Limitations in the Quantum Operations to Achieve the Envariance Symmetry in Bipartite System

We discuss the pivotal role of unitary operations in quantum symmetries and thermodynamic control, emphasizing the concept of entanglement-assisted invariance (envariance). Focusing on the bipartite system, by examining maximally envariant states, we show that local non-unitary completely positive and trace-preserving (CPTP) maps cannot achieve the envariance condition for such states, nor can they enable environment-assisted shortcuts to adiabaticity. Furthermore, we present that if a unitary operation acting on one Hilbert subspace satisfies the envariance condition, the corresponding operation on the other subspace must be a uniquely defined unitary. These findings highlight the limitations of quantum operations in nonequilibrium thermodynamic control of open quantum systems and provide deeper insights into the fundamental symmetries underlying quantum physics.