BrauerStates - Revision history

Nathaniel: Updated to v0.51

2014-11-12T17:25:03Z

Updated to v0.51

Nathaniel: /* Four-qubit Brauer States */ typo

2014-11-10T00:26:49Z

Four-qubit Brauer States: typo

Nathaniel at 13:05, 7 November 2014

2014-11-07T13:05:15Z

Nathaniel at 20:41, 6 November 2014

2014-11-06T20:41:50Z

Nathaniel: Created page with "{{Function |name=BrauerStates |desc=Produces all Brauer states |rel=MaxEntangled |cat=List of functions#Special_states,_vectors,_and_operators|Special states, vectors, a..."

2014-11-06T20:13:00Z

Created page with "{{Function |name=BrauerStates |desc=Produces all Brauer states |rel=MaxEntangled |cat=List of functions#Special_states,_vectors,_and_operators|Special states, vectors, a..."

New page

{{Function
|name=BrauerStates
|desc=Produces all Brauer states
|rel=[[MaxEntangled]]
|cat=[[List of functions#Special_states,_vectors,_and_operators|Special states, vectors, and operators]]
|upd=November 6, 2014
|v=0.50}}
<tt>'''BrauerStates'''</tt> is a [[List of functions|function]] that returns all "Brauer" states: state that are the $k$-fold tensor product of the [[MaxEntangled|standard pure maximally-entangled state]]. Note that there are $(2k)!/(k!\cdot 2^k)$ such states, since this is the number of ways of choose $k$ pairs out of $2k$ objects (here, each pair corresponds to two subsystems that are maximally-entangled). Note that the states returned are unnormalized (i.e., all of their entries are 0 or 1, rather than being scaled so that the norm of each state is 1) and sparse.

==Syntax==
* <tt>B = BrauerStates(K,N)</tt>

==Argument descriptions==
* <tt>K</tt>: Half of the number of parties (i.e., the states that this function computes will live in $2K$-partite space).
* <tt>N</tt>: The dimension of each local subsystem (i.e., the states that this function computes will live in $(\mathbb{C}^N)^{\otimes 2K}$).

==Examples==
===Four-qubit Brauer States===
The following code generates a matrix whose columns are all Brauer states on 4 qubits:
<syntaxhighlight>
>> full(BrauerStates(2,2))

ans =

1 1 1
0 0 0
0 0 0
1 0 0
0 0 0
0 1 0
0 0 1
0 0 0
0 0 0
0 0 1
0 1 0
0 0 0
1 0 0
0 0 0
0 0 0
1 1 1
</syntaxhighlight>

Indeed, the first column of the output above the the state that is maximally-entangled between qubits 1 and 2, and maximally-entangled between qubits 3 and 4. The second column is the state that is maximally-entangled between qubits 1 and 3 and between qubits 2 and 4. Finally, the third column is the state that is maximally-entangled between qubits 1 and 4 and between qubits 2 and 3.

==Notes==
In general, the output of this function will be a $N^{2K}$-by-$(2K)!/(K!\cdot 2^K)$ matrix.

{{SourceCode|name=BrauerStates}}

@@ Line 8: / Line 8: @@
 <tt>'''BrauerStates'''</tt> is a [[List of functions|function]] that returns all "Brauer" states: state that are the $k$-fold tensor product of the [[MaxEntangled|standard pure maximally-entangled state]]. Note that there are $(2k)!/(k!\cdot 2^k)$ such states, since this is the number of ways of choose $k$ pairs out of $2k$ objects (here, each pair corresponds to two subsystems that are maximally-entangled). Note that the states returned are unnormalized (i.e., all of their entries are 0 or 1, rather than being scaled so that the norm of each state is 1) and sparse.
-Brauer states are interesting because they are exactly the states that are invariant under the action of $X^{\otimes 2k}$ for every [http://en.wikipedia.org/wiki/Orthogonal_matrix real orthogonal matrix] $X$.
+Brauer states are interesting because they span the subspace that is invariant under the action of $X^{\otimes 2k}$ for every [http://en.wikipedia.org/wiki/Orthogonal_matrix real orthogonal matrix] $X$.
 ==Syntax==

@@ Line 4: / Line 4: @@
 |rel=[[MaxEntangled]]
 |cat=[[List of functions#Special_states,_vectors,_and_operators|Special states, vectors, and operators]]
-|upd=November 6, 2014
+|upd=November 12, 2014
-|v=0.50}}
+|v=0.51}}
-<tt>'''BrauerStates'''</tt> is a [[List of functions|function]] that returns all "Brauer" states: state that are the $k$-fold tensor product of the [[MaxEntangled|standard pure maximally-entangled state]]. Note that there are $(2k)!/(k!\cdot 2^k)$ such states, since this is the number of ways of choose $k$ pairs out of $2k$ objects (here, each pair corresponds to two subsystems that are maximally-entangled). Note that the states returned are unnormalized (i.e., all of their entries are 0 or 1, rather than being scaled so that the norm of each state is 1) and sparse.
+<tt>'''BrauerStates'''</tt> is a [[List of functions|function]] that returns all "Brauer" states: state that are the $p$-fold tensor product of the [[MaxEntangled|standard pure maximally-entangled state]]. Note that there are $(2p)!/(p!\cdot 2^p)$ such states, since this is the number of ways of choose $p$ pairs out of $2p$ objects (here, each pair corresponds to two subsystems that are maximally-entangled). Note that the states returned are unnormalized (i.e., all of their entries are 0 or 1, rather than being scaled so that the norm of each state is 1) and sparse.
-Brauer states are interesting because they span the subspace that is invariant under the action of $X^{\otimes 2k}$ for every [http://en.wikipedia.org/wiki/Orthogonal_matrix real orthogonal matrix] $X$.
+Brauer states are interesting because they span the subspace that is invariant under the action of $X^{\otimes 2p}$ for every [http://en.wikipedia.org/wiki/Orthogonal_matrix real orthogonal matrix] $X$.
 ==Syntax==
-* <tt>B = BrauerStates(K,N)</tt>
+* <tt>B = BrauerStates(D,P)</tt>
 ==Argument descriptions==
-* <tt>K</tt>: Half of the number of parties (i.e., the states that this function computes will live in $2K$-partite space).
+* <tt>D</tt>: The dimension of each local subsystem.
-* <tt>N</tt>: The dimension of each local subsystem (i.e., the states that this function computes will live in $(\mathbb{C}^N)^{\otimes 2K}$).
+* <tt>P</tt>: Half of the number of parties (i.e., the states that this function computes will live in $(\mathbb{C}^D)^{\otimes 2P}$).
 ==Examples==
@@ Line 46: / Line 46: @@
 ==Notes==
-In general, the output of this function will be a $N^{2K}$-by-$(2K)!/(K!\cdot 2^K)$ matrix.
+* In general, the output of this function will be a $N^{2P}$-by-$(2P)!/(P!\cdot 2^P)$ matrix.
 {{SourceCode|name=BrauerStates}}

@@ Line 43: / Line 43: @@
 </syntaxhighlight>
-Indeed, the first column of the output above the the state that is maximally-entangled between qubits 1 and 2, and maximally-entangled between qubits 3 and 4. The second column is the state that is maximally-entangled between qubits 1 and 3 and between qubits 2 and 4. Finally, the third column is the state that is maximally-entangled between qubits 1 and 4 and between qubits 2 and 3.
+Indeed, the first column of the output above is the state that is maximally-entangled between qubits 1 and 2, and maximally-entangled between qubits 3 and 4. The second column is the state that is maximally-entangled between qubits 1 and 3 and between qubits 2 and 4. Finally, the third column is the state that is maximally-entangled between qubits 1 and 4 and between qubits 2 and 3.
 ==Notes==