SymmetricInnerExtension
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| SymmetricInnerExtension | |
| Determines whether or not an operator has a symmetric inner extension | |
| Other toolboxes required | cvx IsPPT IsPSD jacobi_poly opt_args PartialTrace PartialTranspose PermutationOperator PermuteSystems sporth SymmetricProjection |
|---|---|
SymmetricInnerExtension is a function that determines whether or not a given positive semidefinite operator is in the cone defined in reference [1] that approximates the set of separable operators from the inside, based on operators with symmetric extensions.
Syntax
- EX = SymmetricInnerExtension(X)
- EX = SymmetricInnerExtension(X,K)
- EX = SymmetricInnerExtension(X,K,DIM)
- EX = SymmetricInnerExtension(X,K,DIM,PPT)
- EX = SymmetricInnerExtension(X,K,DIM,PPT,TOL)
- [EX,WIT] = SymmetricInnerExtension(X,K,DIM,PPT,TOL)
Argument descriptions
Input arguments
- X: A positive semidefinite operator.
- K (optional, default 2): The number of copies of the second subsystem in the desired symmetric extension.
- DIM (optional, by default has both subsystems of equal dimension): A 1-by-2 vector containing the dimensions of the two subsystems that X acts on.
- PPT (optional, default 1): A flag (either 1 or 0) that indicates whether or not the desired symmetric extension must have positive partial transpose.
- TOL (optional, default eps^(1/4)): The numerical tolerance used throughout this script. It is recommended that this is left at the default value unless numerical problems arise and the script has difficulty determining whether or not X has a symmetric inner extension.
Output arguments
- EX: A flag (either 1 or 0) indicating that X does or does not have a symmetric inner extension of the desired type. A value of 1 means that X is separable.
- WIT (optional): A witness that verifies that the answer provided by EX is correct. If EX = 1 (i.e., X is in the "inner" symmetric extension cone) then WIT is a symmetric extension of the operator \(\sigma_{AB}\) from [1], and thus acts as a witness that verifies that EX = 1 is correct. If X is not in this cone (i.e., EX = 0) then WIT is an operator with trace(WIT*X) = -1 but trace(WIT*Y) >= 0 for all operators in the described cone. Note that WIT may not be an entanglement witness!
Examples
Please add some examples.
References
- ↑ 1.0 1.1 M. Navascués, M. Owari, and M. B. Plenio. Complete Criterion for Separability Detection. Physical Review Letters, 103:160404, 2009. E-print: arXiv:0906.2735 [quant-ph]