Single-particle nonlocality and entanglement with the vacuum

Björk, Gunnar; Jönsson, Per; Sánchez-Soto, Luis L.

doi:10.1103/physreva.64.042106

Cited by 49 publications

(48 citation statements)

References 41 publications

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“…In this representation, the above entanglement is not apparent, and indeed it has been argued that nonlocality cannot be a single-particle effect [10] (although see Ref. [11]). As a second example, consider a two-particle state where Alice has one particle and Bob the other.…”

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confidence: 99%

Entanglement of Indistinguishable Particles Shared between Two Parties

2003

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Using an operational definition we quantify the entanglement, EP, between two parties who share an arbitrary pure state of N indistinguishable particles. We show that EP ≤ EM, where EM is the bipartite entanglement calculated from the mode-occupation representation. Unlike EM, EP is super-additive. For example, EP = 0 for any single-particle state, but the state |1 |1 , where both modes are split between the two parties, has EP = 1/2. We discuss how this relates to quantum correlations between particles, for both fermions and bosons.PACS numbers: 03.65. Ta, Entanglement lies at the heart of quantum mechanics, and is profoundly important in quantum information (QI) [1]. It might be thought that there is nothing new to be said about bipartite entanglement if the shared state |Ψ AB is pure. In ebits, the entanglement is simply [2]Here S(ρ) is the binary von Neumann entropy −Tr [ρ log 2 ρ], and (since we will use unnormalized kets)However, in the context of indistinguishable particles, a little consideration reveals a less than clear situation, which has been the subject of recent controversy [3,4,5,6,7]. Consider, for example, a single particle in an equal superposition of being with Alice and with Bob. In the mode-occupation, or Fock, representation, the state isHere we are following the conventions of writing Alice's occupation number(s) followed by Bob's, separated by a comma, and of omitting any modes that are unoccupied. On the face of it, this is an entangled state with one ebit, and such a state has been argued to show nonlocality of a single photon [8,9]. However, the particle's wavefunction, in the co-ordinate representation, is of the formwhere the subscripts indicate where the wavepackets are localized in co-ordinate (x) space. In this representation, the above entanglement is not apparent, and indeed it has been argued that nonlocality cannot be a single-particle effect [10] (although see Ref.[11]). As a second example, consider a two-particle state where Alice has one particle and Bob the other. In the mode-occupation picture, the state is |1, 1 , which appears unentangled. But since these are identical particles, the wavefunction must be symmetrized asfor bosons and fermions respectively. This has the appearance of an entangled state. Finally, consider another two-particle state, but this time where the two particles are prepared and shared as in the first example, but in different modes. In the Fock representation, this state is entangled:(|0, 1 + |1, 0 )(|0, 1 + |1, 0 ) = |00, 11 + |01, 10 + |10, 01 + |11, 00 .The corresponding wavefunctionalso has the appearance of an entangled state. In this Letter we give an operational definition of entanglement between two parties who share an arbitrary pure state of indistinguishable particles. Applying this to the three states introduced above yields an entanglement (in ebits) of 0, 0, and 1/2 respectively.To justify these (non-obvious) answers, we proceed as follows. First we define precisely what we mean when we use the term particle. Next we review two pre...

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confidence: 99%

Entanglement of Indistinguishable Particles Shared between Two Parties

2003

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“…Accordingly, it is absolutely irrelevant which irreducible representation we select for our explicit calculations. These remarks apply, in particular, to all the papers on entanglement with vacuum and nonlocality of a single photon where the calculations were performed in the ∞-representation [17,18,19,20,21,22,23,24,25] (see however [26]). …”

Section: Discussionmentioning

confidence: 99%

“…This is true, but the problem with the B-representation is that the third subsystem does not have a natural decomposition into subsystems [8]. The distinction between the vacuum |0 and 'a vacuum |0 at one side of the beam-splitter' is undefined in this representation [9] since the vacuum is here unique, and corresponds to the vector (17). An entanglement with a unique vacuum must be trivial.…”

Section: B-representation Calculationmentioning

confidence: 99%

Degree of entanglement as a physically ill-posed problem: The case of entanglement with vacuum

Pawłowski

Czachor

2006

Phys. Rev. A

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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 ∞-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.

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“…This came as a surprise, because single particle states show no signs of entanglement when expressed as a simple superposition of wavefunctions, and initially there were doubts as to whether single-particle quantum nonlocality is genuine. Subsequent theoretical [7][8][9][10][11][12][13] and experimental [14][15][16][17][18][19][20] investigations confirmed that single-particle QN is indeed genuine and clarified that the entanglement lies between the spatial modes, rather than between the particles (a clear and concise summary can be found in Ref. [10]).…”

Section: Introductionmentioning

confidence: 96%

Single-photon quantum nonlocality: Violation of the Clauser-Horne-Shimony-Holt inequality using feasible measurement setups

et al. 2017

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We investigate quantum nonlocality of a single-photon entangled state under feasible measurement techniques consisting of on-off and homodyne detections along with unitary operations of displacement and squeezing. We test for a potential violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality, in which each of the bipartite party has a freedom to choose between 2 measurement settings, each measurement yielding a binary outcome. We find that single-photon quantum nonlocality can be detected when two or less of the 4 total measurements are carried out by homodyne detection. The largest violation of the CHSH inequality is obtained when all four measurements are squeezed-and-displaced on-off detections. We test robustness of violations against imperfections in on-off detectors and single-photon sources, finding that the squeezed-and-displaced measurement schemes perform better than the displacement-only measurement schemes.

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Single-particle nonlocality and entanglement with the vacuum

Cited by 49 publications

References 41 publications

Entanglement of Indistinguishable Particles Shared between Two Parties

Entanglement of Indistinguishable Particles Shared between Two Parties

Degree of entanglement as a physically ill-posed problem: The case of entanglement with vacuum

Single-photon quantum nonlocality: Violation of the Clauser-Horne-Shimony-Holt inequality using feasible measurement setups

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