Deciding Whether a C-Q Channel Preserves a Bit is QCMA-Complete

TL;DR

This paper proves that determining if a classical-quantum channel preserves a classical bit is QCMA-complete, revealing the computational complexity of a fundamental problem in quantum information theory.

Contribution

It establishes QCMA-completeness for the bit-preservation problem and provides a matrix analysis-based characterization of optimal witnesses.

Findings

The bit-preservation problem is QCMA-complete.

Optimal witnesses are computational basis states or |+>, |-> states.

Concise proofs of QCMA-completeness are provided.

Abstract

We prove that deciding whether a classical-quantum (C-Q) channel can exactly preserve a single classical bit is QCMA-complete. This "bit-preservation" problem is a special case of orthogonality-constrained optimization tasks over C-Q channels, in which one seeks orthogonal input states whose outputs have small or large Hilbert-Schmidt overlap after passing through the channel. Both problems can be cast as biquadratic optimization with orthogonality constraints. Our main technical contribution uses tools from matrix analysis to give a complete characterization of the optimal witnesses: computational basis states for the minimum, and |+>, |-> over a single basis pair for the maximum. Using this characterization, we give concise proofs of QCMA-completeness for both problems.