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arXiv:quant-ph/0405108 · Journal of Physics A Mathematical and General

Entanglement and tensor product decomposition for two fermions

Paweł Caban, Krzysztof Podlaski, Jakub Rembieliński, Kordian A. Smoliński, Zbigniew Walczak

Published Jan 27, 2005• 46 citations
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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.

DOI
10.1088/0305-4470/38/6/l02

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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.
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.
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.
We study characterization of separable (classically correlated) states for composite systems of distinguishable fermions. In the computation of entanglement formation for such systems where located subsystems are coupled by the canonical anticommutation relations (not by tensor product), the state decompositions to be taken should respect the univalence superselection rule. (The usual entanglement formation taking all the state decompositions measures non-separability of states between a given subsystem and its commutant and cannot detect non-separability between the CAR pair under our consideration properly.) We prove that any fermion hopping terms always induce non-separability. This feature contrasts with the case of tensor product systems where the states with bosonic hopping correlation may or may not be separable. If we transform a given bipartite fermion system into a tensor product system by Jordan-Klein-Wigner transformation, then any separable state for the former is also separable for the latter. We provide a class of U(1) gauge invariant mixed states that are non-separable for the former, however, separable for the latter. ∗ Mailing address:
We study separable (i.e., classically correlated) states for composite systems of spinless fermions that are distinguishable. For a proper formulation of entanglement formation for such systems, the state decompositions for mixed states should respect the univalence superselection rule. Fermion hopping always induces non-separability, while states with bosonic hopping correlation may or may not be separable. Under the Jordan–Klein–Wigner transformation from a given bipartite fermion system into a tensor product one, any separable state for the former is also separable for the latter. There are, however, U (1)-gauge invariant states that are non-separable for the former but separable for the latter.
Quantum computing is fundamentally based on the mathematical structure of complex Hilbert spaces, linear operators, tensor products, and unitary transformations. Among these concepts, tensor products provide the formal mechanism for constructing the state space of composite quantum systems, whereas linear and unitary operators describe quantum transformations, measurements, and computational gates. This article examines the mathematical relationship between linear operators and tensor products and analyzes their role in the formation, manipulation, and characterization of quantum entanglement. Particular attention is given to two-qubit systems, separable and entangled states, Schmidt decomposition, controlled operations, and the role of non-factorizable operators in quantum circuits. The analysis demonstrates that tensor products perform a dual function: they provide the mathematical representation of composite systems and enable the construction of local operations acting on individual subsystems. At the same time, genuinely entangling transformations, such as the controlled-NOT operation, cannot in general be represented as a single tensor product of independent local operators. Recent research further demonstrates that tensor-product and operator-Schmidt decompositions can be used to characterize operator entanglement, reduce circuit complexity, and construct efficient representations of quantum processes. The results establish that linear algebra and tensor-product structures are not merely formal tools but constitute the mathematical foundation connecting quantum entanglement with quantum computation.