Old and New Concepts of Physics

Bogoliubov Transformations and Entanglement of Two Fermions

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

Published Jul 3, 2007• 1 citation

Abstract

We show that Bogoliubov transformations, widely used in quantum field theory, can be also useful in quantum information theory. Namely, we show that the problem of the choice of tensor product decomposition in the system of two fermions can be analysed with the help of Bogoliubov transformations of creation and annihilation operators.

DOI: 10.2478/v10005-007-0017-8

Related papers

Entanglement and tensor product decomposition for two fermions

Paweł Caban, Krzysztof Podlaski, Jakub Rembieliński +2 more

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.

arXiv:quant-ph/0405108Journal of Physics A Mathematical and GeneralJan 27, 200546 citations

Tensor Product Decomposition, Entanglement, and Bogoliubov Transformations for Two Fermion System

Paweł Caban, Krzysztof Podlaski, Jakub Rembieliński +2 more

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.

Open Systems & Information DynamicsMay 31, 20053 citations

Entanglement in systems of indistinguishable fermions

Mari Carmen Bañuls, J. I. Cirac, Michael M. Wolf

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.

Journal of Physics Conference SeriesJun 1, 200917 citations

Tensor operators for quantum groups and applications

V. Rittenberg, Manfred Scheunert

Tensor operators are defined for Hopf algebras. If the Hopf algebra is quasitriangular, the properly defined tensor product of tensor operators is a tensor operator. In one application, the problem of defining invariant correlation functions for the Uq (sl(2))-symmetric spin 1/2 Heisenberg chain is considered. In another application, it is shown that suitably modified bosonic and fermionic q-creation and annihilation operators are tensor operators of Uq (sl(n)).

Journal of Mathematical PhysicsFeb 1, 199251 citations

APPLICATION OF LINEAR OPERATORS AND TENSOR PRODUCTS IN QUANTUM ENTANGLEMENT AND QUANTUM COMPUTING

Sevinchova Charosxon Maxmudjonovna, Worldly Knowledge Publishing Centre

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.

HSR (London)Oct 8, 20260 citations

Exploring the entanglement of free spin- , spin-1 and spin-2 fields *

Zhi Hui Yang, Ling-Yan Hung

Abstract In this study, we explore the entanglement of free spin- <?CDATA $ \displaystyle\frac{1}{2} $?> , spin-1, and spin-2 fields. We start with an example involving Majorana fields in 1+1 and 2+1 dimensions. Subsequently, we perform the Bogoliubov transformation and express the vacuum state with a particle pair state in the configuration space, which is used to calculate the entropy. This clearly demonstrates that the entanglement entropy originates from the particles across the boundary. Finally, we generalize this method to free spin-1 and spin-2 fields. These higher free massless spin fields have well-known complications owing to gauge redundancy. We deal with the redundancy by gauge-fixing in the light-cone gauge. We show that this gauge provides a natural tensor product structure in the Hilbert space, while surrendering explicit Lorentz invariance. We also use the Bogoliubov transformation to calculate the entropy. The area law emerges naturally by this method.

Chinese Physics CApr 25, 20190 citations