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Entropy
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Quantum Probability, Statistics and Control
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Quantum Probability, Statistics and Control

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Quantum Information".

Deadline for manuscript submissions: closed (31 May 2020) | Viewed by 10410

Special Issue Editor

Prof. Holger F. Hofmann

E-Mail Website
Guest Editor
Graduate School of Advanced Sciences of Matter, Hiroshima University, Higashi Hiroshima 739-8530, Japan
Interests: quantum optics; quantum information

Special Issue Information

Dear Colleagues,

Quantum physics is about to move into a new stage as the technical possibilities of controlling quantum systems are giving us access to more and more details of the statistical features described by quantum interferences and the associated Hilbert space algebra. These developments provide us with a unique opportunity to vastly enhance our understanding of quantum physics by exploring the fundamental relations between quantum states and quantum measurements. The aim of this special issue is to invite scientists to share both theoretical and experimental results on the statistics of quantum systems at the ultimate limits of control. In particular, any work relating to quantum measurement, uncertainty relations, entanglement and other non-classical correlations, and quantum tomography and related tests of states and processes will be highly welcome. More generally, any scientific contributions relating to fundamental or practical aspects of quantum systems will be considered.

Prof. Holger F. Hofmann
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Published Papers (4 papers)

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Research

18 pages, 498 KiB  
Article
Uncertainty Relation for Errors Focusing on General POVM Measurements with an Example of Two-State Quantum Systems
by Jaeha Lee and Izumi Tsutsui
Entropy 2020, 22(11), 1222; https://doi.org/10.3390/e22111222 - 27 Oct 2020
Cited by 2 | Viewed by 3028
Abstract
A novel uncertainty relation for errors of general quantum measurement is presented. The new relation, which is presented in geometric terms for maps representing measurement, is completely operational and can be related directly to tangible measurement outcomes. The relation violates the naïve bound [...] Read more.
A novel uncertainty relation for errors of general quantum measurement is presented. The new relation, which is presented in geometric terms for maps representing measurement, is completely operational and can be related directly to tangible measurement outcomes. The relation violates the naïve bound ℏ/2 for the position-momentum measurement, whilst nevertheless respecting Heisenberg’s philosophy of the uncertainty principle. The standard Kennard–Robertson uncertainty relation for state preparations expressed by standard deviations arises as a corollary to its special non-informative case. For the measurement on two-state quantum systems, the relation is found to offer virtually the tightest bound possible; the equality of the relation holds for the measurement performed over every pure state. The Ozawa relation for errors of quantum measurements will also be examined in this regard. In this paper, the Kolmogorovian measure-theoretic formalism of probability—which allows for the representation of quantum measurements by positive-operator valued measures (POVMs)—is given special attention, in regard to which some of the measure-theory specific facts are remarked along the exposition as appropriate. Full article
(This article belongs to the Special Issue Quantum Probability, Statistics and Control)
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18 pages, 457 KiB  
Article
Adaptive State Fidelity Estimation for Higher Dimensional Bipartite Entanglement
by Jun-Yi Wu
Entropy 2020, 22(8), 886; https://doi.org/10.3390/e22080886 - 12 Aug 2020
Cited by 1 | Viewed by 2016
Abstract
An adaptive method for quantum state fidelity estimation in bipartite higher dimensional systems is established. This method employs state verifier operators which are constructed by local POVM operators and adapted to the measurement statistics in the computational basis. Employing this method, the state [...] Read more.
An adaptive method for quantum state fidelity estimation in bipartite higher dimensional systems is established. This method employs state verifier operators which are constructed by local POVM operators and adapted to the measurement statistics in the computational basis. Employing this method, the state verifier operators that stabilize Bell-type entangled states are constructed explicitly. Together with an error operator in the computational basis, one can estimate the lower and upper bounds on the state fidelity for Bell-type entangled states in few measurement configurations. These bounds can be tighter than the fidelity bounds derived in [Bavaresco et al., Nature Physics (2018), 14, 1032–1037], if one constructs more than one local POVM measurements additional to the measurement in the computational basis. Full article
(This article belongs to the Special Issue Quantum Probability, Statistics and Control)
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19 pages, 549 KiB  
Article
Generic Entanglement Entropy for Quantum States with Symmetry
by Yoshifumi Nakata and Mio Murao
Entropy 2020, 22(6), 684; https://doi.org/10.3390/e22060684 - 19 Jun 2020
Cited by 1 | Viewed by 2987
Abstract
When a quantum pure state is drawn uniformly at random from a Hilbert space, the state is typically highly entangled. This property of a random state is known as generic entanglement of quantum states and has been long investigated from many perspectives, ranging [...] Read more.
When a quantum pure state is drawn uniformly at random from a Hilbert space, the state is typically highly entangled. This property of a random state is known as generic entanglement of quantum states and has been long investigated from many perspectives, ranging from the black hole science to quantum information science. In this paper, we address the question of how symmetry of quantum states changes the properties of generic entanglement. More specifically, we study bipartite entanglement entropy of a quantum state that is drawn uniformly at random from an invariant subspace of a given symmetry. We first extend the well-known concentration formula to the one applicable to any subspace and then show that 1. quantum states in the subspaces associated with an axial symmetry are still highly entangled, though it is less than that of the quantum states without symmetry, 2. quantum states associated with the permutation symmetry are significantly less entangled, and 3. quantum states with translation symmetry are as entangled as the generic one. We also numerically investigate the phase-transition behavior of the distribution of generic entanglement, which indicates that the phase transition seems to still exist even when random states have symmetry. Full article
(This article belongs to the Special Issue Quantum Probability, Statistics and Control)
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16 pages, 273 KiB  
Article
What Does the Operator Algebra of Quantum Statistics Tell Us about the Objective Causes of Observable Effects?
by Holger F. Hofmann
Entropy 2020, 22(6), 638; https://doi.org/10.3390/e22060638 - 09 Jun 2020
Cited by 2 | Viewed by 2043
Abstract
Quantum physics can only make statistical predictions about possible measurement outcomes, and these predictions originate from an operator algebra that is fundamentally different from the conventional definition of probability as a subjective lack of information regarding the physical reality of the system. In [...] Read more.
Quantum physics can only make statistical predictions about possible measurement outcomes, and these predictions originate from an operator algebra that is fundamentally different from the conventional definition of probability as a subjective lack of information regarding the physical reality of the system. In the present paper, I explore how the operator formalism accommodates the vast number of possible states and measurements by characterizing its essential function as a description of causality relations between initial conditions and subsequent observations. It is shown that any complete description of causality must involve non-positive statistical elements that cannot be associated with any directly observable effects. The necessity of non-positive elements is demonstrated by the uniquely defined mathematical description of ideal correlations which explains the physics of maximally entangled states, quantum teleportation and quantum cloning. The operator formalism thus modifies the concept of causality by providing a universally valid description of deterministic relations between initial states and subsequent observations that cannot be expressed in terms of directly observable measurement outcomes. Instead, the identifiable elements of causality are necessarily non-positive and hence unobservable. The validity of the operator algebra therefore indicates that a consistent explanation of the various uncertainty limited phenomena associated with physical objects is only possible if we learn to accept the fact that the elements of causality cannot be reconciled with a continuation of observable reality in the physical object. Full article
(This article belongs to the Special Issue Quantum Probability, Statistics and Control)
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