InducedSchattenNorm

InducedSchattenNorm is a function that computes a randomized lower bound of the induced p&rarr;q Schatten norm of a superoperator, defined as follows :
 * $$\|\Phi\|_{p\rightarrow q} := \max\big\{\|\Phi(X)\|_q : \|X\|_p = 1 \big\},$$

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)</tt>

Input arguments

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

Output arguments

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

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: