Degree of entanglement as a physically illposed problem: The case of entanglement with vacuum
Abstract
We analyze an example of a photon in a superposition of different modes, and ask what is the degree of their entanglement with a vacuum. The problem turns out to be illposed since we do not know which representation of the algebra of canonical commutation relations (CCR) to choose for the field quantization. Once we make a choice, we can solve the question of entanglement unambiguously. So the difficulty is not with mathematics, but with physics of the problem. In order to make the discussion explicit we analyze from this perspective a popular argument based on a photon leaving a beam splitter and interacting with two twolevel atoms. We first solve the problem algebraically in the Heisenberg picture, without any assumption about the form of representation of CCR. Then we take the {infinity}representation and show in two ways that in twomode states the modes are maximally entangled with the vacuum, but singlemode states are not entangled. Next we repeat the analysis in terms of the representation of CCR taken from Berezin's book and show that twomode states do not involve the modevacuum entanglement. Finally, we switch to a family of reducible representations of CCR recently investigated in the context ofmore »
 Authors:
 Katedra Fizyki Teoretycznej i Metod Matematycznych, Politechnika Gdanska, 80952 Gdansk (Poland)
 Publication Date:
 OSTI Identifier:
 20787071
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Physical Review. A; Journal Volume: 73; Journal Issue: 4; Other Information: DOI: 10.1103/PhysRevA.73.042111; (c) 2006 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 74 ATOMIC AND MOLECULAR PHYSICS; ALGEBRA; ATOMS; BEAM SPLITTING; COMMUTATION RELATIONS; ENERGY LEVELS; GROUP THEORY; HEISENBERG PICTURE; OPTICS; PHOTONS; QUANTIZATION; QUANTUM ENTANGLEMENT; QUANTUM MECHANICS; SWITCHES
Citation Formats
Pawlowski, Marcin, and Czachor, Marek. Degree of entanglement as a physically illposed problem: The case of entanglement with vacuum. United States: N. p., 2006.
Web. doi:10.1103/PHYSREVA.73.0.
Pawlowski, Marcin, & Czachor, Marek. Degree of entanglement as a physically illposed problem: The case of entanglement with vacuum. United States. doi:10.1103/PHYSREVA.73.0.
Pawlowski, Marcin, and Czachor, Marek. Sat .
"Degree of entanglement as a physically illposed problem: The case of entanglement with vacuum". United States.
doi:10.1103/PHYSREVA.73.0.
@article{osti_20787071,
title = {Degree of entanglement as a physically illposed problem: The case of entanglement with vacuum},
author = {Pawlowski, Marcin and Czachor, Marek},
abstractNote = {We analyze an example of a photon in a superposition of different modes, and ask what is the degree of their entanglement with a vacuum. The problem turns out to be illposed since we do not know which representation of the algebra of canonical commutation relations (CCR) to choose for the field quantization. Once we make a choice, we can solve the question of entanglement unambiguously. So the difficulty is not with mathematics, but with physics of the problem. In order to make the discussion explicit we analyze from this perspective a popular argument based on a photon leaving a beam splitter and interacting with two twolevel atoms. We first solve the problem algebraically in the Heisenberg picture, without any assumption about the form of representation of CCR. Then we take the {infinity}representation and show in two ways that in twomode states the modes are maximally entangled with the vacuum, but singlemode states are not entangled. Next we repeat the analysis in terms of the representation of CCR taken from Berezin's book and show that twomode states do not involve the modevacuum entanglement. Finally, we switch to a family of reducible representations of CCR recently investigated in the context of field quantization, and show that the entanglement with the vacuum is present even for singlemode states. Still, the degree of entanglement here is difficult to estimate, mainly because there are N+2 subsystems, with N unspecified and large.},
doi = {10.1103/PHYSREVA.73.0},
journal = {Physical Review. A},
number = 4,
volume = 73,
place = {United States},
year = {Sat Apr 15 00:00:00 EDT 2006},
month = {Sat Apr 15 00:00:00 EDT 2006}
}

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