Fermionic quantum theory and superselection rules for operational probabilistic theories

Abstract

The anticommutation of Fermionic fields raises the problem of simulating the evolution of Fermionic systems by means of commuting quantum systems, say qubits. We tackle the issue considering local Fermionic modes as the elementary systems of an operational probabilistic theory. We show that the locality of Fermionic operations, namely operations on systems that are not causally connected must commute, implies the parity superselection rule, which inhibits the superposition of states with an even and an odd number of excitations. Accordingly we derive the largest probabilistic theory compatible with the parity superselection constraint and show that it lacks two distinctive traits of quantum theory, the local tomography and the monogamy of the entanglement. We generalize the notion of superselection rule to general probabilistic theories as sets of linear constraints on the convex set of states and prove a trade-off between the cardinality of the superselection rule and the degree of holism of the resulting theory. Within this scenario the Fermionic quantum theory and the real quantum theory can be regarded as superselected versions of the usual quantum theory. These results are published in Ref. [7] http://iopscience.iop.org/0295-5075/107/2/20009 and Ref. [8] http://www.worldscientific.com/doi/abs/10.1142/S0217751X14300257. In the last three decades the relation between Fermionic systems and other quantum systems has been throughly investigated from both the computational and the physical point of view. In particular the puzzling anti-commuting nature of the Fermionic systems casts a shadow on the possibility of simulating the physical evolution of a bunch of Fermionic systems by means of commuting quantum systems—say qubits. This issue was raised by R. P. Feynman in 1982 [9], when in his seminal work on physical computation he wondered about the possibility of simulating Fermions by local quantum systems in interaction—what we would call nowadays a quantum computer. The problem is that of encoding the evolution of Fermionic fields onto localized quantum systems. A well-known encoding of N Fermionic systems into N qubits is given by the Jordan-Wigner transform (JWT) [14]. Such an encoding, based on the identification between the Fock space of N Fermions and the Hilbert space of N qubits, provides a -algebra isomorphism between the Fermionic anticommuting algebra and the commuting algebra of qubits. Such a correspondence has been a valuable instrument in modern solid state physics for solving the one dimensional XY spin-chains [18, 15] and then for the understanding of superconductivity and the quantum Hall effect. Moreover, a time-adaptive JWT has been introduced in Ref. [19], which allows to contract Fermionic unitary circuits with the same complexity as for the corresponding spin model. In quantum information science the JWT has been used to extend to the Fermionic case notions as entanglement [1], the entropic area law [24], and universal computation [5]. More recently the JWT, which originally regards one dimensional chains of spin-1=2 systems, has been generalized to any spin [3] and lattice [13] dimension. Despite its computational power, the JWT fails to solve completely the issue established by Feynman: physically local Fermionic operations are mapped into nonlocal quantum ones and viceversa. As noticed by many authors this can lead to ambiguities in defining the partial trace [16, 4, 17, 10], and in assessing