# InducedSchattenNorm

InducedSchattenNorm | |

Computes a lower bound of the induced p→q Schatten norm of a superoperator | |

Other toolboxes required | none |
---|---|

Related functions | DiamondNorm InducedMatrixNorm SchattenNorm |

Function category | Norms |

Usable within CVX? | no |

` InducedSchattenNorm` is a function that computes a randomized lower bound of the induced p→q Schatten norm of a superoperator, defined as follows

^{[1]}: \[\|\Phi\|_{p\rightarrow q} := \max\big\{\|\Phi(X)\|_q : \|X\|_p = 1 \big\},\] where \[\|X\|_{p} := \left(\sum_i\sigma_i(X)^p\right)^{1/p}\] is the Schatten p-norm.

When `p = q = 1`, this is the induced trace norm that comes up frequently in quantum information theory (and whose stabilization is the diamond norm). In the `p = q = Inf` case, this is usually called the operator norm of \(\Phi\), which comes up frequently in operator theory.

The lower bound is found via the algorithm described here, which starts with a random input matrix and performs a local optimization based on that starting matrix.

## Syntax

`NRM = InducedSchattenNorm(PHI,P)``NRM = InducedSchattenNorm(PHI,P,Q)``NRM = InducedSchattenNorm(PHI,P,Q,DIM)``NRM = InducedSchattenNorm(PHI,P,Q,DIM,TOL)``NRM = InducedSchattenNorm(PHI,P,Q,DIM,TOL,X0)``[NRM,X] = InducedSchattenNorm(PHI,P,Q,DIM,TOL,X0)`

## Argument descriptions

### Input arguments

`PHI`: A superoperator to have its induced Schatten (`P`→`Q`)-norm computed, specified as either a Choi matrix or a cell array of Kraus operators.`P`: A real number ≥ 1, or`Inf`.`Q`(optional, default equals`P`): A real number ≥ 1, or`Inf`.`DIM`(optional): A 1-by-2 vector containing the input and output dimensions of`PHI`, in that order. Not required if`PHI`'s input and output spaces have the same dimension or if it is provided as a cell array of Kraus operators.`TOL`(optional, default equals`sqrt(eps)`): Numerical tolerance used throughout the script.`X0`(optional, default is randomly-generated): An input matrix to start the numerical search from.

### Output arguments

`NRM`: A lower bound on the norm of`X`.`X`(optional): A matrix with`SchattenNorm(X,P) = 1`such that`SchattenNorm(ApplyMap(X,PHI),Q) = NRM`(i.e., an input matrix that attains the local maximum that was found).

## Examples

### A difference of unitaries channel

If \(\Phi(X) = X - UXU^\dagger\), then the induced trace norm (i.e., Schatten 1-norm) of \(\Phi\) is the diameter of the smallest circle that contains the eigenvalues of \(U\). The following code verifies that this is indeed a lower bound in one special case:

```
>> U = [1 1;-1 1]/sqrt(2);
>> Phi = {eye(2),eye(2); U,-U};
>> InducedSchattenNorm(Phi,1)
ans =
1.4142
>> lam = eig(U)
lam =
0.7071 + 0.7071i
0.7071 - 0.7071i
>> abs(lam(1) - lam(2))
ans =
1.4142
```

## Source code

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

## References

- ↑ J. Watrous. Notes on super-operator norms induced by Schatten norms.
*Quantum Information & Computation*, 5(1):58–68, 2005. E-print: arXiv:quant-ph/0411077