Statistical consistency of quantum-classical hybrids

Abstract

After formulating a no-go theorem for perfect quantum-classical hybrid systems, a new consistency requirement based on standard statistical considerations is noted. It is shown that such requirement is not fulfilled by the mean-field approach, nor by the statistical ensemble approach. Further unusual features of the latter scheme are pointed out.Comment: 15 pages, no figure

Hybrid classical-quantum formulations ask for hybrid notions

“…Again, this formulation bypasses the bracket problem as the definition of observables goes beyond being a tensor product of observables in the quantum and classical sectors. In the presence of interaction the distinction of which is the classical and which is the quantum sector is blurred [28]. This approach has the same problem, or characteristic, as the previous one: A statistical mixture has much more information than a standard density matrix.…”

“…In this setting the construction of a hybrid theory would involve the construction of a hybrid bracket. As suggested by what is known in the literature, the straightforward construction of such a bracket could be obstructed if one requires certain conditions [28]. However, one could certainly do it in an extended version, such as the Sudarshan analysis in the previous sections, but at the cost of having many different possible constructions.…”

“…In this section we want to briefly discuss the "statistical consistency problem" of hybrid formulations raised recently by Salcedo [28] (in this section we follow his analysis, although slightly changing the presentation). To describe this issue in its simplest terms, let us consider a CQ theory in which the quantum system is described as a classical system with all the information of the quantum state contained in a set of symplectic pairs (X i ,K i ) [12,13].…”

“…Moreover, Salcedo [28] has recently shown that the evolution of a statistical mixture of configurations would evolve differently for different representations of the initial density matrix of the quantum sector. This is so because a distribution function in configuration space has more information than a density matrix.…”

“…For instance, the dynamics is dictated by a bracket failing to satisfy Jacobi's identity and Leibniz's rule. Under reasonable hypotheses it has been shown that there is no Lie bracket for a semiquantized theory of this form [21,[25][26][27][28]. If one permits the bracket to be of non-Lie type, it can be shown that the theory will lose the positivity of its density matrix [2,21] (positive operators could have nonpositive expectation values).…”