scholarly journals Complete Pick positivity and unitary invariance

2010 ◽  
Vol 200 (2) ◽  
pp. 149-162
Author(s):  
Angshuman Bhattacharya ◽  
Tirthankar Bhattacharyya
Keyword(s):  
Unitary Invariance

scholarly journals Integrable systems with unitary invariance from non-stretching geometric curve flows in the Hermitian symmetric space Sp(n)/U(n)

2015 ◽  
Vol 38 ◽  
pp. 1560071 ◽  
Author(s):  
Stephen C. Anco ◽  
Esmaeel Asadi ◽  
Asieh Dogonchi
Keyword(s):  
Symmetric Space ◽  
Integrable Systems ◽  
Hermitian Symmetric Space ◽  
Complex Matrix ◽  
Curve Flows ◽  
Frame Method ◽  
Parallel Frame ◽  
Natural Way ◽  
Sg Equation ◽  
Unitary Invariance

A moving parallel frame method is applied to geometric non-stretching curve flows in the Hermitian symmetric space Sp(n)/U(n) to derive new integrable systems with unitary invariance. These systems consist of a bi-Hamiltonian modified Korteweg-de Vries equation and a Hamiltonian sine-Gordon (SG) equation, involving a scalar variable coupled to a complex vector variable. The Hermitian structure of the symmetric space Sp(n)/U(n) is used in a natural way from the beginning in formulating a complex matrix representation of the tangent space 𝔰𝔭(n)/𝔲(n) and its bracket relations within the symmetric Lie algebra (𝔲(n), 𝔰𝔭(n)).

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.

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 Unitary invariance and spectral variation

1987 ◽  
Vol 95 ◽  
pp. 43-68 ◽  
Author(s):  
Rajendra Bhatia ◽  
John A.R. Holbrook
Keyword(s):  
Spectral Variation ◽  
Unitary Invariance
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