SymmetricExtension

SymmetricExtension is a function that determines whether or not a given positive semidefinite operator has a symmetric extension. This function is extremely useful for showing that quantum states are entangled (see the Examples section). Various types of symmetric extensions (such as Bosonic and/or PPT extensions) can be looked for by specifying optional arguments in the function.

Syntax

 * EX = SymmetricExtension(X)
 * EX = SymmetricExtension(X,K)
 * EX = SymmetricExtension(X,K,DIM)
 * EX = SymmetricExtension(X,K,DIM,PPT)
 * EX = SymmetricExtension(X,K,DIM,PPT,BOS)
 * EX = SymmetricExtension(X,K,DIM,PPT,BOS,TOL)
 * [EX,WIT] = SymmetricExtension(X,K,DIM,PPT,BOS,TOL)

Input arguments

 * X</tt>: A positive semidefinite operator.
 * K</tt> (optional, default 2): The number of copies of the second subsystem in the desired symmetric extension.
 * DIM</tt> (optional, by default has both subsystems of equal dimension): A 1-by-2 vector containing the dimensions of the two subsystems that X</tt> acts on.
 * PPT</tt> (optional, default 0): A flag (either 1 or 0) that indicates whether or not the desired symmetric extension must have positive partial transpose.
 * BOS</tt> (optional, default 0): A flag (either 1 or 0) that indicates whether or not the desired symmetric extension must be Bosonic (i.e., have its range contained within the symmetric subspace).
 * TOL</tt> (optional, default eps^(1/4)</tt>): 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</tt> has a symmetric extension.

Output arguments

 * EX</tt>: A flag (either 1 or 0) indicating that X</tt> does or does not have a symmetric extension of the desired type.
 * WIT</tt> (optional): A witness that verifies that the answer provided by EX</tt> is correct. If EX = 1</tt> (i.e., X</tt> has a symmetric extension) then WIT</tt> is such a symmetric extension. If <tt>EX = 0</tt> (i.e., no symmetric extension exists) then WIT is an entanglement witness with <tt>trace(WIT*X) = -1</tt> but <tt>trace(WIT*Y) >= 0</tt> for all symmetrically extendable <tt>Y</tt>.

2-qubit symmetric extension
It is known that a 2-qubit state $\rho_{AB}$ has a (not necessarily PPT) symmetric extension if and only if ${\rm Tr}(\rho_B^2) \geq {\rm Tr}(\rho_{AB}^2) - 4\sqrt{\det(\rho_{AB})}$. The following code verifies that one such state does indeed have a symmetric extension.