Annotation: dorsal/arxiv

Authors	Marcin Pawlowski, Marek Czachor
Categories	quant-ph
ArXiv ID	quant-ph/0512253
URL	https://arxiv.org/abs/quant-ph/0512253
DOI	10.1103/PhysRevA.73.042111
Journal	Phys. Rev. A 73, 042111 (2006)

Authors

Marcin Pawlowski, Marek Czachor

Abstract

We analyze an example of a photon in superposition of different modes, and ask what is the degree of their entanglement with vacuum. The problem turns out to be ill-posed since we do not know which representation of the algebra of canonical commutation relations (CCR) to choose for 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 two-level atoms. We first solve the problem algebraically in Heisenberg picture, without any assumption about the form of representation of CCR. Then we take the $\infty$-representation and show in two ways that in two-mode states the modes are maximally entangled with vacuum, but single-mode states are not entangled. Next we repeat the analysis in terms of the representation of CCR taken from Berezin's book and show that two-mode states do not involve the mode-vacuum 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 vacuum is present even for single-mode states. Still, the degree of entanglement is here difficult to estimate, mainly because there are $N+2$ subsystems, with $N$ unspecified and large.

{ "annotation_id": "2404b356-cf8e-4a2a-ab15-06de0d350246", "date_created": "2026-03-02T18:02:23.786000Z", "date_modified": "2026-03-02T18:02:23.786000Z", "file_hash": "58b9aea96c41f7417c1bbfb4746d6b3776a737e950f3718f9e00e4e1d5fd551b", "private": false, "record": { "abstract": "We analyze an example of a photon in superposition of different modes, and\nask what is the degree of their entanglement with vacuum. The problem turns out\nto be ill-posed since we do not know which representation of the algebra of\ncanonical commutation relations (CCR) to choose for field quantization. Once we\nmake a choice, we can solve the question of entanglement unambiguously. So the\ndifficulty is not with mathematics, but with physics of the problem. In order\nto make the discussion explicit we analyze from this perspective a popular\nargument based on a photon leaving a beam splitter and interacting with two\ntwo-level atoms. We first solve the problem algebraically in Heisenberg\npicture, without any assumption about the form of representation of CCR. Then\nwe take the $\\infty$-representation and show in two ways that in two-mode\nstates the modes are maximally entangled with vacuum, but single-mode states\nare not entangled. Next we repeat the analysis in terms of the representation\nof CCR taken from Berezin\u0027s book and show that two-mode states do not involve\nthe mode-vacuum entanglement. Finally, we switch to a family of reducible\nrepresentations of CCR recently investigated in the context of field\nquantization, and show that the entanglement with vacuum is present even for\nsingle-mode states. Still, the degree of entanglement is here difficult to\nestimate, mainly because there are $N+2$ subsystems, with $N$ unspecified and\nlarge.", "arxiv_id": "quant-ph/0512253", "authors": [ "Marcin Pawlowski", "Marek Czachor" ], "categories": [ "quant-ph" ], "doi": "10.1103/PhysRevA.73.042111", "journal_ref": "Phys. Rev. A 73, 042111 (2006)", "title": "Degree of entanglement as a physically ill-posed problem: The case of entanglement with vacuum", "url": "https://arxiv.org/abs/quant-ph/0512253" }, "schema_id": "dorsal/arxiv", "source": { "execution_id": "d10ecfdd-1349-4b19-9dc3-9867e3688024", "id": "arXiv Dataset IDs", "type": "Model", "variant": "snapshot-2026-03-01", "version": "0.1.0" }, "user_id": 1000002 }