corresponding probabilities are determined by the measuring devices employed. If the measurements are highly accurate, the virtual paths become “real”, and the mean values of a quantity (a functional) are directly related to the frequencies with which the paths are traveled. If the measurements are highly inaccurate, the mean (weak) values are expressed in terms of the relative probabilities’ amplitudes. For pre- and post-selected systems they are bound to take arbitrary values, depending on the chosen transition. This is a direct consequence of the uncertainty principle, which forbids one from distinguishing between interfering alternatives, while leaving the interference between them intact. Full article
(This article belongs to the Special Issue Mathematics of Quantum Uncertainty)
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Open AccessArticle
Quantum Incompatibility in Collective Measurements
Mathematics 2016, 4(3), 54; https://doi.org/10.3390/math4030054 - 10 Sep 2016
Cited by 2
Abstract
We study the compatibility (or joint measurability) of quantum observables in a setting where the experimenter has access to multiple copies of a given quantum system, rather than performing the experiments on each individual copy separately. We introduce the index of incompatibility as [...] Read more.
We study the compatibility (or joint measurability) of quantum observables in a setting where the experimenter has access to multiple copies of a given quantum system, rather than performing the experiments on each individual copy separately. We introduce the index of incompatibility as a quantifier of incompatibility in this multi-copy setting, as well as the notion of the compatibility stack representing various compatibility relations present in a given set of observables. We then prove a general structure theorem for multi-copy joint observables and use it to prove that all abstract compatibility stacks with three vertices have realizations in terms of quantum observables. Full article
(This article belongs to the Special Issue Mathematics of Quantum Uncertainty)
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Open AccessArticle
Role of Measurement Incompatibility and Uncertainty in Determining Nonlocality
Mathematics 2016, 4(3), 52; https://doi.org/10.3390/math4030052 - 15 Aug 2016
Cited by 6
Abstract
It has been recently shown that measurement incompatibility and fine grained uncertainty—a particular form of preparation uncertainty relation—are deeply related to the nonlocal feature of quantum mechanics. In particular, the degree of measurement incompatibility in a no-signaling theory determines the bound on the [...] Read more.
It has been recently shown that measurement incompatibility and fine grained uncertainty—a particular form of preparation uncertainty relation—are deeply related to the nonlocal feature of quantum mechanics. In particular, the degree of measurement incompatibility in a no-signaling theory determines the bound on the violation of Bell-CHSH inequality, and a similar role is also played by (fine-grained) uncertainty along with steering, a subtle non-local phenomenon. We review these connections, along with comments on the difference in the roles played by measurement incompatibility and uncertainty. We also discuss why the toy model of Spekkens (Phys. Rev. A 75, 032110 (2007)) shows no nonlocal feature even though steering is present in this theory. Full article
(This article belongs to the Special Issue Mathematics of Quantum Uncertainty)
Open AccessFeature PaperArticle
Preparational Uncertainty Relations for N Continuous Variables
Mathematics 2016, 4(3), 49; https://doi.org/10.3390/math4030049 - 19 Jul 2016
Cited by 6
Abstract
A smooth function of the second moments of N continuous variables gives rise to an uncertainty relation if it is bounded from below. We present a method to systematically derive such bounds by generalizing an approach applied previously to a single continuous variable. [...] Read more.
A smooth function of the second moments of N continuous variables gives rise to an uncertainty relation if it is bounded from below. We present a method to systematically derive such bounds by generalizing an approach applied previously to a single continuous variable. New uncertainty relations are obtained for multi-partite systems that allow one to distinguish entangled from separable states. We also investigate the geometry of the “uncertainty region” in the N ( 2 N + 1 ) -dimensional space of moments. It is shown to be a convex set, and the points on its boundary are found to be in one-to-one correspondence with pure Gaussian states of minimal uncertainty. For a single degree of freedom, the boundary can be visualized as one sheet of a “Lorentz-invariant” hyperboloid in the three-dimensional space of second moments. Full article
(This article belongs to the Special Issue Mathematics of Quantum Uncertainty)
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Open AccessArticle
Sharing of Nonlocality of a Single Member of an Entangled Pair of Qubits Is Not Possible by More than Two Unbiased Observers on the Other Wing
Mathematics 2016, 4(3), 48; https://doi.org/10.3390/math4030048 - 16 Jul 2016
Cited by 13
Abstract
We address the recently posed question as to whether the nonlocality of a single member of an entangled pair of spin 1 / 2 particles can be shared among multiple observers on the other wing who act sequentially and independently of each other. [...] Read more.
We address the recently posed question as to whether the nonlocality of a single member of an entangled pair of spin 1 / 2 particles can be shared among multiple observers on the other wing who act sequentially and independently of each other. We first show that the optimality condition for the trade-off between information gain and disturbance in the context of weak or non-ideal measurements emerges naturally when one employs a one-parameter class of positive operator valued measures (POVMs). Using this formalism we then prove analytically that it is impossible to obtain violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality by more than two Bobs in one of the two wings using unbiased input settings with an Alice in the other wing. Full article
(This article belongs to the Special Issue Mathematics of Quantum Uncertainty)
Open AccessArticle
Uncertainty Relations for Quantum Coherence
Mathematics 2016, 4(3), 47; https://doi.org/10.3390/math4030047 - 16 Jul 2016
Cited by 19
Abstract
Coherence of a quantum state intrinsically depends on the choice of the reference basis. A natural question to ask is the following: if we use two or more incompatible reference bases, can there be some trade-off relation between the coherence measures in different [...] Read more.
Coherence of a quantum state intrinsically depends on the choice of the reference basis. A natural question to ask is the following: if we use two or more incompatible reference bases, can there be some trade-off relation between the coherence measures in different reference bases? We show that the quantum coherence of a state as quantified by the relative entropy of coherence in two or more noncommuting reference bases respects uncertainty like relations for a given state of single and bipartite quantum systems. In the case of bipartite systems, we find that the presence of entanglement may tighten the above relation. Further, we find an upper bound on the sum of the relative entropies of coherence of bipartite quantum states in two noncommuting reference bases. Moreover, we provide an upper bound on the absolute value of the difference of the relative entropies of coherence calculated with respect to two incompatible bases. Full article
(This article belongs to the Special Issue Mathematics of Quantum Uncertainty)
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Open AccessArticle
Exact Discrete Analogs of Canonical Commutation and Uncertainty Relations
Mathematics 2016, 4(3), 44; https://doi.org/10.3390/math4030044 - 28 Jun 2016
Cited by 2
Abstract
An exact discretization of the canonical commutation and corresponding uncertainty relations are suggested. We prove that the canonical commutation relations of discrete quantum mechanics, which is based on standard finite difference, holds for constant wave functions only. In this paper, we use the [...] Read more.
An exact discretization of the canonical commutation and corresponding uncertainty relations are suggested. We prove that the canonical commutation relations of discrete quantum mechanics, which is based on standard finite difference, holds for constant wave functions only. In this paper, we use the recently proposed exact discretization of derivatives, which is based on differences that are represented by infinite series. This new mathematical tool allows us to build sensible discrete quantum mechanics based on the suggested differences and includes the correct canonical commutation and uncertainty relations. Full article
(This article belongs to the Special Issue Mathematics of Quantum Uncertainty)
Open AccessFeature PaperArticle
Entropic Uncertainty Relations for Successive Generalized Measurements
Mathematics 2016, 4(2), 41; https://doi.org/10.3390/math4020041 - 07 Jun 2016
Cited by 6
Abstract
We derive entropic uncertainty relations for successive generalized measurements by using general descriptions of quantum measurement within two distinctive operational scenarios. In the first scenario, by merging two successive measurements into one we consider successive measurement scheme as a method to perform an [...] Read more.
We derive entropic uncertainty relations for successive generalized measurements by using general descriptions of quantum measurement within two distinctive operational scenarios. In the first scenario, by merging two successive measurements into one we consider successive measurement scheme as a method to perform an overall composite measurement. In the second scenario, on the other hand, we consider it as a method to measure a pair of jointly measurable observables by marginalizing over the distribution obtained in this scheme. In the course of this work, we identify that limits on one’s ability to measure with low uncertainty via this scheme come from intrinsic unsharpness of observables obtained in each scenario. In particular, for the Lüders instrument, disturbance caused by the first measurement to the second one gives rise to the unsharpness at least as much as incompatibility of the observables composing successive measurement. Full article
(This article belongs to the Special Issue Mathematics of Quantum Uncertainty)
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Open AccessFeature PaperArticle
Measurement Uncertainty for Finite Quantum Observables
Mathematics 2016, 4(2), 38; https://doi.org/10.3390/math4020038 - 02 Jun 2016
Cited by 8
Abstract
Measurement uncertainty relations are lower bounds on the errors of any approximate joint measurement of two or more quantum observables. The aim of this paper is to provide methods to compute optimal bounds of this type. The basic method is semidefinite programming, which [...] Read more.
Measurement uncertainty relations are lower bounds on the errors of any approximate joint measurement of two or more quantum observables. The aim of this paper is to provide methods to compute optimal bounds of this type. The basic method is semidefinite programming, which we apply to arbitrary finite collections of projective observables on a finite dimensional Hilbert space. The quantification of errors is based on an arbitrary cost function, which assigns a penalty to getting result x rather than y, for any pair ( x , y ) . This induces a notion of optimal transport cost for a pair of probability distributions, and we include an Appendix with a short summary of optimal transport theory as needed in our context. There are then different ways to form an overall figure of merit from the comparison of distributions. We consider three, which are related to different physical testing scenarios. The most thorough test compares the transport distances between the marginals of a joint measurement and the reference observables for every input state. Less demanding is a test just on the states for which a “true value” is known in the sense that the reference observable yields a definite outcome. Finally, we can measure a deviation as a single expectation value by comparing the two observables on the two parts of a maximally-entangled state. All three error quantities have the property that they vanish if and only if the tested observable is equal to the reference. The theory is illustrated with some characteristic examples. Full article
(This article belongs to the Special Issue Mathematics of Quantum Uncertainty)
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Open AccessFeature PaperArticle
SIC-POVMs and Compatibility among Quantum States
Mathematics 2016, 4(2), 36; https://doi.org/10.3390/math4020036 - 01 Jun 2016
Cited by 9
Abstract
An unexpected connection exists between compatibility criteria for quantum states and Symmetric Informationally Complete quantum measurements (SIC-POVMs). Beginning with Caves, Fuchs and Schack’s "Conditions for compatibility of quantum state assignments", I show that a qutrit SIC-POVM studied in other contexts enjoys additional interesting [...] Read more.
An unexpected connection exists between compatibility criteria for quantum states and Symmetric Informationally Complete quantum measurements (SIC-POVMs). Beginning with Caves, Fuchs and Schack’s "Conditions for compatibility of quantum state assignments", I show that a qutrit SIC-POVM studied in other contexts enjoys additional interesting properties. Compatibility criteria provide a new way to understand the relationship between SIC-POVMs and mutually unbiased bases, as calculations in the SIC representation of quantum states make clear. This, in turn, illuminates the resources necessary for magic-state quantum computation, and why hidden-variable models fail to capture the vitality of quantum mechanics. Full article
(This article belongs to the Special Issue Mathematics of Quantum Uncertainty)
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Open AccessArticle
Tight State-Independent Uncertainty Relations for Qubits
Mathematics 2016, 4(1), 8; https://doi.org/10.3390/math4010008 - 24 Feb 2016
Cited by 25
Abstract
The well-known Robertson–Schrödinger uncertainty relations have state-dependent lower bounds, which are trivial for certain states. We present a general approach to deriving tight state-independent uncertainty relations for qubit measurements that completely characterise the obtainable uncertainty values. This approach can give such relations for [...] Read more.
The well-known Robertson–Schrödinger uncertainty relations have state-dependent lower bounds, which are trivial for certain states. We present a general approach to deriving tight state-independent uncertainty relations for qubit measurements that completely characterise the obtainable uncertainty values. This approach can give such relations for any number of observables, and we do so explicitly for arbitrary pairs and triples of qubit measurements. We show how these relations can be transformed into equivalent tight entropic uncertainty relations. More generally, they can be expressed in terms of any measure of uncertainty that can be written as a function of the expectation value of the observable for a given state. Full article
(This article belongs to the Special Issue Mathematics of Quantum Uncertainty)
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Review

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Open AccessFeature PaperReview
Uncertainty Relations and Possible Experience
Mathematics 2016, 4(2), 40; https://doi.org/10.3390/math4020040 - 03 Jun 2016
Cited by 3
Abstract
The uncertainty principle can be understood as a condition of joint indeterminacy of classes of properties in quantum theory. The mathematical expressions most closely associated with this principle have been the uncertainty relations, various inequalities exemplified by the well known expression regarding position [...] Read more.
The uncertainty principle can be understood as a condition of joint indeterminacy of classes of properties in quantum theory. The mathematical expressions most closely associated with this principle have been the uncertainty relations, various inequalities exemplified by the well known expression regarding position and momentum introduced by Heisenberg. Here, recent work involving a new sort of “logical” indeterminacy principle and associated relations introduced by Pitowsky, expressable directly in terms of probabilities of outcomes of measurements of sharp quantum observables, is reviewed and its quantum nature is discussed. These novel relations are derivable from Boolean “conditions of possible experience” of the quantum realm and have been considered both as fundamentally logical and as fundamentally geometrical. This work focuses on the relationship of indeterminacy to the propositions regarding the values of discrete, sharp observables of quantum systems. Here, reasons for favoring each of these two positions are considered. Finally, with an eye toward future research related to indeterminacy relations, further novel approaches grounded in category theory and intended to capture and reconceptualize the complementarity characteristics of quantum propositions are discussed in relation to the former. Full article
(This article belongs to the Special Issue Mathematics of Quantum Uncertainty)
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