Complete Set Of Invariants Research Articles

A complete set of local invariants for a family of multipartite mixed states

We study the equivalence of quantum states under local unitary transformations by using the singular value decomposition. A complete set of invariants under local unitary transformations is presented for several classes of tripartite mixed states in composite systems. Two density matrices in the same class are equivalent under local unitary transformations if and only if all these invariants have equal values for these density matrices.

Local invariants for a finite multipartite quantum system

For a finite-dimensional multipartite quantum system a finite complete set of invariants under local unitary transformations is described using Lie algebraic tools.

Invariants for a Class of Nongeneric Three-QubitStates

We investigate the equivalence of quantum states under localunitary transformations. A complete set of invariants under localunitary transformations are presented for a class of non-genericthree-qubit mixed states. It is shown that two such states in thisclass are locally equivalent if and only if all these invariantshave equal values for them.

CLIFFORD CLASSES OF PRIMITIVE CENTRAL SIMPLE G-ALGEBRAS

The problem of determining a complete set of invariants for characterizing the Clifford class of a central simple G-algebra over a field K is studied under the assumption that the algebra is simple ring. This is successful in the case of interior central simple G-algebras over K, and also when the inertia subgroup of the G-algebra is a direct factor of G. As an application of the latter, it is shown that if G is a finite abelian group, L is a Galois extension of K with Galois group G, and E is a G-algebra isomorphic to EndKV for some KG-module V, then the skew group ring of G over L ⊗K E is a split matrix algebra over K.

Local Invariants for a Class of Tripartite Quantum MixedStates

We study the equivalence of tripartite mixed states under localunitary transformations. The nonlocal properties for a class oftripartite quantum states inℂK⊗ℂM⊗ℂNcomposite systems are investigated and a complete set ofinvariants under local unitary transformations for these states ispresented. It is shown that two of these states are locallyequivalent if and only if all these invariants have the same values.

EQUIVALENCE OF TRIPARTITE QUANTUM STATES UNDER LOCAL UNITARY TRANSFORMATIONS

The equivalence of tripartite pure states under local unitary transformations is investigated. The nonlocal properties for a class of tripartite quantum states in ℂK ⊗ ℂM ⊗ ℂN composite systems are investigated and a complete set of invariants under local unitary transformations for these states is presented. It is shown that two of these states are locally equivalent if and only if all these invariants have the same values.

Local invariants for a class of mixed states

We investigate the equivalence of quantum states under local unitary transformations. A complete set of invariants under local unitary transformations is presented for a class of mixed states. It is shown that two states in this class are locally equivalent if and only if all these invariants have equal values for them.

Nonlocal properties and local invariants for bipartite systems

The nonlocal properties for a kind of generic N-dimensional bipartite quantum systems are investigated. A complete set of invariants under local unitary transformations is presented. It is shown that two generic density matrices are locally equivalent if and only if all these invariants have equal values in these density matrices.

Trigonometry of the quantum state space, geometric phases and relative phases

A complete set of invariants for three states in the quantum space of states is obtained together with a complete set of relationships linking them. This is done in a way that preserves the self-duality of and leads to a clear geometric description of invariants (distances, lateral phases; Hermitian angles, angular phases; and two purely triangular phases). Some of these invariants appear here for the first time. Symplectic area (and hence the triangle geometric phase) is proportional to a 'mixed phase excess', thus extending to the relation 'area-angular excess' in the real sphere. The new triangle lateral phases provide a description, intrinsic to , of relative phases in a superposition. This approach also provides closed expressions for the triangle holonomy associated with the usual Fubini–Study metric in , as well as many other expressions for similar 'loop' operators along the triangle, including closed and exact expressions for the triangle Aharonov–Anandan geometric phase.

Using Invariants to Solve the Equivalence Problem for Ordinary Differential Equations

Two given ordinary differential equations (ODEs) are called equivalent if one can be transformed into the other by a change of variables. The equivalence problem consists of two parts: deciding equivalence and determining a transformation that connects the ODEs. Our motivation for considering this problem is to translate a known solution of an ODE to solutions of ODEs which are equivalent to it, thus allowing a systematic use of collections of solved ODEs. In general, the equivalence problem is considered to be solved when a complete set of invariants has been found. In practice, using invariants to solve the equivalence problem for a given class of ODEs may require substantial computational effort. Using Tresse's invariants for second order ODEs as a starting point, we present an algorithmic method to solve the equivalence problem for the case of no or one symmetry. The method may be generalized in principle to a wide range of ODEs for which a complete set of invariants is known. Considering Emden-Fowler Equations as an example, we derive algorithmically equivalence criteria as well as special invariants yielding equivalence transformations.

The complex representation of algebraic curves and its simple exploitation for pose estimation and invariant recognition

Representations are introduced for handling 2D algebraic curves (implicit polynomial curves) of arbitrary degree in the scope of computer vision applications. These representations permit fast, accurate pose-independent shape recognition under Euclidean transformations with a complete set of invariants, and fast accurate pose-estimation based on all the polynomial coefficients. The latter is accomplished by a centering of a polynomial based on its coefficients, followed by rotation estimation by decomposing polynomial coefficient space into a union of orthogonal subspaces for which rotations within two-dimensional subspaces or identity transformations within one-dimensional subspaces result from rotations in x, y measured-data space. Angles of these rotations in the two-dimensional coefficient subspaces are proportional to each other and are integer multiples of the rotation angle in the x, y data space. By recasting this approach in terms of a complex variable, i.e., x+iy=z, and complex polynomial-coefficients, further conceptual and computational simplification results. Application to shape-based indexing into databases is presented to illustrate the usefulness and the robustness of the complex representation of algebraic curves.

Limits of generalized state space systems under proportional and derivative feedback

In this paper we study the high-gain feedback classification problem for generalized state space systems. We solve this problem for proportional and derivative feedback transformations of regularizable systems, i.e., we give necessary and sufficient conditions for a regularizable system to be a limit of a given system under high-gain proportional and derivative feedback. We also derive a new complete set of invariants for proportional feedback equivalence and specify a set of necessary conditions for a system to be the limit of another system under these feedback transformations. The necessary conditions are sufficient for arbitrary state space systems and for controllable singular systems.

On complete systems of invariants for small graphs

We define a small graph to be one with n ≤ 6 nodes. The celebrated Graph Isomorphism Problem (GIP) consists in deciding whether or not two given graphs are isomorphic, i.e., whether there is a bijective mapping from the nodes of one graph onto those of the other such that adjacency is preserved. An interesting algorithmic approach to graph isomorphism problem uses the “code” (sometimes called a complete system of invariants). Following this approach, two graphs are isomorphic if and only if they have the same code. We propose several complete sets of invariants to settle the GIP for small graphs. Note that no complete system of invariants for a graph is known, except for those that are equivalent to the entire adjacency matrix or list of adjacencies.

Feedback invariants for linear dynamical systems over a principal ideal domain

This paper studies the action of the feedback group F n,m on m-input, n-dimensional reachable linear dynamical systems over a principal ideal domain R. For such a 2-dimensional system ∑ a complete set of invariants is given which characterizes the feedback class of ∑. In particular it is characterized, in terms of these invariants, when ∑ has the feedback cyclization property. The particular cases R = Z and R = R[ X] are studied in some detail. Finally, when n is arbitrary, the feedback classification is given for the class of reachable systems ∑ = ( F, G) such that G is a matrix with at least n − 1 invariant factors equal to one.

Affine differential geometry of surfaces in ?4

Employing the method of moving frames, i.e. Cartan's algorithm, we find a complete set of invariants for nondegenerate oriented surfacesM2 in ℝ4 relative to the action of the general affine group on ℝ4. The invariants found include a normal bundle, a quadratic form onM2 with values in the normal bundle, a symmetric connection onM2 and a connection on the normal bundle. Integrability conditions for these invariants are also determined. Geometric interpretations are given for the successive reductions to the bundle of affine frames overM2, obtained by using the method of moving frames, that lead to the aforementioned invariants. As applications of these results we study a class of surfaces known as harmonic surfaces, finding for them a complete set of invariants and their integrability conditions. Further applications involve the study of homogeneous surfaces; these are surfaces which are fixed by a group of affine transformations that act transitively on the surface. All homogeneous harmonic surfaces are determined.

Translation Invariants for Periodic Denjoy-Carleman Classes

The Denjoy-Carleman classes in real ${C^\infty }(\mathbb {R}/\mathbb {Z})$ on which the derivative sequence ${f^{(n)}}(x)$ at any point is a complete set of invariants are exactly the ones on which the integrals of products of derivatives ${f^{({n_1})}} \cdots {f^{({n_r})}}$ are a complete set of invariants up to translation.

Translation invariants for periodic Denjoy-Carleman classes

The Denjoy-Carleman classes in real C ∞ ( R / Z ) {C^\infty }(\mathbb {R}/\mathbb {Z}) on which the derivative sequence f ( n ) ( x ) {f^{(n)}}(x) at any point is a complete set of invariants are exactly the ones on which the integrals of products of derivatives f ( n 1 ) ⋯ f ( n r ) {f^{({n_1})}} \cdots {f^{({n_r})}} are a complete set of invariants up to translation.

Synergism between layer cracking and delaminations in multidirectional laminates of carbon-fibre-reinforced epoxy

Photographs of different crank modes are shown giving insight into the damage behaviour of composites fabricated of T300/914C plies of 0·125 mm thickness. An increase in the damage tolerance was observed for laminates stacked by quasi-unidirectional [ ±2] angle plies. For cross-ply laminates under tension fatigue loading, test results, supplied for the spacings of layer cracks, are used to generate formulae for the residual layer stiffnesses. As an extension to the common energy release rate (ERR) concept an ERR tensor is established. The first invariant of this tensor and its mean values describe the energy release rate and its partitioning into different crack modes, respectively. From the complete set of invariants a first estimation of a crack condition for the delamination growth is proposed which includes the effect of fibre orientation.

Controllability and observability indices for linear time-invariant systems: A new approach

This paper is devoted to an investigation of the controllability and observability indices for linear time-invariant discrete-time systems. These indices are here introduced at an intrinsic level in the context of the behavioral approach to systems theory. For the class of the complete controllable systems it is proven that the controllability indices can be interpreted as a minimal and complete set of invariants with respect to a very natural systems equivalence. Autonomous phenomena are also discussed.

Simultaneous reduction of a complex skew matrix and a Hermitian matrix

A canonical form and a complete set of invariants are obtained for the simultaneous reduction of two complex matrices, one of which is skew symmetric and the other Hermitian.