scholarly journals A Knot Theoretic Extension of the Bloch Sphere Representation for Qubits in Hilbert Space and Its Application to Contextuality and Many-Worlds Theories

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1135
Author(s):  
Stefan Heusler ◽  
Paul Schlummer ◽  
Malte S. Ubben
Keyword(s):  
Information Theory ◽  
Hilbert Space ◽  
Quantum Information ◽  
Knot Theory ◽  
Quantum Physics ◽  
Quantum Information Theory ◽  
Entangled States ◽  
Theoretic Model ◽  
Bloch Sphere ◽  
Many Worlds

We argue that the usual Bloch sphere is insufficient in various aspects for the representation of qubits in quantum information theory. For example, spin flip operations with the quaternions I J K = e 2 π i 2 = − 1 and J I K = + 1 cannot be distinguished on the Bloch sphere. We show that a simple knot theoretic extension of the Bloch sphere representation is sufficient to track all unitary operations for single qubits. Next, we extend the Bloch sphere representation to entangled states using knot theory. As applications, we first discuss contextuality in quantum physics—in particular the Kochen-Specker theorem. Finally, we discuss some arguments against many-worlds theories within our knot theoretic model of entanglement. The key ingredients of our approach are symmetries and geometric properties of the unitary group.

scholarly journals A general formulation based on algebraic spinors for the quantum computation

2020 ◽  
Vol 17 (14) ◽  
pp. 2050206
Author(s):  
Marco A. S. Trindade ◽  
Sergio Floquet ◽  
J. David M. Vianna
Keyword(s):  
Information Theory ◽  
Hilbert Space ◽  
Quantum Information ◽  
Quantum Computation ◽  
Quantum Teleportation ◽  
Clifford Algebras ◽  
Quantum Information Theory ◽  
Quantum Gates ◽  
Entangled States ◽  
Space Formulation

In this work, we explore the structure of Clifford algebras and the representations of the algebraic spinors in quantum information theory. Initially, we present a general formulation through elements of minimal left ideals in tensor products of Clifford algebras. Posteriorly, we perform some applications in quantum computation: qubits, entangled states, quantum gates, representations of the braid group, quantum teleportation, Majorana operators and supersymmetry. Finally, we discuss advantages compared to standard Hilbert space formulation.

scholarly journals IMPOSSIBILITY OF PARTIAL SWAPPING OF QUANTUM INFORMATION

2007 ◽  
Vol 05 (04) ◽  
pp. 605-609 ◽  
Author(s):  
I. CHAKRABARTY
Keyword(s):  
Information Theory ◽  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Information ◽  
Higher Dimensions ◽  
Quantum Information Theory ◽  
Quantum States ◽  
Two Dimensional ◽  
Bloch Sphere ◽  
Two Parameters

It is a well–known fact that a quantum state |ψ(θ, ϕ)〉 is represented by a point on the Bloch sphere, characterized by two parameters θ and ϕ. Here in this work, we find out another impossible operation in quantum information theory. We name this impossibility as 'Impossibility of partial swapping of quantum information'. By this we mean that if two unknown quantum states are given at the input port, there exists no physical process, consistent with the principles of quantum mechanics, by which we can partially swap either of the two parameters θ and ϕ between these two quantum states. In this work, we provided the impossibility proofs for the qubits (i.e. the quantum states taken from two-dimensional Hilbert space) and this impossible operation can be shown to hold in higher dimensions also.

QUANTUM GATES AND PROTOCOLS WITH SPHERICAL WIGNER FUNCTIONS

2005 ◽  
Vol 03 (03) ◽  
pp. 501-509
Author(s):  
ORSOLYA KÁLMÁN ◽  
MIHÁLY G. BENEDICT
Keyword(s):  
Information Theory ◽  
Distribution Function ◽  
Quantum Information ◽  
Wigner Function ◽  
Quantum Information Theory ◽  
Quantum Gates ◽  
Bloch Sphere ◽  
Dense Coding ◽  
Space Formulation ◽  
Quasiprobability Distribution

The fundamental concepts and operations of quantum information theory are considered in the framework of a phase space formulation of quantum mechanics, where the states of one or several qubits are represented by a specific continuous quasiprobability distribution function on the Bloch sphere or on its generalizations. The function we use is the spherical Wigner function. It is shown that the usual transformations of quantum information theory are certain rotations or more general transformations of this Wigner function. We show that the standard teleportation and dense coding protocols can be appropriately formulated in terms of the Wigner function.

scholarly journals JENSEN–SHANNON DIVERGENCE AS A MEASURE OF THE DEGREE OF ENTANGLEMENT

2008 ◽  
Vol 06 (supp01) ◽  
pp. 715-720 ◽  
Author(s):  
A. P. MAJTEY ◽  
A. BORRAS ◽  
M. CASAS ◽  
P. W. LAMBERTI ◽  
A. PLASTINO
Keyword(s):  
Information Theory ◽  
Hilbert Space ◽  
Quantum Information ◽  
Quantum Information Theory ◽  
Quantum States ◽  
Jensen Shannon Divergence ◽  
Degree Of Entanglement

The notion of distance in Hilbert space is relevant in many scenarios. In particular, "distances" between quantum states play a central role in quantum information theory. An appropriate measure of distance is the quantum Jensen Shannon divergence (QJSD) between quantum states. Here we study this distance as a geometrical measure of entanglement and apply it to different families of states.

scholarly journals Entropy Production in Quantum is Different

Entropy ◽  
2019 ◽  
Vol 21 (9) ◽  
pp. 854 ◽  
Author(s):  
Mohammad H. Ansari ◽  
Alwin van Steensel ◽  
Yuli V. Nazarov
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Quantum Physics ◽  
Quantum Information Theory ◽  
Nonlinear Dependence ◽  
Erwin Schrödinger ◽  
Quantum Wave Function ◽  
Quantum Wave ◽  
Mere Existence ◽  
Physical Phenomena

Currently, ‘time’ does not play any essential role in quantum information theory. In this sense, quantum information theory is underdeveloped similarly to how quantum physics was underdeveloped before Erwin Schrödinger introduced his famous equation for the evolution of a quantum wave function. In this review article, we cope with the problem of time for one of the central quantities in quantum information theory: entropy. Recently, a replica trick formalism, the so-called ‘multiple parallel world’ formalism, has been proposed that revolutionizes entropy evaluation for quantum systems. This formalism is one of the first attempts to introduce ‘time’ in quantum information theory. With the total entropy being conserved in a closed system, entropy can flow internally between subsystems; however, we show that this flow is not limited only to physical correlations as the literature suggest. The nonlinear dependence of entropy on the density matrix introduces new types of correlations with no analogue in physical quantities. Evolving a number of replicas simultaneously makes it possible for them to exchange particles between different replicas. We will summarize some of the recent news about entropy in some example quantum devices. Moreover, we take a quick look at a new correspondence that was recently proposed that provides an interesting link between quantum information theory and quantum physics. The mere existence of such a correspondence allows for exploring new physical phenomena as the result of controlling entanglement in a quantum device.

QALO-MOR: Improved antlion optimizer based on quantum information theory for model order reduction

2021 ◽  
pp. 1-11
Author(s):  
Rosy Pradhan ◽  
Mohammad Rafique Khan ◽  
Prabir Kumar Sethy ◽  
Santosh Kumar Majhi
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Real World ◽  
Model Order Reduction ◽  
Order Reduction ◽  
Quantum Information Theory ◽  
Hunting Behavior ◽  
Model Order ◽  
Ant Lion Optimization ◽  
Real World Problems

The field of optimization science is proliferating that has made complex real-world problems easy to solve. Metaheuristics based algorithms inspired by nature or physical phenomena based methods have made its way in providing near-ideal (optimal) solutions to several complex real-world problems. Ant lion Optimization (ALO) has inspired by the hunting behavior of antlions for searching for food. Even with a unique idea, it has some limitations like a slower rate of convergence and sometimes confines itself into local solutions (optima). Therefore, to enhance its performance of classical ALO, quantum information theory is hybridized with classical ALO and named as QALO or quantum theory based ALO. It can escape from the limitations of basic ALO and also produces stability between processes of explorations followed by exploitation. CEC2017 benchmark set is adopted to estimate the performance of QALO compared with state-of-the-art algorithms. Experimental and statistical results demonstrate that the proposed method is superior to the original ALO. The proposed QALO extends further to solve the model order reduction (MOR) problem. The QALO based MOR method performs preferably better than other compared techniques. The results from the simulation study illustrate that the proposed method effectively utilized for global optimization and model order reduction.

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