Quantum information effects

We study the two dual quantum information effects to manipulate the amount of information in quantum computation: hiding and allocation. The resulting type-and-effect system is fully expressive for irreversible quantum computing, including measurement. We provide universal categorical constructions that semantically interpret this arrow metalanguage with choice, starting with any rig groupoid interpreting the reversible base language. Several properties of quantum measurement follow in general, and we translate (noniterative) quantum flow charts into our language. The semantic constructions turn the category of unitaries between Hilbert spaces into the category of completely positive trace-preserving maps, and they turn the category of bijections between finite sets into the category of functions with chosen garbage. Thus they capture the fundamental theorems of classical and quantum reversible computing of Toffoli and Stinespring.

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BASICS OF QUANTUM MECHANICS, GEOMETRIZATION AND SOME APPLICATIONS TO QUANTUM INFORMATION

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887808003156 ◽

2008 ◽

Vol 05 (06) ◽

pp. 989-1032 ◽

Cited By ~ 13

Author(s):

JESÚS CLEMENTE-GALLARDO ◽

GIUSEPPE MARMO

Keyword(s):

Quantum Mechanics ◽

Quantum Information ◽

Hilbert Spaces ◽

Functional Independence ◽

Practical Application ◽

Finite Dimensional ◽

Entanglement Witnesses ◽

Geometric Formulation

In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schrödinger framework from this perspective and provide a description of the Weyl–Wigner construction. Finally, after reviewing the basics of the geometric formulation of quantum mechanics, we apply the methods presented to the most interesting cases of finite dimensional Hilbert spaces: those of two, three and four level systems (one qubit, one qutrit and two qubit systems). As a more practical application, we discuss the advantages that the geometric formulation of quantum mechanics can provide us with in the study of situations as the functional independence of entanglement witnesses.

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Notes on super-operator norms induced by Schatten norms

Quantum Information and Computation ◽

10.26421/qic5.1-6 ◽

2005 ◽

Vol 5 (1) ◽

pp. 57-67

Author(s):

J. Watrous

Keyword(s):

Quantum Information ◽

Hilbert Spaces ◽

Positive Semidefinite ◽

Linear Mapping ◽

Completely Positive ◽

Basic Properties ◽

Finite Dimensional ◽

Operator Norms ◽

Definition Of

Let $\Phi$ be a super-operator, i.e., a linear mapping of the form $\Phi:\mathrm{L}(\mathcal{F})\rightarrow\mathrm{L}(\mathcal{G})$ for finite dimensional Hilbert spaces $\mathcal{F}$ and $\mathcal{G}$. This paper considers basic properties of the super-operator norms defined by $\|\Phi\|_{q\rightarrow p}= \sup\{\|\Phi(X)\|_p/\|X\|_q\,:\,X\not=0\}$, induced by Schatten norms for $1\leq p,q\leq\infty$. These super-operator norms arise in various contexts in the study of quantum information. In this paper it is proved that if $\Phi$ is completely positive, the value of the supremum in the definition of $\|\Phi\|_{q\rightarrow p}$ is achieved by a positive semidefinite operator $X$, answering a question recently posed by King and Ruskai~\cite{KingR04}. However, for any choice of $p\in [1,\infty]$, there exists a super-operator $\Phi$ that is the {\em difference} of two completely positive, trace-preserving super-operators such that all Hermitian $X$ fail to achieve the supremum in the definition of $\|\Phi\|_{1\rightarrow p}$. Also considered are the properties of the above norms for super-operators tensored with the identity super-operator. In particular, it is proved that for all $p\geq 2$, $q\leq 2$, and arbitrary $\Phi$, the norm $\|\Phi \|_{q\rightarrow p}$ is stable under tensoring $\Phi$ with the identity super-operator, meaning that $\|\Phi \|_{q\rightarrow p} = \|\Phi \otimes I\|_{q\rightarrow p}$. For $1\leq p < 2$, the norm $\|\Phi\|_{1\rightarrow p}$ may fail to be stable with respect to tensoring $\Phi$ with the identity super-operator as just described, but $\|\Phi\otimes I\|_{1\rightarrow p}$ is stable in this sense for $I$ the identity super-operator on $\mathrm{L}(\mathcal{H})$ for $\op{dim}(\mathcal{H}) = \op{dim}(\mathcal{F})$. This generalizes and simplifies a proof due to Kitaev \cite{Kitaev97} that established this fact for the case $p=1$.

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Interfacing Continuous and Discrete Hilbert Spaces in Quantum Optics and Quantum Information

10.18130/v3s27s ◽

2015 ◽

Author(s):

Niranjan Sridhar

Keyword(s):

Quantum Information ◽

Quantum Optics ◽

Hilbert Spaces

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TOWARDS NONCOMMUTATIVE COMPUTING

International Journal of Modern Physics B ◽

10.1142/s0217979200001965 ◽

2000 ◽

Vol 14 (22n23) ◽

pp. 2451-2454

Author(s):

G. F. MASCARI

Keyword(s):

Information Processing ◽

Quantum Information ◽

Category Theory ◽

Quantum Information Processing ◽

Mathematical Structures ◽

Structure And Dynamics ◽

Physical Experiments ◽

Higher Dimensional ◽

Higher Category Theory ◽

Categorical Constructions

This paper presents first steps of an approach to quantum information processing in the framework of higher category theory from a noncommutative mathematics perspective. The aim is to provide a unifying theory for the structure and dynamics of composite quantum information processing systems, such that states, evolution, entanglement, decoherence are modeled by abstract categorical constructions and vice versa new mathematical structures arising from higher dimensional algebra could be "tested" as computational schemes and possibly realized by physical experiments.

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ENTANGLEMENT DISTRIBUTION WITH GLOBAL CONTROL IN A STAR-SHAPED MULTI-SPLITTER

International Journal of Quantum Information ◽

10.1142/s0219749906001992 ◽

2006 ◽

Vol 04 (03) ◽

pp. 551-561 ◽

Cited By ~ 1

Author(s):

M. PATERNOSTRO ◽

H. McANENEY ◽

M. S. KIM

Keyword(s):

Information Processing ◽

Quantum Information ◽

Local Control ◽

Hilbert Spaces ◽

Quantum Information Processing ◽

Electromechanical Systems ◽

Global Control ◽

Infinite Dimensional ◽

Entanglement Distribution

Distributed quantum information processing (QIP) is a promising way to bypass problems due to unwanted interactions between elements. However, this strategy presupposes the engineering of protocols for remote processors. In many of them, pairwise entanglement is a key resource. We study a model which distributes entanglement among elements of a delocalized network without local control. The model is efficient both in finite- and infinite-dimensional Hilbert spaces. We suggest a setup of electromechanical systems to implement our proposal.

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Estimates for compression norms and additivity violation in quantum information

International Journal of Mathematics ◽

10.1142/s0129167x15500020 ◽

2015 ◽

Vol 26 (01) ◽

pp. 1550002 ◽

Cited By ~ 2

Author(s):

Benoît Collins ◽

Motohisa Fukuda ◽

Ping Zhong

Keyword(s):

Probability Theory ◽

Quantum Information ◽

Hilbert Spaces ◽

Free Probability ◽

Singular Values ◽

Random Subspace ◽

Quantum Channels ◽

Free Probability Theory ◽

Output Entropy ◽

Super Convergence

The free contraction norm (or the (t)-norm) was introduced by Belinschi, Collins and Nechita as a tool to compute the typical location of the collection of singular values associated to a random subspace of the tensor product of two Hilbert spaces. In turn, it was used again by them in order to obtain sharp bounds for the violation of the additivity of the minimum output entropy (MOE) for random quantum channels with Bell states. This free contraction norm, however, is difficult to compute explicitly. The purpose of this note is to give a good estimate for this norm. Our technique is based on results of super convergence in the context of free probability theory. As an application, we give a new, simple and conceptual proof of the violation of the additivity of the MOE.

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Physical entity as quantum information

10.31235/osf.io/pef9z ◽

2020 ◽

Author(s):

Vasil Dinev Penchev

Keyword(s):

Quantum Information ◽

Final Analysis ◽

Physical Reality ◽

Physical Entity ◽

Classical Information ◽

Scientific Epistemology ◽

Concept Of Truth ◽

Fundamental Theorems ◽

In Virtue Of ◽

Physical Quantities

Quantum mechanics was reformulated as an information theory involving ageneralized kind of information, namely quantum information, in the end of the last century.Quantum mechanics is the most fundamental physical theory referring to all claiming to bephysical. Any physical entity turns out to be quantum information in the final analysis. Aquantum bit is the unit of quantum information, and it is a generalization of the unit of classicalinformation, a bit, as well as the quantum information itself is a generalization of classicalinformation. Classical information refers to finite series or sets while quantum information, toinfinite ones. Quantum information as well as classical information is a dimensionless quantity.Quantum information can be considered as a “bridge” between the mathematical and physical.The standard and common scientific epistemology grants the gap between the mathematicalmodels and physical reality. The conception of truth as adequacy is what is able to transfer“over” that gap. One should explain how quantum information being a continuous transitionbetween the physical and mathematical may refer to truth as adequacy and thus to the usualscientific epistemology and methodology. If it is the overall substance of anything claiming to bephysical, one can question how different and dimensional physical quantities appear. Quantuminformation can be discussed as the counterpart of action. Quantum information is what isconserved, action is what is changed in virtue of the fundamental theorems of Emmy Noether(1918). The gap between mathematical models and physical reality, needing truth as adequacyto be overcome, is substituted by the openness of choice. That openness in turn can beinterpreted as the openness of the present as a different concept of truth recollectingHeidegger’s one as “unconcealment” (ἀλήθεια). Quantum information as what is conserved canbe thought as the conservation of that openness.

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