scholarly journals QUANTUM INFORMATION, ENTANGLEMENT AND ENTROPY

2020 ◽  
Author(s):  
Siddharth Sharma
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Information ◽  
Density Matrix ◽  
Pythagorean Theorem ◽  
Quantum Entropy ◽  
Density Matrices ◽  
Relative Quantum ◽  
Axiomatic Quantum Mechanics ◽  
Measure Entropy

In this review I had given an introduction to axiomatic Quantum Mechanics, Quantum Information and its measure entropy. Also, I had given an introduction to Entanglement and its measure as the minimum of relative quantum entropy of a density matrix of a Hilbert space with respect to all separable density matrices of the same Hilbert space. I also discussed geodesic in the space of density matrices, their orthogonality and Pythagorean theorem of density matrix.

A THEOREM OF SOLÉR, THE THEORY OF SYMMETRY AND QUANTUM MECHANICS

2012 ◽  
Vol 09 (02) ◽  
pp. 1260005 ◽  
Author(s):  
GIANNI CASSINELLI ◽  
PEKKA LAHTI
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Logic ◽  
Hilbert Spaces ◽  
Classical Problem ◽  
Partial Solution ◽  
Space Realization ◽  
Symmetry Transformations ◽  
Axiomatic Quantum Mechanics

A classical problem in axiomatic quantum mechanics is deducing a Hilbert space realization for a quantum logic that admits a vector space coordinatization of the Piron–McLaren type. Our aim is to show how a theorem of M. Solér [Characterization of Hilbert spaces by orthomodular spaces, Comm. Algebra23 (1995) 219–243.] can be used to get a (partial) solution of this problem. We first derive a generalization of the Wigner theorem on symmetry transformations that holds already in the Piron–McLaren frame. Then we investigate which conditions on the quantum logic allow the use of Solér's theorem in order to obtain a Hilbert space solution for the coordinatization problem.

TOWARD A FULL RECONSTRUCTION OF DENSITY MATRIX BY ON/OFF MEASUREMENTS

2009 ◽  
Vol 07 (supp01) ◽  
pp. 27-32 ◽  
Author(s):  
G. BRIDA ◽  
M. GENOVESE ◽  
M. GRAMEGNA ◽  
P. TRAINA ◽  
E. PREDAZZI ◽  
...  
Keyword(s):  
Quantum Mechanics ◽  
Quantum Information ◽  
Density Matrix ◽  
Foundations Of Quantum Mechanics ◽  
Quantum Tomography ◽  
Local Oscillator ◽  
Homodyne Detection ◽  
Short Pulses ◽  
Phase Measurements ◽  
Full Reconstruction

The knowledge of density matrix is fundamental for several applications, ranging from quantum information to the foundations of quantum mechanics and quantum optics. Nevertheless, quantum tomography based on homodyne detection is a rather complicated technique when applied to short pulses in photocounting regime. In this paper, we present an experimental work addressed to test an innovative scheme for a full reconstruction of the density matrix by using on/off detection coupled to phase measurements respect to a local oscillator.

scholarly journals Distinguishability of non-orthogonal density matrices does not imply violations of the second law

2019 ◽  
Author(s):  
PierGianLuca Porta Mana
Keyword(s):  
Hilbert Space ◽  
Density Matrix ◽  
Thermodynamic Analysis ◽  
Second Law Of Thermodynamics ◽  
The Other ◽  
Second Law ◽  
Density Matrices ◽  
Matrix Space ◽  
Von Neumann ◽  
Quantum Thermodynamic

The hypothetical possibility of distinguishing preparations described by non-orthogonal density matrices does not necessarily imply a violation of the second law of thermodynamics, as was instead stated by von Neumann. On the other hand, such a possibility would surely mean that the particular density-matrix space (and related Hilbert space) adopted would not be adequate to describe the hypothetical new experimental facts. These points are shown by making clear the distinction between physical preparations and the density matrices which represent them, and then comparing a "quantum" thermodynamic analysis given by Peres with a "classical" one given by Jaynes.

scholarly journals From linear algebra to quantum information

2021 ◽  
pp. 032-047
Author(s):  
Yu LW ◽  
Wang NL ◽  
Kanemitsu S
Keyword(s):  
Functional Analysis ◽  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Information ◽  
Linear Algebra ◽  
Smooth Transition ◽  
Quantum Computers ◽  
Hermite Operator ◽  
Hermite Matrix ◽  
Hermite Operators

Anticipating the realization of quantum computers, we propose the most reader-friendly exposition of quantum information and qubits theory. Although the latter lies within framework of linear algebra, it has some fl avor of quantum mechanics and it would be easier to get used to special symbols and terminologies. Quantum mechanics is described in the language of functional analysis: the state space (the totality of all states) of a quantum system is a Hilbert space over the complex numbers and all mechanical quantities are taken as Hermite operators. Hence some basics of functional analysis is necessary. We make a smooth transition from linear algebra to functional analysis by comparing the elements in these theories: Hilbert space vs. fi nite dimensional vector space, Hermite operator vs. linear map given by a Hermite matrix. Then from Newtonian mechanics to quantum mechanics and then to the theory of qubits. We elucidate qubits theory a bit by accommodating it into linear algebra framework under these precursors.

scholarly journals Problem of the direct quantum-information transformation of chemical substance

2020 ◽  
Author(s):  
Vasil Dinev Penchev
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Information ◽  
Special Kind ◽  
Chemical Substance ◽  
Complex Hilbert Space ◽  
Chemical Transformations ◽  
The Novel ◽  
Direct Transformation ◽  
Matter Energy

Arthur Clark and Michael Kube–McDowell (“The Triger”, 2000) suggested the sci-fi idea about the direct transformation from a chemical substance to another by the action of a newly physical, “Trigger” field. Karl Brohier, a Nobel Prize winner, who is a dramatic persona in the novel, elaborates a new theory, re-reading and re-writing Pauling’s “The Nature of the Chemical Bond”; according to Brohier: “Information organizes and differentiates energy. It regularizes and stabilizes matter. Information propagates through matter-energy and mediates the interactions of matter-energy.” Dr Horton, his collaborator in the novel replies: “If the universe consists of energy and information, then the Trigger somehow alters the information envelope of certain substances –“.“Alters it, scrambles it, overwhelms it, destabilizes it” Brohier adds.There is a scientific debate whether or how far chemistry is fundamentally reducible to quantum mechanics. Nevertheless, the fact that many essential chemical properties and reactions are at least partly representable in terms of quantum mechanics is doubtless. For the quantum mechanics itself has been reformulated as a theory of a special kind of information, quantum information, chemistry might be in turn interpreted in the same terms.Wave function, the fundamental concept of quantum mechanics, can be equivalently defined as a series of qubits, eventually infinite. A qubit, being defined as the normed superposition of the two orthogonal subspaces of the complex Hilbert space, can be interpreted as a generalization of the standard bit of information as to infinite sets or series. All “forces” in the Standard model, which are furthermore essential for chemical transformations, are groups [U(1),SU(2),SU(3)] of the transformations of the complex Hilbert space and thus, of series of qubits.One can suggest that any chemical substances and changes are fundamentally representable as quantum information and its transformations. If entanglement is interpreted as a physical field, though any group above seems to be unattachable to it, it might be identified as the “Triger field”. It might cause a direct transformation of any chemical substance by from a remote distance. Is this possible in principle?

Qubits and the Framework of Quantum Mechanics

2006 ◽  
Author(s):  
Phillip Kaye ◽  
Raymond Laflamme ◽  
Michele Mosca
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Computing ◽  
Quantum Information ◽  
State Space ◽  
Quantum Physics ◽  
Complex Hilbert Space ◽  
The State ◽  
Mathematical Framework ◽  
Infinite Dimensional

In this section we introduce the framework of quantum mechanics as it pertains to the types of systems we will consider for quantum computing. Here we also introduce the notion of a quantum bit or ‘qubit’, which is a fundamental concept for quantum computing. At the beginning of the twentieth century, it was believed by most that the laws of Newton and Maxwell were the correct laws of physics. By the 1930s, however, it had become apparent that these classical theories faced serious problems in trying to account for the observed results of certain experiments. As a result, a new mathematical framework for physics called quantum mechanics was formulated, and new theories of physics called quantum physics were developed in this framework. Quantum physics includes the physical theories of quantum electrodynamics and quantum field theory, but we do not need to know these physical theories in order to learn about quantum information. Quantum information is the result of reformulating information theory in this quantum framework. We saw in Section 1.6 an example of a two-state quantum system: a photon that is constrained to follow one of two distinguishable paths. We identified the two distinguishable paths with the 2-dimensional basis vectors and then noted that a general ‘path state’ of the photon can be described by a complex vector with |α0|2 +|α1|2 = 1. This simple example captures the essence of the first postulate, which tells us how physical states are represented in quantum mechanics. Depending on the degree of freedom (i.e. the type of state) of the system being considered, H may be infinite-dimensional. For example, if the state refers to the position of a particle that is free to occupy any point in some region of space, the associated Hilbert space is usually taken to be a continuous (and thus infinite-dimensional) space. It is worth noting that in practice, with finite resources, we cannot distinguish a continuous state space from one with a discrete state space having a sufficiently small minimum spacing between adjacent locations. For describing realistic models of quantum computation, we will typically only be interested in degrees of freedom for which the state is described by a vector in a finite-dimensional (complex) Hilbert space.

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.

scholarly journals Choice, Infinity, and Negation: Both Set-Theory and Quantum-Information Viewpoints to Negation

2020 ◽  
Author(s):  
Vasil Dinev Penchev
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Information ◽  
Natural Number ◽  
Set Theory ◽  
Peano Arithmetic ◽  
Complex Hilbert Space ◽  
Heyting Arithmetic ◽  
Infinite Set

The concepts of choice, negation, and infinity are considered jointly. The link is the quantity of information interpreted as the quantity of choices measured in units of elementary choice: a bit is an elementary choice between two equally probable alternatives. “Negation” supposes a choice between it and confirmation. Thus quantity of information can be also interpreted as quantity of negations. The disjunctive choice between confirmation and negation as to infinity can be chosen or not in turn: This corresponds to set-theory or intuitionist approach to the foundation of mathematics and to Peano or Heyting arithmetic. Quantum mechanics can be reformulated in terms of information introducing the concept and quantity of quantum information. A qubit can be equivalently interpreted as that generalization of “bit” where the choice is among an infinite set or series of alternatives. The complex Hilbert space can be represented as both series of qubits and value of quantum information. The complex Hilbert space is that generalization of Peano arithmetic where any natural number is substituted by a qubit. “Negation”, “choice”, and “infinity” can be inherently linked to each other both in the foundation of mathematics and quantum mechanics by the meditation of “information” and “quantum information”.

scholarly journals The isomorphism of Minkowski space and the separable complex Hilbert space and its physical interpretation

2020 ◽  
Author(s):  
Vasil Dinev Penchev
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Information ◽  
Special Relativity ◽  
Minkowski Space ◽  
Reference Frame ◽  
Physical Interpretation ◽  
Complex Hilbert Space ◽  
Axiom Of Choice ◽  
The Axiom Of Choice

An isomorphism is built between the separable complex Hilbert space (quantum mechanics) and Minkowski space (special relativity) by meditation of quantum information (i.e. qubit by qubit). That isomorphism can be interpreted physically as the invariance between a reference frame within a system and its unambiguous counterpart out of the system. The same idea can be applied to Poincaré’s conjecture (proved by G. Perelman) hinting another way for proving it, more concise and meaningful physically. Mathematically, the isomorphism means the invariance to choice, the axiom of choice, well-ordering, and well-ordering “theorem” (or “principle”) and can be defined generally as “information invariance”.

Sign in / Sign up

Export Citation Format

Share Document