WITNESS MESS provides procedures for witnessing genuine multi-partite
entanglement from a subset of marginalia both in multi-partite qubit systems
and multi-mode continuous variable Gaussian systems.

Prerequisites

  A suitable semi-definite optimizer is required. 

  - MOSEK is a commercial optimization software. Free personal licenses are
    provided to members of academia. It is preferred for its unmatched speed
    and versatility. 

  - CVXOPT is an open source convex optimization software. It can be used as
    well, however, it is considerably slower than MOSEK.

Installation

  As of release 0.4.2 it is possible to retrieve the latest release of the 
  package from PyPi by running 'pip install --user witnessmess' 

Examples

  See the examples directory. Both discrete and continuous variable examples
  are provided.

Documentation

  Genuine multipartite entanglement can be witnessed with a pair of functions
  that determine optimal entanglement witnesses for particular, either discrete
  or (Gaussian) continuous variable, quantum states.

  Please note that the definition of covariance matrices follows the
  conventions 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 respect
    to the discrete variable multi-qudit state given by density_matrix.

    : density_matrix 
      defines the state, the matrix must be in Kronecker form

    : component_dims 
      defines a list of component dimensions,
      for example a 3-qubit state has [ 2, 2, 2 ]

    : use_pairs_list 
      defines the list of two-body marginalia the witness is
      permitted 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 respect
    to the continuous variable multi-mode state given by covariance_matrix.

    : covariance_matrix 
      defines the state, the matrix must be in X-P interleaved form

    : mode_count 
      defines the number of modes

    : use_pairs_list 
      defines the list of two-body marginalia the witness is
      permitted to use, if unset the entire covariance_matrix is used

  In addition there is a number of helper functions for both discrete and
  continuous variable states.

  - dm_is_physical (density_matrix)
    
    Determines whether the density matrix represents a physical state by
    testing if the matrix is Hermitian, positive semi-definite and its trace
    sufficiently close to one.

    : density_matrix 
      defines 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 by
    testing if the matrix is symmetric and satisfies uncertainty relations.

    : covariance_matrix
      defines the matrix to be checked, must be in X-P interleaved form

    : mode_count
      defines 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_count
      defines a number of modes of the resulting random state

    : spectral_factor
      defines the interval (1, 1 + spectral_factor) symplectic eigenvalues
      are randomly chosen from

References

  The recipes for semi-definite programs are discussed in papers that are
  referenced in the code where appropriate. States used in examples were taken
  from different papers and are referenced as well. The same applies to the
  conditions, criteria and recipes for construction of various mathematical
  objects (such as Gell-Mann matrices) used in the implementation.

  The implementation references the following papers.

  - Multiparticle entanglement as an emergent phenomenon
    https://doi.org/10.1103/PhysRevA.93.020104

  - Proving genuine multiparticle entanglement from separable nearest-neighbor marginals
    https://doi.org/10.1103/PhysRevA.98.062102

  - Verifying genuine multipartite entanglement of the whole from its separable parts
    https://doi.org/10.1364/QIM.2019.F5A.45

  - Optimal entanglement witnesses for continuous-variable systems
    https://doi.org/10.1088/1367-2630/8/4/051

  - Continuous Variable Quantum Information: Gaussian States and Beyond
    https://doi.org/10.1142/s1230161214400010

  - Certifying emergent genuine multipartite entanglement with a partially blind witness
    https://doi.org/10.1103/PhysRevA.106.062410

  - Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms
    https://doi.org/10.1103/PhysRevA.49.1567

  - Peres-Horodecki Separability Criterion for Continuous Variable Systems
    https://doi.org/10.1103/PhysRevLett.84.2726

  - Two-element generation of the symplectic group
    https://doi.org/10.2307/1993590

  - Bloch vectors for qudits
    https://doi.org/10.1088/1751-8113/41/23/235303