TraceNorm: Difference between revisions
Jump to navigation
Jump to search
→Examples: Added CVX example |
|||
| (One intermediate revision by the same user not shown) | |||
| Line 1: | Line 1: | ||
{{Function | {{Function | ||
|name=TraceNorm | |name=TraceNorm | ||
|desc=Computes the | |desc=Computes the trace norm of an operator | ||
|rel=[[kpNorm]]<br />[[KyFanNorm]]<br />[[SchattenNorm]] | |rel=[[kpNorm]]<br />[[KyFanNorm]]<br />[[SchattenNorm]] | ||
|cat=[[List of functions#Norms|Norms]] | |cat=[[List of functions#Norms|Norms]] | ||
|upd=October 22, 2014 | |upd=October 22, 2014 | ||
| | |cvx=yes (convex)}} | ||
<tt>'''TraceNorm'''</tt> is a [[List of functions|function]] that computes the | <tt>'''TraceNorm'''</tt> is a [[List of functions|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== | ==Syntax== | ||
| Line 27: | Line 27: | ||
===Can be used with CVX=== | ===Can be used with CVX=== | ||
This function can be used in the objective function or constraints of a | 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 [[SwapOperator|swap operator]]: | ||
<syntaxhighlight> | <syntaxhighlight> | ||
>> cvx_begin sdp quiet | >> cvx_begin sdp quiet | ||
Latest revision as of 16:36, 24 December 2014
| 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 =
1.0000Can 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 =
-0.3333Source code
Click here to view this function's source code on github.