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

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.