# SkOperatorNorm

SkOperatorNorm | |

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

Other toolboxes required | CVX |
---|---|

Related functions | IsBlockPositive SkVectorNorm |

Function category | Norms |

Usable within CVX? | no |

` SkOperatorNorm` is a function that computes 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.

## Syntax

`LB = SkOperatorNorm(X)``LB = SkOperatorNorm(X,K)``LB = SkOperatorNorm(X,K,DIM)``LB = SkOperatorNorm(X,K,DIM,STR)``LB = SkOperatorNorm(X,K,DIM,STR,TARGET)``LB = SkOperatorNorm(X,K,DIM,STR,TARGET,TOL)``[LB,~,UB] = SkOperatorNorm(X,K,DIM,STR,TARGET,TOL)``[LB,LWIT,UB,UWIT] = SkOperatorNorm(X,K,DIM,STR,TARGET,TOL)`

## Argument descriptions

### Input arguments

`X`: A square matrix acting on bipartite space. Generally,`X`should be positive semidefinite – the bounds produced otherwise are quite poor.`K`(optional, default 1): A positive integer.`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.`STR`(optional, default 2): An integer that determines how hard the script should work to compute the lower and upper bounds (`STR = -1`means that the script won't stop working until the bounds match each other). Other valid values are`0, 1, 2, 3, ...`. In practice, if`STR >= 4`then most computers will run out of memory and/or the sun will explode before computation completes.`TARGET`(optional, default -1): A value that you wish to prove that the norm is above or below. If, at any point while this script is running, it proves that`LB >= TARGET`or that`UB <= TARGET`, then the script will immediately abort and return the best lower bound and upper bound computed up to that point. This is a time-saving feature that can be avoided by setting`TARGET = -1`.`TOL`(optional, default`eps^(3/8)`): The numerical tolerance used throughout the script.

### Output arguments

`LB`: A lower bound of the S(k)-operator norm of`X`.`LWIT`: A witness that verifies that`LB`is indeed a lower bound of the S(k)-operator norm of`X`. More specifically,`LWIT`is a unit vector such that`SchmidtRank(LWIT,DIM) <= K`and`LWIT'*X*LWIT = LB`.`UB`: An upper bound of the S(k)-operator norm of`X`.`UWIT`: A witness that verifies that`UB`is indeed an upper bound of the S(k)-operator norm of`X`. More specifically,`UWIT`is a feasible point of the dual semidefinite program presented in Section 5.2.3 of^{[3]}.

## Examples

### Exact computation in small dimensions

When `X` lives in $M_2 \otimes M_2$, $M_2 \otimes M_3$, or $M_3 \otimes M_2$ (i.e., when `prod(DIM) <= 6`), the script is guaranteed to compute the exact value of $\|X\|_{S(1)}$:

```
>> X = [5 1 1 1;1 1 1 1;1 1 1 1;1 1 1 1]/8;
>> SkOperatorNorm(X)
ans =
0.7286
```

The fact that this computation is correct is illustrated in Example 5.2.11 of ^{[3]}, where it was shown that the S(1)-norm is exactly $(3 + 2\sqrt{2})/8 \approx 0.7286$. However, if we were still unconvinced, we could request witnesses that verify that 0.7286 is both a lower bound and an upper bound of the S(1)-norm:

```
>> [lb,lwit,ub,uwit] = SkOperatorNorm(X)
lb =
0.7286
lwit =
0.8536 + 0.0000i
0.3536 - 0.0000i
0.3536 + 0.0000i
0.1464
ub =
0.7286
uwit =
0.0518 + 0.0000i -0.0625 + 0.0000i -0.0625 - 0.0000i -0.1250 - 0.0000i
-0.0625 - 0.0000i 0.3018 + 0.0000i 0.0000 + 0.0000i -0.0625 - 0.0000i
-0.0625 + 0.0000i 0.0000 - 0.0000i 0.3018 + 0.0000i -0.0625 + 0.0000i
-0.1250 + 0.0000i -0.0625 + 0.0000i -0.0625 - 0.0000i 0.3018 + 0.0000i
>> lwit'*X*lwit % verify that the lower bound is correct
ans =
0.7286 + 0.0000i
>> norm(X + uwit) % verify that the upper bound is correct
ans =
0.7286
```

### Only interested in the lower and upper bounds; not the witnesses

If all you want are the lower and upper bounds, but don't require the witnesses `LWIT` and `UWIT`, you can use code like the following. Note that in this case, $\|X\|_{S(1)}$ is computed exactly, as the lower and upper bound are equal (though this will not happen for all `X`!). However, all we know about $\|X\|_{S(2)}$ is that it lies in the interval [0.3522, 0.3546]. It is unsurprising that $\|X\|_{S(3)}$ is the usual operator norm of `X`, since this is always the case when `K >= min(DIM)`.

```
>> X = RandomDensityMatrix(9);
>> [lb,~,ub] = SkOperatorNorm(X,1)
lb =
0.2955
ub =
0.2955
>> [lb,~,ub] = SkOperatorNorm(X,2)
lb =
0.3522
ub =
0.3546
>> [lb,~,ub] = SkOperatorNorm(X,3)
lb =
0.3770
ub =
0.3770
>> norm(X)
ans =
0.3770
```

## Source code

Click on "expand" to the right to view the MATLAB source code for this function.

## 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]
- ↑
^{3.0}^{3.1}N. Johnston. Norms and Cones in the Theory of Quantum Entanglement. PhD thesis, University of Guelph, 2012. E-print: arXiv:1207.1479 [quant-ph]