scholarly journals On Classes of Local Unitary Transformations

2012 ◽  
Vol 19 (02) ◽  
pp. 283-292
Author(s):  
Naihuan Jing
Keyword(s):  
Homogeneous Spaces ◽  
Unitary Groups ◽  
Mixed States ◽  
Density Matrices ◽  
Double Cosets ◽  
Unitary Transformations ◽  
One To One ◽  
Local Unitary Invariance ◽  
Unitary Invariance

We give a one-to-one correspondence between classes of density matrices under local unitary invariance and the double cosets of unitary groups. We show that the interrelationship among classes of local unitary equivalent multi-partite mixed states is independent from the actual values of the eigenvalues and only depends on the multiplicities of the eigenvalues. The interpretation in terms of homogeneous spaces of unitary groups is also discussed.

scholarly journals Interacting Subsystems and Their Molecular Ensembles

2020 ◽  
pp. 25-30 ◽  
Author(s):  
Roman F. Nalewajski
Keyword(s):  
Molecular System ◽  
Partition Problem ◽  
Mixed States ◽  
Density Matrices ◽  
Information Theoretic ◽  
Density Operators ◽  
Division Rule ◽  
Pure Quantum State ◽  
Stockholder Atoms ◽  
Ensemble Representations

The molecular density-partition problem is reexamined and the information-theoretic (IT) justification of the stockholder division rule is summarized. The ensemble representations of the promolecular and molecular mixed states of constituent atoms are identified and the electron probabilities in the isoelectronic stockholder atoms-in-molecules (AIM) are used to define the molecular-orbital ensembles for the bonded Hirshfeld atoms. In the pure quantum state of the whole molecular system its interacting (entangled) fragments are described by the subsystem density operators, with the subsystem physical properties being generated by the partial traces involving the fragment density matrices.

scholarly journals Quantum Harmonic Analysis of the Density Matrix

Quanta ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 74 ◽  
Author(s):  
Maurice A. De Gosson
Keyword(s):  
Density Matrix ◽  
Harmonic Analysis ◽  
Quantum Mechanical ◽  
Mixed States ◽  
Density Matrices ◽  
Planck’S Constant ◽  
Planck's Constant

We will study rigorously the notion of mixed states and their density matrices. We will also discuss the quantum-mechanical consequences of possible variations of Planck's constant h. This review has been written having in mind two readerships: mathematical physicists and quantum physicists. The mathematical rigor is maximal, but the language and notation we use throughout should be familiar to physicists.Quanta 2018; 7: 74–110.

Bipartite entanglement of decohered mixed states generated from maximally entangled cluster states

2021 ◽  
Vol 36 (03) ◽  
pp. 2150010
Author(s):  
Mostafa Mansour ◽  
Saeed Haddadi
Keyword(s):  
Mixed State ◽  
Entangled States ◽  
Entangled State ◽  
Mixed States ◽  
Density Matrices ◽  
Bipartite Entanglement ◽  
Cluster States ◽  
Maximally Entangled State ◽  
Qubit System ◽  
Maximally Entangled States

In this work, we investigate the bipartite entanglement of decohered mixed states generated from maximally entangled cluster states of [Formula: see text] qubits physical system. We introduce the disconnected cluster states for an ensemble of [Formula: see text] non-interacting qubits and we give the corresponding separable density matrices. The maximally entangled states can be generated from disconnected cluster states, by assuming that the dynamics of the multi-qubit system is governed by a quadratic Hamiltonian of Ising type. When exposed to a local noisy interaction with the environment, the multi-qubit system evolves from its initial pure maximally entangled state to a decohered mixed state. The decohered mixed states generated from bipartite, tripartite and multipartite maximally entangled cluster states are explicitly expressed and their bipartite entanglements are investigated.

ENTANGLEMENT OF FORMATION FOR A CLASS OF SPECIAL QUANTUM STATES

2007 ◽  
Vol 05 (03) ◽  
pp. 343-352 ◽  
Author(s):  
HUI ZHAO ◽  
ZHI-XI WANG
Keyword(s):  
Lower Bound ◽  
Large Class ◽  
Special Kind ◽  
Quantum States ◽  
The Other ◽  
High Dimensional ◽  
Entangled States ◽  
Mixed States ◽  
Density Matrices ◽  
Entanglement Of Formation

The entanglement of formation for a class of high-dimensional quantum mixed states is investigated. A special kind of D-computable states is defined and the lower bound of entanglement of formation for a large class of density matrices whose decompositions lie in these D-computable quantum states is obtained. Moreover we present a kind of construction for this special state which is defined by a class of special matrices with two non-zero different eigenvalues and the other eigenvalues are zero. Making use of the D-computable we construct a class of bound entangled states.

k-uniform maximally mixed states from multi-qudit phase states

2019 ◽  
Vol 34 (19) ◽  
pp. 1950151 ◽  
Author(s):  
Mostafa Mansour ◽  
Mohammed Daoud
Keyword(s):  
Special Kind ◽  
Mixed States ◽  
Density Matrices ◽  
Finite Dimensional ◽  
Heisenberg Type ◽  
Multipartite System ◽  
Multipartite States ◽  
Phase Operators

This paper concerns the construction of k-uniform maximally mixed multipartite states by using the formalism of phase states for finite dimensional systems (qudits). The k-uniform states are a special kind of entangled (n)-qudits states, such that after tracing out arbitrary (n[Formula: see text]k) subsystems, the remaining (k) subsystems are maximally mixed. We recall some basic elements about unitary phase operators of a multi-qudit system and we give the corresponding separable density matrices. Evolved density matrices arise when qudits of the multipartite system are allowed to interact via an Hamiltonian of Heisenberg type. The expressions of maximally mixed states are explicitly derived from multipartite evolved phase states and their properties are discussed.

scholarly journals A Gramian approach to entanglement in bipartite finite dimensional systems: the case of pure states

2020 ◽  
Vol 20 (13&14) ◽  
pp. 1081-1108
Author(s):  
Roman Gielerak ◽  
Marek Sawerwain
Keyword(s):  
Quantitative Measure ◽  
Point Of View ◽  
Gram Matrix ◽  
Matrix Structure ◽  
Matrix Approach ◽  
Mixed States ◽  
Density Matrices ◽  
Finite Dimensional ◽  
Pure States ◽  
Reduced Density

It has been observed that the reduced density matrices of bipartite qudit pure states possess a Gram matrix structure. This observation has opened a possibility of analysing the entanglement in such systems from the purely geometrical point of view. In particular, a new quantitative measure of an entanglement of the geometrical nature, has been proposed. Using the invented Gram matrix approach, a version of a non-linear purification of mixed states describing the system analysed has been presented.

Quantum statistical mechanics

2021 ◽  
pp. 179-216
Author(s):  
James P. Sethna
Keyword(s):  
Black Holes ◽  
Statistical Mechanics ◽  
Neutron Stars ◽  
White Dwarfs ◽  
Quantum Statistical Mechanics ◽  
Bose Condensation ◽  
Mixed States ◽  
Density Matrices ◽  
Materials Properties ◽  
Quantum Statistical

Quantum statistical mechanics governs metals, semiconductors, and neutron stars. Statistical mechanics spawned Planck’s invention of the quantum, and explains Bose condensation, superfluids, and superconductors. This chapter briefly describes these systems using mixed states, or more formally density matrices, and introducing the properties of bosons and fermions. We discuss in unusual detail how useful descriptions of metals and superfluids can be derived by ignoring the seemingly important interactions between their constituent electrons and atoms. Exercises explore how gregarious bosons lead to superfluids and lasers, how unsociable fermions explain transitions between white dwarfs, neutron stars, and black holes, how one calculates materials properties in semiconductors, insulators, and metals, and how statistical mechanics can explain the collapse of the quantum wavefunction during measurement.

AN EFFICIENT SEPARABILITY CRITERION FOR n-PARTITE ARBITRARILY DIMENSIONAL QUANTUM STATES

2011 ◽  
Vol 09 (04) ◽  
pp. 1101-1112
Author(s):  
YINXIANG LONG ◽  
DAOWEN QIU ◽  
DONGYANG LONG
Keyword(s):  
Hilbert Space ◽  
Quantum State ◽  
Pure State ◽  
Coefficient Matrix ◽  
Quantum States ◽  
Mixed States ◽  
Density Matrices ◽  
Separability Criterion ◽  
Dimensional Hilbert Space ◽  
Pure States

In this paper, we obtain an efficient separability criterion for bipartite quantum pure state systems, which is based on the two-order minors of the coefficient matrix corresponding to quantum state. Then, we generalize this criterion to multipartite arbitrarily dimensional pure states. Our criterion is directly built upon coefficient matrices, but not density matrices or observables, so it has the advantage of being computed easily. Indeed, to judge separability for an arbitrary n-partite pure state in a d-dimensional Hilbert space, it only needs at most O(d) times operations of multiplication and comparison. Our criterion can be extended to mixed states. Compared with Yu's criteria, our methods are faster, and can be applied to any quantum state.

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