Package: witnessmess
WITNESS MESS provides procedures for witnessing genuine multi-partiteentanglement from a subset of marginalia both in multi-partite qubit systemsand multi-mode continuous variable Gaussian systems.PrerequisitesA suitable semi-definite optimizer is required.- MOSEK is a commercial optimization software. Free personal licenses areprovided to members of academia. It is preferred for its unmatched speedand versatility.- CVXOPT is an open source convex optimization software. It can be used aswell, however, it is considerably slower than MOSEK.InstallationAs of release 0.4.2 it is possible to retrieve the latest release of thepackage from PyPi by running 'pip install --user witnessmess'ExamplesSee the examples directory. Both discrete and continuous variable examplesare provided.DocumentationGenuine multipartite entanglement can be witnessed with a pair of functionsthat determine optimal entanglement witnesses for particular, either discreteor (Gaussian) continuous variable, quantum states.Please note that the definition of covariance matrices follows theconventions from (https://doi.org/10.1088/1367-2630/8/4/051).- dm_optimal_witness (density_matrix, component_dims, use_pairs_list)Produces an optimal genuine multipartite entanglement witness with respectto the discrete variable multi-qudit state given by density_matrix.: density_matrixdefines the state, the matrix must be in Kronecker form: component_dimsdefines a list of component dimensions,for example a 3-qubit state has [ 2, 2, 2 ]: use_pairs_listdefines the list of two-body marginalia the witness ispermitted to use, if unset the entire density_matrix is used- cm_optimal_witness (covariance_matrix, mode_count, use_pairs_list)Produces an optimal genuine multipartite entanglement witness with respectto the continuous variable multi-mode state given by covariance_matrix.: covariance_matrixdefines the state, the matrix must be in X-P interleaved form: mode_countdefines the number of modes: use_pairs_listdefines the list of two-body marginalia the witness ispermitted to use, if unset the entire covariance_matrix is usedIn addition there is a number of helper functions for both discrete andcontinuous variable states.- dm_is_physical (density_matrix)Determines whether the density matrix represents a physical state bytesting if the matrix is Hermitian, positive semi-definite and its tracesufficiently close to one.: density_matrixdefines the matrix to be checked, must be in Kronecker form- cm_is_physical (covariance_matrix, mode_count)Determines whether the covariance matrix represents a physical state bytesting if the matrix is symmetric and satisfies uncertainty relations.: covariance_matrixdefines the matrix to be checked, must be in X-P interleaved form: mode_countdefines the number of modes of the state,yes, it could be inferred from covariance_matrix but, alas, it is not- cm_build_random (mode_count, spectral_factor)Constructs a random bona-fide covariance matrix.: mode_countdefines a number of modes of the resulting random state: spectral_factordefines the interval (1, 1 + spectral_factor) symplectic eigenvaluesare randomly chosen fromReferencesThe recipes for semi-definite programs are discussed in papers that arereferenced in the code where appropriate. States used in examples were takenfrom different papers and are referenced as well. The same applies to theconditions, criteria and recipes for construction of various mathematicalobjects (such as Gell-Mann matrices) used in the implementation.The implementation references the following papers.- Multiparticle entanglement as an emergent phenomenonhttps://doi.org/10.1103/PhysRevA.93.020104- Proving genuine multiparticle entanglement from separable nearest-neighbor marginalshttps://doi.org/10.1103/PhysRevA.98.062102- Verifying genuine multipartite entanglement of the whole from its separable partshttps://doi.org/10.1364/QIM.2019.F5A.45- Optimal entanglement witnesses for continuous-variable systemshttps://doi.org/10.1088/1367-2630/8/4/051- Continuous Variable Quantum Information: Gaussian States and Beyondhttps://doi.org/10.1142/s1230161214400010- Certifying emergent genuine multipartite entanglement with a partially blind witnesshttps://doi.org/10.1103/PhysRevA.106.062410- Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal formshttps://doi.org/10.1103/PhysRevA.49.1567- Peres-Horodecki Separability Criterion for Continuous Variable Systemshttps://doi.org/10.1103/PhysRevLett.84.2726- Two-element generation of the symplectic grouphttps://doi.org/10.2307/1993590- Bloch vectors for quditshttps://doi.org/10.1088/1751-8113/41/23/235303