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.

Lie groups and unitary transformations in ligand field theory

1983 ◽  
Vol 23 (1) ◽  
pp. 169-183 ◽  
Author(s):  
Chia-Chung Sun ◽  
Yan-De Han ◽  
Be-Fu Li ◽  
Qian-Shu Li
Keyword(s):  
Field Theory ◽  
Lie Groups ◽  
Ligand Field ◽  
Ligand Field Theory ◽  
Unitary Transformations

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.

Uncertainty Relation: From Inequality to Equality

1997 ◽  
Vol 52 (1-2) ◽  
pp. 49-52 ◽  
Author(s):  
Georg Süssmann
Keyword(s):  
Phase Space ◽  
Quantum State ◽  
Uncertainty Relation ◽  
The Other ◽  
Mixed States ◽  
Von Neumann ◽  
Other Hand ◽  
Pure Quantum State ◽  
Minimum Uncertainty

Abstract The uncertainty area δ (p, q): - [∫ W(p, q)2 dp dq] - 1 is proposed in place of δ p • δ q, and it is shown that each pure quantum state is a minimum uncertainty state in this sense: δ (p, q) = 2 π ħ. For mixed states, on the other hand, δ(p, q) > 2π ħ. In a phase space of 2F(=6N) dimensions, S: = k B • log[δF (p,q)/(2 π ħ)F] whit δF (p,q):= [∫ W(p, q)2 dF p dF q]-1 is considered as an alternative to von Neumann`s entropy S̃:= kB • trc [ρ̂ log (ρ̂-1)].

scholarly journals Least number of n-periodic points of self-maps of $$PSU(2)\times PSU(2)$$

2021 ◽  
Vol 24 (1) ◽  
Author(s):  
Jerzy Jezierski
Keyword(s):  
Lie Groups ◽  
Explicit Form ◽  
Homotopy Class ◽  
Unitary Group ◽  
Compact Manifold ◽  
Periodic Points ◽  
Necessary Condition ◽  
Minimal Realization ◽  
Compact Lie Groups

AbstractLet $$f:M\rightarrow M$$ f : M → M be a self-map of a compact manifold and $$n\in {\mathbb {N}}$$ n ∈ N . In general, the least number of n-periodic points in the smooth homotopy class of f may be much bigger than in the continuous homotopy class. For a class of spaces, including compact Lie groups, a necessary condition for the equality of the above two numbers, for each iteration $$f^n$$ f n , appears. Here we give the explicit form of the graph of orbits of Reidemeister classes $$\mathcal {GOR}(f^*)$$ GOR ( f ∗ ) for self-maps of projective unitary group PSU(2) and of $$PSU(2)\times PSU(2)$$ P S U ( 2 ) × P S U ( 2 ) satisfying the necessary condition. The structure of the graphs implies that for self-maps of the above spaces the necessary condition is also sufficient for the smooth minimal realization of n-periodic points for all iterations.

Topological aspects of the projective unitary group

1970 ◽  
Vol 68 (1) ◽  
pp. 57-60 ◽  
Author(s):  
D. J. Simms
Keyword(s):  
Hilbert Space ◽  
Topological Group ◽  
Projective Space ◽  
Unitary Group ◽  
Principal Bundle ◽  
Strong Operator Topology ◽  
Complex Hilbert Space ◽  
Unitary Representations ◽  
Strong Operator ◽  
Unitary Transformations

1. Introduction. The group U(H) of unitary transformations of a complex Hilbert space H, endowed with its strong operator topology, is of interest in the study of unitary representations of a topological group. The unitary transformations of H induce a group U(Ĥ) of transformations of the associated projective space Ĥ. The projective unitary group U(Ĥ) with its strong operator topology is used in the study of projective (ray) representations. U(Ĥ) is, as a group, the quotient of U(H) by the subgroup S1 of scalar multiples of the identity. In this paper we prove that the strong operator toplogy of U(Ĥ) is in fact the quotient of the strong operator topology on U(H). This is related to the fact that U(H) is a principal bundle over U(Ĥ) with fibre S.

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