Majorizes - Revision history

Nathaniel at 15:45, 29 September 2014

2014-09-29T15:45:56Z

Nathaniel: /* Majorization criterion for separability */

2014-03-04T17:59:48Z

Majorization criterion for separability

Nathaniel: /* Majorization criterion for separability */

2014-03-04T16:25:44Z

Majorization criterion for separability

Nathaniel at 15:12, 4 March 2014

2014-03-04T15:12:26Z

Nathaniel: /* Examples */

2014-03-04T14:59:15Z

Examples

Nathaniel: Created page with "{{Function |name=Majorizes |desc=Determines whether or not a vector or matrix majorizes another |upd=March 4, 2014 |v=1.00}} `'''Majorizes'''` is a [[List of functions|..."

2014-03-04T14:48:18Z

Created page with "{{Function |name=Majorizes |desc=Determines whether or not a vector or matrix majorizes another |upd=March 4, 2014 |v=1.00}} <tt>'''Majorizes'''</tt> is a [[List of functions|..."

New page

{{Function
|name=Majorizes
|desc=Determines whether or not a vector or matrix majorizes another
|upd=March 4, 2014
|v=1.00}}
<tt>'''Majorizes'''</tt> is a [[List of functions|function]] that determines whether or not one vector or matrix [http://en.wikipedia.org/wiki/Majorization weakly majorizes] another vector or matrix. That is, given d-dimensional vectors $A$ and $B$, it checks whether or not

: <math> \sum_{i=1}^k a_i^{\downarrow} \geq \sum_{i=1}^k b_i^{\downarrow} \quad \text{for } k=1,\dots,d,</math>

where $a^{\downarrow}_i$ and $b^{\downarrow}_i$ are the elements of $A$ and $B$, respectively, sorted in decreasing order.

In the case of matrices, it is said that $A$ majorizes $B$ if the vector of $A$'s singular values majorizes the vector of $B$'s singular values. If the two vectors or matrices are of different sizes, the smaller one is padded with zeros appropriately so that they are comparable.

==Syntax==
* <tt>M = Majorizes(A,B)</tt>

==Argument descriptions==
* <tt>A</tt>: Either a vector or a matrix.
* <tt>B</tt>: Either a vector or a matrix.

==Examples==
===A simple example===
It is straightforward to see that the vector $(3,0,0)$ majorizes the vector $(1,1,1)$, which we can verify as follows:
<pre>
>> Majorizes([3,0,0],[1,1,1])

ans =

1
</pre>

===Bipartite LOCC===
A well-known result of Neilsen<ref>M. A. Neilsen. Conditions for a class of entanglement transformations. Phys. Rev. Lett., 83:439, 1999.</ref> says that a bipartite pure state $|\psi\rangle \in \mathbb{C}^m \otimes \mathbb{C}^n$ can be converted into another state $|\phi\rangle$ via [[LOCC]] if and only if the Schmidt coefficients of $|\phi\rangle$ majorize the Schmidt coefficients of $|\psi\rangle$. Thus we can determine whether or not we can convert $|\psi\rangle$ to $|\phi\rangle$ via LOCC as follows:
<pre>
>> phi = RandomStateVector(9); % generate two random states in C^3 \otimes C^3
>> psi = RandomStateVector(9);
>> Majorizes(SchmidtDecomposition(phi),SchmidtDecomposition(psi))

ans =

0
</pre>

The above code shows that the conversion $|\psi\rangle \stackrel{LOCC}{\rightarrow} |\phi\rangle$ is impossible.

==References==
<references />

@@ Line 3: / Line 3: @@
 |desc=Determines whether or not a vector or matrix majorizes another
 |upd=March 4, 2014
-|v=1.00}}
+|cat=[[List of functions#Miscellaneous|Miscellaneous]]
 <tt>'''Majorizes'''</tt> is a [[List of functions|function]] that determines whether or not one vector or matrix [http://en.wikipedia.org/wiki/Majorization weakly majorizes] another vector or matrix. That is, given d-dimensional vectors $A$ and $B$, it checks whether or not
@@ Line 22: / Line 23: @@
 ===A simple example===
 It is straightforward to see that the vector $(3,0,0)$ majorizes the vector $(1,1,1)$, which we can verify as follows:
-<pre>
+<syntaxhighlight>
 >> Majorizes([3,0,0],[1,1,1])
@@ Line 28: / Line 29: @@
-</pre>
+</syntaxhighlight>
 ===Bipartite LOCC===
 A well-known result of Nielsen<ref>M. A. Nielsen. Conditions for a class of entanglement transformations. ''Phys. Rev. Lett.'', 83:439, 1999.</ref> says that a bipartite pure state $|\psi\rangle \in \mathbb{C}^m \otimes \mathbb{C}^n$ can be converted into another state $|\phi\rangle$ via [[LOCC]] if and only if the squared Schmidt coefficients of $|\phi\rangle$ majorize the squared Schmidt coefficients of $|\psi\rangle$. Thus we can determine whether or not we can convert $|\psi\rangle$ to $|\phi\rangle$ via LOCC as follows:
-<pre<noinclude></noinclude>>
+<syntaxhighlight>
->> phi = [[RandomStateVector|RandomStateVector(9)]]; % generate two random states in C^3 \otimes C^3
+>> phi = RandomStateVector(9); % generate two random states in C^3 \otimes C^3
 >> psi = RandomStateVector(9);
->> Majorizes([[SchmidtDecomposition|SchmidtDecomposition(phi)]].^2,SchmidtDecomposition(psi).^2)
+>> Majorizes(SchmidtDecomposition(phi).^2,SchmidtDecomposition(psi).^2)
 ans =
-</pre<noinclude></noinclude>>
+</syntaxhighlight>
 The above code shows that the conversion $|\psi\rangle \stackrel{LOCC}{\rightarrow} |\phi\rangle$ is impossible. On the other hand, the following code shows that the maximally-entangled pure state can be converted to a random pure state via LOCC:
-<pre<noinclude></noinclude>>
+<syntaxhighlight>
 >> phi = RandomStateVector(9);
->> psi = [[MaxEntangled|MaxEntangled(3)]]; % maximally-entangled state in C^3 \otimes C^3
+>> psi = MaxEntangled(3); % maximally-entangled state in C^3 \otimes C^3
 >> Majorizes(SchmidtDecomposition(phi).^2,SchmidtDecomposition(psi).^2)
@@ Line 51: / Line 52: @@
-</pre<noinclude></noinclude>>
+</syntaxhighlight>
 ===Majorization criterion for separability===
 The [[majorization criterion]] says that every separable state $\rho_{AB}$ is such that each of $\rho_A$ and $\rho_B$ majorize $\rho_{AB}$<ref>M. A. Nielsen and J. Kempe. Separable states are more disordered globally than locally. ''Phys. Rev. Lett.'', 86:5184, 2001. E-print: [http://arxiv.org/abs/quant-ph/0011117 arXiv:quant-ph/0011117]</ref>. Thus the following code constructs the maximally-entangled quantum state and uses the majorization criterion to verify that it is entangled:
-<pre<noinclude></noinclude>>
+<syntaxhighlight>
 >> v = MaxEntangled(3);
 >> rho = v*v';
->> Majorizes([[PartialTrace|PartialTrace(rho)]],rho)
+>> Majorizes(PartialTrace(rho),rho)
 ans =
-</pre<noinclude></noinclude>>
+</syntaxhighlight>
 Note that the majorization criterion is strictly weaker than the partial transpose criterion (and even the [[reduction criterion]]<ref>T. Hiroshima. Majorization criterion for distillability of a bipartite quantum state. ''Phys. Rev. Lett.'', 91:057902, 2003. E-print: [http://arxiv.org/abs/quant-ph/0303057 arXiv:quant-ph/0303057]</ref>) and thus those alternate tests should be used in most situations instead.
 ==References==
 <references />

@@ Line 65: / Line 65: @@
 </pre<noinclude></noinclude>>
-Note that the majorization criterion is strictly weaker than the partial transpose criterion (and even the [[reduction criterion]] <ref>T. Hiroshima. Majorization criterion for distillability of a bipartite quantum state. ''Phys. Rev. Lett.'', 91:057902, 2003. E-print: [http://arxiv.org/abs/quant-ph/0303057 arXiv:quant-ph/0303057]</ref>) and thus those alternate tests should be used in most situations instead.
+Note that the majorization criterion is strictly weaker than the partial transpose criterion (and even the [[reduction criterion]]<ref>T. Hiroshima. Majorization criterion for distillability of a bipartite quantum state. ''Phys. Rev. Lett.'', 91:057902, 2003. E-print: [http://arxiv.org/abs/quant-ph/0303057 arXiv:quant-ph/0303057]</ref>) and thus those alternate tests should be used in most situations instead.
 ==References==
 <references />

@@ Line 58: / Line 58: @@
 >> v = MaxEntangled(3);
 >> rho = v*v';
->> Majorizes(PartialTrace(rho),rho)
+>> Majorizes([[PartialTrace|PartialTrace(rho)]],rho)
 ans =

@@ Line 31: / Line 31: @@
 ===Bipartite LOCC===
-A well-known result of Nielsen<ref>M. A. Nielsen. Conditions for a class of entanglement transformations. Phys. Rev. Lett., 83:439, 1999.</ref> says that a bipartite pure state $|\psi\rangle \in \mathbb{C}^m \otimes \mathbb{C}^n$ can be converted into another state $|\phi\rangle$ via [[LOCC]] if and only if the squared Schmidt coefficients of $|\phi\rangle$ majorize the squared Schmidt coefficients of $|\psi\rangle$. Thus we can determine whether or not we can convert $|\psi\rangle$ to $|\phi\rangle$ via LOCC as follows:
+A well-known result of Nielsen<ref>M. A. Nielsen. Conditions for a class of entanglement transformations. ''Phys. Rev. Lett.'', 83:439, 1999.</ref> says that a bipartite pure state $|\psi\rangle \in \mathbb{C}^m \otimes \mathbb{C}^n$ can be converted into another state $|\phi\rangle$ via [[LOCC]] if and only if the squared Schmidt coefficients of $|\phi\rangle$ majorize the squared Schmidt coefficients of $|\psi\rangle$. Thus we can determine whether or not we can convert $|\psi\rangle$ to $|\phi\rangle$ via LOCC as follows:
 <pre<noinclude></noinclude>>
 >> phi = [[RandomStateVector|RandomStateVector(9)]]; % generate two random states in C^3 \otimes C^3
@@ Line 52: / Line 52: @@
 </pre<noinclude></noinclude>>
 ==References==
 <references />

Majorizes - Revision history

Nathaniel at 15:45, 29 September 2014

Nathaniel: /* Majorization criterion for separability */

Nathaniel: /* Majorization criterion for separability */

Nathaniel at 15:12, 4 March 2014

Nathaniel: /* Examples */

Nathaniel: Created page with "{{Function |name=Majorizes |desc=Determines whether or not a vector or matrix majorizes another |upd=March 4, 2014 |v=1.00}} '''Majorizes''' is a [[List of functions|..."

Nathaniel: Created page with "{{Function |name=Majorizes |desc=Determines whether or not a vector or matrix majorizes another |upd=March 4, 2014 |v=1.00}} `'''Majorizes'''` is a [[List of functions|..."