リケラボ論文検索は、全国の大学リポジトリにある学位論文・教授論文を一括検索できる論文検索サービスです。

リケラボ 全国の大学リポジトリにある学位論文・教授論文を一括検索するならリケラボ論文検索大学・研究所にある論文を検索できる

リケラボ 全国の大学リポジトリにある学位論文・教授論文を一括検索するならリケラボ論文検索大学・研究所にある論文を検索できる

大学・研究所にある論文を検索できる 「Quantum estimation theory for continuous data」の論文概要。リケラボ論文検索は、全国の大学リポジトリにある学位論文・教授論文を一括検索できる論文検索サービスです。
リケラボ

コピーが完了しました

URLをコピーしました

論文の公開元へ論文の公開元へ
書き出し

Quantum estimation theory for continuous data

久良, 尚任 東京大学 DOI:10.15083/0002004681

2022.06.22

概要

Measurement is a crucial step for the understanding of the world we live in. We struggle to acquire accurate pieces of information from the observation under inevitable noises. Probability theory and statistics have developed various techniques to distinguish signals from noises, while they have revealed fundamental limits on the accuracy which cannot be surpassed due to the inherent stochastic noises.

In quantum mechanics, measurement is a more intricate process than just a probabilistic process. It often does not obey the rules of classical statistics as typified by the uncertainty relation and the violation of Bell’s inequality. It can be a bliss, however, since quantum measurement does not necessarily respect the limits set by the classical statistics. This is the case when both the target system and the measurement probes involve quantumness in certain senses. Quantum metrology is a field of study to exploit the benefit of such quantum measurements: With a classical resource, the measurement error is subject to the standard quantum limit (SQL) of O(N −1/2 ) where N generically represents the amount of resource. With a quantum resource, it can be lower than the SQL, where the new limit of O(N −1 ) called the Heisenberg limit can be reached.

Although the distinction between the SQL and the Heisenberg limit represents a huge impact of quantum measurements, existing theories of quantum metrology does not cover all realistic problems. Continuous data is an example — an important example seen in many situations. To be more concrete, continuously monitored signals vary over time, and images from telescopes and microscopes spread in two-dimensional areas. Interesting phenomena emerge as functional structures in such continuous data as characteristic peaks in the signal or unusual holes in the image.

Such continuous data can be regarded as functions, and can be handled by advanced mathematics. In particular, due to its infinite degrees of freedom, estimation of an unknown function cannot be done without assuming certain regularity. This regularity can be defined in function analysis as a bound on the vector norm of the function, which must appropriately be chosen to perform analytic calculations on the error bound. Theory of signal processing offers important techniques as well, including the wavelet transform that is useful for the edge detection.

In this thesis, we explore the theory of quantum estimation on continuous data. We especially study when and how quantum metrology still benefits for the estimation on systems with unknown functions. Our studies consist of three parts: a preliminary study on multiparameter estimation, a fundamental study on function estimation, and its application to edge detection.

In the preliminary study, we compare the sequential and parallel schemes of multiparameter metrology, which is based on the quantum limits upon general Hamiltonian models derived in the author’s master thesis. After a series of operator-algebraic arithmetics, we show the error bounds on these estimation schemes are both O(md/T), where m is the number of parameters, d is the dimension of the base Hilbert space, and T is the total time cost. We note that the total time counts the multiplicity of the parallel scheme, and therefore quantum metrology can be parallelized without any cost except for an overhead factor independent of the size of the Hamiltonian model.

In the fundamental study, we derive the theoretical limits on function estimation with quantum metrology incorporated. We consider an unknown function φ(x) in a coherent quantum system as a unitary gate with phase factor e iφ(x) varying continuously in space, and the accuracy of the function estimation is measured as the root-mean-square error. We define the regularity of the function by a modified q-H¨older norm, where q is the number that determines the degree of the smoothness. The theoretical limits are derived by reducing the problem to the multiparameter estimation, which we can treat with the results of the preliminary study. The error bound is of O(N −q/(2q+1)) with classical scheme of O(N −q/(q+1)) with quantum scheme, which can be regarded as the SQL and the Heisenberg limit on function estimation. While they scale slower than the corresponding limits on parameter estimation, the latter can be reproduced from the former by formally taking the smooth limit q → ∞. We have also investigated the methodology to achieve the quantum limits. Notably, one may choose either position- or momentum-localized states for the probes: The local linear smoothing can be used for the localized states; the postselected quantum tomography can be used for the momentum-localized states. Both strategies saturate the SQL or the Heisenberg limit, except that function estimation by momentum-localized states is ensured only for 0 < q ≤ 1.

In the last part of our study, we apply the function estimation to the realistic problem of edge detection. Continuing with the position-dependent phase-shift gate, we consider a modified version of ghost imaging as a tool for edge detection. In this strategy, one lets wave packets undergo the phase-shift gate, when the wavelet transform can be measured by detecting the momentum of the output state. The edge of a fixed lengthscale can be detected as the extrema of the momentum. We have shown that the probability distribution of the momentum is subject to a quasi-Markovian process. By this, the estimation error can be decomposed into two parts, each contributed by the momentum and the position. As a result, the error bounds can be determined from the positionmomentum uncertainty relation. Imaging strategy with multi-mode wave packets can be analyzed in the same way, and the Heisenberg-limited edge detection is found to be performed by Gaussian beams with negative quantum correlations in momentum. Notably, although the quantum limits for a single wavelet scale are the same as those of the multiparameter estimation, it is consistent with the function estimation when all wavelet scales are taken into consideration.

Through the studies above, we connect multiparameter quantum metrology to the new theory of the function estimation, and then to its application for the edge detection. These constitute a series of theories on how far quantum mechanics can be exploited for the better estimation of continuous data in the real world.

論文の公開元へ

関連論文

関連論文をもっと見る

参考文献

[1] D. Deutsch and R. Jozsa. “Rapid Solution of Problems by Quantum Computation.” Proc. R. Soc. A 439, 553–558 (1992).

[2] P. W. Shor. “Algorithms for quantum computation: Discrete logarithms and factoring.” In Proc. 35th Ann. Symp. on Fundamentals of Computer Science. IEEE, 124–134 (1994).

[3] L. K. Grover. “A fast quantum mechanical algorithm for database search.” In Proc. 28th Ann. ACM Symp. on Theory of computing. ACM, 212–219 (1996).

[4] P. W. Shor and J. Preskill. “Simple Proof of Security of the BB84 Quantum Key Distribution Protocol.” Phys. Rev. Lett. 85, 441–444 (2000).

[5] N. Gisin et al. “Quantum cryptography.” Rev. Mod. Phys. 74, 145–195 (2002).

[6] M. Hillery, V. Buˇzek, and A. Berthiaume. “Quantum secret sharing.” Phys. Rev. A 59, 1829–1834 (1999).

[7] V. Giovannetti, S. Lloyd, and L. Maccone. “Quantum Metrology.” Phys. Rev. Lett. 96, 010401 (2006).

[8] V. Giovannetti, S. Lloyd, and L. Maccone. “Advances in quantum metrology.” Nat. Photonics 5, 222 (2011).

[9] P. Bouyer and M. A. Kasevich. “Heisenberg-limited spectroscopy with degenerate Bose-Einstein gases.” Phys. Rev. A 56, R1083–R1086 (1997).

[10] M. D. Vidrighin et al. “Joint estimation of phase and phase diffusion for quantum metrology.” Nat. Commun. 5, 3532 (2014).

[11] M. J. Holland and K. Burnett. “Interferometric detection of optical phase shifts at the Heisenberg limit.” Phys. Rev. Lett. 71, 1355–1358 (1993).

[12] M. de Burgh and S. D. Bartlett. “Quantum methods for clock synchronization: Beating the standard quantum limit without entanglement.” Phys. Rev. A 72, 042301 (2005).

[13] E. M. Kessler et al. “Heisenberg-Limited Atom Clocks Based on Entangled Qubits.” Phys. Rev. Lett. 112, 190403 (2014).

[14] M. Fraas. “An Analysis of the Stationary Operation of Atomic Clocks.” Commun. in Math. Phys. 348, 363–393 (2016).

[15] C. M. Caves. “Quantum-mechanical noise in an interferometer.” Phys. Rev. D 23, 1693–1708 (1981).

[16] R. Demkowicz-Dobrza´nski, M. Jarzyna, and J. Ko lody´nski. “Quantum Limits in Optical Interferometry.” Progr. Opt. 60, 345–435 (2015).

[17] R. Schnabel et al. “Quantum metrology for gravitational wave astronomy.” Nat. Commun. 1, 121 (2010).

[18] D. W. Berry and H. M. Wiseman. “Adaptive phase measurements for narrowband squeezed beams.” Phys. Rev. A 73, 063824 (2006).

[19] R. Demkowicz-Dobrza´nski, J. Ko lody´nski, and M. Gut,˘a. “The elusive Heisenberg limit in quantum-enhanced metrology.” Nat. Commun. 3, 1063 (2012).

[20] A. W. Chin, S. F. Huelga, and M. B. Plenio. “Quantum Metrology in NonMarkovian Environments.” Phys. Rev. Lett. 109, 233601 (2012).

[21] R. Chaves et al. “Noisy Metrology beyond the Standard Quantum Limit.” Phys. Rev. Lett. 111, 120401 (2013).

[22] S. Deffner and E. Lutz. “Quantum Speed Limit for Non-Markovian Dynamics.” Phys. Rev. Lett. 111, 010402 (2013).

[23] A. Smirne et al. “Ultimate Precision Limits for Noisy Frequency Estimation.” Phys. Rev. Lett. 116, 120801 (2016).

[24] M. Szczykulska, T. Baumgratz, and A. Datta. “Multi-parameter quantum metrology.” Adv. Phys. X 1, 621–639 (2016).

[25] C. Macchiavello. “Optimal estimation of multiple phases.” Phys. Rev. A 67, 062302 (2003).

[26] H. Imai and A. Fujiwara. “Geometry of optimal estimation scheme for SU(D) channels.” J. Phys. A 40, 4391–4400 (2007).

[27] N. Kura. “Geometrical analysis on multi-parameter Hamiltonian problems and their time complexity.” Master’s thesis, University of Tokyo (2017).

[28] S.-i. Amari. Differential-Geometrical Methods in Statistics. Springer New York (1985).

[29] H. Cram´er. Mathematical methods of statistics. Princeton University Press (1999).

[30] H. Yuan. “Sequential Feedback Scheme Outperforms the Parallel Scheme for Hamiltonian Parameter Estimation.” Phys. Rev. Lett. 117, 160801 (2016).

[31] N. Kura and M. Ueda. “Finite-error metrological bounds on multiparameter Hamiltonian estimation.” Phys. Rev. A 97, 012101 (2018).

[32] L. Galleani and P. Tavella. “Detection and identification of atomic clock anomalies.” Metrologia 45, S127–S133 (2008).

[33] The LIGO Scientific Collaboration. “A gravitational wave observatory operating beyond the quantum shot-noise limit.” Nat. Phys. 7, 962 (2011).

[34] The EHT Collaboration. “First M87 Event Horizon Telescope Results. IV. Imaging the Central Supermassive Black Hole.” Astrophys. J. 875, L4 (2019).

[35] The EHT Collaboration. “First M87 Event Horizon Telescope Results. III. Data Processing and Calibration.” Astrophys. J. 875, L3 (2019).

[36] M. D. Donsker. “Justification and extension of Doob’s heuristic approach to the Kolmogorov-Smirnov theorems.” Ann. Math. Stat. 23, 277–281 (1952).

[37] G. Wahba. “Optimal Convergence Properties of Variable Knot, Kernel, and Orthogonal Series Methods for Density Estimation.” Ann. Stat. 3, 15–29 (1975).

[38] M. P. Wand and M. C. Jones. Kernel smoothing. Chapman and Hall/CRC (1994).

[39] D. W. Berry and H. M. Wiseman. “Adaptive quantum measurements of a continuously varying phase.” Phys. Rev. A 65, 043803 (2002).

[40] D. W. Berry, M. J. W. Hall, and H. M. Wiseman. “Stochastic Heisenberg Limit: Optimal Estimation of a Fluctuating Phase.” Phys. Rev. Lett. 111, 113601 (2013).

[41] D. W. Berry et al. “Quantum Bell-Ziv-Zakai Bounds and Heisenberg Limits for Waveform Estimation.” Phys. Rev. X 5, 031018 (2015).

[42] M. Tsang. “Subdiffraction incoherent optical imaging via spatial-mode demultiplexing.” New J. Phys. 19, 023054 (2017).

[43] H. T. Dinani and D. W. Berry. “Adaptive estimation of a time-varying phase with a power-law spectrum via continuous squeezed states.” Phys. Rev. A 95, 063821 (2017).

[44] J.-F. Cardoso. “Blind signal separation: statistical principles.” Proc. IEEE 86, 2009–2025 (1998).

[45] J. O. Berger. Statistical decision theory and Bayesian analysis. Springer Science & Business Media (2013).

[46] J. Canny. “A Computational Approach to Edge Detection.” IEEE Trans. Pattern Analysis and Machine Intelligence 8, 679–698 (1986).

[47] S. Geman and D. Geman. “Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images.” IEEE Trans. Pattern Analysis and Machine Intelligence PAMI-6, 721–741 (1984).

[48] A. Cochocki and R. Unbehauen. Neural networks for optimization and signal processing. John Wiley & Sons, Inc. (1993).

[49] S. Xie and Z. Tu. “Holistically-Nested Edge Detection.” In The IEEE International Conference on Computer Vision (ICCV) (2015).

[50] S. Mallat and S. Zhong. “Signal characterization from multiscale edges.” In Proc.. 10th International Conference on Pattern Recognition. i, 891–896 (1990).

[51] S. Zhong and S. Mallat. “Characterization of Signals from Multiscale Edges.” IEEE Trans. Pattern Analysis and Machine Intelligence 14, 710–732 (1992).

[52] Y. Bromberg, O. Katz, and Y. Silberberg. “Ghost imaging with a single detector.” Phys. Rev. A 79, 053840 (2009).

[53] G. Brida, M. Genovese, and I. Ruo Berchera. “Experimental realization of sub-shot-noise quantum imaging.” Nat. Photonics 4, 227 (2010).

[54] S. R. Ghaleh et al. “Improved edge detection in computational ghost imaging by introducing orbital angular momentum.” Appl. Opt. 57, 9609–9614 (2018).

[55] S. Zacks. Parametric statistical inference: basic theory and modern approaches. Elsevier (2014).

[56] M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press (2011).

[57] D. Petz. “Monotone metrics on matrix spaces.” Linear Algebra Appl. 244, 81–96 (1996).

[58] P. Gibilisco and T. Isola. “On the characterisation of paired monotone metrics.” Ann. Inst. Stat. Math. 56, 369–381 (2004).

[59] A. Fujiwara and H. Nagaoka. “Quantum Fisher metric and estimation for pure state models.” Phys. Letters A 201, 119–124 (1995).

[60] A. Hayashi, T. Hashimoto, and M. Horibe. “Reexamination of optimal quantum state estimation of pure states.” Phys. Rev. A 72, 032325 (2005).

[61] V. S. Varadarajan. Lie groups, Lie algebras, and their representations. Springer Science & Business Media (2013).

[62] M.-D. Choi. “Completely positive linear maps on complex matrices.” Linear Algebra Appl. 10, 285–290 (1975).

[63] A. Jamio lkowski. “Linear transformations which preserve trace and positive semidefiniteness of operators.” Rep. Math. Phys. 3, 275–278 (1972).

[64] B. E. Sagan. The symmetric group: representations, combinatorial algorithms, and symmetric functions. Springer Science & Business Media (2013).

[65] P. C. Humphreys et al. “Quantum Enhanced Multiple Phase Estimation.” Phys. Rev. Lett. 111, 070403 (2013).

[66] W. F. Stinespring. “Positive functions on C*-algebras.” Proc. Am. Math. Soc. 2, 211–216 (1955).

[67] G. Chiribella, G. M. D’Ariano, and P. Perinotti. “Quantum Circuit Architecture.” Phys. Rev. Lett. 101, 60401 (2008).

[68] B. L. Higgins et al. “Demonstrating Heisenberg-limited unambiguous phase estimation without adaptive measurements.” New J. Phys. 11, 073023 (2009).

[69] W. S. Cleveland. “Robust Locally Weighted Regression and Smoothing Scatterplots.” J. Am. Stat. Assoc. 74, 829–836 (1979).

[70] A. Schaum. “Theory and Design of Local Interpolators.” CVGIP: Graphical Models and Image Processing 55, 464–481 (1993).

[71] E. H. W. Meijering, K. J. Zuiderveld, and M. A. Viergever. “Image reconstruction by convolution with symmetrical piecewise nth-order polynomial kernels.” IEEE Trans. Image Processing 8, 192–201 (1999).

[72] T. Blu, P. Thevenaz, and M. Unser. “Complete parameterization of piecewise-polynomial interpolation kernels.” IEEE Trans. Image Processing 12, 1297–1309 (2003).

[73] G. H. Hardy. “Weierstrass’s non-differentiable function.” Trans. Am. Math. Soc. 17, 301–325 (1916).

[74] H. L. Van Trees. Detection, estimation, and modulation theory. John Wiley & Sons (2004).

[75] Y. K. Belayev. “CONTINUITY AND HOLDER’S CONDITIONS FOR SAMPLE FUNCTIONS OF STATIONARY GAUSSIAN PROCESSES.” In Proc. 4th Berkeley Symp. Math. Statist. and Prob. 23–33 (1961).

[76] N. Kono. “On the modulus of continuity of sample functions of Gaussian processes.” J. Math. Kyoto Univ. 10, 493–536 (1970).

[77] P. Bocchieri and A. Loinger. “Quantum Recurrence Theorem.” Phys. Rev. 107, 337–338 (1957).

[78] A. Peres. “Recurrence Phenomena in Quantum Dynamics.” Phys. Rev. Lett. 49, 1118 (1982).

[79] H. Chernoff. “A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations.” Ann. Math. Stat. 23, 493–507 (1952).

[80] W. Hoeffding. “Probability Inequalities for Sums of Bounded Random Variables.” J. Am. Stat. Assoc. 58, 13–30 (1963).

[81] A. Fujiwara. “Strong consistency and asymptotic efficiency for adaptive quantum estimation problems.” J. Phys. A 39, 12489 (2006).

[82] E. Nitzan, T. Routtenberg, and J. Tabrikian. “A new class of Bayesian cyclic bounds for periodic parameter estimation.” IEEE Trans. Signal Processing (2016).

[83] S. Prossdorf. “Convergence of Fourier series of H¨older continuous functions.” Mathematische Nachrichten 69, 7–14 (1975).

[84] C. J. Stone. “Optimal Rates of Convergence for Nonparametric Estimators.” Ann. Stat. 8, 1348–1360 (1980).

[85] B. L. Higgins et al. “Entanglement-free Heisenberg-limited phase estimation.” Nature 450, 393–396 (2007).

[86] D. V. Strekalov et al. “Observation of Two-Photon “Ghost” Interference and Diffraction.” Phys. Rev. Lett. 74, 3600–3603 (1995).

[87] R. S. Bennink, S. J. Bentley, and R. W. Boyd. ““Two-Photon” Coincidence Imaging with a Classical Source.” Phys. Rev. Lett. 89, 113601 (2002).

[88] R. L. Hudson. “When is the wigner quasi-probability density non-negative?” Rep. Math. Phys. 6, 249–252 (1974).

[89] P. Dallaire, C. Besse, and B. Chaib-draa. “An approximate inference with Gaussian process to latent functions from uncertain data.” Neurocomputing 74, 1945–1955 (2011).

[90] M. I. Kolobov. “The spatial behavior of nonclassical light.” Rev. Mod. Phys. 71, 1539–1589 (1999).

[91] A. Christ et al. “Probing multimode squeezing with correlation functions.” New J. Phys. 13, 033027 (2011).

[92] D. F. Smirnov and A. S. Troshin. “New phenomena in quantum optics: photon antibunching, sub-Poisson photon statistics, and squeezed states.” Soviet Phys. Uspekhi 30, 851–874 (1987).

[93] P. M. Anisimov et al. “Quantum Metrology with Two-Mode Squeezed Vacuum: Parity Detection Beats the Heisenberg Limit.” Phys. Rev. Lett. 104, 103602 (2010).

[94] Z. Y. Ou, J.-K. Rhee, and L. J. Wang. “Photon bunching and multiphoton interference in parametric down-conversion.” Phys. Rev. A 60, 593–604 (1999).

[95] O. A. Ivanova et al. “Multiphoton correlations in parametric downconversion and their measurement in the pulsed regime.” Quantum Electronics 36, 951–956 (2006).

[96] B. Blauensteiner et al. “Photon bunching in parametric down-conversion with continuous-wave excitation.” Phys. Rev. A 79, 63846 (2009).

[97] D. F. Walls. “Squeezed states of light.” Nature 306, 141 (1983).

[98] N. B. Grosse et al. “Measuring Photon Antibunching from Continuous Variable Sideband Squeezing.” Phys. Rev. Lett. 98, 153603 (2007).

[99] A. Prochazka et al. Signal analysis and prediction. Springer Science & Business Media (2013).

[100] J. Rice. “Bandwidth Choice for Nonparametric Regression.” Ann. Stat. 12, 1215–1230 (1984).

[101] F. Faa di Bruno. “Sullo sviluppo delle funzioni.” Annali di scienze matematiche e fisiche 6, 479–480 (1855).

[102] L. Encinas and J. Masqu´e. “A short proof of the generalized Fa`a di bruno’s formula.” Appl. Math. Lett. 16, 975–979 (2003).

[103] L. Carlitz. “Eulerian Numbers and Polynomials.” Math. Magazine 32, 247– 260 (1959).

[104] T. K. Petersen. Eulerian numbers. Ed. by T. K. Petersen. Springer New York (2015).

参考文献をもっと見る

全国の大学の
卒論・修論・学位論文
を
一発検索！

この論文の関連論文を見る