Sum of squares method improves quantum message transmission bounds quickly
A Sum-of-Squares Hierarchy with Quadratic Convergence for Quantum Channel Coding
Information Theory
Summary
Sending classical messages through a quantum channel can be very hard to do perfectly. The authors study how to estimate the best possible success rate for sending messages using a quantum channel. They develop a new mathematical method that gives better and faster approximations than earlier techniques. Their approach works for any number of messages and improves error rates significantly, especially compared to guessing randomly.
What this means in practice
- •For quantum communication engineers: Calculate tighter upper bounds on message decoding success rates for quantum communication devices to optimize their performance.
- •For signal processing developers: Use improved error estimates in quantum state discrimination tasks to design better signal detection algorithms in noisy environments.
A theory result. No direct application yet.
Authors
Hoang Ta, Hoang Anh Tran
Abstract
Computing the optimal success probability for transmitting classical messages through a single use of a quantum channel is NP-hard, even for two messages. An existing semidefinite programming hierarchy based on symmetric extensions provides convergent upper bounds with an a priori error estimate that decays as the inverse square root of the extension level. In this work, we construct a Hermitian sum-of-squares hierarchy for an arbitrary number of messages and prove quadratic convergence in its level. The error bound is proportional to the advantage over random guessing. Our approach combines state-discrimination duality with positive polynomial kernels on products of spheres to construct feasible polynomial dual certificates. For binary messages, the resulting bounds give a multiplicative approximation from above of the trace-norm contraction coefficient.