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].

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A classification of idempotents in Clifford algebras C p,q is presented. It is shown that using isomorphisms between Clifford algebras C p,q and appropriate matrix rings, it is possible to classify idempotents in any Clifford algebra into continuous families. These families include primitive idempotents used to generate minimal one sided ideals in Clifford algebras. Some low dimensional examples are discussed.

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We consider the two-fermion system whose states are subjected to the superselection rule forbidding the superposition of states with fermionic and bosonic statistics. This implies that separable states are described only by diagonal density matrices. Moreover, we find the explicit formula for the entanglement of formation, which in this case cannot be calculated properly using Wootters's concurrence. We also discuss 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. Finally, we show that there exist states which are separable with respect to all tensor product decompositions of the underlying Hilbert space.