Relativistically invariant encoding of quantum information revisited

TL;DR

The paper tackles the problem of encoding quantum information in a way that is invariant under Lorentz (and more generally Poincaré) transformations. It advances the theory by (i) providing a rigorous, distribution-agnostic proof of invariance for Decoherence-Free Subspaces and Decoherence-Free Subsystems, (ii) extending invariant encoding to include pairwise helicity arising from the pairwise little group, and (iii) introducing a non-equal-momentum encoding scheme based on fixed total momentum that unifies and generalizes previous equal-momentum approaches. The authors develop a general multiplicity-space framework and apply Schur-Weyl duality and Poincaré representation theory to construct invariant encodings for massive and massless particles, including dyons, and discuss the implications for dense quantum information encoding. They also outline future directions, such as extending the framework to curved spacetime and non-unitary or SLOCC-type symmetry actions, highlighting practical relevance for space-based quantum communication and reference-frame–independent protocols.

Abstract

In this work, we provide a detailed analysis of the issue of encoding of quantum information which is invariant with respect to arbitrary Lorentz transformations. We significantly extend already known results and provide compliments where necessary. In particular, we introduce novel schemes for invariant encoding which utilize so-called pair-wise helicity -- a physical parameter characterizing pairs of electric-magnetic charges. We also introduce new schemes for ordinary massive and massless particles based on states with fixed total momentum, in contrast to all protocols already proposed, which assumed equal momenta of all the particles involved in the encoding scheme. Moreover, we provide a systematic discussion of already existing protocols and show directly that they are invariant with respect to Lorentz transformations drawn according to any distribution, a fact which was not manifestly shown in previous works.

Figures (1)

• Figure 1: Schematic representation of bits (qubits) encoded in pair-wise helicity and relativistically invariant encoding of bits (qubits). The pair of particles denoted as $1$ and $2$ with charges ($e_1=1,g_1=0$) and ($e_2=0,g_2=1$) respectively encodes one of the logical values while pair of particles $3$ and $4$ with the charges reversed encodes the second logical value in pair-wise helicity. The cell created by considering these two pairs of particles has sum of pair-wise helicities equal to 0. The same holds for the cell created with particles $5,6,7,8$ as configuration in this cell is just a permutation of the indices in pairs which can only result in sign change of pair-wise helicities based on \ref{['eq:pair_wise_helicity']}. This allows for relativistically invariant encoding of bits (qubits). Addition of further cells preserves the property of the system that the sum of pair-wise helicities is equal to 0. This is implied by the fact that addition of an extra pair in the system is associated with appearance of pair-wise helicities between previous pairs and the new ones in the form presented by dashed lines or dot-dashed lines or with the indices permuted inside pairs. In all those situations, the sum of introduced pair-wise helicities is equal to 0.