TraceNorm

TraceNorm
Computes the trace norm of an operator
Other toolboxes required	none
Related functions	kpNorm KyFanNorm SchattenNorm
Function category	Norms
Usable within CVX?	yes (convex)

TraceNorm is a function that computes the trace norm of an operator (i.e., the sum of its singular values). It works with both full and sparse matrices.

Syntax

• NRM = TraceNorm(X)

Argument descriptions

• X: An operator to have its trace norm computed.

Examples

Trace norm of a density matrix

Density matrices all have trace norm equal to 1:

>> rho = RandomDensityMatrix(100);
>> TraceNorm(rho)

ans =

    1.0000

Can be used with CVX

This function can be used in the objective function or constraints of a CVX optimization problem. For example, the following code computes the minimum value of $\mathrm{Tr}(S\rho)$ over all density matrices $\rho$ with the property that the trace norm of their realignment is no larger than 1 (this set of state approximates the set of separable states), where $S$ is the swap operator:

>> cvx_begin sdp quiet
   variable rho(9,9) hermitian;
   minimize trace(rho*SwapOperator(3));
   subject to
       trace(rho) == 1;
       rho >= 0;
       TraceNorm(Realignment(rho)) <= 1;
   cvx_end
   cvx_optval

cvx_optval =

   -0.3333

Source code

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