Difference between revisions of "InducedSchattenNorm"
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| − | <tt>'''InducedSchattenNorm'''</tt> is a [[List of functions|function]] that computes a randomized lower bound of the induced p→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→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> | : <math>\|\Phi\|_{p\rightarrow q} := \max\big\{\|\Phi(X)\|_q : \|X\|_p = 1 \big\},</math> | ||
where | where | ||
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is the Schatten p-norm. | 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 | + | 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. | 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== | ==Examples== | ||
===A difference of unitaries channel=== | ===A difference of unitaries channel=== | ||
| − | If | + | 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> | <syntaxhighlight> | ||
>> U = [1 1;-1 1]/sqrt(2); | >> U = [1 1;-1 1]/sqrt(2); | ||
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{{SourceCode|name=InducedSchattenNorm}} | {{SourceCode|name=InducedSchattenNorm}} | ||
| + | |||
| + | ==References== | ||
| + | <references /> | ||
Latest revision as of 02:36, 1 August 2023
| 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 =
1.4142Source 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