Difference between revisions of "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

Latest revision as of 02:36, 1 August 2023

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

[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

[1]

@@ Line 12: / Line 12: @@
 is the Schatten p-norm.
-When <tt>p = q = 1</tt>, this is the induced trace norm that comes up frequently in quantum information theory (and whose stabilization is the [[DiamondNorm|diamond norm]]). In the <tt>p = q = Inf</tt> case, this is usually called the operator norm of $\Phi$, which comes up frequently in operator theory.
+When <tt>p = q = 1</tt>, this is the induced trace norm that comes up frequently in quantum information theory (and whose stabilization is the [[DiamondNorm|diamond norm]]). In the <tt>p = q = Inf</tt> case, this is usually called the operator norm of <math>\Phi</math>, which comes up frequently in operator theory.
 The lower bound is found via the algorithm described [http://www.njohnston.ca/2016/01/how-to-compute-hard-to-compute-matrix-norms/ here], which starts with a random input matrix and performs a local optimization based on that starting matrix.
@@ Line 39: / Line 39: @@
 ==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:
+If <math>\Phi(X) = X - UXU^\dagger</math>, then the induced trace norm (i.e., Schatten 1-norm) of <math>\Phi</math> is the diameter of the smallest circle that contains the eigenvalues of <math>U</math>. The following code verifies that this is indeed a lower bound in one special case:
 <syntaxhighlight>
 >> U = [1 1;-1 1]/sqrt(2);