While single boson entanglement is known to be equivalent to standard non-identical particle entanglement, the use of fermionic entanglement is constrained by a parity superselection rule. Nevertheless we show that within the framework of second quantization, where local modes play the role of qubits, the analogy with ordinary entanglement holds: two entangled spinless fermion modes can be used to teleport a single local fermion mode and to perform standard dense coding. Entanglement concentration and dilution are possible for fermions. We clarify the meaning of our results by discussing the connection with quantum reference frames and conversion between spinless fermion entanglement and spinless boson entanglement.
While single boson entanglement is known to be equivalent to standard non-identical particle entanglement, the use of fermionic entanglement is constrained by a parity superselection rule. Nevertheless we show that within the framework of second quantization, where local modes play the role of qubits, the analogy with ordinary entanglement holds: two entangled spinless fermion modes can be used to teleport a single local fermion mode and to perform standard dense coding. Entanglement concentration and dilution are possible for fermions. We clarify the meaning of our results by discussing the connection with quantum reference frames and conversion between spinless fermion entanglement and spinless boson entanglement.
Quantum teleportation is considered a basic primitive in many quantum information processing tasks and has been experimentally confirmed in various photonic and matter-based setups. Here, we consider teleportation of quantum information encoded in modes of a fermionic field. In fermionic systems, superselection rules lead to a more differentiated picture of entanglement and teleportation. In particular, one is forced to distinguish between single-mode entanglement swapping and qubit teleportation with or without authentication via Bell inequality violation, as we discuss here in detail. We focus on systems subject to parity superselection where the particle number is not fixed, and contrast them with systems constrained by particle number superselection which are relevant for possible practical implementations. Finally, we analyze the consequences for the operational interpretation of fermionic mode entanglement and examine the usefulness of so-called mixed maximally entangled states for teleportation.
The characterization of entanglement is a fundamental issue for Quantum Information Theory. But the definition of entanglement depends on the notion of locality, and thus on the tensor product structure of the state space of the composite system. This notion is affected by the presence of superselection rules that restrict the accessible Hilbert space to a direct sum of subspaces. Indistinguishability of particles imposes one such restriction, namely to totally symmetric or totally antisymmetric states. The entanglement can in this case be defined with respect to partitions of modes in the second quantization formalism. For fermionic systems the Fock space of m modes is isomorphic to the space of m qubits, but the action of creation and annihilation operators is not local, due to their anticommutation. Conservation of the parity of fermion number imposes another relevant superselection rule. It requires that local physical observables commute with the local parity operator. Taking into account the considerations above, it is possible to define the set of separable states or equivalently the concept of entanglement for fermionic systems in a number of ways. Here we analyze systematically these possibilities and the relation among the various sets of separable states. We also discuss the behavior of the different classes when taking several copies of the state, as well as the characterization of the sets in terms of the usual criteria regarding the tensor product.
Quantum Teleportation is a very useful scheme for transferring quantum information. Given that the quantum information is encoded in a state of a system of distinguishable particles, and given that the shared bi-partite entangled state is also that of a system of distinguishable particles, the optimal teleportation fidelity of the shared state is known to be (Fmaxd+1)/(d+1) with Fmax being the `maximal singlet fraction' of the shared state. However, Parity Superselection Rule (PSSR) in Fermionic Quantum Theory (FQT) puts constraint on the allowed set of physical states and operations, and thereby, leads to a different notion of Quantum entanglement - locally accessible and locally inaccessible (topological correlation). In the present work, we derive an expression for the optimal teleportation fidelity of locally accessible entanglement preservation, given that the quantum information to be teleported is encoded in fermionic modes of dimension 2N×2N using 2N×2N-dim shared fermionic resource between the sender and receiver. To get the optimal teleportation fidelity in FQT, we introduce PSSR restricted twirling operations and establish fermionic state-channel isomorphism. Remarkably, we notice that the structure of the canonical form of twirl invariant fermionic shared state differs from that of the isotropic state -- the corresponding canonical invariant form for teleportation in Standard Quantum Theory (SQT). In this context, we also introduce restricted Clifford twirling operation that constitute the Unitary 2-design in case of FQT for experimentally validating such optimal average fidelity. Finally, we discuss the preservation of locally inaccessible entanglement for a class of fermionic teleportation channel.
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.
Abstract The operator algebra of fermionic modes is isomorphic to that of qubits, the difference between them is twofold: the embedding of subalgebras corresponding to mode subsets and multiqubit subsystems on the one hand, and the parity superselection in the fermionic case on the other. We discuss these two fundamental differences extensively, and illustrate these through the Jordan–Wigner representation in a coherent, self-contained, pedagogical way, from the point of view of quantum information theory. Our perspective leads us to develop useful new tools for the treatment of fermionic systems, such as the fermionic (quasi-)tensor product, fermionic canonical embedding, fermionic partial trace, fermionic products of maps and fermionic embeddings of maps. We formulate these by direct, easily applicable formulas, without mode permutations, for arbitrary partitionings of the modes. It is also shown that fermionic reduced states can be calculated by the fermionic partial trace, containing the proper phase factors. We also consider variants of the notions of fermionic mode correlation and entanglement, which can be endowed with the usual, local operation based motivation, if the parity superselection rule is imposed. We also elucidate some other fundamental points, related to joint map extensions, which make the parity superselection inevitable in the description of fermionic systems.
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 [1,2] the issue of retaining locality of Fermionic operations considering local Fermionic modes as the elementary systems of an operational probabilistic theory. The locality of Fermionic operations, namely operations on systems that are not causally connected must commute, implies the parity superselection rule that inhibits the superposition of states with an even and an odd number of excitations. As a result the Fermionic theory lacks two distinctive traits of quantum theory, the local tomography and the monogamy of the entanglement.