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Cited by 511Quantum Information and CryptographyQuantum Mechanics and ApplicationsQuantum optics and atomic interactions

Nonmaximally Entangled States: Production, Characterization, and Utilization

Open access
White, Andrew G., James, Daniel F. V., Eberhard, Philippe H., Kwiat, Paul G.·1999·Physical Review Letters
DOI 10.1103/physrevlett.83.3103
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Abstract

Using a spontaneous-down-conversion photon source, we produce true nonmaximally entangled states, i.e., without the need for postselection. The degree and phase of entanglement are readily tunable, and are characterized both by a standard analysis using coincidence minima, and by quantum state tomography of the two-photon state. Using the latter, we experimentally reconstruct the reduced density matrix for the polarization. Finally, we use these states to measure the Hardy fraction, obtaining a result that is $122\ensuremath{\sigma}$ from any local-realistic result.

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Written from the paper’s full text.

TL;DR

The paper demonstrates production of true non-maximally entangled photon pairs via spontaneous downconversion, without post-selection, with tunable degree and phase of entanglement. It characterizes these states by coincidence minima and quantum state tomography, then uses them to measure a Hardy fraction that violates local realism by 122 σ.

Key findings

  • Non-maximally entangled states were produced with degree of entanglement ε = tan χ, set by rotating the pump polarization, and phase φ, set by a UV quarter-wave plate.
  • Coincidence minima for a variety of states and analyzer settings fell in the range 0.2–1.0%, indicating very high purities; the maximally entangled case had visibility 97.8 ± 0.1%, while the source had previously attained visibilities better than 99%.
  • Quantum state tomography reconstructed the reduced polarization density matrix using 16 analyzer settings, showing controllable variation in both ε and φ.
  • The maximum measured Hardy fraction was 9.2% (24222 counts) at ε = 0.470, with analysis angles α = 55.2° and β = 72.1° and corresponding minima of 0.43%, 0.52%, and 0.49% (1132, 1372, and 1279 counts); this Hardy fraction is 122 σ larger than any local realistic result.

Methods

  • The source used two thin, adjacent Type-I beta-barium-borate (BBO) crystals with perpendicular optic axes, pumped by a 351 nm laser; pump polarization angle χ sets ε = tan χ, and a UV quarter-wave plate sets φ.
  • Detection used adjustable quarter- and half-wave plates and polarizing beamsplitters for polarization analysis, 702 nm ± 2.5 nm interference filters, single-photon counters, and coincidence detection with a 5.27 ns window; accidental coincidences were negligible at ∼0.4 s⁻¹.
  • States were characterized by coincidence-minima analysis and by two-photon Stokes-parameter tomography with 16 analyzer settings, reconstructing the density matrix from M⁻¹·P.

Limitations

  • The paper states there are small uncertainties in the final reconstructed density matrix and notes that measurement drift and uncertainty are discussed elsewhere; it also notes a slight deviation from ideal ε = tan χ because pump power at the second crystal is slightly attenuated by absorption in the first.
  • The paper states that the density matrix cannot be uniquely decomposed into pure and mixed submatrices to gain a direct measure of state purity, and that developing tomography for the full hyperentangled density matrix remains a considerable challenge.
  • Evident from the text: small |HV⟩ and |VH⟩ components appear even in the maximally entangled case, likely from analyzer/source axis misalignment or non-orthogonal source optic axes, and these components become proportionally more important as the state becomes less entangled.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1103/physrevlett.83.3103

Cites

30 of 30 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

9 of 34 cited sources checked · 2 supported · 1 flagged · accuracy not yet measured

  • Premisecitation
    § Non-maximally entangled states: production, characterization and utilization · p. 1

    Entanglement from spontaneous downconversion is maximally entangled, with ε = 1.

    “This entanglement is of a specific and limited kind: the states are maximally entangled, e.g., ( | HV 〉 ± ε | VH 〉 ) / √ 1 + | ε | 2 , where H and V respectively represent the horizontal and vertical polarizations of two separated photons, and ε = 1.”
    Quote in the paperSubstantial claim1 of 1 sources checked
  • Premisecitation
    § Non-maximally entangled states: production, characterization and utilization · p. 1

    Non-maximally entangled states have been shown to reduce the required detector efficiencies for loophole-free tests of Bell inequalities.

    “Non-maximally entangled states have been shown to reduce the required detector efficiencies for loophole-free tests of Bell inequalities [13]”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [13] source

      Section 1.2 (p. 4) says that every entangled pure state violates some Bell-type inequality, but it does not discuss detector efficiency. Searches for detector efficiency, detection loopholes, and non-maximally entangled states in Bell tests found no passage addressing whether such states reduce the efficiency required for a loophole-free test. (LLM, conf=0.76)

      Experimental — accuracy not yet measured
  • Premisecitation
    § Non-maximally entangled states: production, characterization and utilization · p. 1

    There is no possibility of varying the intrinsic degree of entanglement ε to produce non-maximally entangled states without compromising the purity of the state (introducing mixture).

    “There is no possibility of varying the intrinsic degree of entanglement , ε [11], to produce non-maximally entangled states without compromising the purity of the state, i.e., introducing mixture [12].”
    Quote in the paperSubstantial claim1 of 2 sources checked
    • cites [11] source

      The experiment description (page unavailable) discusses preparing an entangled photon pair, but does not discuss varying its intrinsic degree of entanglement. The passage on an impure teleported state explains how to adapt a measurement analysis; it does not say that producing a non-maximally entangled state necessarily introduces mixture. Searches for non-maximal entanglement and purity found no passage establishing the claimed constraint. (LLM, conf=0.76)

      Experimental — accuracy not yet measured
    • cites [12] source
  • Premisecitation
    § Non-maximally entangled states: production, characterization and utilization · p. 1

    Spontaneous optical parametric downconversion in a nonlinear crystal is the cleanest and most accessible source of multi-particle entanglement to date.

    “To date, the 'cleanest' and most accessible source of such entanglement arises from the process of spontaneous optical parametric downconversion in a nonlinear crystal (for a review, see [10]).”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [10] source

      The Experimental Realization section (p. 3) says entangled photon pairs can readily be generated by spontaneous parametric down-conversion in a nonlinear crystal, supporting the accessibility of the method for producing two-particle entanglement. The paper does not call it the cleanest source; its introduction (p. 1) also says entanglement of more than two quanta had not yet been produced, so it does not support the claim if “multi-particle” means more than two. (LLM, conf=0.88)

      Experimental — accuracy not yet measured
  • Findingcitation
    § Non-maximally entangled states: production, characterization and utilization · p. 2

    For a polarization angle of χ with respect to the vertical, the output state is |ψ> = (|HH> + ε e^{iφ}|VV>)/√(1+ε^2), where ε = tan χ.

    “For a polarization angle of χ with respect to the vertical, the output state is, | ψ 〉 = ( | HH 〉 + ε e i φ | VV 〉 ) / √ 1 + ε 2 , where the degree of entanglement, ε = tan χ [21].”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • The authors state this as an assumption: The two downconversion processes have coherence and high spatial overlap.
  • May combine two claims. 6
  • Classified as probably not a substantial claim. 12

  • May combine two claims.

    Premisecitation
    § Non-maximally entangled states: production, characterization and utilization · p. 1

    In those post-selection experiments, the underlying state is factorizable and so is not truly entangled.

    “However, the underlying state is factorizable, and so is not truly entangled.”
    Quote in the paperSubstantial claim1 of 2 sources checked
    • cites [18] source

      On p. 1, the paper calls the earlier photon and massive-particle experiments studies of entangled states and says their processes were selectable rather than deterministic. It does not say their underlying states were factorizable or that the states were not truly entangled; its distinction is whether entanglement is generated on demand. (LLM, conf=0.87)

      Checked by a reasoning model — may be wrong.

      Experimental — accuracy not yet measured
    • cites [19] source

  • Classified as probably not a substantial claim.

    Premisecitation
    § Non-maximally entangled states: production, characterization and utilization · p. 1

    Several optical experiments have controllably generated non-maximally entangled states via post-selection, i.e., selective measurement of a product state, after the state had been produced.

    “there have been several optical experiments where non-maximally entangled states were controllably generated via post-selection , i.e., selective measurement of a product state, after the state had been produced [18,19].”
    Quote in the paperSubstantial claim1 of 2 sources checked
    • cites [18] source

      The introduction (p. 1) says earlier photon-entanglement experiments involved random processes and were selectable rather than deterministic. It does not say those experiments controllably generated non-maximally entangled states, or describe post-selection as selective measurement of a product state after production. (LLM, conf=0.85)

      Experimental — accuracy not yet measured
    • cites [19] source

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingcitation
    § Non-maximally entangled states: production, characterization and utilization · p. 2

    With this source, the authors recently attained visibilities of better than 99% for maximally entangled states analyzed by keeping one analyzer fixed and varying the other.

    “with this source we recently attained visibilities of better than 99% [20]”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [20] source

  • May combine two claims.

    Findingown data
    § Non-maximally entangled states: production, characterization and utilization · p. 2

    Experimentally, coincidence minima were found for a variety of states and analyzer settings, and fell in the range 0.2-1.0%, indicating very high purities.

    “Coincidence minima were found for a variety of states and analyzer settings, and fell in the range 0.2-1.0%, indicating very high purities.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Findingown data
    § Non-maximally entangled states: production, characterization and utilization · p. 3

    At that point the analysis angles were α=55.2° and β=72.1°.

    “the analysis angles at this point were α = 55 . 2 ◦ and β = 72 . 1 ◦”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Study design and conductown data
    § Non-maximally entangled states: production, characterization and utilization · p. 2

    A coincidence window of 5.27 ns was sufficient to capture true coincidences; the resulting rate of accidental coincidences was negligible (~0.4 s^-1), so no corrections were necessary.

    “A coincidence window of 5.27 ns was sufficient to capture true coincidences; since the resulting rate of accidental coincidences was negligible ( ∼ 0.4s -1 ), no corrections for this were necessary.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § Non-maximally entangled states: production, characterization and utilization · p. 1

    States with a fixed degree of entanglement ε ≃ 4/3 have been deterministically generated in ion traps.

    “States with a fi xed degree of entanglement, ε ≃ 4 / 3, have been deterministically generated in ion traps [17]”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [17] source

  • Classified as probably not a substantial claim.

    Premisecitation
    § Non-maximally entangled states: production, characterization and utilization · p. 2

    Tomography is also possible with discrete variables.

    “Tomography is also possible with discrete variables [25].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [25] source

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § Non-maximally entangled states: production, characterization and utilization · p. 3

    A statistical measure composed of coincidence measurements from a variety of analyzer settings was first proposed by Bell.

    “a statistical measure composed of coincidence measurements from a variety of analyzer settings, was first proposed by Bell [5]”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [5] source

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § Non-maximally entangled states: production, characterization and utilization · p. 2

    Stokes parameters characterize the mean polarization of a classical beam.

    “the Stokes parameters, which characterize the mean polarization of a classical beam [26]”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [26]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § Non-maximally entangled states: production, characterization and utilization · p. 2

    For continuous variables, tomography has been implemented in both quantum and atom optics.

    “For continuous variables, tomography has been implemented in both quantum [23] and atom [24] optics.”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [23] source
    • cites [24] source

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § Non-maximally entangled states: production, characterization and utilization · p. 3

    All tests have found that, modulo some physically reasonable assumptions, nature does indeed violate local realism in accordance with quantum mechanics.

    “All tests have found that, modulo some physically reasonable assumptions, nature does indeed violate local realism in accordance with quantum mechanics.”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [15] source
    • cites [20] source

  • Classified as probably not a substantial claim. May combine two claims.

    Premisecitation
    § Non-maximally entangled states: production, characterization and utilization · p. 3

    More generally, P12(β,-β) > P12(α,-α) + P12(β,-α⊥) + P12(α⊥,-β), which is the inequality to be tested experimentally.

    “More generally, P 12 ( β, -β ) > P 12 ( α, -α ) + P 12 ( β, -α ⊥ ) + P 12 ( α ⊥ , -β ), which is the inequality to be tested experimentally [16].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [16] source

  • Classified as probably not a substantial claim. May combine two claims.

    Study design and conductown data
    § Non-maximally entangled states: production, characterization and utilization · p. 3

    The data points are normalized coincidence rates measured at ±β, with an inherent polarization analyzer uncertainty of 0.5°.

    “The data points are normalized coincidence rates measured at ± β , (to within 0 . 5 ◦ , the inherent uncertainty in our polarization analyzers).”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Premisecitation
    § Non-maximally entangled states: production, characterization and utilization · p. 3

    Hardy proposed an 'all-or-nothing' test of local realism.

    “Recently Hardy proposed an 'all-or-nothing' test of local realism [14,15].”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [14] source
    • cites [15] source

  • May combine two claims.

    Interpretationreasoning
    § Non-maximally entangled states: production, characterization and utilization · p. 3

    The ~1% probability of measuring |HV> and |VH> terms signifies either that the axes of the analysis systems were not perfectly aligned with the axes of the source, or that the optic axes of the source were not perfectly orthogonal.

    “the ∼ 1% probability of measuring | HV 〉 and | VH 〉 terms signifies that either the axes of the analysis systems were not perfectly aligned with the axes of the source ('horizontal' at the analyzer is rotated with respect to 'horizontal' at the source), or that the optic axes of the source were not perfectly orthogonal”
    Quote in the paperNot a substantial claim
16 of 16

  • The authors state this as an assumption.

    The two downconversion processes have coherence and high spatial overlap.

    “Given the coherence and high spatial overlap between these two processes”

  • The paper relies on this without support.

    One application of such states is testing local realism.

    “As mentioned earlier, one application of such states is testing local realism.”
2 of 2

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • The original Hardy proposal was experimentally demonstrated by Torgerson et al. (1995) and White et al. (1999), with results fed into inequalities derived for imperfections under the fair sampling assumption.

    “The original proposal was experimentally demonstrated by Torgerson et al. (1995) and White et al.(1999)); as in the GHZ-type experiments, the results of the experiments were fed into specific inequalities, specially derived to take into account experimental imperfections (under the fair sampling assumption, of course).”

    From Multiphoton entanglement and interferometry (2012) · cites this paper as White et al. (1999)

    On p. 3, White et al. describe an experimental test of Hardy’s proposal, give the inequality used, and report a Hardy fraction 122σ above the local-realistic bound. Footnote [16] on p. 4 says experimental realizations require inequalities because measurements have uncertainties. The copy does not specify the fair-sampling assumption or say that the inequality was specially derived to account for experimental imperfections. (LLM, conf=0.85)

    Experimental — accuracy not yet measured

  • Using the theory of measurement of qubits, including tomographic reconstruction from 16 polarization measurements together with maximum-likelihood numerical optimization, one can obtain and reconstruct the non-negative definite density operator.

    “It has been shown theoretically and experimentally [56], [57], [58] that using the theory of the measurement of qubits, which includes a tomographic reconstruction of the DMOS due to the linear set of 16 polarization-measurements together with the numerical optimization method, the so-called maximum-likelihood-technique (MLT), one can obtain and reconstruct the non-negative definite density operat”

    From Nonlocal realism tests and quantum state tomography in Sagnac-based type-II polarization-entanglement SPDC-source (2021) · cites this paper as [57]

    Split into 2 atoms — 0 confirmed, 2 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

    Experimental — accuracy not yet measured

  • The complexity of the transformation only scales linearly with the number of photons in the state, in contrast to the exponential scaling in the number of measurements required in usual tomography.

    “Importantly, the complexity of the transformation only scales linearly with the number of photons in the state, in contrast to the exponential scaling in the number of measurements required in usual tomography.36”

    From Scalable on-chip quantum state tomography (2018) · cites this paper as [36]

    The tomography discussion (p. 2) says a full analysis requires d²−1 parameters and gives 3 and 15 analyzer settings for single- and two-photon states, supporting the contrast with the growing measurement demands of tomography. It does not describe the cited transformation or say that its complexity scales linearly with photon number. (LLM, conf=0.87)

    Experimental — accuracy not yet measured

  • Experimentally, quantum state tomography was used for two-, three-, four-photon polarization states and also hyperentangled photon pairs.

    “Experimentally, this technique was used for two-, three-, four-photon polarization states and also hyperentangled photon pairs (White et al. 1999;Resch et al. 2005;Walther et al. 2005b;Barreiro et al. 2005).”

    From Multiphoton entanglement and interferometry (2012) · cites this paper as White et al. 1999

  • Parametric down-conversion has been serving as the main workhorse for the multiphoton experiments reviewed here; up to eight entangled photons have been observed.

    “Parametric down-conversion (Kwiat et al. 1995;White et al. 1999) has been serving as the main workhorse for the multiphoton experiments reviewed here; up to eight entangled photons have been observed (Yao et al. 2007a).”

    From Multiphoton entanglement and interferometry (2012) · cites this paper as White et al. 1999

Cites

Flagged first, then by citations

1 of the 31 references couldn’t be matched to a known paper.

Research works printed but not linked: 1

  1. G. G. Stokes, 1852, Trans. Cambridge Philos. Soc., 9, 399

Cited by

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Related

Authors
White, A. G., James, D. F. V., Eberhard, P. H., & Kwiat, P. G.
Affiliations
  • Los Alamos National Laboratory (US)
  • Lawrence Berkeley National Laboratory (US)
ORCID
Venue
Physical Review Letters
Publisher
American Physical Society (APS)
Published
1999-10-18
Volume
83
Issue
16
Pages
3103-3107
ISSN
  • 0031-9007 (ISSN-L)
  • 1079-7114
DOI
10.1103/physrevlett.83.3103
OpenAlex
W2073659633
Semantic Scholar
615ec387673818556d60ca4756136618cb2e005a
Cited by
511
References
31
Citations by year
2026: 7 · 2025: 15 · 2024: 18 · 2023: 11 · 2022: 17 · 2021: 19 · 2020: 14 · 2019: 16 · 2018: 15 · 2017: 21 · 2016: 14 · 2015: 18 · 2014: 17 · 2013: 18 · 2012: 21
FWCI
23.61
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical Review Letters (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum optics and atomic interactions
Keywords
non-maximally entangled states, spontaneous parametric down-conversion, two-photon entanglement, polarization entanglement, quantum state tomography, reduced density matrix, Hardy inequality, local realism

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