SymmetricExtension

SymmetricExtension
Determines whether or not an operator has a symmetric extension
Other toolboxes required	cvx
Related functions	IsPPT IsSeparable SymmetricInnerExtension SymmetricProjection
Function category	Entanglement and separability

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 can be looked for – by default, the extension that is looked for must be Bosonic and have positive partial transpose, but both of these restrictions can be removed.

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)

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 0): A flag (either 1 or 0) that indicates whether or not the desired symmetric extension must have positive partial transpose.
• BOS (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 (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 extension.

Output arguments

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

Examples

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.

>> rho = [1 0 0 -1;0 1 1/2 0;0 1/2 1 0;-1 0 0 1];
>> [trace(PartialTrace(rho)^2), trace(rho^2) - 4*sqrt(det(rho))] % if the first number is >= the second number, rho has a symmetric extension

ans =

    8.0000    6.5000

>> SymmetricExtension(rho) % verify that rho has a symmetric extension

ans =

     1

Notes

If your goal is to detect entanglement in an operator, then it is always in your best interest to set the optional arguments PPT and BOS to be 1. Setting BOS = 1 increases the effectiveness of the entanglement test without any computational overhead at all. Setting PPT = 1 slows down the computation quite a bit, but increases the effectiveness as an entanglement test considerably.

Source code

Click here to view this function's source code on github.