InducedSchattenNorm - Revision history

Nathaniel: LaTeX fix

2023-08-01T02:36:33Z

LaTeX fix

2023-08-01T02:35:33Z

LaTeX fix

2016-01-12T03:25:02Z

added reference

2016-01-12T02:40:07Z

added external link

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 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.

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 ==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);

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 |upd=January 11, 2016
 |cvx=no}}
-<tt>'''InducedSchattenNorm'''</tt> is a [[List of functions|function]] that computes a randomized lower bound of the induced p&rarr;q [[SchattenNorm|Schatten norm]] of a superoperator, defined as follows:
+<tt>'''InducedSchattenNorm'''</tt> is a [[List of functions|function]] that computes a randomized lower bound of the induced p&rarr;q [[SchattenNorm|Schatten norm]] of a superoperator, defined as follows <ref>J. Watrous. Notes on super-operator norms induced by Schatten norms. ''Quantum Information & Computation'', 5(1):58–68, 2005. E-print: [http://arxiv.org/abs/quant-ph/0411077 arXiv:quant-ph/0411077]</ref>:
 : <math>\|\Phi\|_{p\rightarrow q} := \max\big\{\|\Phi(X)\|_q : \|X\|_p = 1 \big\},</math>
 where
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 {{SourceCode|name=InducedSchattenNorm}}

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 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.
-The lower bound is found via the algorithm described here (link to be filled in), which starts with a random input matrix and performs a local optimization based on that starting matrix.
+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.
 ==Syntax==