# SchmidtDecomposition

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 Other toolboxes required SchmidtDecomposition Computes the Schmidt decomposition of a bipartite vector none IsProductVectorOperatorSchmidtDecompositionSchmidtRank Entanglement and separability

SchmidtDecomposition is a function that computes the Schmidt decomposition of a bipartite vector. The user may specify how many terms in the Schmidt decomposition they wish to be returned.

## Syntax

• S = SchmidtDecomposition(VEC)
• S = SchmidtDecomposition(VEC,DIM)
• S = SchmidtDecomposition(VEC,DIM,K)
• [S,U,V] = SchmidtDecomposition(VEC,DIM,K)

## Argument descriptions

### Input arguments

• VEC: A bipartite vector (e.g., a pure quantum state) to have its Schmidt decomposition computed.
• DIM (optional, by default has both subsystems of equal dimension): A 1-by-2 vector containing the dimensions of the subsystems that VEC lives on.
• K (optional, default 0): A flag that determines how many terms in the Schmidt decomposition should be computed. If K = 0 then all terms with non-zero Schmidt coefficients are computed. If K = -1 then all terms (including zero Schmidt coefficients) are computed. If K > 0 then the K terms with largest Schmidt coefficients are computed.

### Output arguments

• S: A vector containing the Schmidt coefficients of VEC.
• U (optional): A matrix whose columns are the left Schmidt vectors of VEC.
• V (optional): A matrix whose columns are the right Schmidt vectors of VEC.

## Examples

The following code returns the Schmidt decomposition of the standard maximally-entangled pure state $\frac{1}{\sqrt{d}}\sum_j|j\rangle\otimes|j\rangle \in \mathbb{C}^d \otimes \mathbb{C}^d$ in the $d = 3$ case:

>> [s,u,v] = SchmidtDecomposition(MaxEntangled(3))

s =

0.5774
0.5774
0.5774

u =

1     0     0
0     1     0
0     0     1

v =

1     0     0
0     1     0
0     0     1