The quantum superposition principle allows adding (superimposing) arbitrary physical states, no matter how “different” they are. This was the starting point for Erwin Schrödinger, back in 1935, to introduce his famous gedankenexperiment about a cat being in a superposition of “dead” and “alive”. If one believes in the universality of quantum theory, there should be no fundamental reason to prevent the existence of such “paradoxical” states. And indeed, once we develop a scalable quantum technology, artificial preparation of macroscopic superposition (MS) will be fairly simple. On the other hand, one may wonder whether a MS naturally exists as a unique ground state of a macroscopic quantum system, directly engineered by physical interactions within the system. The affirmative answer to this question would open up the possibility to prepare MS by simply cooling the system to its ground state.
In their recent work, Borivoje Dakić (IQOQI) and Milan Radonjić (University of Vienna) explored the question of what kind of macroscopic superpositions can(not) naturally exist as unique ground states of local Hamiltonians. They define a typical MS with respect to the measurement of a macroscopic observable that exhibits the distribution of outcomes with two well-resolvable regions which correspond to the semiclassical states that constitute the MS. An archetypal example is the celebrated GHZ (Greenberger-Horne-Zeilinger) state composed of N spins in a coherent superposition of two macroscopically distinguishable states (all spins pointing “up” + all spins pointing “down”). Clearly, the measurement of magnetization (along “up”/”down” direction) reveals the distribution with two sharp and macroscopically distinct peaks which correspond to the two semiclassical states (“all up” and “all down”) that embody MS. The key observation is that no physical (local) Hamiltonian can couple the semiclassical components of the GHZ state, meaning that the Hamiltonian matrix element linking these two states is exactly zero for any interaction of finite order (i.e. involving a fixed number of particles). By providing a simple lemma that the energy gap is upper-bounded by such a matrix element, they show that the GHZ-like states can be, at best, only degenerate ground states of local Hamiltonians. In such a case, cooling the system to the ground state will result in an incoherent (classical) mixture of the semiclassical components. In a more general setting, the authors prove that this is generic behavior for all physical Hamiltonians (with finite order of interaction). Under the assumption that the ground state of the system is a “typical” MS state, they show that the energy gap must vanish as the number N of particles increases for all finite-order Hamiltonians (see Figure 1). This in turn implies that a MS can only be a degenerate ground state of physical Hamiltonians. For that reason, cooling the system towards zero temperature will result in a classical mixture instead of a coherent MS.
The general conclusion nicely complies with the previous investigations of ground states of various physical systems and demonstrates that more sophisticated experimental techniques are needed for the preparation of a genuine MS. In addition, the presented methods and results find applications in quantum marginal related problems and adiabatic quantum computing.