# Sk iterate

sk_iterate | |

Computes a lower bound of the S(k)-norm of an operator | |

Other toolboxes required | none |
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Related functions | SchmidtDecomposition SchmidtRank SkOperatorNorm |

Function category | Helper functions |

This is a helper function that only exists to aid other functions in QETLAB. If you are an end-user of QETLAB, you likely will never have a reason to use this function. |

` sk_iterate` is a function that iteratively computes a lower bound on the S(k)-norm of an operator

^{[1]}

^{[2]}: $$ \|X\|_{S(k)} := \sup_{|v\rangle , |w\rangle } \Big\{ \big| \langle w| X |v \rangle \big| : SR(|v \rangle), SR(|v \rangle) \leq k, \big\||v \rangle\big\| = \big\||w \rangle\big\| = 1 \Big\}, $$ where $SR(\cdot)$ refers to the Schmidt rank of a pure state. The method used to compute this lower bound is described here.

## Syntax

`SK = sk_iterate(X)``SK = sk_iterate(X,K)``SK = sk_iterate(X,K,DIM)``SK = sk_iterate(X,K,DIM,TOL)``SK = sk_iterate(X,K,DIM,TOL,V0)``[SK,V] = sk_iterate(X,K,DIM,TOL,V0)`

## Argument descriptions

### Input arguments

`X`: A square positive semidefinite matrix to have its S(k)-norm bounded.`K`(optional, default 1): A positive integer, the Schmidt rank to optimize over.`DIM`(optional, by default has both subsystems of equal dimension): A 1-by-2 vector containing the dimensions of the subsystems that`X`acts on.`TOL`(optional, default \(10^{-5}\)): The numerical tolerance used when determining whether or not the iterative procedure has converged.`V0`(optional, default is randomly-generated): The vector to begin the iterative procedure from.

### Output arguments

`V`(optional): A vector with Schmidt rank at most`K`such that`V'*X*V == SK`.

## Examples

### A two-qubit example

In ^{[3]}, it was shown that the density matrix
$$
\rho = \frac{1}{8}\begin{bmatrix}5 & 1 & 1 & 1\\1 & 1 & 1 & 1\\1 & 1 & 1 & 1\\1 & 1 & 1 & 1\end{bmatrix}
$$
has S(1)-norm equal to $(3+2\sqrt{2})/8 \approx 0.7286$. The following code shows that this quantity is indeed a lower bound of the S(1)-norm:

```
>> rho = [5 1 1 1;1 1 1 1;1 1 1 1;1 1 1 1]/8;
>> sk_iterate(rho)
ans =
0.7286
```

## Source code

Click here to view this function's source code on github.

## References

- ↑ N. Johnston and D. W. Kribs. A Family of Norms With Applications in Quantum Information Theory.
*J. Math. Phys.*, 51:082202, 2010. E-print: arXiv:0909.3907 [quant-ph] - ↑ N. Johnston and D. W. Kribs. A Family of Norms With Applications in Quantum Information Theory II. Quantum Information & Computation, 11(1 & 2):104–123, 2011. E-print: arXiv:1006.0898 [quant-ph]
- ↑ N. Johnston.
*Norms and Cones in the Theory of Quantum Entanglement*. PhD thesis, University of Guelph, 2012. E-print: arXiv:1207.1479 [quant-ph]