Symplectic Operators Research Articles

Nonlocality for continuous variable system with local symplectic operation

We study Bell's theorem for two-mode squeezed state with realizable operations in experiment. For the purpose, we suggest the Bell–CHSH operator with photon presence measurement using symplectic operation and displacement. The symplectic operation can be decomposed into phase shifter and squeezing operation in single mode. These operations are realizable experimentally in quantum optics. As a result, we obtain a larger degree of quantum nonlocality by local symplectic operation and displacement.

On Restricted Unitary Cayley Graphs and Symplectic Transformations Modulo $n$

We present some observations on a restricted variant of unitary Cayley graphs modulo $n$, and implications for a decomposition of elements of symplectic operators over the integers modulo $n$. We define quadratic unitary Cayley graphs $G_n$, whose vertex set is the ring ${\Bbb Z}_n$, and where residues $a,b$ modulo $n$ are adjacent if and only if their difference is a quadratic residue. By bounding the diameter of such graphs, we show an upper bound on the number of elementary operations (symplectic scalar multiplications, symplectic row swaps, and row additions or subtractions) required to decompose a symplectic matrix over ${\Bbb Z}_n$. We also characterize the conditions on $n$ for $G_n$ to be a perfect graph.

On the [formula omitted] polar decomposition of matrices

We present new results on the ϕ J polar decomposition of matrices. We show that every symplectic matrix may be written as a product of symplectic operation matrices. We present a simple form attained under symplectic equivalence, which makes it easy to determine if a matrix does not have a ϕ J polar decomposition. We also determine the rank 4 matrices with ϕ J polar decomposition.

Odd Hamiltonian structure for supersymmetric Sawada–Kotera equation

We study the supersymmetric N = 1 hierarchy connected with the Lax operator of the supersymmetric Sawada–Kotera equation. This operator produces the physical equations as well as the exotic equations with odd time. The odd Bi-Hamiltonian structure for the N = 1 supersymmetric Sawada–Kotera equation is defined. The product of the symplectic and implectic Hamiltonian operator gives us the recursion operator. In that way we prove the integrability of the supersymmetric Sawada–Kotera equation in the sense that it has the Bi-Hamiltonian structure. The so-called “quadratic” Hamiltonian operator of even order generates the exotic equations while the “cubic” odd Hamiltonian operator generates the physical equations.

Symplectic Dirac operators on Hermitian symmetric spaces

We describe the shape of the symplectic Dirac operators on Hermitian symmetric spaces. For this, we consider these operators as families of operators that can be handled more easily than the original ones.

Topology of broken Lefschetz fibrations and near-symplectic four-manifolds

The topology of broken Lefschetz fibrations is studied by means of handle decompositions. We consider a slight generalization of round handles and describe the handle diagrams for all that appear in dimension four. We establish simplified handlebody and monodromy representations for a certain subclass of broken Lefschetz fibrations and pencils, showing that all near-symplectic closed 4-manifolds can be supported by such objects, paralleling a result of Auroux, Donaldson and Katzarkov. Various constructions of broken Lefschetz fibrations and a generalization of the symplectic fiber sum operation to the near-symplectic setting are given. Extending the study of Lefschetz fibrations, we detect certain constraints on the symplectic fiber sum operation to result in a 4-manifold with nontrivial Seiberg�Witten invariant, as well as the self-intersection numbers that sections of broken Lefschetz fibrations can acquire.

Lenard scheme for two-dimensional periodic Volterra chain

We prove that for compatible weakly nonlocal Hamiltonian and symplectic operators, hierarchies of infinitely many commuting local symmetries and conservation laws can be generated under some easily verified conditions no matter whether the generating Nijenhuis operators are weakly nonlocal or not. We construct a recursion operator of the two-dimensional periodic Volterra chain from its Lax representation and prove that it is a Nijenhuis operator. Furthermore we show that this system is a (generalized) bi-Hamiltonian system. Rather surprisingly, the product of its weakly nonlocal Hamiltonian and symplectic operators gives rise to the square of the recursion operator.

INTEGRABLE SYSTEMS IN SYMPLECTIC GEOMETRY

AbstractQuaternionic vector mKDV equations are derived from the Cartan structure equation in the symmetric space=Sp(n+1)/Sp(1) ×Sp(n). The derivation of the soliton hierarchy utilizes a moving parallel frame and a Cartan connection 1-form ω related to the Cartan geometry onmodelled on$(\mk{sp}_{n+1}, \mk{sp}_{1}\,{\times}\, \mk{sp}_{n})$. The integrability structure is shown to be geometrically encoded by a Poisson–Nijenhuis structure and a symplectic operator.

Symplectic Dirac Operator and its Generalization

At an infinitesimal level, we will give a classification of 1 st order invariant differential operators acting on fields defined over contact projective geometries and having values in higher symplectic spinors. These fields are symplectic analogues of ordinary spinor fields in Riemannian geometry (the orthogonal case). In particular, we shall present symplectic analogues of Dirac, twistor and Rarita-Schwinger operators and other higher spinor operators in the realm of parabolic geometries of contact projective type.

Classification of 1st order symplectic spinor operators over contact projective geometries

We give a classification of 1st order invariant differential operators acting between sections of certain bundles associated to Cartan geometries of the so-called metaplectic contact projective type. These bundles are associated via representations, which are derived from the so-called higher symplectic (sometimes also called harmonic or generalized Kostant) spinor modules. Higher symplectic spinor modules are arising from the Segal–Shale–Weil representation of the metaplectic group by tensoring it by finite dimensional modules. We show that for all pairs of the considered bundles, there is at most one 1st order invariant differential operator up to a complex multiple and give an equivalence condition for the existence of such an operator. Contact projective analogues of the well known Dirac, twistor and Rarita–Schwinger operators appearing in Riemannian geometry are special examples of these operators.

The Transformation Law and Trace Formula on Theta Series Under Siegel Modular Group

The purposes of this article are to discuss the symplectic transformation laws on theta series and to give some explicit formulas for the trace of the symplectic operator.

A Birkhoff Contraction Formula with Applications to Riccati Equations

In this paper we show that the symplectic Hamiltonian operators on a Hilbert space give rise to linear fractional transformations on the open convex cone of positive definite operators that contract a natural invariant Finsler metric, the Thompson or part metric, on the convex cone. More precisely, the constants of contraction for the Hamiltonian operators satisfy the classical Birkhoff formula: the Lipschitz constant for the corresponding linear fractional transformations on the cone of positive definite operators is equal to the hyperbolic tangent of one fourth the diameter of the image. By means of the close connections between Hamilitonian operators and Riccati equations, this result and the associated machinery are applied to obtain convergence results for discrete algebraic Riccati equations and Riccati differential equations.

Symplectic splitting operator methods for the time-dependent Schrödinger equation

We present a family of symplectic splitting methods especially tailored to solve numerically the time-dependent Schrodinger equation. When discretized in time, this equation can be recast in the form of a classical Hamiltonian system with a Hamiltonian function corresponding to a generalized high-dimensional separable harmonic oscillator. The structure of the system allows us to build highly efficient symplectic integrators at any order. The new methods are accurate, easy to implement, and very stable in comparison with other standard symplectic integrators.

The Symplectic Semigroup and Riccati Differential Equations

In this paper, we study close connections that exist between the Riccati operator (differential) equation that arises in linear control systems and the symplectic group and its subsemigroup of symplectic Hamiltonian operators. A canonical triple factorization is derived for the symplectic Hamiltonian operators, and their closure under multiplication is deduced from this property. This semigroup of Hamiltonian operators, which we call the symplectic semigroup, is studied from the viewpoint of Lie semigroup theory, and resulting consequences for the theory of the Riccati equation are delineated. Among other things, these developments provide an elementary proof for the existence of a solution of the Riccati equation for all t ? 0 under rather general hypotheses.

Nonlinear Squeezed States and Multiplication Rule of Nonlinear SqueezingOperators Gained via Nonlinear Coherent State Representation and IWOP Technique**Theproject supported by National Natural Science Foundation of China under GrantNo. 10475056 and the President Foundation of the Chinese Academy of Sciences

Using the nonlinear coherent state representation we derive nonlinearsqueezed states and the multiplication rule of nonlinear squeezingoperators. We find that the symplectic matrices multiplication rule innonlinear coherent state projection operator representation maps into themultiplication rule of successive nonlinear squeezing operators. Thetechnique of integral within an ordered product of operators plays anessential role in deriving the multiplication rule.

Why nonlocal recursion operators produce local symmetries: new results and applications

It is well known that integrable hierarchies in (1+1) dimensions are local while the recursion operators that generate them usually contain nonlocal terms. We resolve this apparent discrepancy by providing simple and universal sufficient conditions for a (nonlocal) recursion operator in (1+1) dimensions to generate a hierarchy of local symmetries. These conditions are satisfied by virtually all known today recursion operators and are much easier to verify than those found in earlier work. We also give explicit formulas for the nonlocal parts of higher recursion operators, Poisson and symplectic structures of integrable systems in (1+1) dimensions. Using these two results we prove, under some natural assumptions, the Maltsev--Novikov conjecture stating that higher Hamiltonian, symplectic and recursion operators of integrable systems in (1+1) dimensions are weakly nonlocal, i.e., the coefficients of these operators are local and these operators contain at most one integration operator in each term.

Improving teleportation of continuous variables by local operations

We study a continuous-variable (CV) teleportation protocol based on a shared entangled state produced by the quantum-nondemolition (QND) interaction of two vacuum states. The scheme utilizes the QND interaction or an unbalanced beam splitter in the Bell measurement. It is shown that in the non-unity gain regime the signal transfer coefficient can be enhanced while the conditional variance product remains preserved by applying appropriate local squeezing operation on sender's part of the shared entangled state. In the unity gain regime it is demonstrated that the fidelity of teleportation can be increased with the help of the local squeezing operations on parts of the shared entangled state that convert effectively our scheme to the standard CV teleportation scheme. Further, it is proved analytically that such a choice of the local symplectic operations minimizes the noise by which the mean number of photons in the input state is increased during the teleportation. Finally, our analysis reveals that the local symplectic operation on sender's side can be integrated into the Bell measurement if the interaction constant of the interaction in the Bell measurement can be adjusted properly.

Pseudorandom operators of the circular ensembles

We demonstrate quantum algorithms to implement pseudorandom operators that closely reproduce statistical properties of random matrices from the three universal classes: unitary, symmetric, and symplectic. Modified versions of the algorithms are introduced for the less experimentally challenging quantum cellular automata. For implementing pseudorandom symplectic operators we provide gate sequences for the unitary part of the time-reversal operator.

A trick loop algebra and a corresponding Liouville integrable hierarchy of evolution equations

A subalgebra of loop algebra A 2 is first constructed, which has its own special feature. It follows that a new Liouville integrable hierarchy of evolution equations is obtained, possessing a tri-Hamiltonian structure, which is proved by us in this paper. Especially, three symplectic operators are constructed directly from recurrence relations. The conjugate operator of a recurrence operator is a hereditary symmetry. As reduction cases of the hierarchy presented in this paper, the celebrated MKdV equation and heat-conduction equation are engendered, respectively. Therefore, we call the hierarchy a generalized MKdV-H system. At last, a high-dimension loop algebra G is constructed by making use of a proper scalar transformation. As a result, a type expanding integrable model of the MKdV-H system is given.

A generalized Boite–Pempinelli–Tu (BPT) hierarchy and its bi-Hamiltonian structure

A subalgebra of loop algebra A ̃ 1 is presented. It follows that a new Liouville integrable hierarchy with 6 potential functions, which possesses a bi-Hamiltonian structure, is obtained. Since it can reduce to Boite–Pempinelli–Tu (BPT) hierarchy, we call it a generalized BPT system. Two symplectic operators of the system obtained are directly derived from recurrence relations, not from a recurrence operator. As a reduction case, the famous KdV-mKdV equation is given.