We know that the Minkowski vacuum corresponds to the thermal state in a Rindler wedge at Unruh temp. But does the thermal state in one Rindler wedge at Unruh temperature uniquely map to the Minkowski vacuum? Or could there be other states in the Minkowski space field theory which also corresponds to thermal state in one of the wedges? In other words, If I am doing field theory in say the left Rindler wedge and the state is specified to be thermal at Unruh temp, does it mean that the state in the Minkowski space field theory is the vacuum, or could it be something else?
One class of states it would map to are the following: $$U_R|0\rangle$$ where $U_R$ denotes an unitary operator in the right Rindler wedge. But it's not clear that they correspond to nice Minkowski states. Are there other states in Minkowski space field theory that, on writing in the Rindler basis and tracing out degrees of freedom from one wedge, would yield the thermal state?