# SymmetricProjection

 Other toolboxes required SymmetricProjection Produces the projection onto the symmetric subspace none AntisymmetricProjectionPermutationOperatorSwapOperator Permutations and symmetry of subsystems

SymmetricProjection is a function that computes the orthogonal projection onto the symmetric subspace of two or more subsystems. The output of this function is always a sparse matrix.

## Syntax

• PS = SymmetricProjection(DIM)
• PS = SymmetricProjection(DIM,P)
• PS = SymmetricProjection(DIM,P,PARTIAL)
• PS = SymmetricProjection(DIM,P,PARTIAL,MODE)

## Argument descriptions

• DIM: The dimension of each of the subsystems.
• P (optional, default 2): The number of subsystems.
• PARTIAL (optional, default 0): If PARTIAL = 1 then PS isn't the orthogonal projection itself, but rather a matrix whose columns form an orthonormal basis for the symmetric subspace (and hence PS*PS' is the orthogonal projection onto the symmetric subspace).
• MODE (optional, default -1): A flag that determines which of two algorithms is used to compute the symmetric projection. If MODE = -1 then this script chooses which algorithm it thinks will be faster based on the values of DIM and P. If you wish to force the script to use a specific one of the algorithms (not recommended!), they are as follows:
• MODE = 0: Computes the symmetric projection by explicitly constructing an orthonormal basis of the symmetric subspace. The details of how to construct such a basis can be found in [1]. This method is typically fast when DIM is small compared to P.
• MODE = 1: Computes the symmetric projection by averaging all P! permutation operators (in the sense of the PermutationOperator function). Because P! grows very quickly, this method is only practical when P is small.

## Examples

### Two subsystems

To compute the symmetric projection on two-qubit space, the following code suffices:

>> SymmetricProjection(2)

ans =

(1,1)       1.0000
(2,2)       0.5000
(3,2)       0.5000
(2,3)       0.5000
(3,3)       0.5000
(4,4)       1.0000

Note that the output of this function is always sparse. If you want a full matrix (not recommended for even moderately large DIM or P), you must explicitly convert it (as in the following example).

### Three subsystems

To compute a matrix whose columns form an orthonormal basis for the symmetric subspace of three-qubit space, set PARTIAL = 1:

>> PS = full(SymmetricProjection(2,3,1))

PS =

1.0000         0         0         0
0         0   -0.5774         0
0         0   -0.5774         0
0         0         0   -0.5774
0         0   -0.5774         0
0         0         0   -0.5774
0         0         0   -0.5774
0    1.0000         0         0

Note that PS is an isometry from the symmetric subspace to the full three-qubit space. In other words, PS'*PS is the identity matrix and PS*PS' is the orthogonal projection onto the symmetric subspace, which we can verify as follows:

>> PS'*PS

ans =

1     0     0     0
0     1     0     0
0     0     1     0
0     0     0     1

>> PS*PS'

ans =

1.0000         0         0         0         0         0         0         0
0    0.3333    0.3333         0    0.3333         0         0         0
0    0.3333    0.3333         0    0.3333         0         0         0
0         0         0    0.3333         0    0.3333    0.3333         0
0    0.3333    0.3333         0    0.3333         0         0         0
0         0         0    0.3333         0    0.3333    0.3333         0
0         0         0    0.3333         0    0.3333    0.3333         0
0         0         0         0         0         0         0    1.0000

## Notes

When using PARTIAL = 1 and MODE = 1, there is no guarantee about what the orthonormal basis of the symmetric subspace will look like. If you want control over this, specify MODE = 0, which forces the basis (i.e., the columns of PS) to be symmetric tensors $$|i_1\rangle \vee |i_1\rangle \vee \cdots \vee |i_p\rangle$$. Since May 2, 2022, the columns of PS are these symmetric tensors in lexicographical order (prior to that date they were returned in a different order).

## References

1. John Watrous. Lecture 21: The quantum de Finetti theorem, Theory of Quantum Information Lecture Notes, 2008.