Nathaniel: CVX info in side box

2015-06-24T13:18:17Z

CVX info in side box

← Older revision		Revision as of 13:18, 24 June 2015
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	\|cat=[[List of functions#Entanglement_measures\|Entanglement measures]]		\|cat=[[List of functions#Entanglement_measures\|Entanglement measures]]
	\|upd=October 22, 2014		\|upd=October 22, 2014
	\|v=~~0.50~~}}		\|cvx=yes (convex)}}
	<tt>'''Negativity'''</tt> is a [[List of functions\|function]] that computes the negativity of a bipartite density matrix, which is defined as follows:		<tt>'''Negativity'''</tt> is a [[List of functions\|function]] that computes the negativity of a bipartite density matrix, which is defined as follows:
	: <math>\mathcal{N}(\rho) := \frac{1}{2}\big( \\|\rho^\Gamma\\|_1 - 1 \big),</math>		: <math>\mathcal{N}(\rho) := \frac{1}{2}\big( \\|\rho^\Gamma\\|_1 - 1 \big),</math>

Nathaniel: Created page with "{{Function |name=Negativity |desc=Computes the negativity of a bipartite density matrix |rel=PartialTranspose
TraceNorm |cat=List of functions#Entanglement_meas..."

2014-10-23T15:58:48Z

Created page with "{{Function |name=Negativity |desc=Computes the negativity of a bipartite density matrix |rel=PartialTranspose<br />TraceNorm |cat=List of functions#Entanglement_meas..."

New page

{{Function
|name=Negativity
|desc=Computes the negativity of a bipartite density matrix
|rel=[[PartialTranspose]]<br />[[TraceNorm]]
|cat=[[List of functions#Entanglement_measures|Entanglement measures]]
|upd=October 22, 2014
|v=0.50}}
<tt>'''Negativity'''</tt> is a [[List of functions|function]] that computes the negativity of a bipartite density matrix, which is defined as follows:
: <math>\mathcal{N}(\rho) := \frac{1}{2}\big( \|\rho^\Gamma\|_1 - 1 \big),</math>
where $\rho^\Gamma$ is the [[PartialTranspose|partial transpose]] of $\rho$ and $\|\cdot\|_1$ is the [[TraceNorm|trace norm]]. Equivalently, the negativity of $\rho$ is the absolute value of the sum of the negative eigenvalues of $\rho^\Gamma$.

==Syntax==
* <tt>NEG = Negativity(RHO)</tt>
* <tt>NEG = Negativity(RHO,DIM)</tt>

==Argument descriptions==
* <tt>RHO</tt>: A bipartite density matrix.
* <tt>DIM</tt> (optional, by default has both subsystems of equal dimension): A specification of the dimensions of the subsystems that <tt>RHO</tt> acts on. <tt>DIM</tt> can be provided in one of two ways:
** If <tt>DIM</tt> is a scalar, it is assumed that the first subsystem has dimension <tt>DIM</tt> and the second subsystem has dimension <tt>length(RHO)/DIM</tt>.
** If $X \in M_{n_1} \otimes M_{n_2}$ then <tt>DIM</tt> should be a row vector containing the dimensions (i.e., <tt>DIM = [n_1, n_2]</tt>).

==Examples==
===PPT states have zero negativity===
States with positive partial transpose have zero negativity. The following code verifies this fact for one particular [[ChessboardState|Chessboard state]]:
<syntaxhighlight>
>> rho = ChessboardState(1,2,3,4,5,6);
>> Negativity(rho)

ans =

1.1102e-16
</syntaxhighlight>

===Can be used with CVX===
This function is convex and can be used in the objective function or constraints of a CVX optimization problem. For example, the following code finds the maximum overlap of a density matrix $\rho$ with the [[MaxEntangled|maximally-entangled pure state]], subject to the constraint that its negativity is no larger than 1/2:
<syntaxhighlight>
>> d = 3;
>> phi = MaxEntangled(d);
>> phi = phi*phi'; % the maximally-entangled pure state, represented as a rank-1 density matrix
>> cvx_begin sdp quiet
variable rho(d^2,d^2) hermitian;
maximize trace(phi*rho);
subject to
trace(rho) == 1;
rho >= 0;
Negativity(rho) <= 1/2;
cvx_end
cvx_optval

cvx_optval =

0.6667
</syntaxhighlight>

{{SourceCode|name=Negativity}}

Negativity - Revision history

Nathaniel: CVX info in side box

Nathaniel: Created page with "{{Function |name=Negativity |desc=Computes the negativity of a bipartite density matrix |rel=PartialTransposeTraceNorm |cat=List of functions#Entanglement_meas..."

Nathaniel: Created page with "{{Function |name=Negativity |desc=Computes the negativity of a bipartite density matrix |rel=PartialTranspose
TraceNorm |cat=List of functions#Entanglement_meas..."