DiamondNorm
DiamondNorm | |
Computes the diamond norm of a superoperator | |
Other toolboxes required | CVX |
---|---|
Related functions | CBNorm InducedSchattenNorm MaximumOutputFidelity |
Function category | Norms |
Usable within CVX? | yes (convex) |
DiamondNorm is a function that computes the diamond norm $\|\Phi\|_\diamond$ of a superoperator $\Phi$.
Contents
Syntax
- DN = DiamondNorm(PHI)
- DN = DiamondNorm(PHI,DIM)
Argument descriptions
- PHI: A superoperator. Should be provided as either a Choi matrix, or as a cell with either 1 or 2 columns (see the tutorial page for more details about specifying superoperators within QETLAB). PHIC will be a cell of Kraus operators if PHI is a cell of Kraus operators, and similarly PHIC will be a Choi matrix if PHI is a Choi matrix.
- DIM (optional, default has input and output spaces of equal dimension): A 1-by-2 vector containing the input and output dimensions of PHI, in that order (equivalently, these are the dimensions of the first and second subsystems of the Choi matrix PHI, in that order). If the input or output space is not square, then DIM's first row should contain the input and output row dimensions, and its second row should contain its input and output column dimensions. DIM is required if and only if PHI has unequal input and output dimensions and is provided as a Choi matrix.
Examples
A completely positive map
If $\Phi$ is completely positive then $\|\Phi\|_{\diamond} = \|\Phi^\dagger(I)\|$, where $I$ is the identity matrix, $\Phi^\dagger$ is the dual map of $\Phi$, and $\|\cdot\|$ is the usual operator norm, which we can verify in a special case via the following code:
A difference of unitaries channel
If $\Phi(X) = X - UXU^\dagger$, then the diamond norm of $\Phi$ is the diameter of the smallest circle that contains the eigenvalues of $U$, which we can verify in a special case via the following code:
Can be used within CVX
The diamond norm is convex (like all norms) and this function can be used in the same way as any other convex function within CVX. Thus you can minimize the diamond norm or use the diamond norm in constraints of CVX optimization problems. For example, the following code finds the closest Pauli channel (i.e., channel with Kraus operators all of which are multiples of Pauli matrices) to a given 2-qubit channel, where the measure of "closest" is the diamond norm:
>> num_qubits = 2; >> H = [1,1;1,-1]/sqrt(2); % Hadamard gate >> Psi = ChoiMatrix({Tensor(H,num_qubits)}); % channel that acts as Hadamard gate on 2 qubits >> cvx_begin sdp quiet variable Phi(4^num_qubits,4^num_qubits) hermitian; variable p(4^num_qubits); minimize DiamondNorm(Phi - Psi) subject to % these constraints force Phi to be a Pauli channel Phi == PauliChannel(p); sum(p) == 1; p >= 0; cvx_end cvx_optval cvx_optval = 1.5000
Source code
Click on "expand" to the right to view the MATLAB source code for this function.
%% DIAMONDNORM Computes the diamond norm of a superoperator
% This function has one required input argument:
% PHI: a superoperator
%
% DN = DiamondNorm(PHI) is the diamond norm of the superoperator PHI. PHI
% should be specified as either a cell with one or two columns of Kraus
% operators, or as a Choi matrix (see online QETLAB tutorial for details
% about specifying superoperators). If PHI is provided as a Choi matrix
% with unequal input and output dimensions, a second argument specifying
% the dimensions should also be provided (see below).
%
% This function has one optional input argument:
% DIM (default has both subsystems of equal dimension)
%
% DN = DiamondNorm(PHI,DIM) is the diamond norm of PHI, as above, where
% DIM is a 1-by-2 vector containing the input and output dimensions of
% PHI, in that order (equivalently, these are the dimensions of the first
% and second subsystems of the Choi matrix PHI, in that order).
%
% URL: http://www.qetlab.com/DiamondNorm
% requires: ApplyMap.m, ComplementaryMap.m, ChoiMatrix.m, CVX
% (http://cvxr.com/cvx/), DualMap.m, iden.m, IsCP.m,
% IsHermPreserving.m, IsPSD.m, KrausOperators.m,
% MaxEntangled.m, opt_args.m, PermuteSystems.m, Swap.m
%
% author: Nathaniel Johnston (nathaniel@njohnston.ca), based on an
% algorithm by John Watrous
% package: QETLAB
% last updated: April 26, 2016
function dn = DiamondNorm(Phi,varargin)
% If PHI is a CVX variable, assume it is already a Choi matrix and get its
% dimensions in a CVX-safe way.
if(isa(Phi,'cvx'))
len_phi = length(Phi);
sqrt_len = round(sqrt(len_phi));
% Set optional argument defaults: dim=sqrt_len
[dim] = opt_args({ [sqrt_len,sqrt_len] },varargin{:});
% If PHI is not a CVX variable, we can get its dimensions a bit more
% robustly and allow Kraus operator input. We can also implement some
% speed-ups like computing the diamond norm of a completely positive map
% exactly.
else
Phi = KrausOperators(Phi,varargin{:});
[dim] = opt_args({ [size(Phi{1,1},2),size(Phi{1,1},1)] },varargin{:});
len_phi = dim(1)*dim(2);
two_cols = (size(Phi,2) > 1);
if(two_cols && ((size(Phi{1,2},2) ~= dim(1) || size(Phi{1,2},1) ~= dim(2))))
error('DiamondNorm:InvalidDim','The input and output spaces of PHI must both be square.');
end
% The diamond norm of a CP map is trivial to compute.
if(~two_cols || IsCP(Phi))
dn = norm(ApplyMap(speye(dim(2)),DualMap(Phi)));
return
end
Phi = ChoiMatrix(Phi);
end
% This SDP has two advantages over the other SDP for the diamond norm:
% it works in a CVX-safe way that allows this function to be used within
% other CVX optimization problems, and it has better numerical accuracy.
% The downside is that it is slightly slower for channels with few Kraus
% operators.
cvx_begin sdp quiet
cvx_precision best;
variable Y0(len_phi,len_phi) hermitian
variable Y1(len_phi,len_phi) hermitian
minimize norm(PartialTrace(Y0,2,dim)) + norm(PartialTrace(Y1,2,dim))
subject to
cons = [Y0,-Phi;-Phi',Y1];
cons + cons' >= 0; % avoid some numerical problems: CVX often thinks things aren't symmetric without this
Y0 >= 0;
Y1 >= 0;
cvx_end
dn = cvx_optval/2;