This is a repository for the article Characterising transformations between quantum objects, 'completeness' of quantum properties, and transformations without a fixed causal order

This code requires:

• QETLAB - a free MATLAB toolbox for quantum entanglement theory.

For the code presented here, we used the following convention:

• Quantum channels are Choi operators in the space L(H_in ⊗ H_out), that is, ordered as (input, output)
• Multipartite channels are Choi operators in the space L(H_in_1 ⊗ H_out_1 ⊗ H_in_2 ⊗ H_out_2 ⊗ ...), that is, ordered as (input1, output1, input2, output2, ...)
• Multipartite processes are Choi operators in the space L(H_in_1 ⊗ H_out_1 ⊗ H_in_2 ⊗ H_out_2 ⊗ ...), that is, ordered as (input1, output1, input2, output2, ...). Processes do not have a common past nor common future, and are the dual affine of multipartite channels (hence, equivalent of a tester with a single tester element)
• A superchannel is a Choi operator in L(H_P ⊗ H_in ⊗ H_out ⊗ H_F) which transforms the Choi operator of a Channel in L(H_in ⊗ H_out) into the Choi operator of a channel in L(H_P ⊗ H_F)
• A multipartite superchannel is a Choi operator in L(H_P ⊗ H_in_1 ⊗ H_out_1 ⊗ H_in_2 ⊗ H_out_2 ⊗ ... ⊗ H_F) which transforms the Choi operator of a Channel in L(H_in_1 ⊗ H_out_1 ⊗ H_in_2 ⊗ H_out_2 ⊗ ...), into the Choi operator of a channel in L(H_P ⊗ H_F)

This repository contains of projectors onto the linear subspace spanned by Choi operators of:

• Quantum Channels: ProjChannel.m
• Quantum Unital Channels: ProjUnitalChannel.m
• Quantum SuperChannels: ProjSuperChannel.m
• Quantum Parallel Channels: ProjParChannel.m
• Quantum Parallel Processes: ProjParProcess.m
• Quantum Parallel SuperChannels: ProjParSuperChannel.m
• Quantum Sequential Channels: ProjSeqChannel.m
• Quantum Sequential Processes: ProjSeqProcess.m
• Quantum Sequential SuperChannels: ProjSeqSuperChannel.m
• Quantum Non-Signalling Channels: ProjNSchannel.m
• Quantum General Processes (Process Matrices): ProjGenProcess.m
• Quantum General SuperChannels (Process with open Past and Future): ProjGenSuperChannel.m

This repository also contains a function which implements the trace-and-replace map:

• Map that partial traces a system and substitute it by the normalised identity: TR.m