scholarly journals Parts and Composites of Quantum Systems

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1031
Author(s):  
Stanley P. Gudder
Keyword(s):  
Tensor Product ◽  
Hilbert Spaces ◽  
Quantum Systems ◽  
Quantum Measurements ◽  
Quantum Channels ◽  
Measurement Models ◽  
Composite Systems ◽  
Finite Dimensional ◽  
Sequential Products

We consider three types of entities for quantum measurements. In order of generality, these types are observables, instruments and measurement models. If α and β are entities, we define what it means for α to be a part of β. This relationship is essentially equivalent to α being a function of β and in this case β can be employed to measure α. We then use the concept to define the coexistence of entities and study its properties. A crucial role is played by a map α^ which takes an entity of a certain type to one of a lower type. For example, if I is an instrument, then I^ is the unique observable measured by I. Composite systems are discussed next. These are constructed by taking the tensor product of the Hilbert spaces of the systems being combined. Composites of the three types of measurements and their parts are studied. Reductions in types to their local components are discussed. We also consider sequential products of measurements. Specific examples of Lüders, Kraus and trivial instruments are used to illustrate various concepts. We only consider finite-dimensional systems in this article. Finally, we mention the role of symmetry representations for groups using quantum channels.

scholarly journals A COMPLETELY ENTANGLED SUBSPACE OF MAXIMAL DIMENSION

2006 ◽  
Vol 04 (02) ◽  
pp. 325-330 ◽  
Author(s):  
B. V. RAJARAMA BHAT
Keyword(s):  
Tensor Product ◽  
Hilbert Spaces ◽  
Product Formula ◽  
Maximal Dimension ◽  
Maximum Dimension ◽  
Dimension Formula ◽  
Finite Dimensional ◽  
Minimum Dimension ◽  
Orthogonal Product ◽  
Product Vector

Consider a tensor product [Formula: see text] of finite-dimensional Hilbert spaces with dimension [Formula: see text], 1 ≤ i ≤ k. Then the maximum dimension possible for a subspace of [Formula: see text] with no non-zero product vector is known to be d1 d2…dk - (d1 + d2 + … + dk + k - 1. We obtain an explicit example of a subspace of this kind. We determine the set of product vectors in its orthogonal complement and show that it has the minimum dimension possible for an unextendible product basis of not necessarily orthogonal product vectors.

scholarly journals Separation of quantum, spatial quantum, and approximate quantum correlations

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 389
Author(s):  
Salman Beigi
Keyword(s):  
Mathematical Model ◽  
Hilbert Spaces ◽  
Quantum Correlations ◽  
Quantum Measurements ◽  
Finite Dimensional ◽  
Nonlocal Correlations ◽  
Local Quantum

Quantum nonlocal correlations are generated by implementation of local quantum measurements on spatially separated quantum subsystems. Depending on the underlying mathematical model, various notions of sets of quantum correlations can be defined. In this paper we prove separations of such sets of quantum correlations. In particular, we show that the set of bipartite quantum correlations with four binary measurements per party becomes strictly smaller once we restrict the local Hilbert spaces to be finite dimensional, i.e., Cq(4,4,2,2)≠Cqs(4,4,2,2). We also prove non-closure of the set of bipartite quantum correlations with four ternary measurements per party, i.e., Cqs(4,4,3,3)≠Cqa(4,4,3,3).

scholarly journals Multiplicities in the Tensor Product of Finite-Dimensional Representations of Discrete Groups

1978 ◽  
Vol 21 (1) ◽  
pp. 17-19
Author(s):  
Dragomir Ž. Djoković
Keyword(s):  
Tensor Product ◽  
Hilbert Spaces ◽  
Unitary Representations ◽  
Vector Spaces ◽  
Closed Field ◽  
Discrete Groups ◽  
Algebraically Closed Field ◽  
Finite Dimensional ◽  
Irreducible Unitary Representations

Let G be a group and ρ and σ two irreducible unitary representations of G in complex Hilbert spaces and assume that dimp ρ= n < ∞. D. Poguntke [2] proved that is a sum of at most n2 irreducible subrepresentations. The case when dim a is also finite he attributed to R. Howe.We shall prove analogous results for arbitrary finite-dimensional representations, not necessarily unitary. Thus let F be an algebraically closed field of characteristic 0. We shall use the language of modules and we postulate that allour modules are finite-dimensional as F-vector spaces. The field F itself will be considered as a trivial G-module.

scholarly journals On the Relation between States and Maps in Infinite Dimensions

2007 ◽  
Vol 14 (04) ◽  
pp. 355-370 ◽  
Author(s):  
Janusz Grabowski ◽  
Marek Kuś ◽  
Giuseppe Marmo
Keyword(s):  
Hilbert Spaces ◽  
Trace Class ◽  
Quantum Systems ◽  
Bounded Operators ◽  
Infinite Dimensions ◽  
Infinite Dimensional ◽  
Continuous Map ◽  
Finite Dimensional ◽  
Trace Class Operators ◽  
Product Realization

Relations between states and maps, which are known for quantum systems in finite-dimensional Hilbert spaces, are formulated rigorously in geometrical terms with no use of coordinate (matrix) interpretation. In a tensor product realization they are represented simply by a permutation of factors. This leads to natural generalizations for infinite-dimensional Hilbert spaces and a simple proof of a generalized Choi Theorem. The natural framework is based on spaces of Hilbert-Schmidt operators [Formula: see text] and the corresponding tensor products [Formula: see text] of Hilbert spaces. It is proved that the corresponding isomorphisms cannot be naturally extended to compact (or bounded) operators, nor reduced to the trace-class operators. On the other hand, it is proven that there is a natural continuous map [Formula: see text] from trace-class operators on [Formula: see text] (with the nuclear norm) into compact operators mapping the space of all bounded operators on [Formula: see text] into trace class operators on [Formula: see text] (with the operator-norm). Also in the infinite-dimensional context, the Schmidt measure of entanglement and multipartite generalizations of state-maps relations are considered in the paper.

scholarly journals Thermalization of small quantum systems: From the zeroth law of thermodynamics

2021 ◽  
Author(s):  
Jiaozi Wang ◽  
Wen-Ge Wang ◽  
Jiao Wang
Keyword(s):  
Quantum System ◽  
Chaotic Systems ◽  
Environmental Temperature ◽  
Quantum Systems ◽  
Necessary Condition ◽  
Environmental System ◽  
Small System ◽  
Composite Systems ◽  
Small Quantum

Abstract Thermalization of isolated quantum systems has been studied intensively in recent years and significant progresses have been achieved. Here, we study thermalization of small quantum systems that interact with large chaotic environments under the consideration of Schrödinger evolution of composite systems, from the perspective of the zeroth law of thermodynamics. Namely, we consider a small quantum system that is brought into contact with a large environmental system; after they have relaxed, they are separated and their temperatures are studied. Our question is under what conditions the small system may have a detectable temperature that is identical with the environmental temperature. This should be a necessary condition for the small quantum system to be thermalized and to have a well-defined temperature. By using a two-level probe quantum system that plays the role of a thermometer, we find that the zeroth law is applicable to quantum chaotic systems, but not to integrable systems.

Superselection Sectors, Pure States, Bose and Fermi Distributions by Completely Positive Quantum-Motions

2005 ◽  
Vol 12 (01) ◽  
pp. 23-35 ◽  
Author(s):  
Klaus Dietz
Keyword(s):  
Hilbert Spaces ◽  
Mixed States ◽  
Completely Positive ◽  
Absolute Value ◽  
Zero Modes ◽  
Finite Dimensional ◽  
Constants Of Motion ◽  
Pure States ◽  
Damped Oscillator

The connection of the operators V, building up the Kossakowski-Lindblad generator, with the asymptotic states of the corresponding completely positive quantum-maps is discussed. Maps leading to decoherence are constructed, the importance of zero-modes in the absolute value [Formula: see text] of V for the generation of pure states from arbitrary mixed states is illustrated. The universal rôle of equipartite states appears when unitary V are chosen. The 'damped oscillator model' is generalized to yield Bose and Fermi distributions as asymptotic states for systems described by a Hamiltonian and other constants of motion. Calculations are performed in finite dimensional Hilbert spaces.

scholarly journals A Gutzwiller trace formula for large hermitian matrices

2017 ◽  
Vol 29 (08) ◽  
pp. 1750027
Author(s):  
Jens Bolte ◽  
Sebastian Egger ◽  
Stefan Keppeler
Keyword(s):  
Time Evolution ◽  
Trace Formula ◽  
Hilbert Spaces ◽  
Semiclassical Approximation ◽  
Integral Operators ◽  
Fourier Integral ◽  
Quantum Systems ◽  
Quantization Condition ◽  
Fourier Integral Operators ◽  
Finite Dimensional

We develop a semiclassical approximation for the dynamics of quantum systems in finite-dimensional Hilbert spaces whose classical counterparts are defined on a toroidal phase space. In contrast to previous models of quantum maps, the time evolution is in continuous time and, hence, is generated by a Schrödinger equation. In the framework of Weyl quantization, we construct discrete, semiclassical Fourier integral operators approximating the unitary time evolution and use these to prove a Gutzwiller trace formula. We briefly discuss a semiclassical quantization condition for eigenvalues as well as some simple examples.

ALTERNATIVE HAMILTONIAN DESCRIPTIONS FOR QUANTUM SYSTEMS AND NON-HERMITIAN OPERATORS WITH REAL SPECTRUM

2002 ◽  
Vol 17 (24) ◽  
pp. 1589-1599 ◽  
Author(s):  
FRANCO VENTRIGLIA
Keyword(s):  
Inverse Problem ◽  
Quantum Mechanics ◽  
Hilbert Spaces ◽  
Quantum Systems ◽  
Theoretical Physics ◽  
Inner Product ◽  
Real Spectrum ◽  
Standard Methods ◽  
Finite Dimensional

Many problems in theoretical physics are very frequently dealt with non-Hermitian operators. Recently there has been a lot of interest in non-Hermitian operators with real spectra. In this paper, by using the inverse problem for quantum systems, we show that, on finite-dimensional Hilbert spaces, all diagonalizable operators with a real spectrum can be made Hermitian with respect to a properly chosen inner product. In particular this allows the use of standard methods of quantum mechanics to analyze non-Hermitian operators with real spectra.

scholarly journals Spectral representations of normal operators in quaternionic Hilbert spaces via intertwining quaternionic PVMs

2017 ◽  
Vol 29 (10) ◽  
pp. 1750034 ◽  
Author(s):  
Riccardo Ghiloni ◽  
Valter Moretti ◽  
Alessandro Perotti
Keyword(s):  
Tensor Product ◽  
Hilbert Spaces ◽  
Resolvent Operator ◽  
Mathematical Object ◽  
Quantum Systems ◽  
Promising Tool ◽  
Normal Operators ◽  
Definition Of ◽  
Spectral Representations ◽  
Quaternionic Formulation

The possibility of formulating quantum mechanics over quaternionic Hilbert spaces can be traced back to von Neumann’s foundational works in the thirties. The absence of a suitable quaternionic version of spectrum prevented the full development of the theory. The first rigorous quaternionic formulation has started only in 2007 with the definition of the spherical spectrum of a quaternionic operator based on a quadratic version of resolvent operator. The relevance of this notion is proved by the existence of a quaternionic continuous functional calculus and a theory of quaternionic semigroups relying upon it. A problem of the quaternionic formulation is the description of composite quantum systems in the absence of a natural tensor product due to non-commutativity of quaternions. A promising tool towards a solution is a quaternionic projection-valued measure (PVM), making possible a tensor product of quaternionic operators with physical relevance. A notion with this property, called intertwining quaternionic PVM, is presented here. This foundational paper aims to investigate the interplay of this new mathematical object and the spherical spectral features of quaternionic generally unbounded normal operators. We discover, in particular, the existence of other spectral notions equivalent to the spherical ones, but based on a standard non-quadratic notion of resolvent operator.

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