scholarly journals From the Jordan Product to Riemannian Geometries on Classical and Quantum States

Entropy ◽  
2020 ◽  
Vol 22 (6) ◽  
pp. 637 ◽  
Author(s):  
Florio M. Ciaglia ◽  
Jürgen Jost ◽  
Lorenz Schwachhöfer
Keyword(s):  
Riemannian Metric ◽  
Riemann Tensor ◽  
Linear Operators ◽  
Metric Tensor ◽  
Jordan Product ◽  
C Algebra ◽  
Finite Dimensional ◽  
Riemannian Geometries ◽  
Gns Construction ◽  
Metric Tensors

The Jordan product on the self-adjoint part of a finite-dimensional C * -algebra A is shown to give rise to Riemannian metric tensors on suitable manifolds of states on A , and the covariant derivative, the geodesics, the Riemann tensor, and the sectional curvature of all these metric tensors are explicitly computed. In particular, it is proved that the Fisher–Rao metric tensor is recovered in the Abelian case, that the Fubini–Study metric tensor is recovered when we consider pure states on the algebra B ( H ) of linear operators on a finite-dimensional Hilbert space H , and that the Bures–Helstrom metric tensors is recovered when we consider faithful states on B ( H ) . Moreover, an alternative derivation of these Riemannian metric tensors in terms of the GNS construction associated to a state is presented. In the case of pure and faithful states on B ( H ) , this alternative geometrical description clarifies the analogy between the Fubini–Study and the Bures–Helstrom metric tensor.

Global Perturbation of Nonlinear Eigenvalues

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Julián López-Gómez ◽  
Juan Carlos Sampedro
Keyword(s):  
Perturbation Theory ◽  
Spectral Parameter ◽  
Classical Theory ◽  
General Setting ◽  
Perturbation Parameter ◽  
Linear Operators ◽  
Fredholm Operators ◽  
Index Zero ◽  
Finite Dimensional ◽  
Nonlinear Eigenvalues

Abstract This paper generalizes the classical theory of perturbation of eigenvalues up to cover the most general setting where the operator surface 𝔏 : [ a , b ] × [ c , d ] → Φ 0 ⁢ ( U , V ) {\mathfrak{L}:[a,b]\times[c,d]\to\Phi_{0}(U,V)} , ( λ , μ ) ↦ 𝔏 ⁢ ( λ , μ ) {(\lambda,\mu)\mapsto\mathfrak{L}(\lambda,\mu)} , depends continuously on the perturbation parameter, μ, and holomorphically, as well as nonlinearly, on the spectral parameter, λ, where Φ 0 ⁢ ( U , V ) {\Phi_{0}(U,V)} stands for the set of Fredholm operators of index zero between U and V. The main result is a substantial extension of a classical finite-dimensional theorem of T. Kato (see [T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Class. Math., Springer, Berlin, 1995, Chapter 2, Section 5]).

scholarly journals Nuclear subalgebras of UHF C*-algebras

1986 ◽  
Vol 29 (1) ◽  
pp. 97-100 ◽  
Author(s):  
R. J. Archbold ◽  
Alexander Kumjian
Keyword(s):  
Tensor Product ◽  
Inductive Limit ◽  
Inductive Limits ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
The Family ◽  
Algebraic Tensor Product

A C*-algebra A is said to be approximately finite dimensional (AF) if it is the inductive limit of a sequence of finite dimensional C*-algebras(see [2], [5]). It is said to be nuclear if, for each C*-algebra B, there is a unique C*-norm on the *-algebraic tensor product A ⊗B [11]. Since finite dimensional C*-algebras are nuclear, and inductive limits of nuclear C*-algebras are nuclear [16];,every AF C*-algebra is nuclear. The family of nuclear C*-algebras is a large and well-behaved class (see [12]). The AF C*-algebras for a particularly tractable sub-class which has been completely classified in terms of the invariant K0 [7], [5].

scholarly journals Boundary problems for Riccati and Lyapunov equations

1986 ◽  
Vol 29 (1) ◽  
pp. 15-21 ◽  
Author(s):  
Lucas Jódar
Keyword(s):  
Boundary Condition ◽  
Boundary Problem ◽  
Transport Theory ◽  
Adjoint Operator ◽  
Linear Operators ◽  
Dimensional Case ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Image Position

The resolution problem of the systemwhere U(t), A, B, D and Uo are bounded linear operators on H and B* denotes the adjoint operator of B, arises in control theory, [9], transport theory, [12], and filtering problems, [3]. The finite-dimensional case has been introduced in [6,7], and several authors have studied the infinite-dimensional case, [4], [13], [18]. A recent paper, [17],studies the finite dimensional boundary problemwhere t ∈[0,b].In this paper we consider the more general boundary problemwhere all operators which appear in (1.2) are bounded linear operators on a separable Hilbert space H. Note that we do not suppose C = −B* and the boundary condition in (1.2) is more general than the boundary condition in (1.1).

scholarly journals CLASSICAL TENSORS AND QUANTUM ENTANGLEMENT I: PURE STATES

2010 ◽  
Vol 07 (03) ◽  
pp. 485-503 ◽  
Author(s):  
P. ANIELLO ◽  
J. CLEMENTE-GALLARDO ◽  
G. MARMO ◽  
G. F. VOLKERT
Keyword(s):  
Hilbert Space ◽  
Symplectic Geometry ◽  
Vector Fields ◽  
Riemannian Metric ◽  
Inner Product ◽  
Tensor Fields ◽  
Metric Tensor ◽  
Separable States ◽  
Pure States ◽  
Complex Projective

The geometrical description of a Hilbert space associated with a quantum system considers a Hermitian tensor to describe the scalar inner product of vectors which are now described by vector fields. The real part of this tensor represents a flat Riemannian metric tensor while the imaginary part represents a symplectic two-form. The immersion of classical manifolds in the complex projective space associated with the Hilbert space allows to pull-back tensor fields related to previous ones, via the immersion map. This makes available, on these selected manifolds of states, methods of usual Riemannian and symplectic geometry. Here, we consider these pulled-back tensor fields when the immersed submanifold contains separable states or entangled states. Geometrical tensors are shown to encode some properties of these states. These results are not unrelated with criteria already available in the literature. We explicitly deal with some of these relations.

ON THE RECOVERY OF A MANIFOLD WITH PRESCRIBED METRIC TENSOR

2003 ◽  
Vol 01 (04) ◽  
pp. 433-453 ◽  
Author(s):  
CRISTINEL MARDARE
Keyword(s):  
Differential Geometry ◽  
Classical Theorem ◽  
Compatibility Conditions ◽  
Metric Tensor ◽  
Distributional Sense ◽  
Metric Tensors

A classical theorem in differential geometry asserts that if a C2-metric tensor satisfies the Riemann compatibility conditions, then it is induced by an immersion. We prove that this theorem still holds true for C1-metric tensors satisfying the Riemann compatibility conditions in a distributional sense.

scholarly journals Infinite dimensional generalizations of Choi’s Theorem

2019 ◽  
Vol 7 (1) ◽  
pp. 67-77
Author(s):  
Shmuel Friedland
Keyword(s):  
Hilbert Spaces ◽  
Quantum Channel ◽  
Trace Class ◽  
Natural Generalization ◽  
Linear Operators ◽  
Completely Positive Map ◽  
Completely Positive ◽  
Bounded Linear ◽  
Positive Map ◽  
Finite Dimensional

Abstract In this paper we give a simple sequence of necessary and sufficient finite dimensional conditions for a positive map between certain subspaces of bounded linear operators on separable Hilbert spaces to be completely positive. These criterions are natural generalization of Choi’s characterization for completely positive maps between pairs of linear operators on finite dimensional Hilbert spaces. We apply our conditions to a completely positive map between two trace class operators on separable Hilbert spaces. A completely positive map μ is called a quantum channel, if it is trace preserving, and μ is called a quantum subchannel if it decreases the trace of a positive operator.We give simple neccesary and sufficient condtions for μ to be a quantum subchannel.We show that μ is a quantum subchannel if and only if it hasHellwig-Kraus representation. The last result extends the classical results of Kraus and the recent result of Holevo for characterization of a quantum channel.

Range-compatible homomorphisms over the field with two elements

2018 ◽  
Vol 34 ◽  
pp. 71-114
Author(s):  
Clément De Seguins Pazzis
Keyword(s):  
Linear Subspace ◽  
Linear Operators ◽  
Vector Spaces ◽  
Operator Spaces ◽  
Affine Subspaces ◽  
Finite Dimensional ◽  
Reflexive Operator ◽  
Recent Classification ◽  
Special Case

Let U and V be finite-dimensional vector spaces over a field K, and S be a linear subspace of the space L(U, V ) of all linear operators from U to V. A map F : S → V is called range-compatible when F(s) ∈ Im s for all s ∈ S. Previous work has classified all the range-compatible group homomorphisms provided that codimL(U,V )S ≤ 2 dim V − 3, except in the special case when K has only two elements and codimL(U,V )S = 2 dim V − 3. This article gives a thorough treatment of that special case. The results are partly based upon the recent classification of vector spaces of matrices with rank at most 2 over F2. As an application, the 2-dimensional non-reflexive operator spaces are classified over any field, and so do the affine subspaces of Mn,p(K) with lower-rank at least 2 and codimension 3.

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