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Theoretical and Mathematical Physics

On the quantization of identical particles

R. MacKenzie, Prasanta Kumar Panigrahi, Manu B. Paranjape, S. Sakhi

Published Jan 1, 1994• 1 citation
Publisher
DOI
10.1007/bf01015124

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arXiv:1901.09535•Physical Review A•May 28, 2019•19 citations
We probe the theoretical connection among three different approaches to analyze the entanglement of identical particles, i.e., the first quantization language (1QL), exterior and elementary-symmetric products (which has the mathematical equivalence to no-labeling approaches), and the algebraic approach based on the GNS construction. Among several methods to quantify entanglement of identical particles, we focus on the computation of reduced density matrix, which can be achieved by the concept of \emph{symmetrized partial trace} defined in 1QL. We show that the symmetrized partial trace corresponds to the interior product between exterior (or elementary symmetric) state vectors, which also corresponds to the subalgebra restriction in the algebraic approach based on GNS representation. By expressing the state of identical particles as exterior and elementary symmetric state vectors, we can also prove that GHJW theorem holds for identical particles.
We suggest a formalism to illustrate the entanglement of identical particles in the first quantization language (1QL). Our 1QL formalism enables one to exploit all the well-established quantum information tools to understand the indistinguishable ones, including the reduced density matrix and familiar entanglement measures. The rigorous quantitative relation between the amount of entanglement and the spatial coherence of particles is possible in this formalism. Our entanglement detection process is a generalization of the entanglement extraction protocol for identical particles with mode splitting proposed by Killoran et al. [Phys. Rev. Lett. 112, 150501 (2014)].