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arXiv:0810.3016

Nilpotent quantum mechanics, qubits, and flavors of entanglement

Andrzej Frydryszak

Published Oct 16, 2008• 4 citations
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Abstract

We address the question of description of qubit system in a formalism based on the nilpotent commuting variables. In this formalism qubits exhibit properties of composite objects being subject of the Pauli exclusion principle, but otherwise behaving boson-like. They are not fundamental particles. In such an approach the classical limit yields the nilpotent mechanics. Using the space of $η$-wavefunctions, generalized Schrödinger equation etc. we study properties of pure qubit systems and also properties of some composed, hybrid models: fermion-qubit, boson-qubit. The fermion-qubit system can be truly supersymmetric, with both SUSY partners having identical spectra. It is new and very interesting that SUSY transformations relate here only nilpotent object. The $η$-eigenfunctions for the qubit-qubit system give the set of Bloch vectors as a natural basis. Then the $η$-formalism is applied to the description of the pure state entanglement. Nilpotent commuting variables were firstly used in this context in [A. Mandilara, et. al., Phys. Rev. A \textbf{74}, 022331 (2006)], we generalize and extend approach presented there. Our main tool for study the entanglement or separability of states are Wronskians of $η$-functions. The known invariants and entanglement monotones for systems of $n=2,3,4$ qubits are expressed in terms of the Wronskians. This approach gives criteria for separability of states and insight into the flavor of entanglement of the system and simplifies description.

DOI
10.48550/arxiv.0810.3016

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We address the question of description of qubit system in a formalism based on the nilpotent commuting variables. In this formalism qubits exhibit properties of composite objects being subject of the Pauli exclusion principle, but otherwise behaving boson-like. They are not fundamental particles. In such an approach the classical limit yields the nilpotent mechanics. Using the space of ηηη-wavefunctions, generalized Schrödinger equation etc. we study properties of pure qubit systems and also properties of some composed, hybrid models: fermion-qubit, boson-qubit. The fermion-qubit system can be truly supersymmetric, with both SUSY partners having identical spectra. It is new and very interesting that SUSY transformations relate here only nilpotent object. The ηηη-eigenfunctions for the qubit-qubit system give the set of Bloch vectors as a natural basis. Then the ηηη-formalism is applied to the description of the pure state entanglement. Nilpotent commuting variables were firstly used in this context in [A. Mandilara, et. al., Phys. Rev. A \textbf{74}, 022331 (2006)], we generalize and extend approach presented there. Our main tool for study the entanglement or separability of states are Wronskians of ηηη-functions. The known invariants and entanglement monotones for systems of n=2,3,4n=2,3,4n=2,3,4 qubits are expressed in terms of the Wronskians. This approach gives criteria for separability of states and insight into the flavor of entanglement of the system and simplifies description.
We develop a generalized quantum mechanical formalism based on the nilpotent commuting variables (η-variables). In the nonrelativistic case such formalism provides natural realization of a two-level system (qubit). Using the space of η-wavefunctions, η-Hilbert space and generalized Schrödinger equation we study properties of pure multiqubit systems and also properties of some composed, hybrid models: fermion–qubit, boson–qubit. The fermion–qubit system can be truly supersymmetric, with both SUSY partners having identical spectra. It is a novel feature that SUSY transformations relate here only nilpotent object. The η-eigenfunctions of the Hamiltonian for the qubit–qubit system give the set of Bloch vectors as a natural basis.
We analyze recently proposed formalisms which use nilpotent variables to describe and/or generalize qubits and notion of entanglement. There are two types of them distinguished by the commutativity and or anti-commutativity of basics variables. While nilpotent commuting variables suit the new description of the qubits and entanglement, the anticommuting do not, but they can be used to describe generalized objects - superqubits. A squbit, in the present context, is a version of superqubit introduced to cure some of problematic properties of the superqubit.
We apply the general formalism of nilpotent polynomials (Mandilara et al 2006 Phys. Rev. A 74 022331) to the problem of pure-state multipartite entanglement classification in four qubits. In addition to establishing contact with the existing results, we explicitly show how the nilpotent formalism naturally suggests constructions of entanglement measures invariant under the required unitary or invertible class of local operations. A candidate measure of four-partite entanglement is also suggested, and its behaviour numerically tested on random pure states.
We propose a general method for introducing extensive characteristics of quantum entanglement. The method relies on polynomials of nilpotent raising operators that create entangled states acting on a reference vacuum state. By introducing the notion of tanglemeter, the logarithm of the state vector represented in a special canonical form and expressed via polynomials of nilpotent variables, we show how this description provides a simple criterion for entanglement as well as a universal method for constructing the invariants characterizing entanglement. We compare the existing measures and classes of entanglement with those emerging from our approach. We derive the equation of motion for the tanglemeter and, in representative examples of up to four-qubit systems, show how the known classes appear in a natural way within our framework. We extend our approach to qutrits and higher-dimensional systems, and make contact with the recently introduced idea of generalized entanglement. Possible future developments and applications of the method are discussed.