量子码的多项式不变量

On Symmetric SL-Invariant Polynomials in Four Qubits

We find the generating set of SL-invariant polynomials in four qubits that are also invariant under permutations of the qubits. The set consists of four polynomials of degrees 2,6,8, and 12, for which we find an elegant expression in the space of critical states. In addition, we show that the Hyperdeterminant in four qubits is the only SL-invariant polynomial (up to powers of itself) that is non-vanishing precisely on the set of generic states.

高秩矩阵的不变恒等式和多项式恒等式

Invariant and polynomial identities for higher rank matrices

我们表现出明确的表达式，在组件，判别式，决定因素，特征多项式和多项式身份矩阵的秩较高。我们定义了置换张量，并根据它们构造了判别式和行列式作为$d$阶的判别式，其中$d$是矩阵的维数。由此得到了高秩矩阵的特征多项式和Cayley-Hamilton定理。

Generalized Alexander Polynomial Invariants

We propose an algorithm which allows to derive the generalized Alexander polynomial invariants of knots and links with the help of the q,p-numbers, appearing in bosonic two-parameter quantum algebra. These polynomials turn into HOMFLY ones by applying special parametrization. The Jones polynomials can be also obtained by using this algorithm.

Alexander Polynomial Invariants of Torus Knots T(n,3) and Chebyshev Polynomials

The explicit formula, which expresses the Alexander polynomials Δ_{n,3}(t) of torus knots T(n,3) as a sum of the Alexander polynomials Δ_{k,2}(t) of torus knots T(k,2), is found. Using this result and those from our previous papers, we express the Alexander polynomials Δ_{n,3}(t) through Chebyshev polynomials. The latter result is extended to general torus knots T(n,l) with n and l coprime.

On the ring of local polynomial invariants for a pair of entangled qubits

The entanglement characteristics of two qubits are encoded in the invariants of the adjoint action of SU(2) x SU(2) group on the space of density matrices defined as the space of positive semi-definite Hermitian matrices. The corresponding ring of polynomial invariants is studied. The special integrity basis for this ring is described and constraints on its elements due to the positive semi-definiteness of density matrices are given explicitly in the form of polynomial inequalities. The suggested basis is characterized by the property that only a minimal number of invariants, namely two primar…

Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials

Two sets of infinitely many exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials are presented. They are derived as the eigenfunctions of shape invariant and thus exactly solvable quantum mechanical Hamiltonians, which are deformations of those for the Wilson and Askey-Wilson polynomials in terms of a degree \ell (\ell=1,2,...) eigenpolynomial. These polynomials are exceptional in the sense that they start from degree \ell\ge1 and thus not constrained by any generalisation of Bochner's theorem.

Rationally extended shape invariant potentials in arbitrary D-dimensions associated with exceptional $X_m$ polynomials

Rationally extended shape invariant potentials in arbitrary D-dimensions are obtained by using point canonical transformation (PCT) method. The bound-state solutions of these exactly solvable potentials can be written in terms of X_m Laguerre or X_m Jacobi exceptional orthogonal polynomials. These potentials are isospectral to their usual counterparts and possess translationally shape invariance property.

Infinitely many shape invariant potentials and new orthogonal polynomials

Three sets of exactly solvable one-dimensional quantum mechanical potentials are presented. These are shape invariant potentials obtained by deforming the radial oscillator and the trigonometric/hyperbolic Pöschl-Teller potentials in terms of their degree \ell polynomial eigenfunctions. We present the entire eigenfunctions for these Hamiltonians (\ell=1,2,...) in terms of new orthogonal polynomials. Two recently reported shape invariant potentials of Quesne and Gómez-Ullate et al's are the first members of these infinitely many potentials.