On the issues arising when defining an X gate for qudits: Extending the Bit-Flip Channel to $d$-dimensional systems

TL;DR

This paper explores the complexities of defining and extending the X gate and flip channels for higher-dimensional quantum systems, revealing multiple inequivalent formulations and their varied effects on entanglement measures.

Contribution

It introduces new formulations of dit flip channels for qudits, extending beyond cyclic models, and analyzes their distinct impacts on entanglement properties.

Findings

Different formulations of dit flip channels affect entanglement differently.

Extended channels generalize the cyclic trit flip channels.

Inequivalence of channel versions influences quantum state entanglement.

Abstract

Given the current interest in quantum information tasks involving higher-dimensional systems, we discuss issues that appear when extending the bit-flip channel to qutrit systems. The difficulties arise from the different interpretations of the Pauli X gate for qubits, leading to three inequivalent formulations. We compared our results with the commonly used cyclic one-parameter trit flip channels and demonstrated that they are particular cases of those more general formulations we present here. Also, we extended these channels to higher-dimensional qudit systems, therefore defining different dit flip channels. Finally, we studied their impact on the Negativity, as an entanglement measure, of qubit-qutrit and 2-qutrit Werner states. In doing so, we showed the inequivalence of these versions, as they affect the states' entanglement in very distinct ways.