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

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

doi:10.1007/s11080-005-5729-8

Cited by 3 publications

(4 citation statements)

References 18 publications

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“…Fermions are also peculiar because they obey a parity superselection rule [208,209,210,211,212,213,214,215,216,217] that prevents the superposition of a bosonic compound and a fermionic system. These properties affects entanglement theory and its applications in quantum information [218,219,220,221,222,223,224,225,226].…”

Section: Mode-partitioningmentioning

confidence: 99%

“…Indeed, unlike identical particles, orthogonal modes can always be individually addressed in experiments [39,40,41]. Several fundamental properties of entanglement-V have been studied [71,120,227,176,177,219,220,228,221,222,151,223,224,229,230,231,154,232,233,234,235,226,203,236,237,238], as well as its experimental detection [239,240,241,242,243,244,245,246,247] and the possibility of practically manipulating it [248,249,250,251,252,253,254,255,256,257,…”

Section: Mode-partitioningmentioning

confidence: 99%

“…Several fundamental properties of entanglement-V have been studied [71,120,227,176,177,219,220,228,221,222,151,223,224,229,230,231,154,232,233,234,235,226,203,236,237,238], as well as its experimental detection [239,240,241,242,243,244,245,246,247] and the possibility of practically manipulating it [248,249,250,251,252,253,254,255,256,257,258]. Furthermore, entanglement-V can be quantified by means of so-called entanglement monotones, such as entanglement entropy [259,260,…”

Section: Mode-partitioningmentioning

confidence: 99%

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Entanglement in indistinguishable particle systems

et al. 2020

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For systems consisting of distinguishable particles, there exists an agreed upon notion of entanglement which is fundamentally based on the possibility of addressing individually each one of the constituent parties. Instead, the indistinguishability of identical particles hinders their individual addressability and has prompted diverse, sometimes discordant definitions of entanglement. In the present review, we provide a comparative analysis of the relevant existing approaches, which is based on the characterization of bipartite entanglement in terms of the behaviour of correlation functions. Such a a point of view provides a fairly general setting where to discuss the presence of non-local effects; it is performed in the light of the following general consistency criteria: i) entanglement corresponds to non-local correlations and cannot be generated by local operations; ii) when, by "freezing" suitable degrees of freedom, identical particles can be effectively distinguished, their entanglement must reduce to the one that holds for distinguishable particles; iii) in absence of other quantum resources, only entanglement can outperform classical information protocols. These three requests provide a setting that allows to evaluate strengths and weaknesses of the existing approaches to indistinguishable particle entanglement and to contribute to the current understanding of such a crucial issue. Indeed, they can be classified into five different classes: four hinging on the notion of particle and one based on that of physical modes. We show that only the latter approach is consistent with all three criteria, each of the others indeed violating at least one of them.

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Section: Mode-partitioningmentioning

confidence: 99%

Section: Mode-partitioningmentioning

confidence: 99%

Section: Mode-partitioningmentioning

confidence: 99%

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Entanglement in indistinguishable particle systems

et al. 2020

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“…We conclude this subsection with some comments on the related literature. The Hamiltonian (175) with ν 12 = ν 34 = ν has been studied for fermion systems in [48,50,52,[154][155][156][157][158], while in the references [159][160][161] the somewhat more general situation where not necessarily ν 12 = ν 34 is being considered. Particular versions of the general Hamiltonian (157) have been studied in [154,155,162].…”

Section: Diagonalizing Two-fermion Hamiltoniansmentioning

confidence: 99%

Nonlinear Bogolyubov-Valatin transformations: Two modes

Scharnhorst

Holten

2011

Annals of Physics

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Extending our earlier study of nonlinear Bogolyubov-Valatin transformations (canonical transformations for fermions) for one fermionic mode, in the present paper we perform a thorough study of general (nonlinear) canonical transformations for two fermionic modes. We find that the Bogolyubov-Valatin group for n = 2 fermionic modes which can be implemented by means of unitary SU (2 n = 4) transformations is isomorphic to SO(6; R)/Z 2 . The investigation touches on a number of subjects. As a novelty from a mathematical point of view, we study the structure of nonlinear basis transformations in a Clifford algebra [specifically, in the Clifford algebra C(0, 4)] entailing (supersymmetric) transformations among multivectors of different grades. A prominent algebraic role in this context is being played by biparavectors (linear combinations of products of Dirac matrices, quadriquaternions, sedenions) and spin bivectors (antisymmetric complex matrices). The studied biparavectors are equivalent to Eddington's E-numbers and can be understood in terms of the tensor product of two commuting copies of the division algebra of quaternions H. From a physical point of view, we present a method to diagonalize any arbitrary two-fermion Hamiltonian. Relying on Jordan-Wigner transformations for two-spin-1 2 and single-spin-3 2 systems, we also study nonlinear spin transformations and the related problem of diagonalizing arbitrary two-spin-1 2 and single-spin-3 2 Hamiltonians. Finally, from a calculational point of view, we pay due attention to explicit parametrizations of SU (4) and SO(6; R) matrices (of respective sizes 4 × 4 and 6 × 6) and their mutual relation. † As a consequence of eq. (17) and the anti-Hermiticity of the operatorsĉ k , the operatorsĉ k [i.e., the basis elements of the paravector space associated with the Clifford algebra C(0, 5)] obey the relation (k, l = −1, . . . , 4)(304) 35 Below we have not indicated that the wedge product relates to a different vector space than in eq. (301), i.e.:

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