Negative absolute temperatures have been recognized for a long time, and the condition required to create them is that the number of possible states of the system corresponding to a specific energy content of the system, decreases as the energy increases, because temperature ultimately is simply the rate at which energy changes per unit of entropy change (under the circumstance that no macroscopic work is done in the process). The classical example is an assembly of spinning subatomic particles within a material in a magnetic field, each capable of spin-up or spin-down orientation. The lowest energy state of the particle subsystem is with all of them spin-down, and only one such state is possible. Higher energy states correspond to mixes of spin-up and spin-down, and, as with tossed coins, the number of possible arrangements increases as more energy is available to place a larger fraction of the particles in the spin-up configuration -- up to a point. The point is when the number of spin-up and spin-down particles are equal, and that is the state of maximum entropy. At that point, its temperature is infinite because the rate of change of energy per unit change in entropy is a division-by-zero expression (the system is at maximum entropy and therefore it does not change when the energy moves from slightly below the midpoint to slightly above), and I understand that one cannot actually reach this point by simply heating the system by conventional means. No matter how hot the flame, the probabilities would always have fewer up particles than down particles. The system can, however, be energized beyond that point by other means, and if it is, the system enters a state of negative temperature. If a system at a certain negative temperature is connected to a like system at a corresponding positive temperature, so that their energies can mingle, the result is not zero, but infinity, because the number of up spins and down spins become equal, which causes maximum entropy. And yes, if the system is in a state of all up spins, it is in a state of minimum entropy, so that it is again at a temperature of zero even though the energy state is different than with them all in down spins. Nevertheless, if the two systems are connected, zero plus zero does not equal zero, because the highly ordered situations would become randomized to maximum disorder again. Not all systems are capable of negative absolute temperature because they do not achieve maximum entropy at a finite energy level. It has to be a rather special system that does.