Abstract

In recent years, there has been an increased interest in the generation of superpositions of coherent states with opposite phases, the so-called photonic Schrödinger cat states. These experiments are challenging, and, so far, cats involving only small photon numbers have been implemented. Here, we propose to consider two-mode squeezed states as examples of Schrödinger cat-like states. For this, we apply criteria that aim to identify macroscopic superpositions in a more general sense. We extend some of these criteria to the two-mode continuous variable regime. Furthermore, we compare the size of states obtained in several experiments and discuss experimental challenges for further improvements. Our results not only promote two-mode squeezed states for exploring quantum effects at the macroscopic level but also provide direct measures to evaluate their usefulness for quantum metrology.

© 2015 Optical Society of America

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References

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  5. G. Björk and P. G. L. Mana, “A size criterion for macroscopic superposition states,” J. Opt. B 6, 429–436 (2004).
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  9. F. Marquardt, B. Abel, and J. Von Delft, “Measuring the size of a quantum superposition of many-body states,” Phys. Rev. A 78, 012109 (2008).
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  11. F. Fröwis and W. Dür, “Measures of macroscopicity for quantum spin systems,” New J. Phys. 14, 093039 (2012).
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  26. X. Su, A. Tan, X. Jia, Q. Pan, C. Xie, and K. Peng, “Experimental demonstration of quantum entanglement between frequency-nondegenerate optical twin beams,” Opt. Lett. 31, 1133–1135 (2006).
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  27. G. Keller, V. D’Auria, N. Treps, T. Coudreau, J. Laurat, and C. Fabre, “Experimental demonstration of frequency-degenerate bright EPR beams with a self-phase-locked OPO,” Opt. Express 16, 9351–9356 (2008).
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    [Crossref]
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    [Crossref]
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    [Crossref]
  35. O. Morin, K. Huang, J. Liu, H. Le Jeannic, C. Fabre, and J. Laurat, “Remote creation of hybrid entanglement between particle-like and wave-like optical qubits,” Nat. Photonics 8, 570–574 (2014).
    [Crossref]
  36. H. Jeong, A. Zavatta, M. Kang, S.-W. Lee, L. S. Costanzo, S. Grandi, T. C. Ralph, and M. Bellini, “Generation of hybrid entanglement of light,” Nat. Photonics 8, 564–569 (2014).
    [Crossref]
  37. P. Sekatski, N. Sangouard, M. Stobińska, F. Bussières, M. Afzelius, and N. Gisin, “Proposal for exploring macroscopic entanglement with a single photon and coherent states,” Phys. Rev. A 86, 060301 (2012).
    [Crossref]
  38. S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439–3443 (1994).
    [Crossref]
  39. E. G. Cavalcanti and M. D. Reid, “Criteria for generalized macroscopic and mesoscopic quantum coherence,” Phys. Rev. A 77, 062108 (2008).
    [Crossref]
  40. F. Fröwis and W. Dür, “Are cloned quantum states macroscopic?” Phys. Rev. Lett. 109, 170401 (2012).
    [Crossref]
  41. While superpositions of coherent states with opposite phases have a negative Wigner representation, here we show that the size of two-mode vacuum squeezed states is basically the same. This shows that states with a positive Wigner representation can be macroscopically quantum.
  42. To illustrate this, consider a state ρϵ in Eq. (6) with f1(x,x¯)=f2(x,x¯)=1 for |x−x¯|<ϵ and zero otherwise. The peculiarity of this state is that f satisfies 〈f′〉=〈f′′〉=0 in such a way that the violation of the Duan–Simon criterion by ρϵ is independent of ϵ unless it is strictly equal to zero. Therefore, ρϵ with ϵ approaching zero is an example of a state that can give an arbitrarily high violation of the Duan criteria, while being arbitrarily close to a separable state.
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    [Crossref]
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  46. T. J. Volkoff and K. B. Whaley, “Measurement- and comparison-based sizes of Schrödinger cat states of light,” Phys. Rev. A 89, 012122 (2014).
    [Crossref]
  47. S. Raeisi, P. Sekatski, and C. Simon, “Coarse graining makes it hard to see micro-macro entanglement,” Phys. Rev. Lett. 107, 250401 (2011).
    [Crossref]
  48. F. Fröwis, M. Van den Nest, and W. Dür, “Certifiability criterion for large-scale quantum systems,” New J. Phys. 15, 113011 (2013).
    [Crossref]
  49. T. Wang, R. Ghobadi, S. Raeisi, and C. Simon, “Precision requirements for observing macroscopic quantum effects,” Phys. Rev. A 88, 062114 (2013).
    [Crossref]
  50. P. Sekatski, N. Gisin, and N. Sangouard, “How difficult is it to prove the quantumness of macroscropic states?” Phys. Rev. Lett. 113, 090403 (2014).
    [Crossref]

2015 (5)

A. Laghaout, J. S. Neergaard-Nielsen, and U. L. Andersen, “Assessments of macroscopicity for quantum optical states,” Opt. Commun. 337, 96–101 (2015).

H. Jeong, M. Kang, and H. Kwon, “Characterizations and quantifications of macroscopic quantumness and its implementations using optical fields,” Opt. Commun. 337, 12–21 (2015).

T. Farrow and V. Vedral, “Classification of macroscopic quantum effects,” Opt. Commun. 337, 22–26 (2015).

F. Fröwis, N. Sangouard, and N. Gisin, “Linking measures for macroscopic quantum states via photon–spin mapping,” Opt. Commun. 337, 2–11 (2015).

F. Fröwis and N. Gisin, “Tighter quantum uncertainty relations follow from a general probabilistic bound,” Phys. Rev. A 92, 012102 (2015).

2014 (5)

T. J. Volkoff and K. B. Whaley, “Measurement- and comparison-based sizes of Schrödinger cat states of light,” Phys. Rev. A 89, 012122 (2014).
[Crossref]

P. Sekatski, N. Gisin, and N. Sangouard, “How difficult is it to prove the quantumness of macroscropic states?” Phys. Rev. Lett. 113, 090403 (2014).
[Crossref]

P. Sekatski, N. Sangouard, and N. Gisin, “Size of quantum superpositions as measured with classical detectors,” Phys. Rev. A 89, 012116 (2014).
[Crossref]

O. Morin, K. Huang, J. Liu, H. Le Jeannic, C. Fabre, and J. Laurat, “Remote creation of hybrid entanglement between particle-like and wave-like optical qubits,” Nat. Photonics 8, 570–574 (2014).
[Crossref]

H. Jeong, A. Zavatta, M. Kang, S.-W. Lee, L. S. Costanzo, S. Grandi, T. C. Ralph, and M. Bellini, “Generation of hybrid entanglement of light,” Nat. Photonics 8, 564–569 (2014).
[Crossref]

2013 (7)

T. Eberle, V. Händchen, and R. Schnabel, “Stable control of 10 dB two-mode squeezed vacuum states of light,” Opt. Express 21, 11546–11553 (2013).
[Crossref]

S. Nimmrichter and K. Hornberger, “Macroscopicity of mechanical quantum superposition states,” Phys. Rev. Lett. 110, 160403 (2013).
[Crossref]

G. Tóth and D. Petz, “Extremal properties of the variance and the quantum Fisher information,” Phys. Rev. A 87, 032324 (2013).
[Crossref]

F. Fröwis, M. Van den Nest, and W. Dür, “Certifiability criterion for large-scale quantum systems,” New J. Phys. 15, 113011 (2013).
[Crossref]

T. Wang, R. Ghobadi, S. Raeisi, and C. Simon, “Precision requirements for observing macroscopic quantum effects,” Phys. Rev. A 88, 062114 (2013).
[Crossref]

N. Bruno, A. Martin, P. Sekatski, N. Sangouard, R. T. Thew, and N. Gisin, “Displacement of entanglement back and forth between the micro and macro domains,” Nat. Phys. 9, 545–548 (2013).
[Crossref]

A. I. Lvovsky, R. Ghobadi, A. Chandra, A. S. Prasad, and C. Simon, “Observation of micro-macro entanglement of light,” Nat. Phys. 9, 541–544 (2013).
[Crossref]

2012 (3)

F. Fröwis and W. Dür, “Are cloned quantum states macroscopic?” Phys. Rev. Lett. 109, 170401 (2012).
[Crossref]

F. Fröwis and W. Dür, “Measures of macroscopicity for quantum spin systems,” New J. Phys. 14, 093039 (2012).
[Crossref]

P. Sekatski, N. Sangouard, M. Stobińska, F. Bussières, M. Afzelius, and N. Gisin, “Proposal for exploring macroscopic entanglement with a single photon and coherent states,” Phys. Rev. A 86, 060301 (2012).
[Crossref]

2011 (2)

C.-W. Lee and H. Jeong, “Quantification of macroscopic quantum superpositions within phase space,” Phys. Rev. Lett. 106, 220401 (2011).
[Crossref]

S. Raeisi, P. Sekatski, and C. Simon, “Coarse graining makes it hard to see micro-macro entanglement,” Phys. Rev. Lett. 107, 250401 (2011).
[Crossref]

2010 (1)

2008 (3)

G. Keller, V. D’Auria, N. Treps, T. Coudreau, J. Laurat, and C. Fabre, “Experimental demonstration of frequency-degenerate bright EPR beams with a self-phase-locked OPO,” Opt. Express 16, 9351–9356 (2008).
[Crossref]

F. Marquardt, B. Abel, and J. Von Delft, “Measuring the size of a quantum superposition of many-body states,” Phys. Rev. A 78, 012109 (2008).
[Crossref]

E. G. Cavalcanti and M. D. Reid, “Criteria for generalized macroscopic and mesoscopic quantum coherence,” Phys. Rev. A 77, 062108 (2008).
[Crossref]

2007 (2)

J. I. Korsbakken, K. B. Whaley, J. Dubois, and J. I. Cirac, “Measurement-based measure of the size of macroscopic quantum superpositions,” Phys. Rev. A 75, 042106 (2007).
[Crossref]

K. Wakui, H. Takahashi, A. Furusawa, and M. Sasaki, “Photon subtracted squeezed states generated with periodically poled KTiOPO4,” Opt. Express 15, 3568–3574 (2007).
[Crossref]

2006 (5)

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[Crossref]

J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
[Crossref]

J. Jing, S. Feng, R. Bloomer, and O. Pfister, “Experimental continuous-variable entanglement from a phase-difference-locked optical parametric oscillator,” Phys. Rev. A 74, 041804 (2006).
[Crossref]

X. Su, A. Tan, X. Jia, Q. Pan, C. Xie, and K. Peng, “Experimental demonstration of quantum entanglement between frequency-nondegenerate optical twin beams,” Opt. Lett. 31, 1133–1135 (2006).
[Crossref]

E. G. Cavalcanti and M. D. Reid, “Signatures for generalized macroscopic superpositions,” Phys. Rev. Lett. 97, 170405 (2006).
[Crossref]

2005 (2)

A. Shimizu and T. Morimae, “Detection of macroscopic entanglement by correlation of local observables,” Phys. Rev. Lett. 95, 090401 (2005).
[Crossref]

A. S. Villar, L. S. Cruz, K. N. Cassemiro, M. Martinelli, and P. Nussenzveig, “Generation of bright two-color continuous variable entanglement,” Phys. Rev. Lett. 95, 243603 (2005).
[Crossref]

2004 (1)

G. Björk and P. G. L. Mana, “A size criterion for macroscopic superposition states,” J. Opt. B 6, 429–436 (2004).
[Crossref]

2003 (1)

W. P. Bowen, R. Schnabel, P. K. Lam, and T. C. Ralph, “Experimental investigation of criteria for continuous variable entanglement,” Phys. Rev. Lett. 90, 043601 (2003).
[Crossref]

2002 (3)

A. J. Leggett, “Testing the limits of quantum mechanics: motivation, state of play, prospects,” J. Phys. Condens. Matter 14, R415–R451 (2002).
[Crossref]

W. Dür, C. Simon, and J. I. Cirac, “Effective size of certain macroscopic quantum superpositions,” Phys. Rev. Lett. 89, 210402 (2002).
[Crossref]

A. Shimizu and T. Miyadera, “Stability of quantum states of finite macroscopic systems against classical noises, perturbations from environments, and local measurements,” Phys. Rev. Lett. 89, 270403 (2002).
[Crossref]

2001 (1)

C. Silberhorn, P. K. Lam, O. Weiß, F. König, N. Korolkova, and G. Leuchs, “Generation of continuous variable Einstein-Podolsky-Rosen entanglement via the Kerr nonlinearity in an optical fiber,” Phys. Rev. Lett. 86, 4267–4270 (2001).
[Crossref]

2000 (3)

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[Crossref]

R. Simon, “Peres-Horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2726–2729 (2000).
[Crossref]

Y. Zhang, H. Wang, X. Li, J. Jing, C. Xie, and K. Peng, “Experimental generation of bright two-mode quadrature squeezed light from a narrow-band nondegenerate optical parametric amplifier,” Phys. Rev. A 62, 023813 (2000).
[Crossref]

1994 (1)

S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439–3443 (1994).
[Crossref]

1980 (1)

A. J. Leggett, “Macroscopic quantum systems and the quantum theory of measurement,” Prog. Theor. Phys. Suppl. 69, 80–100 (1980).
[Crossref]

1935 (1)

E. Schrödinger, “Die gegenwärtige Situation in der Quantenmechanik,” Naturwissenschaften 23, 807–812 (1935).
[Crossref]

Abel, B.

F. Marquardt, B. Abel, and J. Von Delft, “Measuring the size of a quantum superposition of many-body states,” Phys. Rev. A 78, 012109 (2008).
[Crossref]

Afzelius, M.

P. Sekatski, N. Sangouard, M. Stobińska, F. Bussières, M. Afzelius, and N. Gisin, “Proposal for exploring macroscopic entanglement with a single photon and coherent states,” Phys. Rev. A 86, 060301 (2012).
[Crossref]

Andersen, U. L.

A. Laghaout, J. S. Neergaard-Nielsen, and U. L. Andersen, “Assessments of macroscopicity for quantum optical states,” Opt. Commun. 337, 96–101 (2015).

Bellini, M.

H. Jeong, A. Zavatta, M. Kang, S.-W. Lee, L. S. Costanzo, S. Grandi, T. C. Ralph, and M. Bellini, “Generation of hybrid entanglement of light,” Nat. Photonics 8, 564–569 (2014).
[Crossref]

Björk, G.

G. Björk and P. G. L. Mana, “A size criterion for macroscopic superposition states,” J. Opt. B 6, 429–436 (2004).
[Crossref]

Bloomer, R.

J. Jing, S. Feng, R. Bloomer, and O. Pfister, “Experimental continuous-variable entanglement from a phase-difference-locked optical parametric oscillator,” Phys. Rev. A 74, 041804 (2006).
[Crossref]

Bowen, W. P.

W. P. Bowen, R. Schnabel, P. K. Lam, and T. C. Ralph, “Experimental investigation of criteria for continuous variable entanglement,” Phys. Rev. Lett. 90, 043601 (2003).
[Crossref]

Braunstein, S. L.

S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439–3443 (1994).
[Crossref]

Bruno, N.

N. Bruno, A. Martin, P. Sekatski, N. Sangouard, R. T. Thew, and N. Gisin, “Displacement of entanglement back and forth between the micro and macro domains,” Nat. Phys. 9, 545–548 (2013).
[Crossref]

Bussières, F.

P. Sekatski, N. Sangouard, M. Stobińska, F. Bussières, M. Afzelius, and N. Gisin, “Proposal for exploring macroscopic entanglement with a single photon and coherent states,” Phys. Rev. A 86, 060301 (2012).
[Crossref]

Cassemiro, K. N.

A. S. Villar, L. S. Cruz, K. N. Cassemiro, M. Martinelli, and P. Nussenzveig, “Generation of bright two-color continuous variable entanglement,” Phys. Rev. Lett. 95, 243603 (2005).
[Crossref]

Cavalcanti, E. G.

E. G. Cavalcanti and M. D. Reid, “Criteria for generalized macroscopic and mesoscopic quantum coherence,” Phys. Rev. A 77, 062108 (2008).
[Crossref]

E. G. Cavalcanti and M. D. Reid, “Signatures for generalized macroscopic superpositions,” Phys. Rev. Lett. 97, 170405 (2006).
[Crossref]

Caves, C. M.

S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439–3443 (1994).
[Crossref]

Chandra, A.

A. I. Lvovsky, R. Ghobadi, A. Chandra, A. S. Prasad, and C. Simon, “Observation of micro-macro entanglement of light,” Nat. Phys. 9, 541–544 (2013).
[Crossref]

Cirac, J. I.

J. I. Korsbakken, K. B. Whaley, J. Dubois, and J. I. Cirac, “Measurement-based measure of the size of macroscopic quantum superpositions,” Phys. Rev. A 75, 042106 (2007).
[Crossref]

W. Dür, C. Simon, and J. I. Cirac, “Effective size of certain macroscopic quantum superpositions,” Phys. Rev. Lett. 89, 210402 (2002).
[Crossref]

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[Crossref]

Costanzo, L. S.

H. Jeong, A. Zavatta, M. Kang, S.-W. Lee, L. S. Costanzo, S. Grandi, T. C. Ralph, and M. Bellini, “Generation of hybrid entanglement of light,” Nat. Photonics 8, 564–569 (2014).
[Crossref]

Coudreau, T.

Cruz, L. S.

A. S. Villar, L. S. Cruz, K. N. Cassemiro, M. Martinelli, and P. Nussenzveig, “Generation of bright two-color continuous variable entanglement,” Phys. Rev. Lett. 95, 243603 (2005).
[Crossref]

D’Auria, V.

Duan, L.-M.

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[Crossref]

Dubois, J.

J. I. Korsbakken, K. B. Whaley, J. Dubois, and J. I. Cirac, “Measurement-based measure of the size of macroscopic quantum superpositions,” Phys. Rev. A 75, 042106 (2007).
[Crossref]

Dür, W.

F. Fröwis, M. Van den Nest, and W. Dür, “Certifiability criterion for large-scale quantum systems,” New J. Phys. 15, 113011 (2013).
[Crossref]

F. Fröwis and W. Dür, “Are cloned quantum states macroscopic?” Phys. Rev. Lett. 109, 170401 (2012).
[Crossref]

F. Fröwis and W. Dür, “Measures of macroscopicity for quantum spin systems,” New J. Phys. 14, 093039 (2012).
[Crossref]

W. Dür, C. Simon, and J. I. Cirac, “Effective size of certain macroscopic quantum superpositions,” Phys. Rev. Lett. 89, 210402 (2002).
[Crossref]

Eberle, T.

Fabre, C.

O. Morin, K. Huang, J. Liu, H. Le Jeannic, C. Fabre, and J. Laurat, “Remote creation of hybrid entanglement between particle-like and wave-like optical qubits,” Nat. Photonics 8, 570–574 (2014).
[Crossref]

G. Keller, V. D’Auria, N. Treps, T. Coudreau, J. Laurat, and C. Fabre, “Experimental demonstration of frequency-degenerate bright EPR beams with a self-phase-locked OPO,” Opt. Express 16, 9351–9356 (2008).
[Crossref]

Farrow, T.

T. Farrow and V. Vedral, “Classification of macroscopic quantum effects,” Opt. Commun. 337, 22–26 (2015).

Feng, S.

J. Jing, S. Feng, R. Bloomer, and O. Pfister, “Experimental continuous-variable entanglement from a phase-difference-locked optical parametric oscillator,” Phys. Rev. A 74, 041804 (2006).
[Crossref]

Fröwis, F.

F. Fröwis, N. Sangouard, and N. Gisin, “Linking measures for macroscopic quantum states via photon–spin mapping,” Opt. Commun. 337, 2–11 (2015).

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F. Fröwis, M. Van den Nest, and W. Dür, “Certifiability criterion for large-scale quantum systems,” New J. Phys. 15, 113011 (2013).
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F. Fröwis and W. Dür, “Are cloned quantum states macroscopic?” Phys. Rev. Lett. 109, 170401 (2012).
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F. Fröwis and W. Dür, “Measures of macroscopicity for quantum spin systems,” New J. Phys. 14, 093039 (2012).
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Ghobadi, R.

T. Wang, R. Ghobadi, S. Raeisi, and C. Simon, “Precision requirements for observing macroscopic quantum effects,” Phys. Rev. A 88, 062114 (2013).
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A. I. Lvovsky, R. Ghobadi, A. Chandra, A. S. Prasad, and C. Simon, “Observation of micro-macro entanglement of light,” Nat. Phys. 9, 541–544 (2013).
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L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
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F. Fröwis, N. Sangouard, and N. Gisin, “Linking measures for macroscopic quantum states via photon–spin mapping,” Opt. Commun. 337, 2–11 (2015).

F. Fröwis and N. Gisin, “Tighter quantum uncertainty relations follow from a general probabilistic bound,” Phys. Rev. A 92, 012102 (2015).

P. Sekatski, N. Gisin, and N. Sangouard, “How difficult is it to prove the quantumness of macroscropic states?” Phys. Rev. Lett. 113, 090403 (2014).
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P. Sekatski, N. Sangouard, and N. Gisin, “Size of quantum superpositions as measured with classical detectors,” Phys. Rev. A 89, 012116 (2014).
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N. Bruno, A. Martin, P. Sekatski, N. Sangouard, R. T. Thew, and N. Gisin, “Displacement of entanglement back and forth between the micro and macro domains,” Nat. Phys. 9, 545–548 (2013).
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P. Sekatski, N. Sangouard, M. Stobińska, F. Bussières, M. Afzelius, and N. Gisin, “Proposal for exploring macroscopic entanglement with a single photon and coherent states,” Phys. Rev. A 86, 060301 (2012).
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H. Jeong, A. Zavatta, M. Kang, S.-W. Lee, L. S. Costanzo, S. Grandi, T. C. Ralph, and M. Bellini, “Generation of hybrid entanglement of light,” Nat. Photonics 8, 564–569 (2014).
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A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
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Hettich, C.

J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
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S. Nimmrichter and K. Hornberger, “Macroscopicity of mechanical quantum superposition states,” Phys. Rev. Lett. 110, 160403 (2013).
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O. Morin, K. Huang, J. Liu, H. Le Jeannic, C. Fabre, and J. Laurat, “Remote creation of hybrid entanglement between particle-like and wave-like optical qubits,” Nat. Photonics 8, 570–574 (2014).
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H. Jeong, M. Kang, and H. Kwon, “Characterizations and quantifications of macroscopic quantumness and its implementations using optical fields,” Opt. Commun. 337, 12–21 (2015).

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Y. Zhang, H. Wang, X. Li, J. Jing, C. Xie, and K. Peng, “Experimental generation of bright two-mode quadrature squeezed light from a narrow-band nondegenerate optical parametric amplifier,” Phys. Rev. A 62, 023813 (2000).
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H. Jeong, M. Kang, and H. Kwon, “Characterizations and quantifications of macroscopic quantumness and its implementations using optical fields,” Opt. Commun. 337, 12–21 (2015).

H. Jeong, A. Zavatta, M. Kang, S.-W. Lee, L. S. Costanzo, S. Grandi, T. C. Ralph, and M. Bellini, “Generation of hybrid entanglement of light,” Nat. Photonics 8, 564–569 (2014).
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König, F.

C. Silberhorn, P. K. Lam, O. Weiß, F. König, N. Korolkova, and G. Leuchs, “Generation of continuous variable Einstein-Podolsky-Rosen entanglement via the Kerr nonlinearity in an optical fiber,” Phys. Rev. Lett. 86, 4267–4270 (2001).
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C. Silberhorn, P. K. Lam, O. Weiß, F. König, N. Korolkova, and G. Leuchs, “Generation of continuous variable Einstein-Podolsky-Rosen entanglement via the Kerr nonlinearity in an optical fiber,” Phys. Rev. Lett. 86, 4267–4270 (2001).
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J. I. Korsbakken, K. B. Whaley, J. Dubois, and J. I. Cirac, “Measurement-based measure of the size of macroscopic quantum superpositions,” Phys. Rev. A 75, 042106 (2007).
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H. Jeong, M. Kang, and H. Kwon, “Characterizations and quantifications of macroscopic quantumness and its implementations using optical fields,” Opt. Commun. 337, 12–21 (2015).

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A. Laghaout, J. S. Neergaard-Nielsen, and U. L. Andersen, “Assessments of macroscopicity for quantum optical states,” Opt. Commun. 337, 96–101 (2015).

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W. P. Bowen, R. Schnabel, P. K. Lam, and T. C. Ralph, “Experimental investigation of criteria for continuous variable entanglement,” Phys. Rev. Lett. 90, 043601 (2003).
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C. Silberhorn, P. K. Lam, O. Weiß, F. König, N. Korolkova, and G. Leuchs, “Generation of continuous variable Einstein-Podolsky-Rosen entanglement via the Kerr nonlinearity in an optical fiber,” Phys. Rev. Lett. 86, 4267–4270 (2001).
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O. Morin, K. Huang, J. Liu, H. Le Jeannic, C. Fabre, and J. Laurat, “Remote creation of hybrid entanglement between particle-like and wave-like optical qubits,” Nat. Photonics 8, 570–574 (2014).
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O. Morin, K. Huang, J. Liu, H. Le Jeannic, C. Fabre, and J. Laurat, “Remote creation of hybrid entanglement between particle-like and wave-like optical qubits,” Nat. Photonics 8, 570–574 (2014).
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C.-W. Lee and H. Jeong, “Quantification of macroscopic quantum superpositions within phase space,” Phys. Rev. Lett. 106, 220401 (2011).
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H. Jeong, A. Zavatta, M. Kang, S.-W. Lee, L. S. Costanzo, S. Grandi, T. C. Ralph, and M. Bellini, “Generation of hybrid entanglement of light,” Nat. Photonics 8, 564–569 (2014).
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Y. Zhang, H. Wang, X. Li, J. Jing, C. Xie, and K. Peng, “Experimental generation of bright two-mode quadrature squeezed light from a narrow-band nondegenerate optical parametric amplifier,” Phys. Rev. A 62, 023813 (2000).
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O. Morin, K. Huang, J. Liu, H. Le Jeannic, C. Fabre, and J. Laurat, “Remote creation of hybrid entanglement between particle-like and wave-like optical qubits,” Nat. Photonics 8, 570–574 (2014).
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F. Marquardt, B. Abel, and J. Von Delft, “Measuring the size of a quantum superposition of many-body states,” Phys. Rev. A 78, 012109 (2008).
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N. Bruno, A. Martin, P. Sekatski, N. Sangouard, R. T. Thew, and N. Gisin, “Displacement of entanglement back and forth between the micro and macro domains,” Nat. Phys. 9, 545–548 (2013).
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A. S. Villar, L. S. Cruz, K. N. Cassemiro, M. Martinelli, and P. Nussenzveig, “Generation of bright two-color continuous variable entanglement,” Phys. Rev. Lett. 95, 243603 (2005).
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A. Shimizu and T. Miyadera, “Stability of quantum states of finite macroscopic systems against classical noises, perturbations from environments, and local measurements,” Phys. Rev. Lett. 89, 270403 (2002).
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J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
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O. Morin, K. Huang, J. Liu, H. Le Jeannic, C. Fabre, and J. Laurat, “Remote creation of hybrid entanglement between particle-like and wave-like optical qubits,” Nat. Photonics 8, 570–574 (2014).
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A. Laghaout, J. S. Neergaard-Nielsen, and U. L. Andersen, “Assessments of macroscopicity for quantum optical states,” Opt. Commun. 337, 96–101 (2015).

J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
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J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
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A. S. Villar, L. S. Cruz, K. N. Cassemiro, M. Martinelli, and P. Nussenzveig, “Generation of bright two-color continuous variable entanglement,” Phys. Rev. Lett. 95, 243603 (2005).
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A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
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J. Jing, S. Feng, R. Bloomer, and O. Pfister, “Experimental continuous-variable entanglement from a phase-difference-locked optical parametric oscillator,” Phys. Rev. A 74, 041804 (2006).
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J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
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A. I. Lvovsky, R. Ghobadi, A. Chandra, A. S. Prasad, and C. Simon, “Observation of micro-macro entanglement of light,” Nat. Phys. 9, 541–544 (2013).
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T. Wang, R. Ghobadi, S. Raeisi, and C. Simon, “Precision requirements for observing macroscopic quantum effects,” Phys. Rev. A 88, 062114 (2013).
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S. Raeisi, P. Sekatski, and C. Simon, “Coarse graining makes it hard to see micro-macro entanglement,” Phys. Rev. Lett. 107, 250401 (2011).
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H. Jeong, A. Zavatta, M. Kang, S.-W. Lee, L. S. Costanzo, S. Grandi, T. C. Ralph, and M. Bellini, “Generation of hybrid entanglement of light,” Nat. Photonics 8, 564–569 (2014).
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P. Sekatski, N. Sangouard, and N. Gisin, “Size of quantum superpositions as measured with classical detectors,” Phys. Rev. A 89, 012116 (2014).
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P. Sekatski, N. Gisin, and N. Sangouard, “How difficult is it to prove the quantumness of macroscropic states?” Phys. Rev. Lett. 113, 090403 (2014).
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N. Bruno, A. Martin, P. Sekatski, N. Sangouard, R. T. Thew, and N. Gisin, “Displacement of entanglement back and forth between the micro and macro domains,” Nat. Phys. 9, 545–548 (2013).
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P. Sekatski, N. Sangouard, M. Stobińska, F. Bussières, M. Afzelius, and N. Gisin, “Proposal for exploring macroscopic entanglement with a single photon and coherent states,” Phys. Rev. A 86, 060301 (2012).
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P. Sekatski, N. Sangouard, and N. Gisin, “Size of quantum superpositions as measured with classical detectors,” Phys. Rev. A 89, 012116 (2014).
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P. Sekatski, N. Gisin, and N. Sangouard, “How difficult is it to prove the quantumness of macroscropic states?” Phys. Rev. Lett. 113, 090403 (2014).
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N. Bruno, A. Martin, P. Sekatski, N. Sangouard, R. T. Thew, and N. Gisin, “Displacement of entanglement back and forth between the micro and macro domains,” Nat. Phys. 9, 545–548 (2013).
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P. Sekatski, N. Sangouard, M. Stobińska, F. Bussières, M. Afzelius, and N. Gisin, “Proposal for exploring macroscopic entanglement with a single photon and coherent states,” Phys. Rev. A 86, 060301 (2012).
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S. Raeisi, P. Sekatski, and C. Simon, “Coarse graining makes it hard to see micro-macro entanglement,” Phys. Rev. Lett. 107, 250401 (2011).
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A. Shimizu and T. Morimae, “Detection of macroscopic entanglement by correlation of local observables,” Phys. Rev. Lett. 95, 090401 (2005).
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A. Shimizu and T. Miyadera, “Stability of quantum states of finite macroscopic systems against classical noises, perturbations from environments, and local measurements,” Phys. Rev. Lett. 89, 270403 (2002).
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C. Silberhorn, P. K. Lam, O. Weiß, F. König, N. Korolkova, and G. Leuchs, “Generation of continuous variable Einstein-Podolsky-Rosen entanglement via the Kerr nonlinearity in an optical fiber,” Phys. Rev. Lett. 86, 4267–4270 (2001).
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T. Wang, R. Ghobadi, S. Raeisi, and C. Simon, “Precision requirements for observing macroscopic quantum effects,” Phys. Rev. A 88, 062114 (2013).
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A. I. Lvovsky, R. Ghobadi, A. Chandra, A. S. Prasad, and C. Simon, “Observation of micro-macro entanglement of light,” Nat. Phys. 9, 541–544 (2013).
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S. Raeisi, P. Sekatski, and C. Simon, “Coarse graining makes it hard to see micro-macro entanglement,” Phys. Rev. Lett. 107, 250401 (2011).
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N. Bruno, A. Martin, P. Sekatski, N. Sangouard, R. T. Thew, and N. Gisin, “Displacement of entanglement back and forth between the micro and macro domains,” Nat. Phys. 9, 545–548 (2013).
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A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
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T. Farrow and V. Vedral, “Classification of macroscopic quantum effects,” Opt. Commun. 337, 22–26 (2015).

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T. Wang, R. Ghobadi, S. Raeisi, and C. Simon, “Precision requirements for observing macroscopic quantum effects,” Phys. Rev. A 88, 062114 (2013).
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C. Silberhorn, P. K. Lam, O. Weiß, F. König, N. Korolkova, and G. Leuchs, “Generation of continuous variable Einstein-Podolsky-Rosen entanglement via the Kerr nonlinearity in an optical fiber,” Phys. Rev. Lett. 86, 4267–4270 (2001).
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H. Jeong, A. Zavatta, M. Kang, S.-W. Lee, L. S. Costanzo, S. Grandi, T. C. Ralph, and M. Bellini, “Generation of hybrid entanglement of light,” Nat. Photonics 8, 564–569 (2014).
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Y. Zhang, H. Wang, X. Li, J. Jing, C. Xie, and K. Peng, “Experimental generation of bright two-mode quadrature squeezed light from a narrow-band nondegenerate optical parametric amplifier,” Phys. Rev. A 62, 023813 (2000).
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J. Opt. B (1)

G. Björk and P. G. L. Mana, “A size criterion for macroscopic superposition states,” J. Opt. B 6, 429–436 (2004).
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J. Phys. Condens. Matter (1)

A. J. Leggett, “Testing the limits of quantum mechanics: motivation, state of play, prospects,” J. Phys. Condens. Matter 14, R415–R451 (2002).
[Crossref]

Nat. Photonics (2)

O. Morin, K. Huang, J. Liu, H. Le Jeannic, C. Fabre, and J. Laurat, “Remote creation of hybrid entanglement between particle-like and wave-like optical qubits,” Nat. Photonics 8, 570–574 (2014).
[Crossref]

H. Jeong, A. Zavatta, M. Kang, S.-W. Lee, L. S. Costanzo, S. Grandi, T. C. Ralph, and M. Bellini, “Generation of hybrid entanglement of light,” Nat. Photonics 8, 564–569 (2014).
[Crossref]

Nat. Phys. (2)

N. Bruno, A. Martin, P. Sekatski, N. Sangouard, R. T. Thew, and N. Gisin, “Displacement of entanglement back and forth between the micro and macro domains,” Nat. Phys. 9, 545–548 (2013).
[Crossref]

A. I. Lvovsky, R. Ghobadi, A. Chandra, A. S. Prasad, and C. Simon, “Observation of micro-macro entanglement of light,” Nat. Phys. 9, 541–544 (2013).
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Naturwissenschaften (1)

E. Schrödinger, “Die gegenwärtige Situation in der Quantenmechanik,” Naturwissenschaften 23, 807–812 (1935).
[Crossref]

New J. Phys. (2)

F. Fröwis and W. Dür, “Measures of macroscopicity for quantum spin systems,” New J. Phys. 14, 093039 (2012).
[Crossref]

F. Fröwis, M. Van den Nest, and W. Dür, “Certifiability criterion for large-scale quantum systems,” New J. Phys. 15, 113011 (2013).
[Crossref]

Opt. Commun. (4)

A. Laghaout, J. S. Neergaard-Nielsen, and U. L. Andersen, “Assessments of macroscopicity for quantum optical states,” Opt. Commun. 337, 96–101 (2015).

H. Jeong, M. Kang, and H. Kwon, “Characterizations and quantifications of macroscopic quantumness and its implementations using optical fields,” Opt. Commun. 337, 12–21 (2015).

T. Farrow and V. Vedral, “Classification of macroscopic quantum effects,” Opt. Commun. 337, 22–26 (2015).

F. Fröwis, N. Sangouard, and N. Gisin, “Linking measures for macroscopic quantum states via photon–spin mapping,” Opt. Commun. 337, 2–11 (2015).

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. A (11)

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Other (3)

While superpositions of coherent states with opposite phases have a negative Wigner representation, here we show that the size of two-mode vacuum squeezed states is basically the same. This shows that states with a positive Wigner representation can be macroscopically quantum.

To illustrate this, consider a state ρϵ in Eq. (6) with f1(x,x¯)=f2(x,x¯)=1 for |x−x¯|<ϵ and zero otherwise. The peculiarity of this state is that f satisfies 〈f′〉=〈f′′〉=0 in such a way that the violation of the Duan–Simon criterion by ρϵ is independent of ϵ unless it is strictly equal to zero. Therefore, ρϵ with ϵ approaching zero is an example of a state that can give an arbitrarily high violation of the Duan criteria, while being arbitrarily close to a separable state.

S. Yu, “Quantum fisher information as the convex roof of variance,” arXiv:1302.5311 (2013).

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Figures (1)

Fig. 1.
Fig. 1.

Bounds on the effective size N eff (blue squares) of two-mode squeezed states obtained from experimental data reported in [2129] using the inequality in Eq. (10). The red triangles indicate the minimal photon number N necessary for a cat state | α + | α to have the same effective size according to Eq. (9). For example, the state reported in [21] has a size N eff 1.2 for which one needs at least a cat state with N 0.2 for the same size.

Equations (36)

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| ψ tms = ( 1 tanh 2 g ) 1 2 e tanh g a 1 a 2 | 00 .
V ψ tms ( X ¯ 1 φ X ¯ 2 ϕ ) = cosh 2 g sinh 2 g cos ( φ + ϕ ) .
V sep ( | a | X ¯ 1 ϕ + 1 a X ¯ 2 Φ ) + V sep ( | a | X ¯ 1 ϕ 1 a X ¯ 2 Φ ) > a 2 [ X ¯ 1 ϕ , X ¯ 1 ϕ ] + 1 a 2 [ X ¯ 2 Φ , X ¯ 2 Φ ] 2 for ϕ ϕ = Φ Φ = π 2 ,
V ψ tms ( X ¯ 1 0 X ¯ 2 0 ) + V ψ tms ( X ¯ 1 π / 2 + X ¯ 2 π / 2 ) = 2 e 2 g .
P σ guess = 1 2 + 1 π arctan ( sinh 2 g 1 + 2 σ 2 cosh 2 g ) .
σ max = 1 + N ( 1 2 + N ) cotan 2 ( 1 2 P σ guess ) 2 + 2 N .
σ max = | α | 2 ( erf 1 ( P σ guess ) ) 2 1 2 .
ρ = p ( x 1 , x 2 ) p * ( x ¯ 1 , x ¯ 2 ) f ( x 1 , x ¯ 1 , x 2 , x ¯ 2 ) | x 1 , x 2 x ¯ 1 , x ¯ 2 | d x 1 d x 2 d x ¯ 1 d x ¯ 2 ,
V ( X ¯ 1 π 2 + X ¯ 2 π 2 ) = V ( X ¯ 1 π 2 + X ¯ 2 π 2 ) | ideal ( x 1 x ¯ 1 + x 2 x ¯ 2 ) 2 f ,
( x 1 x ¯ 1 + x 2 x ¯ 2 ) 2 f V ( X ¯ 1 π 2 + X ¯ 2 π 2 ) .
x C = 1 V ψ tms ( X ¯ 1 π 2 + X ¯ 2 π 2 ) .
N eff ( ρ ) = 1 2 n max θ F ρ ( X θ ) ,
N eff ( ρ ) 1 V ρ ( X ¯ 1 π / 2 + X ¯ 2 π / 2 ) .
ρ = F ( x 1 , x 2 , x ¯ 1 , x ¯ 2 ) | x 1 , x 2 x ¯ 1 , x ¯ 2 | d x⃗ .
F ( x 1 , x 2 , x ¯ 1 , x ¯ 2 ) = p ( x 1 , x 2 ) p * ( x ¯ 1 , x ¯ 2 ) f ( x 1 , x 2 , x ¯ 1 , x ¯ 2 ) ,
p 1 = F ( x 1 , x 2 , x ¯ 1 , x ¯ 2 ) p e i p ( x 1 x ¯ 1 ) 2 π δ ( x 2 x ¯ 2 ) d p d x⃗ , p 1 2 = F ( x 1 , x 2 , x ¯ 1 , x ¯ 2 ) p 2 e i p ( x 1 x ¯ 1 ) 2 π δ ( x 2 x ¯ 2 ) d p d x⃗ , p 1 p 2 = F ( x 1 , x 2 , x ¯ 1 , x ¯ 2 ) p 1 p 2 e i p 1 ( x 1 x ¯ 1 ) 2 π × e i p 2 ( x 2 x ¯ 2 ) 2 π d p 1 d p 2 d x⃗ .
p 1 = ( i x 1 x ¯ 1 ) F ( x 1 , x 2 , x ¯ 1 , x ¯ 2 ) | x 1 = x ¯ 1 d x 1 d x 2 , p 1 2 = ( i x 1 x ¯ 1 ) 2 F ( x 1 , x 2 , x ¯ 1 , x ¯ 2 ) | x 1 = x ¯ 1 d x 1 d x 2 , p 1 p 2 = ( i x 2 x ¯ 2 ) ( i x 1 x ¯ 1 ) , F ( x 1 , x 2 , x ¯ 1 , x ¯ 2 ) | x 1 = x ¯ 1 , x 2 = x ¯ 2 d x 1 d x 2 .
p 1 = p 1 f 1 , p 1 2 = p 1 2 f 1 x 1 x ¯ 1 2 f , p 1 2 = p 2 2 f 1 x 2 x ¯ 2 2 f , p 1 p 2 = p 1 p 2 f 1 x 1 x ¯ 1 x 2 x ¯ 2 f ,
D [ f ] = | p ( x 1 , x 2 ) | 2 D [ f ] ( x 1 , x 2 ) d x 1 d x 2 .
V ( p 1 + p 2 ) = V ( p 1 + p 2 ) f 1 ( x 1 x ¯ 1 + x 2 x ¯ 2 ) 2 f .
( x 1 x ¯ 1 + x 2 x ¯ 2 ) 2 f V ( p 1 + p 2 ) .
H A = i χ ( a b a b ) ,
O ( η , μ , κ ) = e κ e i ( η a + μ b ) e i ( η * a + μ * b )
tr loss U d t O ( η , μ , κ ) U d t = O ( η + ( μ * χ η λ ) d t , μ + ( η * χ μ λ ) d t , κ ( μ * η * + μ η ) χ d t ) ,
{ η ˙ ( t ) = χ μ * ( t ) λ η ( t ) μ ˙ ( t ) = χ η * ( t ) λ μ ( t ) κ ( t ) = χ 0 t ( η * ( s ) μ * ( s ) + η ( s ) μ ( s ) ) d s + κ ( 0 ) .
( η ( t ) μ * ( t ) ) = exp ( ( λ χ χ λ ) t ) ( η 0 μ 0 * ) , ( μ ( t ) η * ( t ) ) = exp ( ( λ χ χ λ ) t ) ( μ 0 η 0 * ) , κ ( t ) = ( η 0 μ 0 * ) · ( 0 t e ( λ χ χ λ ) 2 s d s ) · ( μ 0 η 0 * ) + κ 0 .
| x θ a x θ a | = 1 2 π d ζ e i ζ ( x ^ θ a x ) d ζ
x ^ θ a = 1 2 ( a e i θ + a e i θ ) ,
| x θ a , y ξ b x θ a , y ξ b | = d ζ d γ ( 2 π ) 2 e i ζ x i γ y × e ζ 2 + γ 2 4 e i ( ζ e i θ 2 a + γ e i ξ 2 b ) e i ( ζ e i θ 2 a + γ e i ξ 2 b ) O 0 ,
p ( x θ a , y ξ b ) = d ζ d γ ( 2 π ) 2 e i ζ x i γ y O t .
p ( x θ a , y ξ b ) = d ζ d γ ( 2 π ) 2 e i ( ζ γ ) · ( x Z ( α , β ) y Γ ( α , γ ) ) e ζ 2 + γ 2 4 exp [ 1 8 ( ζ γ ) T · ( ( 1 + e 2 t ( λ + χ ) λ + χ 1 + e 2 t ( χ λ ) λ χ ) e i ( θ + ξ ) ( 1 + e 2 t ( χ λ ) λ χ 1 + e 2 t ( λ + χ ) λ + χ ) e i ( θ + ξ ) ( 1 + e 2 t ( χ λ ) λ χ 1 + e 2 t ( λ + χ ) λ + χ ) ( 1 + e 2 t ( λ + χ ) λ + χ 1 + e 2 t ( χ λ ) λ χ ) ) · ( ζ γ ) ] .
p ( x θ a , y ξ b ) = r r + 4 π e 1 4 ( x Z ( α , β ) y Γ ( α , β ) ) · M · ( x Z ( α , β ) y Γ ( α , β ) ) ,
r = ( λ + χ e 2 t ( λ + χ ) 4 ( λ + χ ) ) 1 ,
r + = ( λ χ e 2 t ( χ λ ) 4 ( λ χ ) ) 1 .
Δ + λ 2 ( λ χ ) Δ λ 2 ( λ + χ ) ,
Δ + χ 2 ( χ λ ) e 2 t ( χ λ ) Δ λ 2 ( λ + χ ) ,

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