InducedSchattenNorm

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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 (PQ)-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

  1. 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