SkOperatorNorm

SkOperatorNorm
Computes the S(k)-norm of an operator
Other toolboxes required	cvx
Related functions	IsBlockPositive SkVectorNorm
Function category	Norms

SkOperatorNorm is a function that computes the S(k)-norm of an operator: $$ \|X\|_{S(k)} := \sup_{|v\rangle , |w\rangle } \Big\{ \big| \langle w| X |v \rangle \big| : SR(|v \rangle), SR(|v \rangle) \leq k, \big\||v \rangle\big\| = \big\||w \rangle\big\| = 1 \Big\}, $$ 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 .

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 =

    0.7286

The fact that this computation is correct is illustrated in Example 5.2.11 of , 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 =

    0.7286

Only 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 =

    0.3770

Source code

Click here to view this function's source code on github.