# Sk iterate

 Other toolboxes required sk_iterate Computes a lower bound of the S(k)-norm of an operator none SchmidtDecompositionSchmidtRankSkOperatorNorm 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

1. 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]
2. 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]
3. N. Johnston. Norms and Cones in the Theory of Quantum Entanglement. PhD thesis, University of Guelph, 2012. E-print: arXiv:1207.1479 [quant-ph]