Bipartite entanglement in fermion systems

We discuss the relation between fermion entanglement and bipartite entanglement. We first show that an exact correspondence between them arises when the states are constrained to have a definite local number parity. Moreover, for arbitrary states in a four-dimensional single-particle Hilbert space, the fermion entanglement is shown to measure the entanglement between two distinguishable qubits defined by a suitable partition of this space. Such entanglement can be used as a resource for tasks like quantum teleportation. On the other hand, this fermionic entanglement provides a lower bound to the entanglement of an arbitrary bipartition, although in this case the local states involved will generally have different number parities. Finally, the fermionic implementation of the teleportation and superdense coding protocols based on qubits with odd and even number parity is discussed, together with the role of the previous types of entanglement.

Figure 1

Depiction of the eight even-number-parity fermion states of four single-particle modes, partitioned such that the two modes on the left of the dashed line belong to Alice, and the two modes on the right to Bob. In the upper row are the states with odd local number parity (one fermion for Alice and one fermion for Bob); in the bottom row, those with even local number parity (Alice and Bob may have zero or two fermions). The states in the bottom row can be formally obtained from those in the top row by performing the particle-hole transformations $c_{A2}^{†}\leftrightarrow c_{A2}$ and $c_{B2}^{†}\leftrightarrow c_{B2}$.

Figure 2

Teleportation protocol with the present fermionic implementation. Each qubit is represented by a pair of fermionic modes having a total occupation number of 1. The control operation can be realized involving just one of the modes of the pair representing the control qubit due to the occupation number constraint, acting when it is occupied, and similarly, the usual measurement in the standard basis can be implemented by measuring just one of these modes. If the pair occupation number constraint is relaxed so that both local number parities coexist, then control and measurement operations involve both modes.