This repository contains all the codes written in MATLAB used to generate results in the paper Witnessing Negative Conditional Entropy, published in Physical Review A. Open access available on arxiv.

It also contains a few quantum functions useful for 2-qubit systems written in sage.

Functions from the following libraries have been used to write the functions in this repository.

• cvx
• cvxquad
• quantinf

cvx and quantinf have already been added in this repository. Please follow the cvxquad link to find install instructions for that package.

You will also require the 'Optimization Toolbox' in MATLAB to run codes in this repository.

The repository primarily contains codes to generate analytical and numerical witnesses to separate a given state from the CVENN class.

Clone the repository: git clone https://github.com/Tinkidinki/cvenn-codes.git

Open matlab in the same directory as all the files are present. If linux terminal is being used:

matlab -nodesktop

To use functions in analytical witness file:

aw = analytical_witness

Define the matrix rho_s as the state you want to separate from the CVENN class, or use the predefined function to generate a Werner like state alpha|phi^+> <phi+| + (1-alpha)I/d^2. Please ensure the state you generate lies outside the CVENN class. Refer to the alpha vs dimensions graph in the paper and pick an alpha such that the state lies outside the class. Eg: alpha = 0.8, d = 5

rho_s = aw.werner_like_state(alpha, dimensions)

Generate a witness for the state:

W_a = aw.witness(rho_s, dimensions)

Check the state against the witness:

check_witness(W_a, rho_s)

Test any other state against the witness to see if states inside CVENN give a positive value.

Using the \rho_s above,

W_na = aw.non_optimal_witness(rho_s, d)

Note that unlike the analytical witness which quickly finds solutions for large dimensions, the numerical witness takes a long time to converge for greater than a 2-qubit system. To find a witness for a 2-qubit or 2-qutrit system, do the following. Note that the 2-qutrit system will take 8-9 minutes to find the closest state.

We would have to use the cvx library to converge to the closest CVENN state. To setup cvx, open MATLAB in the cvx folder of this repository and run cvx_setup in MATLAB.

Now open MATLAB in the root of the repository. To add all the functions in cvxquad to the MATLAB path, run:

addpath("cvxquad")

Once again, either define rho_s, the state you wish to separate from CVENN yourself, or generate a Werner like state.

rho_s = aw.werner_like_state(alpha, dimensions) % where dimensions = 2 or 3

Find the numerical witness:

W_n = numerical_witness(rho_s, dimensions)

Check the state against the witness

check_witness(W_n, rho_s)

Check other states against this witness

check_witness(W_n, x)

If w is the generated witness, and d is the dimensions, to see a graph of how your generated witness performs on Werner like states, wp = aw.witness_performance_on_werner(w,d)

Look at the graph of the actual quantum conditional entropy values vs the witness's output.

The distance of a state from a CVENN witness is one of the ways in which entanglement of a state is quantified. Comparing how the analytical and numerical witness perform as such a measure could be an interesting project. Note that the numerical witness finds distance from the closest CVENN state, whereas the analytical witness finds distance from any state at the boundary. (Read paper for more details).

Please contact the author Mahathi Vempati at [email protected] for any queries, or discussion.