Code to accompany: Reversing unknown quantum transformations: A universal protocol for inverting general unitary operations

Marco Túlio Quintino, Qingxiuxiong Dong, Atsushi Shimbo, Akihito Soeda, and Mio Murao

This repository provides the codes used to obtain Table I and Table II in the article Reversing unknown quantum transformations: A universal protocol for inverting general unitary operations, Phys. Rev. Lett. 123, 210502 (2019), arXiv:1810.06944

Marco Túlio Quintino, Qingxiuxiong Dong, Atsushi Shimbo, Akihito Soeda, and Mio Murao

All codes are written in MATLAB and require:

• CVX - a Matlab-based convex modeling framework

We have also used functions of QETLAB (QETLAB - A MATLAB Toolbox for Quantum Entanglement), but all used functions can be found in the subfolder QETLAB_used_functions.

It has been tested on Matlab R2018A, and CVX 2.1

The main scripts of this repository are:

• run_unitary_inverse: Obtain the maximal success probability for parallel, sequential, and general (potentially with indefinite causal order) protocols to transform 'k' uses of a 'd'-dimensional unitary into its inverse.
• run_unitary_transpose: Obtain the maximal success probability for parallel, sequential, and general (potentially with indefinite causal order) protocols to transform 'k' uses of a 'd'-dimensional unitary into its transpose.

The scripts run_unitary_inverse and run_unitary_transpose make use of the following subfunctions:

• find_dimension: Determine the dimension of the linear space spanned by 'k' copies of 'd' dimension unitary channels

• is_general_protocol: Determine if a supermap represents a general superchannel

• is_sequential_protocol.m: Determine if a supermap represents a sequential superchannel

• is_superchannel1use.m: Determine if a supermap represents a superchannel with a single use of the input-operation

• maxp_general.m: Find the maximal success probability of transforming k uses of a list of unitary operations into another list of unitaries in a general protocol

• maxp_parallel.m: Find the maximal success probability of transforming k uses of a list of unitary operations into another list of unitaries in a parallel protocol

• maxp_sequential.m: Find the maximal success probability of transforming k uses of a list of unitary operations into another list of unitaries in a sequential protocol

• unitary_linear_independent.m: Find the number of linearly independent unitary channels contained in a list of unitaries