Efficient Computation of the Quantum Rate-Distortion Function

About

A MATLAB implementation of an inexact mirror descent algorithm to compute the quantum rate-distortion function for an input quantum density matrix $\rho$ with purification $|\psi\rangle$, positive distortion observable $\Delta$, and distortion multiplier $\kappa\geq0$

$$\min_{\mathcal{N}} \quad I(\rho, \mathcal{N}) + \kappa \langle \Delta , (\mathcal{N} \otimes \text{Id.})(|\psi\rangle\langle\psi|) \rangle, \qquad \text{subj. to} \quad \mathcal{N}\in\mathsf{CPTP}$$

where $I(\rho, \mathcal{N})$ is the quantum mutual information, $\mathsf{CPTP}$ is the set of completely positive trace preserving quantum channels. In particular, for entanglement fidelity distortion $\Delta=\text{Id.} - |\psi\rangle\langle\psi|$, we provide an efficient implementation of the algorithm which uses symmetry reduction to significantly reduce the dimension of dimension of the optimization problem.

Usage

The code is completely standadlone, and should run without any additional packages. The two main functions users should use are

• src/solveQrd.m: Computes the quantum rate-distortion function for arbitrary distortion observable.
• src/solveEfQrd.m: Computes the quantum rate-distortion function specifically for entanglement fidelity distortion. Uses symmetry reduction to significantly improve computational efficiency.

An example of how to use these functions is as follows:

A = [0.2 0; 0 0.8]; 					% Define input state
kappa = 1.0;						% Define distortion mutliplier
[rate, distortion, info] = solveEfQrd(A, kappa); 	% Solve for rate-distortion

Other examples of how to use the function can be found in examples. These include

• examples/entanglementFidelity.m: Solves the entanglement fidelity rate-distortion for a random input density matrix.
• examples/computeCurve.m: Traces out the entire entanglement fidelity rate-distortion curve for a random input density matrix.
• examples/compareMethods.m: Demonstration of various possible methods to solve for the entanglement fidelity rate-distortion.

Citation

This code is based on the work the paper here: https://quantum-journal.org/papers/q-2024-04-09-1314/. If you find our work useful, please cite us using

@article{he2024efficient,
  title={Efficient Computation of the Quantum Rate-Distortion Function},
  author={He, Kerry and Saunderson, James and Fawzi, Hamza},
  journal={Quantum},
  volume={8},
  pages={1314},
  year={2024},
  publisher={Verein zur F{\"o}rderung des Open Access Publizierens in den Quantenwissenschaften},
  doi = {10.22331/q-2024-04-09-1314},
}