Generalized no-flipping theorem as a consequence of commutation of spacelike operations

A quantum flipping device is a theoretical apparatus designed to transform an input quantum state, represented by the density operator $\ensuremath{\rho}$ acting on a $d$-dimensional Hilbert space $\mathcal{H}$, into an output state $\ensuremath{\sigma}$ residing within the subspace orthogonal to $\ensuremath{\rho}$. While the no-flipping theorem establishes the impossibility of such a device for qubits (two-dimensional systems), its extension to higher-dimensional systems ($d>2$) allows for the possibility of such devices due to the increased space available for the output state. In this paper, we derive an impossibility theorem for generalized flipping devices, applicable to higher-dimensional systems, based on a causal principle, namely, commutation of spacelike operations. Given a bipartite quantum state shared between two parties Alice and Bob, this principle prohibits the resulting state from depending on the time order of Alice's and Bob's spacelike separated local operations applied on the respective parts of the bipartite systems. Importantly, unlike the argument applied to qubits, our approach does not rely on the unitary dynamics postulate of quantum theory. Furthermore, we investigate the implications of violating the principle of commutation of spacelike operations and demonstrate that such violations do not necessarily entail a violation of the no-signaling principle.