Difference between revisions of "BrauerStates"
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<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 $k$-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 | + | 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== | ==Syntax== |
Revision as of 13:05, 7 November 2014
BrauerStates | |
Produces all Brauer states | |
Other toolboxes required | none |
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Related functions | MaxEntangled |
Function category | Special states, vectors, and operators |
BrauerStates is a function that returns all "Brauer" states: state that are the $k$-fold tensor product of the 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 span the subspace that is invariant under the action of $X^{\otimes 2k}$ for every real orthogonal matrix $X$.
Syntax
- B = BrauerStates(K,N)
Argument descriptions
- K: Half of the number of parties (i.e., the states that this function computes will live in $2K$-partite space).
- N: 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:
>> 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
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.
Source code
Click here to view this function's source code on github.