Abstract

We discuss several methods to produce superpositions of optical coherent states (also known as “cat states”). Cat states have remarkable properties that could allow them to be powerful tools for quantum information processing and metrology. A number of proposals for how one can produce cat states have appeared in the literature in recent years. We describe these proposals and present a new simulation and analysis of them incorporating practical issues such as photon loss, detector inefficiency, and limited strength of nonlinear interactions. We also examine how each would perform in a realistic experiment.

© 2008 Optical Society of America

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2008 (1)

H. Vahlbruch, M. Mehmet, N. Lastzka, B. Hage, S. Chelkowski, A. Franzen, S. Gossler, K. Danzmann, and R. Schnabel, “Observation of squeezed light with 10 dB quantum noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[CrossRef] [PubMed]

2007 (3)

Y. Takeno, M. Yukawa, H. Yonezawa, and A. Furusawa, “Observation of −9 dB quadrature squeezing with improvement of phase stability in homodyne measurement,” Opt. Express 15, 4321-4327 (2007).
[CrossRef] [PubMed]

A. I. Lvovsky, W. Wasilewski, and K. Banaszek, “Decomposing a pulsed optical parametric amplifier into independent squeezers,” J. Mod. Opt. 54, 721-733 (2007).
[CrossRef]

P. P. Rohde, W. Mauerer, and C. Silberhorn, “Spectral structure and decompositions of optical states, and their applications,” New J. Phys. 9, 91 (2007).
[CrossRef]

2006 (6)

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A 73, 063819 (2006).
[CrossRef]

E. Waks, E. Diamanti, and Y. Yamamoto, “Generation of photon number states,” New J. Phys. 8, 4 (2006).
[CrossRef]

R. D. Somma, J. Chiaverini, and D. J. Berkeland, “Lower bounds for the fidelity of entangled state preparation,” Phys. Rev. A 74, 052302 (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] [PubMed]

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

S. L. Braunstein, “Squeezing as an irreducible resource,” Phys. Rev. A 71, 055801 (2006).
[CrossRef]

2005 (10)

A. B. U'Ren, C. Silberhorn, R. Erdmann, K. Banaszek, W. P. Grice, I. A. Walmsley, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146-161 (2005).

M. G. Raymer, J. Noh, K. Banaszek, and I. A. Walmsley, “Pure-state single-photon wave-packet generation by parametric down conversion in a distributed microcavity,” Phys. Rev. A 72, 023825 (2005).
[CrossRef]

H. Jeong, A. P. Lund, and T. C. Ralph, “Production of superpositions of coherent states in traveling optical fields with inefficient photon detection,” Phys. Rev. A 72, 013801 (2005).
[CrossRef]

H. Jeong, “Using weak nonlinearity under decoherence for macroscopic entanglement generation and quantum computation,” Phys. Rev. A 72, 034305 (2005).
[CrossRef]

Y. A. Vlasov, M. O'Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

E. Dulkeith, S. J. McNab, and Y. A. Vlasov, “Mapping the optical properties of slap-type two-dimensional photonic crystal waveguides,” Phys. Rev. B 72, 115102 (2005).
[CrossRef]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633-673 (2005).
[CrossRef]

D. Rosenberg, A. E. Lita, A. J. Miller, and S. W. Nam, “Noise-free, high-efficiency, photon-number-resolving detectors,” Phys. Rev. A 71, 061803 (2005).
[CrossRef]

M. S. Kim, E. Park, P. L. Knight, and H. Jeong, “Nonclassicality of a photon-subtracted Gaussian field,” Phys. Rev. A 71, 043805 (2005).
[CrossRef]

S. Olivares and M. G. A. Paris, “Squeezed Fock state by inconclusive photon subtraction,” J. Opt. B: Quantum Semiclassical Opt. 7, S616-S621 (2005).
[CrossRef]

2004 (8)

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
[CrossRef] [PubMed]

S. Glancy, H. M. Vasconcelos, and T. C. Ralph, “Transmission of optical coherent state qubits,” Phys. Rev. A 70, 022317 (2004).
[CrossRef]

A. P. Lund, H. Jeong, T. C. Ralph, and M. S. Kim, “Conditional production of superpositions of coherent states with inefficient photon detection,” Phys. Rev. A 70, 020101 (2004).
[CrossRef]

H. Jeong, M. S. Kim, T. C. Ralph, and B. S. Ham, “Generation of macroscopic superposition states with small nonlinearity,” Phys. Rev. A 70, 061801 (2004).
[CrossRef]

E. Waks, E. Diamanti, B. C. Sanders, S. D. Bartlett, and Y. Yamamoto, “Direct observation of non-classical photon statistics in parametric downconversion,” Phys. Rev. Lett. 92, 113602 (2004).
[CrossRef] [PubMed]

M. Soljacic and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211-219 (2004).
[CrossRef] [PubMed]

P. Grangier, B. Sanders, and J. Vuckovic, “Focus on single photons on demand,” New J. Phys. 6, 85-100 (2004).
[CrossRef]

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Pulsed homodyne measurements of femtosecond squeezed pulses generated by single-pass parametric deamplification,” Opt. Lett. 29, 1267-1269 (2004).
[CrossRef] [PubMed]

2003 (3)

E. Waks, K. Inoue, W. D. Oliver, E. Diamanti, and Y. Yamamoto, “High efficiency photon number detection for quantum information processing,” IEEE J. Quantum Electron. 9, 1502-1511 (2003).
[CrossRef]

T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy, “Quantum computation with optical coherent states,” Phys. Rev. A 68, 042319 (2003).
[CrossRef]

A. Auffeves, P. Maioli, T. Meunier, S. Gleyzes, G. Nogues, M. Brune, J.-M. Raimond, and S. Haroche, “Entanglement of a mesoscopic field with an atom induced by photon graininess in a cavity,” Phys. Rev. Lett. 91, 230405 (2003).
[CrossRef] [PubMed]

2002 (1)

T. C. Ralph, “Coherent superposition states as quantum rulers,” Phys. Rev. A 65, 042313 (2002).
[CrossRef]

2001 (2)

H. Jeong, M. S. Kim, and J. Lee, “Quantum information processing for a coherent superposition state via a mixed entangled coherent channel,” Phys. Rev. A 64, 052308 (2001).
[CrossRef]

W. P. Grice, A. B. U'Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A 64, 063815 (2001).
[CrossRef]

2000 (1)

M. D. Lukin and A. Imamoglu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419-1422 (2000).
[CrossRef] [PubMed]

1999 (4)

P. T. Cochrane, G. J. Milburn, and W. J. Munro, “Macroscopically distinct quantum-superposition states as a bosonic code for amplitude damping,” Phys. Rev. A 59, 2631-2634 (1999).
[CrossRef]

C. C. Gerry, “Generation of optical macroscopic quantum superposition states via state reduction with a Mach-Zender interferometer containing a Kerr medium,” Phys. Rev. A 59, 4095-4098 (1999).
[CrossRef]

S. Takeuchi, J. Kim, Y. Yamamoto, and H. Hogue, “Development of a high-quantum-efficiency single-photon counting system,” Appl. Phys. Lett. 74, 1063-1065 (1999).
[CrossRef]

J. Kim, S. Takeuchi, and Y. Yamamoto, “Multiphoton detection using visible light photon counter,” Appl. Phys. Lett. 74, 902-904 (1999).
[CrossRef]

1997 (3)

M. Dakna, T. Anhut, T. Opatrný, L. Knöll, and D.-G. Welsch, “Generating Schrödinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184-3194 (1997).
[CrossRef]

D. Vitali and P. Tombesi, “Generation and detection of linear superpositions of classically distinguishable states of a radiation mode,” Int. J. Mod. Phys. B 11, 2119-2140 (1997).
[CrossRef]

D. Vitali, P. Tombesi, and P. Grangier, “Conditional Schrödinger cats generation and detection by quantum non-demolition measurements,” Appl. Phys. B 64, 249-257 (1997).
[CrossRef]

1996 (3)

H. Schmidt and A. Imamoglu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. 21, 1936-1938 (1996).
[CrossRef] [PubMed]

P. Tombesi and D. Vitali, “All-optical model for the generation and the detection of macroscopic quantum coherence,” Phys. Rev. Lett. 77, 411-415 (1996).
[CrossRef] [PubMed]

M. Brune, E. Hagley, J. Dreyer, X. Maître, A. Maali, C. Wunderlich, J. M. Raimond, and S. Haroche, “Observing the progressive decoherence of the meter in a quantum measurement,” Phys. Rev. Lett. 77, 4887-4890 (1996).
[CrossRef] [PubMed]

1994 (1)

L. Krippner, W. J. Munro, and M. D. Reid, “Transient macroscopic quantum superposition states in degenerate parametric oscillation: calculations in the large-quantum-noise limit using the positive p representation,” Phys. Rev. A 50, 4330-4338 (1994).
[CrossRef] [PubMed]

1993 (1)

K. S. Lee, M. S. Kim, S. D. Lee, and V. Buzek, “Squeezing properties of multicomponent superposition states of light,” J. Korean Phys. Soc. 26, 197-204 (1993).

1992 (2)

M. D. Reid and B. Yurke, “Effect of bistability and superpositions on quantum statistics in degenerate parametric oscillation,” Phys. Rev. A 46, 4131-4137 (1992).
[CrossRef] [PubMed]

E. S. Polzik, J. Carri, and H. J. Kimble, “Spectroscopy with squeezed light,” Phys. Rev. Lett. 68, 3020-3023 (1992).
[CrossRef] [PubMed]

1991 (2)

A. LaPorta and R. E. Slusher, “Squeezing limits at high parametric gains,” Phys. Rev. A 44, 2013-2022 (1991).
[CrossRef]

T. Gantsog and R. Tanas, “Discrete superpositions of coherent states and phase properties of elliptically polarized light propagating in a Kerr medium,” Quantum Opt. 3, 33-48 (1991).
[CrossRef]

1990 (2)

B. Yurke, W. Schleich, and D. F. Walls, “Quantum superpositions generated by quantum nondemolition measurements,” Phys. Rev. A 42, 1703-1711 (1990).
[CrossRef] [PubMed]

S. Song, C. M. Caves, and B. Yurke, “Generation of superpositions of classically distinguishable quantum states from optical back-action evasion,” Phys. Rev. A 41, 5261-5264 (1990).
[CrossRef] [PubMed]

1988 (1)

M. Wolinsky and H. J. Carmichael, “Quantum noise in the parametric oscillator: from squeezed states to coherent-state superpositions,” Phys. Rev. Lett. 60, 1836-1839 (1988).
[CrossRef] [PubMed]

1987 (3)

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1985 (1)

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H. Vahlbruch, M. Mehmet, N. Lastzka, B. Hage, S. Chelkowski, A. Franzen, S. Gossler, K. Danzmann, and R. Schnabel, “Observation of squeezed light with 10 dB quantum noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
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Appl. Phys. Lett. (2)

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J. Opt. Soc. Am. B (2)

Laser Phys. (1)

A. B. U'Ren, C. Silberhorn, R. Erdmann, K. Banaszek, W. P. Grice, I. A. Walmsley, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146-161 (2005).

Nat. Mater. (1)

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Nature (1)

Y. A. Vlasov, M. O'Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65-69 (2005).
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New J. Phys. (3)

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E. Waks, E. Diamanti, and Y. Yamamoto, “Generation of photon number states,” New J. Phys. 8, 4 (2006).
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Opt. Express (1)

Opt. Lett. (2)

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Although Eq. appears in , they discussed neither the relationship between fidelity and cat amplitude nor the cat's phase's dependence on the homodyne measurement result.

I. Fushman and J. Vuckovic, “Analysis of a quantum nondemolition measurement scheme based on Kerr nonlinearity in photonic crystal waveguides,” arXiv.org e-Print archive, quant-ph/0603150v1, 16 March 2006, http://arxiv.org/abs/quant-ph/0603150v1.

Figure appears in , but Figs. are new in this paper.

P. P. Rhode and A. P. Lund, “Practical effects in cat state breeding,” arXiv.org e-Print archive, quant-ph/0702064vl, 7 February 2007, http://arxiv.org/abs/quant-ph/0702064vl.

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Figures (27)

Fig. 1
Fig. 1

Plot of F + ( α , η ) , the fidelity of a cat state Ψ + ( α ) that has suffered from some decoherence by passing through a medium of transmissivity η. The plot includes curves for even cats with α = 1 (solid curve), α = 2 (small dashes), α = 4 (medium dashes), α = 10 (large dashes).

Fig. 2
Fig. 2

(a) Fidelity of the state produced using the Kerr effect in the presence of loss when our goal is to produce a cat of amplitude β = 1 (solid curve), 2 (long dashes), 3 (medium dashes), and 4 (short dashes). The amplitude of the input coherent state has been optimized to give the maximum fidelity. (b) Same as (a), zoomed in on small γ χ .

Fig. 3
Fig. 3

Diagram of the scheme to make cats using the backaction evasion measurement. S ̂ 12 ( r ) is a nondegenerate down conversion crystal, which creates squeezed light in modes 1 and 2 at frequency ω. The dashed line is a beam splitter with transmissivity T. The triangle is a photon counter, and S ̂ 1 ( s ) is a degenerate downconversion crystal that squeezes mode 1. Not shown are a light beam or beams used to pump the downconversion processes and mirrors to redirect the light.

Fig. 4
Fig. 4

Diagram of the improved scheme to make cats using the backaction evasion measurement. It is similar to that in Fig. 3 except that S ̂ 1 ( r ) has been moved to the beginning of the network. Not shown are a light beam or beams used to pump the down conversion processes and mirrors to redirect the light.

Fig. 5
Fig. 5

Diagram of our simplified “backaction evasion” scheme to make cats. Modes 1 and 2 begin in the vacuum state, and are transformed to squeezed states by S ̂ 1 ( r ) and S ̂ 2 ( s ) . The two squeezed states meet at the beam splitter with transmissivity T. We then count the number of photons in mode 2, obtaining result m. Mode 1 then contains ψ m , which should be similar to a cat state. Not shown are a light beam or beams used to pump the downconversion processes and mirrors to redirect the light.

Fig. 6
Fig. 6

Schematic for the generation of cat states by means of a conditional photon number measurement on a beam splitter. The downconversion S ̂ 1 ( r ) creates the single mode squeezed state in mode 1. It is input into one port of a variable transmissivity T beam splitter with mode 2 containing a vacuum state. A definite measurement of m photons on one output port of the beam splitter prepares the state ψ m , which is a good approximation to a cat state.

Fig. 7
Fig. 7

Plot of the fidelity of the state ψ m with Ψ + ( 2 ) versus λ T for m = 0 (solid curve), m = 2 (small dashes), m = 4 (medium dashes), m = 6 (large dashes). The upper horizontal axis shows the number of decibels of squeezing required when T = 1 is used.

Fig. 8
Fig. 8

Plot of the fidelity of the state Ψ m with Ψ + ( α ) for α = 1 (diamonds), 2 (stars), and 3 (squares). Notice that only even m are represented, because odd m events give a fidelity of zero. For each point we have numerically optimized λ T to give the maximum fidelity.

Fig. 9
Fig. 9

Plot of the product λ T that maximizes the fidelity shown in Fig. 8.

Fig. 10
Fig. 10

Plots of the probability to detect m photons after sending a squeezed vacuum state with squeezing parameter λ through a beam splitter with transmissivity T. Each plot shows the probability obtained using the product λ T that maximizes the fidelity shown in Fig. 9. Plot (a) shows the case in which λ = 0.99 and the probability is high. Plot (b) shows the case in which λ = T . Plot (c) shows the case in which squeezing is decreased and T = 0.99 .

Fig. 11
Fig. 11

Probability to detect m = 2 (solid curve), 4 (small dashes), and 6 (long dashes) photons as a function of squeezing (shown with both λ and in decibels). In each case T is adjusted so that λ T gives the best fidelity to produce a cat state with α = 2 .

Fig. 12
Fig. 12

Fidelity to make a cat state with α = 2 as a function of λ when using detectors with efficiency η = 0.9 . The solid curve shows the m = 2 case, the small dashed curve shows the m = 4 case, and the large dashed curve shows the m = 6 case. For each combination of λ and m, the beam splitter transitivity has been adjusted to give the maximum fidelity. These maximum fidelities are equal to the maximum fidelities achievable perfect detectors are used.

Fig. 13
Fig. 13

Probability to detect m = 2 (solid curve), m = 4 (small dashes), and m = 6 (long dashes) as a function of λ when detectors with efficiency η = 0.9 are used. For each combination of λ and m, the beam splitter transitivity has been adjusted to give the maximum fidelity, as shown in Fig. 12.

Fig. 14
Fig. 14

Beam splitter transitivity T that gives the largest maximum fidelity in Fig. 12 as a function of λ. The solid curve shows the m = 2 case, the small dashed curve shows the m = 4 case, and the large dashed curve shows the m = 6 case.

Fig. 15
Fig. 15

Model of photon subtraction scheme including impurity in the initial squeezed state, inefficiency of the photon counter, and dark counts. The beam splitters with transmissivities ν and η model loss from the squeezed state and inefficiency in the photon counter. Modes 2 and 4 are lost to the environment, mode 5 contains “dark counts,” and the output of mode 1 contains ρ d ( m ) , which should be similar to a cat state.

Fig. 16
Fig. 16

Fidelity between an odd cat state of amplitude α and a squeezed single photon as a function of α.

Fig. 17
Fig. 17

Squeezing r required to maximize the fidelity between a squeezed single photon and an odd cat of amplitude α.

Fig. 18
Fig. 18

Fidelity of the odd cat state with α = 1 2 with squeezed single photon state made from a photon source whose inefficiency is p.

Fig. 19
Fig. 19

Schematic of the nondeterministic amplification process to grow cat state amplitude. Two small cat states at modes 1 and 2 and a coherent state in mode 3 are manipulated with linear optics. A larger amplitude cat is produced in the output of mode 2 when photons are detected in both modes 1 and 3.

Fig. 20
Fig. 20

Success probabilities for a single attempt at the process to make a cat from two kittens, depicted in Fig. 19. We show probabilities for the input fields of two identical small odd cat states (solid curve), two identical small even cats (dashed curve), and even and odd small cats (dotted curve).

Fig. 21
Fig. 21

Fidelity of a cat with α = 2 produced from kittens with α i = 2 as a function of the average number of dark counts in the detectors. The detectors have efficiencies η = 0.88 on the solid line and η = 0.8 on the dashed line.

Fig. 22
Fig. 22

Fidelity of a generated cat with α = 2 produced from kittens with α i = 2 as a function of the detectors’ efficiency η. The detectors have mean dark counts d = 10 4 on the solid curve and d = 10 2 on the dashed curve.

Fig. 23
Fig. 23

Schematic of the process to generate a cat state using small Kerr nonlinearity, a beam splitter, and a homodyne detection of the x ̂ -quadrature.

Fig. 24
Fig. 24

Fidelity F of the cat state produced by a weak Kerr effect when an x = 0 measurement is obtained versus the generated cat amplitude i α . The solid curve shows the case when N = 20 (one tenth of the interaction required by using the Kerr effect directly), the small dashes show N = 12 , and the long dashes show N = 8 .

Fig. 25
Fig. 25

Fidelity of the generated cat state against the measurement outcome x for N = 20 . The solid curve shows generated cat amplitude α = 20 i , and the dashed curve shows α = 10 i . The fidelity has been maximized over the phase ϕ of the generated cat state.

Fig. 26
Fig. 26

Cat state phase ϕ that maximizes the fidelity shown in Fig. 25. The solid curve shows generated cat amplitude α = 20 i , and the dashed curve shows α = 10 i .

Fig. 27
Fig. 27

Diagram of a scheme to produce cat states using a cross-phase modulating Kerr medium. The dashed lines are beam splitters with transmissivity equal to 1 2 , and the solid bold lines are mirrors. The triangles are photon detectors.

Tables (4)

Tables Icon

Table 1 Fidelities for Making Cats with Backaction Evasion a

Tables Icon

Table 2 Fidelities for Making Cats with the Improved Backaction Scheme a

Tables Icon

Table 3 Fidelities for Making Cats with the Simplified Backaction Scheme a

Tables Icon

Table 4 Comparison of the Features of Cat Making Schemes

Equations (83)

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a ̂ α = α α ,
α = e α 2 2 n = 0 α n n ! n .
Ψ ± ( α ) = 1 N ± ( α ) ( α ± α ) .
Ψ ± ( α ) = e α 2 2 N ± ( α ) n = 0 α n n ! ( ( 1 ) n ± 1 ) n .
x ̂ = 1 2 ( a ̂ + a ̂ ) ,
p ̂ = i 2 ( a ̂ a ̂ ) .
Ψ ± ( α ) ( x ) = π 1 4 N ± ( α ) ( e ( 1 2 ) ( x + 2 α ) 2 ± e ( 1 2 ) ( x 2 α ) 2 ) .
Ψ + ( α ) ( p ) = 2 π 1 4 N ± ( α ) e p 2 2 cos ( 2 p α ) ,
Ψ ( α ) ( p ) = 2 π 1 4 N ± ( α ) e p 2 2 i sin ( 2 p α ) .
Ψ ϕ ( α ) = 1 N ϕ ( α ) ( α + e i ϕ α ) .
ψ 1 ϕ 2 B ̂ ( T ) ψ 1 ϕ 2 ,
B ̂ ( T ) = e arccos ( T ) ( a ̂ 1 a ̂ 2 a ̂ 1 a ̂ 2 ) .
α 1 β 2 α T β 1 T 1 β T + α 1 T 2 .
P η ( m ) = n = m ( n m ) η m ( 1 η ) n m P ( n ) ,
p d ( q ) = d q e d q ! .
P d ( m ) = x = 0 m P η ( x ) p d ( m x ) = x = 0 m p d ( m x ) n = x ( n x ) η x ( 1 η ) n x P ( n ) .
log 10 ( D ) = 3.226 η + 1.206 ,
α η 1 α 1 η 2 + α η 1 α 1 η 2 ,
ρ ± ( α , η ) = ( 1 P ± ) Ψ ± ( α η ) Ψ ± ( α η ) + P ± Ψ [ ( α η ) Ψ ( α η ) ] ,
P ± = 1 2 N ( α η ) N ± ( α ) ( 1 e 2 α 2 ( 1 η ) ) ,
F ± ( α , η ) = Ψ ± ( α ) ρ ± ( α , η ) Ψ ± ( α ) .
F ± ( α , η ) = ( 1 P ± ) 4 e α 2 ( 1 + η ) N ± ( α ) N ± ( α η ) ( e α 2 η ± e α 2 η ) 2 .
H K = ω n ̂ + χ n ̂ 2 ,
β Ψ π 2 ( β ) = 1 2 ( β + i β ) .
n ̂ ( t ) = n ̂ ( 0 ) e γ t ,
d ρ ̂ d t = i χ [ n ̂ 2 , ρ ̂ ] + γ 2 [ a ̂ ρ ̂ , a ̂ ] + γ 2 [ a ̂ , ρ ̂ a ̂ ] .
Q ( a ) = 1 π a ρ ̂ a .
Q loss ( a ) = e a 2 q , p = 0 ( a β * ) q ( a * β ) p q ! p ! ( i p 2 q 2 ) exp [ π γ ( p + q ) 4 χ ] exp [ a 2 ( γ χ i p q e γ 2 χ + 2 i ( p q ) γ χ + 2 i ( p q ) ) ] ,
F = π d 2 a P cat ( a ) Q loss ( a ) ,
ρ ̂ = d 2 a a P ( a ) .
P cat ( a ) = 1 2 m , n = 0 β n ( β * ) m m ! n ! ( 1 + ( 1 ) n + m + i e 2 β 2 [ ( 1 ) n ( 1 ) m ] ) n a n m ( a * ) m δ 2 ( a ) ,
n ( I ) = n 1 + n 2 I = n 1 + n 2 ( ω n ̂ A eff T ) ,
χ = ω 2 n 2 A eff T ,
H int = χ ( a ̂ 1 a ̂ 2 + a ̂ 1 a 2 ̂ ) 2 ,
e i π 4 0 1 Ψ ( β ) 2 + Ψ + ( i β ) 1 0 2 .
H int = χ n ̂ 1 n ̂ 2 ,
Ψ + ( α ) 1 β 2 Ψ ( α ) 1 β 2 .
H = i g 2 ( a ̂ s 2 a ̂ p a ̂ s 2 b ̂ p ) + i E ( a ̂ p a ̂ p ) + H loss ,
S ̂ 12 ( r ) 0 1 , 0 2 = e r ( a ̂ 1 a ̂ 2 a ̂ 1 a ̂ 2 ) 0 1 , 0 2 .
ψ sq ( x 1 , x 2 ) = x 1 , x 2 S ̂ 12 ( r ) 0 1 , 0 2 = 1 π exp [ 1 2 e 2 r ( x 1 2 x 2 2 ) 2 1 2 e 2 r ( x 1 2 + x 2 2 ) 2 ] .
ϕ m ( x 2 ) = e x 2 2 H m ( x 2 ) 2 m m ! π ,
S ̂ 1 ( s ) = exp [ s 2 ( a ̂ 1 2 ( a ̂ 1 ) 2 ) ] .
ψ m ( x 1 ) = e s 2 P ( m ) d x 2 ϕ m ( x 2 ) ψ sq ( T x 1 e s + 1 T x 2 , 1 T x 1 e s T x 2 ) ,
F = d x 1 Ψ ± ( α ) ( x ) ψ m ( x 1 ) 2 .
ψ sq ( x 1 ) ψ sq ( x 2 ) = e r 2 + s 2 π exp [ ( x 1 e r ) 2 2 ( x 2 e s ) 2 2 ] .
ψ m ( x 1 ) = d x 2 ϕ m ( x 2 ) ψ sq ( T x 1 + 1 T x 2 ) ψ sq ( 1 T x 1 T x 2 ) ,
Number of decibels = 10 log 10 e 2 s 8.69 s
ψ sq = e r ( a 1 ̂ 2 ( a 1 ̂ ) 2 ) 2 0 = ( 1 λ 2 ) 1 4 n = 0 ( 2 n ) ! n ! ( λ 2 ) n 2 n ,
ψ m = 1 N m n = 0 c n , m ( λ T 2 ) ( n + m ) 2 n ,
c n , m = ( n + m ) ! ( 1 + ( 1 ) n + m ) ( n ! Γ ( n + m 2 + 1 ) )
n ¯ = 1 N m n = 0 n c n , m 2 ( λ T 2 ) n + m .
ψ m = 1 N m n = 0 ( 2 n + m ) ! ( λ T 2 ) n + m 2 ( n + m 2 ) ! ( 2 n ) ! 2 n .
ψ m 0 + λ T 1 + m 2 2 + .
P ( m ) = 1 λ 2 1 ( λ T ) 2 [ λ 2 T ( 1 T ) 1 ( λ T ) 2 ] m × l = 0 Int [ m 2 ] m ! ( m 2 l ) ! l ! 2 ( 2 λ T ) 2 l .
ρ η ( m ) = 1 P η ( m ) n = m P ( n ) ( n m ) η m ( 1 η ) n m Ψ n Ψ n .
v x = 1 ν 2 + ν e 2 r 2 ,
v p = 1 ν 2 + ν e 2 r 2 .
S ̂ ( r ) 1 = n = 0 ( tanh r ) n ( 2 n + 1 ) ! ( cosh r ) 3 2 2 n n ! 2 n + 1 ,
F ( r , α ) = Ψ ( α ) S ̂ ( r ) 1 2 = 2 α 2 exp [ α 2 ( tanh r 1 ) ] ( cosh r ) 3 ( 1 exp [ 2 α 2 ] ) .
p 0 0 + ( 1 p ) 1 1 ,
p S ̂ ( r ) 0 0 S ̂ ( r ) + ( 1 p ) S ̂ ( r ) 1 1 S ̂ ( r ) .
T = α 2 α 2 + β 2 .
γ 3 = 2 α β α 2 + β 2 3 .
B ̂ 13 ( 1 2 ) B ̂ 12 ( T ) Ψ ϕ ( α ) 1 Ψ φ ( β ) 2 γ 3 [ γ 2 1 ( 1 A 2 + e i ( φ + ϕ ) A 2 ) γ 2 3 + e i ϕ 2 γ 1 α 2 β 2 A 2 0 3 + e i ϕ 0 1 α 2 β 2 A 2 2 γ 3 ] ,
P φ , ϕ ( α , β ) = ( 1 e [ 2 α 2 β 2 ( α 2 + β 2 ) ] ) 2 [ 1 + cos ( φ + ϕ ) e 2 ( α 2 + β 2 ) ] 2 ( 1 + cos ( φ ) e 2 α 2 ) ( 1 + cos ( ϕ ) e 2 β 2 ) ,
ρ η ( l 1 , l 3 ) = 1 P η ( l 1 , l 3 ) n 1 = l 1 n 3 = l 3 P ( n 1 , n 3 ) ( n 1 l 1 ) ( n 3 l 3 ) η l 1 ( 1 η ) n 1 l 1 × η l 3 ( 1 η ) n 3 l 3 ψ ( n 1 , n 3 ) ψ ( n 1 , n 3 ) ,
ρ d ( m 1 , m 3 ) = 1 P d ( m 1 , m 3 ) l 1 = 0 m 1 l 3 = 0 m 3 P η ( l 1 , l 3 ) p d ( m 1 l 1 ) p d ( m 3 l 3 ) ρ η ( l 1 , l 3 ) ,
ρ accept = m 1 = 1 m 3 = 1 P d ( m 1 , m 3 ) ρ d ( m 1 , m 3 ) m 1 = 1 m 3 = 1 P d ( m 1 , m 3 ) .
ψ 1 = e ( α i 2 2 ) n = 0 α i n e i ϕ n n ! n 1 ,
ψ N 1 n = 1 N C n , N α i e 2 i n π N 1 ,
n = 1 N C n , N ( e 2 i n π N ) k = e i π k 2 N .
C n , N = 1 N k = 0 N 1 ( 1 ) k exp [ i π k N ( 2 n k ) ] .
n = 1 N C n , N α i e 2 i n π N 2 1 α i e 2 i n π N 2 2 .
ψ N ( x ) 1 = n = 1 N C n , N ( x ) α i e 2 i n π N 2 1 ,
C n , N ( x ) = N x C n , N 2 x α i e 2 i n π N 2 2 ,
F ( α i , N , x ) = max ϕ [ Ψ ϕ ( α i 2 ) ψ N ( x ) 2 ] .
α β = e β α 2 2 e ( α * β α β * ) 2 ,
F = max ϕ [ N ϕ ( α i ) 2 N x 2 n = 1 N C n , N ( x ) { exp [ α i 2 2 ( 1 + e 2 i n π N ) ] + e i ϕ exp [ α i 2 2 ( 1 e 2 i n π N ) ] } 2 ] .
P ( α i , N , δ ) = δ d x Tr 12 [ ψ n 1212 ψ n x 11 x ] = δ d x n , m = 1 N α i e 2 i n π N 2 x x α i e 2 i m π N 2 exp [ α i 2 2 ( 1 e 2 i ( m n ) π N ) ] ,
1 2 ( α , 1 , 0 123 + α , 0 , 1 123 ) .
1 2 ( α e i ϕ , 1 , 0 123 + α , 0 , 1 123 ) ,
1 2 ( α e i ϕ , 1 , 0 123 + α e i ϕ , 0 , 1 123 α , 1 , 0 123 + α , 0 , 1 123 ) .
1 2 ( α e i ϕ 1 ± α 1 ) .

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