Quantum coherence marks a deviation from classical physics, and has been studied as a resource for metrology and quantum computation. Finding reliable and effective methods for assessing its presence is then highly desirable. Coherence witnesses rely
on measuring observables whose outcomes can guarantee that a state is not diagonal in a known reference basis. Here we experimentally measure a novel type of coherence witness that uses pairwise state comparisons to identify superpositions in a basis-independent way. Our experiment uses a single interferometric set-up to simultaneously measure the three pairwise overlaps among three single-photon states via Hong-Ou-Mandel tests. Besides coherence witnesses, we show the measurements also serve as a Hilbert-space dimension witness. Our results attest to the effectiveness of pooling many two-state comparison tests to ascertain various relational properties of a set of quantum states.
Coisotropic algebras consist of triples of algebras for which a reduction can be defined and unify in a very algebraic fashion coisotropic reduction in several settings. In this paper we study the theory of (formal) deformation of coisotropic algebra
s showing that deformations are governed by suitable coisotropic DGLAs. We define a deformation functor and prove that it commutes with reduction. Finally, we study the obstructions to existence and uniqueness of coisotropic algebras and present some geometric examples.
In this paper we propose a reduction scheme for multivector fields phrased in terms of $L_infty$-morphisms. Using well-know geometric properties of the reduced manifolds we perform a Taylor expansion of multivector fields, which allows us to built up
a suitable deformation retract of DGLAs. We first obtained an explicit formula for the $L_infty$-Projection and -Inclusion of generic DGLA retracts. We then applied this formula to the deformation retract that we constructed in the case of multivector fields on reduced manifolds. This allows us to obtain the desired reduction $L_infty$-morphism. Finally, we perfom a comparison with other reduction procedures.
Photon indistinguishability plays a fundamental role in information processing, with applications such as linear-optical quantum computation and metrology. It is then necessary to develop appropriate tools to quantify the amount of this resource in a
multiparticle scenario. Here we report a four-photon experiment in a linear-optical interferometer designed to simultaneously estimate the degree of indistinguishability between three pairs of photons. The interferometer design dispenses with the need of heralding for parametric down-conversion sources, resulting in an efficient and reliable optical scheme. We then use a recently proposed theoretical framework to quantify genuine four-photon indistinguishability, as well as to obtain bounds on three unmeasured two-photon overlaps. Our findings are in high agreement with the theory, and represent a new resource-effective technique for the characterization of multiphoton interference.
Coisotropic reduction from Poisson geometry and deformation quantization is cast into a general and unifying algebraic framework: we introduce the notion of coisotropic triples of algebras for which a reduction can be defined. This allows to construc
t also a notion of bimodules for such triples leading to bicategories of bimodules for which we have a reduction functor as well. Morita equivalence of coisotropic triples of algebras is defined as isomorphism in the ambient bicategory and characterized explicitly. Finally, we investigate the classical limit of coisotropic triples of algebras and their bimodules and show that classical limit commutes with reduction in the bicategory sense.
In this short note we prove an equivariant version of the formality of multidiffirential operators for a proper Lie group action. More precisely, we show that the equivariant Hochschild-Kostant-Rosenberg quasi-isomorphism between the cohomology of th
e equivariant multidifferential operators and the complex of equivariant multivector fields extends to an $L_infty$-quasi-isomorphism. We construct this $L_infty$-quasi-isomorphism using the $G$-invariant formality constructed by Dolgushev. This result has immediate consequences in deformation quantization, since it allows to obtain a quantum moment map from a classical momentum map with respect to a $G$-invariant Poisson structure.
In this paper we introduce a notion of quantum Hamiltonian (co)action of Hopf algebras endowed with Drinfeld twist structure (resp., 2-cocycles). First, we define a classical Hamiltonian action in the setting of Poisson Lie groups compatible with the
2-cocycle stucture and we discuss a concrete example. This allows us to construct, out of the classical momentum map, a quantum momentum map in the setting of Hopf coactions and to quantize it by using Drinfeld approach.
In this short note we describe an alternative global version of the twisting procedure used by Dolgushev to prove formality theorems. This allows us to describe the maps of Fedosov resolutions, which are key factors of the formality morphisms, in ter
ms of a twist of the fiberwise quasi-isomorphisms induced by the local formality theorems proved by Kontsevich and Shoikhet. The key point consists in considering $L_infty$-resolutions of the Fedosov resolutions obtained by Dolgushev and an adapted notion of Maurer-Cartan element. This allows us to perform the twisting of the quasi-isomorphism intertwining them in a global manner.
In this paper we provide a quantization via formality of Poisson actions of a triangular Lie algebra $(mathfrak g,r)$ on a smooth manifold $M$. Using the formality of polydifferential operators on Lie algebroids we obtain a deformation quantization o
f $M$ together with a quantum group $mathscr{U}_hbar(mathfrak{g})$ and a map of associated DGLAs. This motivates a definition of quantum action in terms of $L_infty$-morphisms which generalizes the one given by Drinfeld.
VB-groupoids and algebroids are vector bundle objects in the categories of Lie groupoids and Lie algebroids respectively, and they are related via the Lie functor. VB-groupoids and algebroids play a prominent role in Poisson and related geometries. A
dditionally, they can be seen as models for vector bundles over singular spaces. In this paper we study their infinitesimal automorphisms, i.e. vector fields on them generating a flow by diffeomorphisms preserving both the linear and the groupoid/algebroid structures. For a special class of VB-groupoids/algebroids coming from representations of Lie groupoids/algebroids, we prove that infinitesimal automorphisms are the same as multiplicative sections of a certain derivation groupoid/algebroid.
