Separation of Orbits under Group Actions with an Application to Quantum Systems

For a given Hilbert space $$\mathcal H$$ , consider the space of self-adjoint projections $$\mathcal P(\mathcal H)$$ . In this paper we study the differentiable structure of a canonical sphere bundle over $$\mathcal P(\mathcal H)$$ given by $$\begin{aligned} \mathcal R=\{\, (P,f)\in \mathcal P(\mathcal H)\times \mathcal H \, : \, Pf=f , \, \Vert f\Vert =1\, \}. \end{aligned}$$ We establish the smooth action on $$\mathcal R$$ of the group of unitary operators of $$\mathcal H$$ , and it thereby turns out that the connected components of $$\mathcal R$$ are homogeneous spaces. Then we study the metric structure of $$\mathcal R$$ by endowing it first with the uniform quotient metric, which is a Finsler metric, and we establish minimality results for the geodesics. These are given by certain one-parameter groups of unitary operators, pushed into $$\mathcal R$$ by the natural action of the unitary group. Then we study the restricted bundle $$\mathcal R_2^+$$ given by considering only the projections in the restricted Grassmannian, locally modeled by Hilbert–Schmidt operators. Therefore we endow $$\mathcal R_2^+$$ with a natural Riemannian metric that can be obtained by declaring that the action of the group is a Riemannian submersion. We study the Levi–Civita connection of this metric and establish a Hopf–Rinow theorem for $$\mathcal R_2^+$$ , again obtaining a characterization of the geodesics as the image of certain one-parameter groups with special speeds.

It is a well-established fact in quantum information theory, that uniform averaging over the collective action of a unitary group on a multipartite quantum state projects the state to a form equivalent to a permutation operator of the subsystems. Hence states equivalent to permutation operators are untouched by collective unitary noise. A trivial observation shows that uniform averaging over permutation operators projects the state into a form with block-diagonal structure equivalent to the one of the collective action of the unitary group. We introduce a name for this property: duality of averaging. The mathematical reason behind this duality is the fact that the collective action of the unitary group on the tensor product state space of a multipartite quantum system and the action of the permutation operations are mutual commutants when treated as matrix algebras. Such pairs of matrix algebras are known as dual reductive pairs. In this work we show, that in the case of finite dimensional quantum systems such duality of averaging holds for any pairs of symmetry groups being dual reductive pairs, regardless of whether they are compact or not, as long as the averaging operation is defined via iterated integral over the Cartan decomposition of the group action. Although our result is very general, we focus much attention on the concrete example of a dual reductive pair consisting of collective action of special linear matrices and permutation operations, which physically corresponds to averaging multipartite quantum states over non-unitary SLOCC-type (Stochastic Local Operations and Classical Communication) operations. In this context we show, that noiseless subsystems known from collective unitary averaging persist in the case of SLOCC averaging in a conditional way: upon postselection to specific invariant subspaces.

We study when a multipartite non--local unitary operation can deterministically or probabilistically simulate another one when local operations of a certain kind ---in some cases including also classical communication--- are allowed. In the case of probabilistic simulation and allowing for arbitrary local operations, we provide necessary and sufficient conditions for the simulation to be possible. Deterministic and probabilistic interconversion under certain kinds of local operations are used to define equivalence relations between gates. In the probabilistic, bipartite case this induces a finite number of classes. In multiqubit systems, however, two unitary operations typically cannot simulate each other with non-zero probability of success. We also show which kind of entanglement can be created by a given non--local unitary operation and generalize our results to arbitrary operators.

We use an algorithm based on the notion of transvectants from classical invariant theory in determining the form of Stanley decomposition of the ring of invariants for the coupled Takens–Bogdanov systems when the Stanley decompositions of the Jordan blocks of the linear part are known at each stage. The set of systems of differential equations that are in normal form with respect to a particular linear part has the structure of a module of equivariants, and is best described by giving a Stanley decomposition of that module.

On equations defining Coincident Root loci

This paper surveys the new, algorithmic theory of moving frames developed by the author and M. Fels. Applications in geometry, computer vision, classical invariant theory, the calculus of variations, and numerical analysis are indicated.

Inspired by Wilson's paper on sectional curvatures of Kähler moduli, we consider a natural Riemannian metric on a hypersurface {f=1} in a real vector space, defined using the Hessian of a homogeneous polynomial f. We give examples to answer a question posed by Wilson about when this metric has nonpositive curvature. Also, we exhibit a large class of polynomials f on R3 such that the associated metric has constant negative curvature. We ask if our examples, together with one example by Dubrovin, are the only ones with constant negative curvature. This question can be rephrased as an appealing question in classical invariant theory, involving the "Clebsch covariant". We give a positive answer for polynomials of degree at most 4, as well as a partial result in any degree.

This paper gives a survey on the relation between Hibi algebras and representation theory. The notion of Hodge algebras or algebras with straightening laws has been proved to be very useful to describe the structure of many important algebras in classical invariant theory and representation theory (Bruns and Herzog 1993; De Concini et al. 1982; Eisenbud 1980; Gonciulea and Lakshmibai 2001; Seshadri 2007). In particular, a special type of such algebras introduced by Hibi (1987) provides a nice bridge between combinatorics and representation theory of classical groups. We will examine certain poset structures of Young tableaux and affine monoids, Hibi algebras in toric degenerations of flag varieties, and their relations to polynomial representations of the complex general linear group.

In quantum state discrimination, the observers are given a quantum system and aim to verify its state from the two or more possible target states. In the local quantum state marking as an extension of quantum state discrimination, there are N composite quantum systems and N possible orthogonal target quantum states. Distant Alice and Bob are asked to correctly mark the states of the given quantum systems via local operations and classical communication. Here we investigate the local state marking with N systems, N = 4, 5, 6, and 7. Therein, Alice and Bob allow for three local operations: measuring the local observable either σ z or σ x simultaneously, and entanglement swapping. It shows that, given arbitrary four systems, Alice and Bob can perform the perfect local quantum state marking. In the N = 5, 6 cases, they can perform perfect local state marking with specific target states. We conjecture the impossibility of the local quantum state marking given any seven target states since Alice and Bob cannot fulfill the task in the simplest case.

Wave-particle duality is one of the basic features of quantum mechanics, giving rise to the use of complex numbers in describing states of quantum systems and their dynamics and interaction. Since the inception of quantum theory, it has been debated whether complex numbers are essential or whether an alternative consistent formulation is possible using real numbers only. Here, we attack this long-standing problem theoretically and experimentally, using the powerful tools of quantum resource theories. We show that, under reasonable assumptions, quantum states are easier to create and manipulate if they only have real elements. This gives an operational meaning to the resource theory of imaginarity. We identify and answer several important questions, which include the state-conversion problem for all qubit states and all pure states of any dimension and the approximate imaginarity distillation for all quantum states. As an application, we show that imaginarity plays a crucial role in state discrimination, that is, there exist real quantum states that can be perfectly distinguished via local operations and classical communication but that cannot be distinguished with any nonzero probability if one of the parties has no access to imaginarity. We confirm this phenomenon experimentally with linear optics, discriminating different two-photon quantum states by local projective measurements. Our results prove that complex numbers are an indispensable part of quantum mechanics.

In this paper we establish a unique factorization theorem for pure quantum states expressed in computational basis. We show that there always exists unique factorization for any given N-qubit pure quantum state in terms of the tensor product of non-factorable or ``prime'' pure quantum states. This result is based on a simple criterion: Given N-qubit pure quantum state in computational basis can be factorized as the tensor product of an m-qubit pure quantum state and an n-qubit pure quantum state, where (m + n) = N, if and only if the rank of the certain associated matrix is equal to one. This simple criterion leads to a factorization algorithm which when applied to an N-qubit pure quantum state factorizes that state into the tensor product of non-factorable or ``prime'' pure quantum states. This paper shows that for any given N-qubit pure quantum state the said factorization always ``exists'' and is ``unique''. We demonstrated our work here on a computational basis. PACS Number: 03.67.Mn, 03.65.Ca, 03.65.Ud

The compatible Grassmannian

Nonlocality without entanglement(NLWE) is a nonlocal phenomenon that occurs in quantum state discrimination of multipartite separable states. In the discrimination of orthogonal separable states, the term NLWE is used when the quantum states cannot be discriminated perfectly by local operations and classical communication. In this case, the occurrence of NLWE is independent of nonzero prior probabilities of quantum states being prepared. Recently, it has been found that the occurrence of NLWE can depend on nonzero prior probabilities in minimum-error discrimination of nonorthogonal separable states. Here, we show that even in optimal unambiguous discrimination, the occurrence of NLWE can depend on nonzero prior probabilities. We further show that NLWE can occur regardless of nonzero prior probabilities, even if only one state can be locally discriminated without error. Our results provide new insights into classifying sets of multipartite quantum states in terms of quantum state discrimination.

Quantum teleportation is one of the most basic quantum protocols, which transfers an unknown quantum state from one location to another through local operation and classical communication by using shared quantum entanglement without physical transfer of the information carrier. And it has been widely used in various quantum information protocols such as entanglement swapping, quantum repeaters, quantum gate teleportation, quantum computation based on measurement, and quantum teleportation networks, which have important application value in quantum computation and quantum information. Quantum teleportation is a naturally bipartite process, in which an unknown quantum state can only be transmitted from one node to another. With the further development of quantum information research, it is necessary to transfer quantum states or quantum information among more and more nodes. Multipartite quantum protocols are expected to form fundamental components for larger-scale quantum communication and computation. A bipartite quantum teleportation should be extended to a multipartite protocol known as a quantum teleportation network. In this paper, a multifunctional quantum teleportation network is proposed theoretically. We first propose a special method of constructing four-partite quantum resources in continuous variables (CVs), and based on this, construct two different types of CV quantum teleportation networks. One type of network contains just one quantum teleportation process consisting of a sender, a receiver and two controllers. In this type of network, the unknown quantum state can be recovered at any other node according to the requirement after the measurement in the input node, which enriches the transfer direction and transfer mode of the unknown quantum state. And meanwhile, the two controllers can control the transfer of a quantum state from the sender to the receiver by restricting the sender and receiver’s access to their information, which makes the quantum teleportation network controllable. The other type of network has two quantum teleportation processes, each containing only a sender, a receiver and no controllers, which increases the number of quantum states that can be transmitted. Then we analyze the dependence of the fidelity of each quantum teleportation network on different physical parameters, and compare the characteristics, advantages and disadvantages among different types of quantum teleportation networks. The scheme for constructing a multifunctional quantum teleportation network in this paper shows some advantages, such as the greater number of quantum nodes, diversity of types, simple operation procedure. And all these advantages provide a broader application prospect for establishing larger and more complex quantum information networks in the future and quicken the pace of the application of quantum information.

, for t ≥ 0, be a strongly continuous Markovian semigroup acting on C(X), where X is a compact Hausdorf space, and let D denote the domain of its infinitesimal generator Z. Suppose D contains a (perhaps finite) family of functions f separating the points of X and satisfying Zf2 = 2fZf. If either (1) there exists δ > 0 such that (Tt f)2∈ D if 0 ≤ t ≤δ for each f in this family; or (1′) for some core D′ of Z, g ∈ D′ implies g2∈ D, then the underlying Markoff process on X is deterministic. That is, there exists a semiflow — a semigroup (under composition) of continuous functions φt from X into X — such that Ttf(x) = f(φt (x)). If the domain D should be an algebra then conditions (1) and (1′) hold trivially. Conversely, if we have a separating family satisfying Zf2 = 2fZf then each of these conditions implies that D is an algebra. It is an open question as to whether these conditions are redundant. If the functions φt are homeomorphisms from X onto X, then of course we have a Markovian group induced by a flow. This result is obtained by first providing general results about the null-space N of the (function-valued) positive semidefinite quadratic form defined by = Z(fg) - fZg - gZf. The set N can be defined for any generator Z of a strongly continuous Markovian semigroup and is equivalently given by N = {f ∈ D| f2∈ D and Zf2 = 2fZf} = {f ∈ D| Tt(f2)-(Ttf)2 is o(t2) in C(X)}. In the general case N is an algebra closed under composition with any C1-function φ from the reals to the reals, and Z(φ[f]) = (Zf)φ′[f] if f ∈ N. This "chain rule" on N (on which Z must act as a derivation) is a special case of a theorem for C2-functions φ which holds more generally for all f in d, viz., Z(φ[f] = (Zf) φ′[f] + ½ φ″[f], Provided Z is a local operator and D is an algebra. In this case the form itself enjoys the relation = φ′ [f] ψ′[g] , for C2functions φ and ψ. Some of the results and their proofs continue to hold when the setting is switched from the commutative C*-algebra C(X) to a general (noncommutative) C*-algebra A. In the norm continuous case we obtain a sharp characterization of Markovian semigroups that are groups: Let Tt = etz , defined for t ≥ 0, be a Markovian semigroup acting on a C*-algebra A that is norm continuous, i.e., ||Tt - I|| ⇒ 0 as t ⇒ 0 +. Assume Z(a2) = a(Za) + (Za) a for some (perhaps finite) set of self-adjoint elements a that generate a Jordan algebra dense among the self-adjoint elements of A. The etz , -∞ < t < ∞, is a group of Markovian operators.