Difference between revisions of "SkOperatorNorm"
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{{Function | {{Function | ||
|name=SkOperatorNorm | |name=SkOperatorNorm | ||
| − | |desc=Computes the | + | |desc=Computes the S(k)-norm of an operator |
|req=[http://cvxr.com/cvx/ cvx] | |req=[http://cvxr.com/cvx/ cvx] | ||
|rel=[[IsBlockPositive]]<br />[[SkVectorNorm]] | |rel=[[IsBlockPositive]]<br />[[SkVectorNorm]] | ||
|cat=[[List of functions#Norms|Norms]] | |cat=[[List of functions#Norms|Norms]] | ||
|upd=September 22, 2014 | |upd=September 22, 2014 | ||
| − | | | + | |cvx=no}} |
| − | <tt>'''SkOperatorNorm'''</tt> is a [[List of functions|function]] that computes the | + | <tt>'''SkOperatorNorm'''</tt> is a [[List of functions|function]] that computes the S(k)-norm of an operator<ref>N. Johnston and D. W. Kribs. A Family of Norms With Applications in Quantum Information Theory. ''J. Math. Phys.'', 51:082202, 2010. E-print: [http://arxiv.org/abs/0909.3907 arXiv:0909.3907] [quant-ph]</ref><ref>N. Johnston and D. W. Kribs. A Family of Norms With Applications in Quantum Information Theory II. Quantum Information & Computation, 11(1 & 2):104–123, 2011. E-print: [http://arxiv.org/abs/1006.0898 arXiv:1006.0898] [quant-ph]</ref>: |
$$ | $$ | ||
\|X\|_{S(k)} := \sup_{|v\rangle , |w\rangle } \Big\{ \big| \langle w| X |v \rangle \big| : SR(|v \rangle), SR(|v \rangle) \leq k, \big\||v \rangle\big\| = \big\||w \rangle\big\| = 1 \Big\}, | \|X\|_{S(k)} := \sup_{|v\rangle , |w\rangle } \Big\{ \big| \langle w| X |v \rangle \big| : SR(|v \rangle), SR(|v \rangle) \leq k, \big\||v \rangle\big\| = \big\||w \rangle\big\| = 1 \Big\}, | ||
$$ | $$ | ||
| − | where $SR(\cdot)$ refers to the [[Schmidt rank]] of a pure state. | + | where $SR(\cdot)$ refers to the [[SchmidtRank|Schmidt rank]] of a pure state. |
==Syntax== | ==Syntax== | ||
| Line 25: | Line 25: | ||
==Argument descriptions== | ==Argument descriptions== | ||
===Input arguments=== | ===Input arguments=== | ||
| − | * <tt>X</tt>: A square matrix acting on | + | * <tt>X</tt>: A square matrix acting on bipartite space. Generally, <tt>X</tt> should be positive semidefinite – the bounds produced otherwise are quite poor. |
* <tt>K</tt> (optional, default 1): A positive integer. | * <tt>K</tt> (optional, default 1): A positive integer. | ||
* <tt>DIM</tt> (optional, by default has both subsystems of equal dimension): A 1-by-2 vector containing the dimensions of the subsystems that <tt>X</tt> acts on. | * <tt>DIM</tt> (optional, by default has both subsystems of equal dimension): A 1-by-2 vector containing the dimensions of the subsystems that <tt>X</tt> acts on. | ||
Revision as of 16:38, 24 December 2014
| SkOperatorNorm | |
| Computes the S(k)-norm of an operator | |
| Other toolboxes required | cvx |
|---|---|
| Related functions | IsBlockPositive SkVectorNorm |
| Function category | Norms |
| Usable within CVX? | no |
SkOperatorNorm is a function that computes the S(k)-norm of an operator[1][2]: $$ \|X\|_{S(k)} := \sup_{|v\rangle , |w\rangle } \Big\{ \big| \langle w| X |v \rangle \big| : SR(|v \rangle), SR(|v \rangle) \leq k, \big\||v \rangle\big\| = \big\||w \rangle\big\| = 1 \Big\}, $$ where $SR(\cdot)$ refers to the Schmidt rank of a pure state.
Syntax
- LB = SkOperatorNorm(X)
- LB = SkOperatorNorm(X,K)
- LB = SkOperatorNorm(X,K,DIM)
- LB = SkOperatorNorm(X,K,DIM,STR)
- LB = SkOperatorNorm(X,K,DIM,STR,TARGET)
- LB = SkOperatorNorm(X,K,DIM,STR,TARGET,TOL)
- [LB,~,UB] = SkOperatorNorm(X,K,DIM,STR,TARGET,TOL)
- [LB,LWIT,UB,UWIT] = SkOperatorNorm(X,K,DIM,STR,TARGET,TOL)
Argument descriptions
Input arguments
- X: A square matrix acting on bipartite space. Generally, X should be positive semidefinite – the bounds produced otherwise are quite poor.
- K (optional, default 1): A positive integer.
- DIM (optional, by default has both subsystems of equal dimension): A 1-by-2 vector containing the dimensions of the subsystems that X acts on.
- STR (optional, default 2): An integer that determines how hard the script should work to compute the lower and upper bounds (STR = -1 means that the script won't stop working until the bounds match each other). Other valid values are 0, 1, 2, 3, .... In practice, if STR >= 4 then most computers will run out of memory and/or the sun will explode before computation completes.
- TARGET (optional, default -1): A value that you wish to prove that the norm is above or below. If, at any point while this script is running, it proves that LB >= TARGET or that UB <= TARGET, then the script will immediately abort and return the best lower bound and upper bound computed up to that point. This is a time-saving feature that can be avoided by setting TARGET = -1.
- TOL (optional, default eps^(3/8)): The numerical tolerance used throughout the script.
Output arguments
- LB: A lower bound of the S(k)-operator norm of X.
- LWIT: A witness that verifies that LB is indeed a lower bound of the S(k)-operator norm of X. More specifically, LWIT is a unit vector such that SchmidtRank(LWIT,DIM) <= K and LWIT'*X*LWIT = LB.
- UB: An upper bound of the S(k)-operator norm of X.
- UWIT: A witness that verifies that UB is indeed an upper bound of the S(k)-operator norm of X. More specifically, UWIT is a feasible point of the dual semidefinite program presented in Section 5.2.3 of [3].
Examples
Exact computation in small dimensions
When X lives in $M_2 \otimes M_2$, $M_2 \otimes M_3$, or $M_3 \otimes M_2$ (i.e., when prod(DIM) <= 6), the script is guaranteed to compute the exact value of $\|X\|_{S(1)}$:
>> X = [5 1 1 1;1 1 1 1;1 1 1 1;1 1 1 1]/8;
>> SkOperatorNorm(X)
ans =
0.7286The fact that this computation is correct is illustrated in Example 5.2.11 of [3], where it was shown that the S(1)-norm is exactly $(3 + 2\sqrt{2})/8 \approx 0.7286$. However, if we were still unconvinced, we could request witnesses that verify that 0.7286 is both a lower bound and an upper bound of the S(1)-norm:
>> [lb,lwit,ub,uwit] = SkOperatorNorm(X)
lb =
0.7286
lwit =
0.8536 + 0.0000i
0.3536 - 0.0000i
0.3536 + 0.0000i
0.1464
ub =
0.7286
uwit =
0.0518 + 0.0000i -0.0625 + 0.0000i -0.0625 - 0.0000i -0.1250 - 0.0000i
-0.0625 - 0.0000i 0.3018 + 0.0000i 0.0000 + 0.0000i -0.0625 - 0.0000i
-0.0625 + 0.0000i 0.0000 - 0.0000i 0.3018 + 0.0000i -0.0625 + 0.0000i
-0.1250 + 0.0000i -0.0625 + 0.0000i -0.0625 - 0.0000i 0.3018 + 0.0000i
>> lwit'*X*lwit % verify that the lower bound is correct
ans =
0.7286 + 0.0000i
>> norm(X + uwit) % verify that the upper bound is correct
ans =
0.7286Only interested in the lower and upper bounds; not the witnesses
If all you want are the lower and upper bounds, but don't require the witnesses LWIT and UWIT, you can use code like the following. Note that in this case, $\|X\|_{S(1)}$ is computed exactly, as the lower and upper bound are equal (though this will not happen for all X!). However, all we know about $\|X\|_{S(2)}$ is that it lies in the interval [0.3522, 0.3546]. It is unsurprising that $\|X\|_{S(3)}$ is the usual operator norm of X, since this is always the case when K >= min(DIM).
>> X = RandomDensityMatrix(9);
>> [lb,~,ub] = SkOperatorNorm(X,1)
lb =
0.2955
ub =
0.2955
>> [lb,~,ub] = SkOperatorNorm(X,2)
lb =
0.3522
ub =
0.3546
>> [lb,~,ub] = SkOperatorNorm(X,3)
lb =
0.3770
ub =
0.3770
>> norm(X)
ans =
0.3770Source code
Click here to view this function's source code on github.
References
- ↑ N. Johnston and D. W. Kribs. A Family of Norms With Applications in Quantum Information Theory. J. Math. Phys., 51:082202, 2010. E-print: arXiv:0909.3907 [quant-ph]
- ↑ N. Johnston and D. W. Kribs. A Family of Norms With Applications in Quantum Information Theory II. Quantum Information & Computation, 11(1 & 2):104–123, 2011. E-print: arXiv:1006.0898 [quant-ph]
- ↑ 3.0 3.1 N. Johnston. Norms and Cones in the Theory of Quantum Entanglement. PhD thesis, University of Guelph, 2012. E-print: arXiv:1207.1479 [quant-ph]