Fundamental limitations on quantum channel manipulation

Code accompanying the manuscript arXiv:2010.11942.

The code below serves as an example of how the measures $R_{\mathbb{O}}$ and $W_{\mathbb{O}}$ can be computed in relevant settings. The implementation requires CVX.


Quantum communication

In the case of communication assisted by no-signalling correlations, $\mathbb{O}$ is the set of replacement (constant) channels. The example code below takes as an input the (unnormalized) Choi matrix J of a channel and a vector dim of the form [dA, dB] indicating the input and output dimensions of the given channel. The dimensions are assumed to be equal if not provided.

Robustness

function [cvx_optval,X] = Rob_NS(J, dim)

if nargin < 2
    dim = [1,1]*sqrt(size(J,1));
end

cvx_begin sdp quiet

variable sigm(dim(2),dim(2)) hermitian semidefinite

J <= kron( eye(dim(1)), sigm )

minimize trace(sigm)

cvx_end

X = kron( eye(dim(1)), sigm );

end

Weight

function [cvx_optval,X] = Weight_NS(J, dim)

if nargin < 2
    dim = [1,1]*sqrt(size(J,1));
end

cvx_begin sdp quiet

variable sigm(dim(2),dim(2)) hermitian semidefinite

J >= kron( eye(dim(1)), sigm )

maximize trace(sigm)

cvx_end

X = kron( eye(dim(1)), sigm );

end

Magic

In the case of magic, we choose $\mathbb{O}$ as the set of completely stabilizer-preserving operations, in which case we can compute the measures by optimizing over the set of stabilizer states. The implementations below take three arguments: the (unnormalized) Choi matrix J of the channel, a matrix T whose columns are all of the stabilizer states in the given dimension (as pure state vectors), and a vector dim of the form [dA, dB] indicating the input and output dimensions of the given channel. The last argument is optional, with the dimension assumed to be equal if not provided. The T matrices for 1-4 qubits are provided below — this allows the computation of the measures for general channels from $n$ to $m$ qubits where $n+m \leq 4$, or for special cases of channels which can be realized through state injection of a state of up to 4 qubits, such as diagonal Clifford gates from the third level of the Clifford hierarchy (see manuscript).

Download auxiliary files:

Robustness

function [cvx_optval, x] = Rob_mag(J, T, dim)

if nargin < 3
    dim = [1,1]*sqrt(size(J,1));
end
k = size(T,2);

cvx_begin sdp quiet

variable x(k) nonnegative
variable t

Z = T*diag(x)*T';

PartialTrace(Z,2,dim) == t*eye(dim(1))

J <= Z

minimize sum(x)/dim(1)

cvx_end

end

Weight

function [cvx_optval, x] = Weight_mag(J, T, dim)

if nargin < 3
    dim = [1,1]*sqrt(size(J,1));
end

k = size(T,2);

cvx_begin sdp quiet

variable x(k) nonnegative
variable t

Z = T*diag(x)*T';

PartialTrace(Z,2,dim) == t*eye(dim(1))

J >= Z

maximize sum(x)/dim(1)

cvx_end

end

To compute the measures for states, it suffices to remove the constraint PartialTrace(Z,2,dim) == t*eye(dim(1)) and remove dim(1) from the objective function (see also the code for state-based measures of magic).