Low degree Lorentz invariant polynomials as potential entanglement invariants for multiple Dirac spinors
arXiv:2105.07503 · doi:10.1016/j.aop.2023.169410
Abstract
A system of multiple spacelike separated Dirac particles is considered and a method for constructing polynomial invariants under the spinor representations of the local proper orthochronous Lorentz groups is described. The method is a generalization of the method used in [Phys. Rev. A {\bf 105}, 032402 (2022), arXiv:2103.07784] for the case of two Dirac particles. All polynomials constructed by this method are identically zero for product states. The behaviour of the polynomials under local unitary evolution that acts unitarily on any subspace defined by fixed particle momenta is described. By design all of the polynomials have invariant absolute values on this kind of subspaces if the evolution is locally generated by zero-mass Dirac Hamiltonians. Depending on construction some polynomials have invariant absolute values also for the case of nonzero-mass or additional couplings. Because of these properties the polynomials are considered potential candidates for describing the spinor entanglement of multiple Dirac particles, with either zero or arbitrary mass or additional couplings. Polynomials of degree 2 and 4 are derived for the cases of three and four Dirac spinors. For three spinors no non-zero degree 2 polynomials are found but 67 linearly independent polynomials of degree 4 are identified. For four spinors 16 linearly independent polynomials of degree 2 are constructed as well as 26 polynomials of degree 4 selected from a much larger number. The relations of these polynomials to the polynomial spin entanglement invariants of three and four non-relativistic spin- particles are described. Moreover, it is described how degree 4 polynomials for five spinors can be constructed and how degree 2 polynomials can be constructed for any even number of spinors.
arXiv commentMain text 19 pages + Appendices 11 pages. Comments are welcome. v2 Minor revision, references added 20+11 pages. v3 25+12 pages. Major revision, text extended and revised to reflect changes in arXiv:2103.07784, references added. v4 Accepted version, Major revision, text extended, figures added, references added 56+20 pages
References in corpus (48)
- Two-Dimensional Gas of Massless Dirac Fermions in Graphene
- Entanglement of Formation of an Arbitrary State of Two Qubits
- Mixed State Entanglement and Quantum Error Correction
- Discovery of a Weyl Fermion Semimetal and Topological Fermi Arcs
- Entanglement of a Pair of Quantum Bits
- Distributed Entanglement
- Discovery of a Three-dimensional Topological Dirac Semimetal, Na3Bi
- Persistent entanglement in arrays of interacting particles
- Observation of a topological 3D Dirac semimetal phase in high-mobility Cd3As2
- Electron-Electron Interactions in Graphene: Current Status and Perspectives
- Experimental Realization of a Three-Dimensional Dirac Semimetal
- Classification of stable three-dimensional Dirac semimetals with nontrivial topology
- Universal state inversion and concurrence in arbitrary dimensions
- Entanglement monotones
- Weyl points and line nodes in gapless gyroid photonic crystals
- Bounding the set of quantum correlations
- Generalized Schmidt decomposition and classification of three-quantum-bit states
- Entanglement of Dirac fields in non-inertial frames
- Exact and Asymptotic Measures of Multipartite Pure State Entanglement
- Fidelity and Concurrence of conjugated states
- Quantum Entanglement of Moving Bodies
- Quantum entropy and special relativity
- Normal forms and entanglement measures for multipartite quantum states
- Classification of multipartite entangled states by multidimensional determinants
- Multi-particle entanglement and its applications to cryptography
- Einstein-Podolsky-Rosen-Bohm experiment with relativistic massive particles
- The polynomial invariants of four qubits
- Multipartite generalisation of the Schmidt decomposition
- Computing Local Invariants of Qubit Systems
- Three-qubit pure-state canonical forms
- Non-local properties of multi-particle density matrices
- On multi-particle entanglement
- On local invariants of pure three-qubit states
- Local symmetry properties of pure 3-qubit states
- Axiomatic quantum field theory in curved spacetime
- Relativistic Einstein-Podolsky-Rosen correlation and Bell's inequality
- Two roles of relativistic spin operators
- Einstein-Podolsky-Rosen correlations of Dirac particles - quantum field theory approach
- Multipartite entanglement in 2 x 2 x n quantum systems
- Algebraic invariants of five qubits
- Imaging emergent heavy Dirac fermions of a topological Kondo insulator
- Quantum Field Theory in Graphene
- Strange behavior of the relativistic Einstein-Podolsky-Rosen correlations
- Speed dependent polarization correlations in QED and entanglement
- A Lorentz Invariant Pairing Mechanism: Relativistic Cooper Pairs
- Bell's inequality with Dirac particles
- Understanding entangled spins in QED
- Candidate entanglement invariants for two Dirac spinors