Comment on “Single-particle entanglement”

Aurélien Drezet*Institute of Physics, Karl-Franzens University, Universitätsplatz 5, A-8010 Graz, Austria

�Received 4 January 2005; published 21 August 2006�

We show that contrarily to what it is claimed in Phys. Rev. A 72, 064306 �2005� entanglement isnot necessary for explaining experimental results associated with the “single particle-vacuum” state�����0�A �1�B+ �1�A �0�B.

DOI: 10.1103/PhysRevA.74.026301 PACS number�s�: 03.67.�a, 03.65.Ta, 32.80.Lg, 07.79.Fc

In �1� van Enk showed that the single particle state ��AB�,which in the Fock space can be written

��AB� � ��0A��1B� + �1A��0B��/�2, �1�

defines an entanglement photon-vacuum �or electron-vacuum� displaying nonlocality �we omit here the otherempty modes not taken into account in the experiments�.This state could be created by considering a single photonwave packet impinging on a 50-50 balanced beam splitter�A and B denote the two remote optical modes at the exits ofthe beam splitter�. The idea of �1� is not new and has beenalready presented in, for example, �2–4�. This stirred up con-troversy and a debate �5–8� on the genuine meaning of non-locality for a single particle. Yet no consensus has been ob-tained despite the fact that entanglement �e.g., quantumcorrelation� needs at least two particles to be detectable andthus physically defined �6�.

However, in �1� Enk �like Hardy �5�� gives a counterar-gument to �6� stating that if one could see a nonclassicaleffect of the state given by Eq. �1� on two remote detectorsinteracting only locally with the photon field then this willprove in turn that ��AB� carries nonlocal properties. The es-sential claim of �1� is indeed that after local interaction of thestate ��AB� with a pair of spatially separated detectors A�Alice� and B �Bob� we have:

UAUB���AB��gA��gB�� = ����0A��0B� , �2�

with ���= ��gA��eB�+ �eA��gB�� /�2,which together with Eq.�1� implies a propagation of entanglement from ��AB� to thetwo-detector �two-particle� state ���. This is indeed consis-tent with the fundamental law of quantum information pro-cessing �9� prohibiting the increase of entanglement underlocal actions made by Alice and Bob. More precisely since

the local evolution operators UA, UB are unitary we deducethat the entanglement measure E���=−TrA��Aln��A��, with�A= 1

2 ��1A�1A � + �0A�0A � �, �9� is conserved and equals ln�2�.The result in Ref. �1� is then that the photon acted nonlo-

cally on the two remote stations since the pair or detectors inA and B share quantum correlations �which could be used to“violate” Bell’s inequality�. This is indeed a counterargumentto �6� because this evidence of nonlocality is obtained even ifit is not possible for both Alice and Bob to know that a

photon has been emitted from the source �5�. However, whilewe agree on the fact that this example reveals a clear anddirect manifestation of nonlocality for a single particle�in the sense previously defined by Hardy �5,10�� we dis-agree nevertheless on the necessity of using the entangledstate given by Eq. �1� to understand this effect.

Our doubt comes from the fact that for a nonrelativisticelectron it exists a first quantized picture which does notshow entanglement in the state given by Eq. �1�. Indeed

��r,t� = Vacuum���+��r���AB� =�A�r,t� + �B�r,t�

�2�3�

defines the first quantized wave function associated with anelectron state ��AB� and separated into two remote wavepackets �A,B�r , t�. Clearly here the entanglement is notpresent anymore.

Opposite a two electrons state such as �1A� �1B� becomesin the first quantized formalism

Vacuum���+��r���+��r���1A��1B�

= �A�r,t��B�r�,t� + �B�r,t��A�r�,t� . �4�

This state is entangled only if the wave packets �A,B are notspatially separated.

Because of this ambiguity it must be noted that for non-relativistic electrons there is no contradiction in supposingthat the entanglement between the two detector is createdduring the interaction between the electron and the atoms Aand B. This is indeed clear since there exists a first quantizedpicture for such systems. In that case we have the globalunitary evolution

UAB� � d3r��r,t��r� � �gA��gB�� = ���� , �5�

where UAB� cannot be separated into local operators acting onA or on B and ��r , t� is given by Eq. �3�. In the nonrelativ-istic regime where the number of particles is conserved thereis no absorption �destruction� of particles but rearrangement�e contains one particle more than g�. For example, �gA,B�could represent, say, a proton and �eA,B� a hydrogen atom.Writing the �spectator� 2 protons state �PA , PB� one can de-fine equivalently a second quantized version of Eq. �5�slightly different from Eq. �2�:

UAUB���AB��PA,PB�� = ��AB��PA,PB� , �6�*Electronic address: aurelien.drezet@uni-graz.at

PHYSICAL REVIEW A 74, 026301 �2006�

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where ��AB�= ��0A� � 1B�+ �1A� � 0B�� /�2 is the electron statefor the hydrogen modes in the hydrogen mode basis. Obvi-ously the existence of these two �nonrelativistic� formalismssuggests strongly that the entanglement in Eq. �1� is purelyformal �symbolical� and not necessary for the understandingof experiments with a single particle.

It should be emphasized that the existence of differentformalisms �one with and the other without entanglement� isnot a paradox since following the orthodox interpretation thewave function is a purely formal tool between two observa-tions �11,12�. The experimental predictions are the same inboth formalisms and there is no contradiction. One can inthis context remark that if in the first quantized formalismthe entanglement is created at the two remote detectors it isoppositely created at the beam splitter in the second quan-tized version of the theory. The increase of entanglement�E=ln�2� is the same in both formalisms and there is noway to distinguish them experimentally.

Finally it should be observed that for relativistic particleslike photons there exists an algebraical representation ofthe second quantized formalism which does not show en-tanglement in the state given by Eq. �1�. This formalism �see�8�� constitutes a good way for dealing with first quantizationby using methods of second quantization and it is particu-larly adapted to problems with a variation of the number ofparticles. Van Enk acknowledges indirectly the existenceof such formalism when he suggests a possible objection

to his conclusion based on the existence of the notation��AB�= �aA

† +aB†� /�2 �Vacuum� which is independent of the

algebraical representation used �13�. He rejected immedi-ately this possibility on the basis that the new system definedby Eq. �2� is composed of atoms which can be described bythe first quantized formalism. His argumentation is here con-nected to the discussion concerning the meaning of a refer-ence frame when the particle detected is not ambiguously theoriginal one �1�. In �1� it is indeed claimed that such anambiguity always exists since any quantum system is alwaysdetected by an amplifying device which is itself analyzed bythe macroscopical observer. The switching from one systemdescribed by ��AB� to another described by ��� would be thusirrelevant and the conclusion obtained in Ref. �1� should betrue. However, it has already been pointed out by Bohr thatsuch discussions only show that “an essential element ofambiguity is involved in ascribing conventional physical at-tributes to atomic objects” �12�. Moreover this means that wecan not be sure that the entanglement is associated with thesystem photon�vacuum described by ��AB� or with the sys-tem detector A� detector B described by ���. This is clearlywhat we deduced from the previous example with non rela-tivistic electrons. We can consequently conclude that ourmain conclusions are still true for photon states, e.g., theentanglement in ��AB���0�A �1�B+ �1�A �0�B is purely formaland not necessary.

�1� S. J. Enk, Phys. Rev. A 72, 064306 �2005�.�2� B. J. Oliver and C. Stroud, Jr., Phys. Lett. A 135, 407 �1989�.�3� S. M. Tan, D. F. Walls, and M. J. Collet, Phys. Rev. Lett. 66,

252 �1991�; B. Hessmo et al., ibid. 92, 180401 �2004�.�4� L. Hardy, Phys. Rev. Lett. 73, 2279 �1994�.�5� E. Santos, Phys. Rev. Lett. 68, 894 �1992�; S. M. Tan, D. F.

Walls, and M. J. Collet, ibid. 68, 895 �1992�; D. M. Green-berger, M. A. Horne, and A. Zeilinger, ibid. 75, 2064 �1995�;L. Vaidman, ibid. 75, 2063 �1995�; L. Hardy, ibid. 75, 2065�1995�.

�6� D. M. Greenberger, M. A. Horne, and A. Zeilinger, in Quan-tum Interferometry II, edited by F. De Martini, G. Deuardo,and Y. Shih �VCH, Weinheim 1997�.

�7� A. Peres, Phys. Rev. Lett. 74, 4571 �1995�.�8� M. Pawlowski and M. Czachor, e-print quant-ph/0507151;

M. Pawlowski and M. Czachor, e-print quant-ph/0512253;F. Berezin, The Method of Second Quantization �AcademicPress, New York, 1966�.

�9� V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, Phys.

Rev. Lett. 78, 2275 �1996�; M. B. Plenio and V. Vedral, Con-temp. Phys. 39, 431 �1998�.

�10� It can be remarked that there is an inherent ambiguity in theterm non locality for single particle used in �1–4�. Indeed inthese experiments one can only speak about the effect of onesingle particle on, say, a pair of detectors A and B which be-come entangled after that. One should thus in reality betterspeak of multiparticles nonlocality induced by a single particle�1–3� or a superposition particle vacuum �4�.

�11� A. Peres, Am. J. Phys. 52, 644 �1984�.�12� N. Bohr, in Albert Einstein Philosopher-Scientist, edited by P.

A. Schilpp �The Library of Living Philosophers, Evanston,1949�, pp. 200–241.

�13� It can be added that in the multivacuum formalism generallyconsidered in quantum optics the operator aA

† and aB† must be

written aA†

� IB and IA � aB† . This shows that the operator

aA†

� IB+ IA � aB† acting on the vacuum �Vacuum�= �0A� � �0B�

is nonlocal. However, this is not valid in the representationdiscussed in �8�.

COMMENTS PHYSICAL REVIEW A 74, 026301 �2006�

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