AbsPPTConstraints - Revision history

Nathaniel: Updated for v0.70

2014-12-16T18:03:04Z

Updated for v0.70

Nathaniel: /* Notes */

2014-11-15T04:17:45Z

Notes

Nathaniel: /* Notes */ Added OEIS entry.

2014-11-15T04:17:33Z

Notes: Added OEIS entry.

Nathaniel: /* Notes */

2014-11-15T04:13:56Z

Notes

Nathaniel: /* Qutrit-qutrit states */ Made example more explicit

2014-11-15T04:12:39Z

Qutrit-qutrit states: Made example more explicit

Nathaniel: /* Notes */ typo

2014-11-15T03:58:55Z

Notes: typo

Nathaniel: /* Can be used with CVX */

2014-11-15T02:40:57Z

Can be used with CVX

Nathaniel at 02:40, 15 November 2014

2014-11-15T02:40:25Z

Nathaniel: Added CVX example

2014-11-15T02:39:07Z

Added CVX example

Nathaniel: Created page with "{{Function |name=AbsPPTConstraints |desc=Builds the eigenvalue matrices that determine whether or not a state is absolutely PPT |rel=IsAbsPPT |cat=List of functions#Ball..."

2014-11-15T01:42:35Z

Created page with "{{Function |name=AbsPPTConstraints |desc=Builds the eigenvalue matrices that determine whether or not a state is absolutely PPT |rel=IsAbsPPT |cat=List of functions#Ball..."

New page

{{Function
|name=AbsPPTConstraints
|desc=Builds the eigenvalue matrices that determine whether or not a state is absolutely PPT
|rel=[[IsAbsPPT]]
|cat=[[List of functions#Ball_of_separability|Ball of separability]]
|upd=November 14, 2014
|v=0.60}}
<tt>'''AbsPPTConstraints'''</tt> is a [[List of functions|function]] that computes the matrices of eigenvalues that determine whether or not a given density matrix $\rho$ is "absolutely PPT" (that is whether or not $U\rho U^\dagger$ has [[IsPPT|positive partial transpose]] for all unitary matrices $U$). The state $\rho$ is absolutely PPT if and only if every matrix that this function produces as output is positive semidefinite. The matrices that characterize when a density matrix is absolutely PPT were derived in <ref name="Hil07">R. Hildebrand. Positive partial transpose from spectra. ''Phys. Rev. A'', 76:052325, 2007. E-print: [http://arxiv.org/abs/quant-ph/0502170 arXiv:quant-ph/0502170]</ref>.

==Syntax==
* <tt>L = AbsPPTConstraints(LAM,DIM)</tt>
* <tt>L = AbsPPTConstraints(LAM,DIM,ESC_IF_NPOS)</tt>
* <tt>L = AbsPPTConstraints(LAM,DIM,ESC_IF_NPOS,LIM)</tt>

==Argument descriptions==
* <tt>LAM</tt>: The eigenvalues of a bipartite density matrix.
* <tt>DIM</tt>: A specification of the dimensions of the subsystems that the density matrix 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(LAM)/DIM</tt>.
** <tt>DIM</tt> can be a 1-by-2 vector containing the dimensions of the local subsystems.
* <tt>ESC_IF_NPOS</tt> (optional, default 0): A flag, either 1 or 0, indicating that the function should stop computing constraint matrices as soon as it finds one that is not positive semidefinite (i.e., as soon as it finds proof that the density matrix in question is not absolutely PPT).
* <tt>LIM</tt> (optional, default 0): A non-negative integer specifying how many constraint matrices should be computed. If this equals 0 then ''all'' constraint matrices will be computed.

==Examples==
===Qutrit-qutrit states===
It was noted in <ref name="Hil07"></ref> that, in the qutrit-qutrit case, the absolutely PPT property is determined by whether or not two 3-by-3 matrices of eigenvalues are positive semidefinite. We can compute these two matrices for a given set of eigenvalues as follows:
<syntaxhighlight>
>> rho = RandomDensityMatrix(9);
>> lam = eig(rho);
>> L = AbsPPTConstraints(lam,3) % compute the two constraint matrices

L =

[3x3 double] [3x3 double]

>> L{1}

ans =

0.0013 -0.3100 -0.2579
-0.3100 0.0679 -0.1159
-0.2579 -0.1159 0.2102

>> L{2}

ans =

0.0013 -0.3100 -0.2381
-0.3100 0.0282 -0.1159
-0.2381 -0.1159 0.2102

>> IsPSD(L{1})

ans =

0 % L{1} is not positive semidefinite, so rho is not absolutely PPT
</syntaxhighlight>

==Notes==
For all practical purposed, this function can only compute all constraint matrices when at least one of the local dimensions is 6 or less. When both local dimensions are 7 or higher, the <tt>LIM</tt> argument should be specified.

Additionally, this function actually computes slightly more constraints than is optimal. It computes the optimal set of constraints when one of the local dimensions is 5 or less, but when the smallest local dimension is 6, it computes 2612 constraint matrices rather than the optimal set of 2608 constraint matrices: the additional 4 matrices are redundant, but do not cause any harm.

{{SourceCode|name=AbsPPTConstraints}}

==References==
<references />

@@ Line 94: / Line 94: @@
 * This function actually computes slightly more constraints than is optimal. It computes the optimal set of constraints when one of the local dimensions is 5 or less, but when the smallest local dimension is 6, it computes 2612 constraint matrices rather than the optimal set of 2608 constraint matrices: the additional 4 matrices are redundant, but do not cause any harm.
-* The optimal number of constraint matrices, when the smaller of the two local dimension is 2, 3, 4, ..., is given by the sequence 1, 2, 10, 114, 2608, ... ([http://oeis.org/A237749 A237749] in the OEIS).
+* The optimal number of constraint matrices, when the smaller of the two local dimensions is 2, 3, 4, ..., is given by the sequence 1, 2, 10, 114, 2608, ... ([http://oeis.org/A237749 A237749] in the OEIS).
 * Absolutely PPT states are sometimes said to be "PPT from spectrum".

← Older revision		Revision as of 04:17, 15 November 2014
Line 93:		Line 93:

	* This function actually computes slightly more constraints than is optimal. It computes the optimal set of constraints when one of the local dimensions is 5 or less, but when the smallest local dimension is 6, it computes 2612 constraint matrices rather than the optimal set of 2608 constraint matrices: the additional 4 matrices are redundant, but do not cause any harm.		* This function actually computes slightly more constraints than is optimal. It computes the optimal set of constraints when one of the local dimensions is 5 or less, but when the smallest local dimension is 6, it computes 2612 constraint matrices rather than the optimal set of 2608 constraint matrices: the additional 4 matrices are redundant, but do not cause any harm.
		+
		+	* The optimal number of constraint matrices, when the smaller of the two local dimension is 2, 3, 4, ..., is given by the sequence 1, 2, 10, 114, 2608, ... ([http://oeis.org/A237749 A237749] in the OEIS).

	* Absolutely PPT states are sometimes said to be "PPT from spectrum".		* Absolutely PPT states are sometimes said to be "PPT from spectrum".

@@ Line 90: / Line 90: @@
 ==Notes==
-* For all practical purposes, this function can only compute all constraint matrices when at least one of the local dimensions is 6 or less. When both local dimensions are 7 or higher, the <tt>LIM</tt> argument should be specified.
+* For all practical purposes, this function can only compute all constraint matrices when at least one of the local dimensions is 6 or less. When both local dimensions are 7 or higher, the <tt>LIM</tt> argument should be specified (otherwise the computation may take days, weeks, or longer).
 * This function actually computes slightly more constraints than is optimal. It computes the optimal set of constraints when one of the local dimensions is 5 or less, but when the smallest local dimension is 6, it computes 2612 constraint matrices rather than the optimal set of 2608 constraint matrices: the additional 4 matrices are redundant, but do not cause any harm.

@@ Line 87: / Line 87: @@
 ==Notes==
-* For all practical purposed, this function can only compute all constraint matrices when at least one of the local dimensions is 6 or less. When both local dimensions are 7 or higher, the <tt>LIM</tt> argument should be specified.
+* For all practical purposes, this function can only compute all constraint matrices when at least one of the local dimensions is 6 or less. When both local dimensions are 7 or higher, the <tt>LIM</tt> argument should be specified.
 * This function actually computes slightly more constraints than is optimal. It computes the optimal set of constraints when one of the local dimensions is 5 or less, but when the smallest local dimension is 6, it computes 2612 constraint matrices rather than the optimal set of 2608 constraint matrices: the additional 4 matrices are redundant, but do not cause any harm.

@@ Line 69: / Line 69: @@
         lam >= 0; % eigenvalues of a density matrix are non-negative
         for j = 1:length(L)
-            L{j} >= 0; % all of the absolute PPT constraints
+            L{j} >= 0; % all of the absolute PPT constraints (there are 10 of them in this case)
         end
         for j = 1:d^2-1
@@ Line 85: / Line 85: @@
 '''IMPORTANT:''' If you use this function within a CVX optmization problem, make sure that you enforce <tt>lam(1) >= lam(2) >= ... >= 0</tt> as one of the constraints in CVX!
 ==Notes==
 * For all practical purposed, this function can only compute all constraint matrices when at least one of the local dimensions is 6 or less. When both local dimensions are 7 or higher, the <tt>LIM</tt> argument should be specified.

@@ Line 9: / Line 9: @@
 ==Syntax==
 * <tt>L = AbsPPTConstraints(LAM,DIM)</tt>
 * <tt>L = AbsPPTConstraints(LAM,DIM,ESC_IF_NPOS)</tt>
@@ Line 14: / Line 15: @@
 ==Argument descriptions==
-* <tt>LAM</tt>: The eigenvalues of a bipartite density matrix.
+* <tt>LAM</tt>: A bipartite density matrix or a vector containing the eigenvalues of a bipartite density matrix.
-* <tt>DIM</tt>: A specification of the dimensions of the subsystems that the density matrix acts on. <tt>DIM</tt> can be provided in one of two ways:
+* <tt>DIM</tt> (optional, default has both subsystems of equal dimension): A specification of the dimensions of the subsystems that the density matrix 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(LAM)/DIM</tt>.
 ** <tt>DIM</tt> can be a 1-by-2 vector containing the dimensions of the local subsystems.

@@ Line 23: / Line 23: @@
 ==Examples==
 ===Qutrit-qutrit states===
-It was noted in <ref name="Hil07"></ref> that, in the qutrit-qutrit case, the absolutely PPT property is determined by whether or not two 3-by-3 matrices of eigenvalues are positive semidefinite. We can compute these two matrices for a given set of eigenvalues as follows:
+In the qutrit-qutrit case, a state $\rho$ with eigenvalues $\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_9 \geq 0$ is absolutely PPT if and only if the following two matrices are positive semidefinite:
 <syntaxhighlight>
 >> rho = RandomDensityMatrix(9);