TraceNorm: Difference between revisions
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Updated function so that it plays well with CVX variables |
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{{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 24: | Line 24: | ||
1.0000 | 1.0000 | ||
</syntaxhighlight> | |||
===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 [[SwapOperator|swap operator]]: | |||
<syntaxhighlight> | |||
>> 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 | |||
</syntaxhighlight> | </syntaxhighlight> | ||
{{SourceCode|name=TraceNorm}} | {{SourceCode|name=TraceNorm}} | ||
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.