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 some of the issues that appear when extending the bit-flip channel to qutrit systems. The difficulties arise from different interpretations of the Pauli X gate for qubits, leading to at least three nonequivalent formulations. We compare our results with the commonly used cyclic one-parameter trit flip channels and demonstrate that they are particular cases of those more general formulations we present here. We also extend these channels to higher-dimensional qudit systems, defining various dit flip channels. Finally, we study their impact on the Negativity, as an entanglement measure, of qubit-qutrit and 2-qutrit Werner states. In doing so, we show the distinctiveness of these versions, as they affect the states' entanglement in very distinct ways.