Entanglement and tensor product decomposition for two fermions

Abstract

Abstract. The problem of the choice of tensor product decomposition in a system of two fermions with the help of Bogoliubov transformations of creation and annihilation operators is discussed. The set of physical states of the composite system is restricted by the superselection rule forbidding the superposition of fermions and bosons. It is shown that the Wootters concurrence is not the proper entanglement measure in this case. The explicit formula for the entanglement of formation is found. This formula shows that the entanglement of a given state depends on the tensor product decomposition of a Hilbert space. It is shown that the set of separable states is narrower than in the two-qubit case. Moreover, there exist states which are separable with respect to all tensor product decompositions of the Hilbert space.PACS numbers: 03.67. Mn, 03.65.Ud Entanglement is the key notion of quantum information theory and plays a significant role in most of its applications. The entanglement of a physical system is always relative to a particular set of experimental capabilities (see, e.g. [1, 2]), which is connected with decompositions of the system into subsystems. From the theoretical point of view this is closely related to possible choices of the tensor product decomposition (TPD) of the Hilbert space of the system. As a consequence the following question arises: How much entangled is a given state with respect to a particular TPD?In the present paper we discuss the problem of the choices of TPD in a system of two fermions, neglecting their spatial degrees of freedom and modifying tensor product in the rings of operators because of anticommuting canonical variables. We show that TPDs are connected with each other by Bogoliubov transformations of creation and annihilation operators. We also study the behavior of the entanglement of the system under these transformations. An importance of such investigation can be illustrated for example by the fact that the Bogoliubov transformations used in derivation of the Unruh effect also lead to the change of entanglement [3]. Different approach to the entanglement in the system of two identical fermions, based on the asymmetric decomposition of the algebra generated by a i , a † i (i = 1, 2) into tensor product of two subalgebras was taken up in [4]. Some aspects of the entanglement for two-fermion system were also discussed in [5].

Multipartite quantum correlations in even and odd spin coherent states

“…Further thought in this direction might be worthwhile in investigating genuine multipartite quantum correlations. Finally, it is interesting to examine the relation between the spin coherent states factorization (23) and the tensor product decomposition of two fermions developed in [53].…”

Characterization of Unruh channel in the context of open quantum systems

“…This arises from the issues related to superselection rules, restricting the superpositions of bosons and fermions [54]. A careful use of the superselection rules is important for determining the class of states permissible for describing a composite system of, for e.g., two fermions [55][56][57][58]. Thus, for e.g., given a fermionic vacuum state |0, 0 , one could generate the states…”

Entanglement in fermionic systems

“…That the presence of superselection rules affects the concept of entanglement has been pointed out and studied in detail in [13,14,15,16,17]. There, the existence of states was shown which are convex combinations of product states but not locally preparable, thus two reasonable definitions of entanglement already differ.…”