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 Virtual Correlations in Single Qutrit

2016 ◽  
Vol 2016 ◽  
pp. 1-5
Author(s):  
Alexey A. Strakhov ◽  
Vladimir I. Man’ko
Keyword(s):  
Mixed State ◽  
Three Dimensional ◽  
Mixed States ◽  
One To One

We construct the positive invertible map of the mixed states of a single qutrit onto the antisymmetrized bipartite qutrit states (quasifermions). It is shown that using this one-to-one correspondence between qutrit states and states of two three-dimensional quasifermions one may attribute hidden entanglement to a single mixed state of qutrit.

Three-qubit Groverian measure

2008 ◽  
Vol 8 (10) ◽  
pp. 925-942
Author(s):  
E.-L. Jung ◽  
M.-R. Hwang ◽  
D. Park ◽  
L. Tamaryan ◽  
S. Tamaryan
Keyword(s):  
Geometric Interpretation ◽  
Qubit State ◽  
Phase Factor ◽  
Analytical Expression ◽  
Near Future ◽  
Analytical Expressions ◽  
Qubit States ◽  
Local Unitary Invariance ◽  
Unitary Invariance

The Groverian measures are analytically computed in various types of three-qubit states. The final results are also expressed in terms of local-unitary invariant quantities in each type. This fact reflects the manifest local-unitary invariance of the Groverian measure. It is also shown that the analytical expressions for various types have correct limits to other types. For some types (type 4 and type 5) we failed to compute the analytical expression of the Groverian measure in this paper. However, from the consideration of local-unitary invariants we have shown that the Groverian measure in type 4 should be independent of the phase factor $\varphi$, which appear in the three-qubit state $|\psi \rangle$. This fact with geometric interpretation on the Groverian measure may enable us to derive the analytical expressions for general arbitrary three-qubit states in near future.

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.

scholarly journals Singular fibres of the Gelfand–Cetlin system on MANI( n )*

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170423 ◽  
Author(s):  
D. Bouloc ◽  
E. Miranda ◽  
N.T. Zung
Keyword(s):  
Lie Groups ◽  
Homogeneous Spaces ◽  
Lagrangian Submanifolds ◽  
Unitary Groups ◽  
Direct Products ◽  
Compact Lie Groups ◽  
Lie Groupoids ◽  
Finite Dimensional ◽  
Adjoint Orbits ◽  
Free Actions

In this paper, we show that every singular fibre of the Gelfand–Cetlin system on co-adjoint orbits of unitary groups is a smooth isotropic submanifold which is diffeomorphic to a two-stage quotient of a compact Lie group by free actions of two other compact Lie groups. In many cases, these singular fibres can be shown to be homogeneous spaces or even diffeomorphic to compact Lie groups. We also give a combinatorial formula for computing the dimensions of all singular fibres, and give a detailed description of these singular fibres in many cases, including the so-called (multi-)diamond singularities. These (multi-)diamond singular fibres are degenerate for the Gelfand–Cetlin system, but they are Lagrangian submanifolds diffeomorphic to direct products of special unitary groups and tori. Our methods of study are based on different ideas involving complex ellipsoids, Lie groupoids and also general ideas coming from the theory of singularities of integrable Hamiltonian systems. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

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.

scholarly journals Parametrizations of degenerate density matrices

2017 ◽  
Vol 29 (08) ◽  
pp. 1750026
Author(s):  
E. Brüning ◽  
S. Nagamachi
Keyword(s):  
Lie Algebra ◽  
Quotient Space ◽  
Homogeneous Spaces ◽  
Density Matrices

It turns out that a parametrization of degenerate density matrices requires a parametrization of [Formula: see text], [Formula: see text] where [Formula: see text] denotes the set of all unitary [Formula: see text]-matrices with complex entries. Unfortunately, the parametrization of this quotient space is quite involved. Our solution does not rely on Lie algebra methods directly, but succeeds through the construction of suitable sections for natural projections, by using techniques from the theory of homogeneous spaces. We mention the relation to the Lie algebra background and conclude with two concrete examples.

scholarly journals CLASSICAL TENSORS AND QUANTUM ENTANGLEMENT II: MIXED STATES

2011 ◽  
Vol 08 (04) ◽  
pp. 853-883 ◽  
Author(s):  
P. ANIELLO ◽  
J. CLEMENTE-GALLARDO ◽  
G. MARMO ◽  
G. F. VOLKERT
Keyword(s):  
Lie Groups ◽  
Quantum State ◽  
Unitary Group ◽  
Functional Dependence ◽  
Invariant Operator ◽  
Mixed States ◽  
Tensor Fields ◽  
Related Function ◽  
Invariant Functions ◽  
Unitary Transformations

Invariant operator-valued tensor fields on Lie groups are considered. These define classical tensor fields on Lie groups by evaluating them on a quantum state. This particular construction, applied on the local unitary group U(n) × U(n), may establish a method for the identification of entanglement monotone candidates by deriving invariant functions from tensors being by construction invariant under local unitary transformations. In particular, for n = 2, we recover the purity and a concurrence related function (Wootters 1998) as a sum of inner products of symmetric and anti-symmetric parts of the considered tensor fields. Moreover, we identify a distinguished entanglement monotone candidate by using a non-linear realization of the Lie algebra of SU(2) × SU(2). The functional dependence between the latter quantity and the concurrence is illustrated for a subclass of mixed states parametrized by two variables.

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