BrauerStates

BrauerStates
Produces all Brauer states
Other toolboxes required	none
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.

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.