Formulation of uncertainty relation between error and disturbance in quantum measurement by using QUantum

Free ebooks ==> www.Ebook777.com Springer Theses Recognizing Outstanding Ph.D Research Yu Watanabe Formulation of Uncertainty Relation Between Error and Disturbance in Quantum Measurement by Using Quantum Estimation Theory www.Ebook777.com Free ebooks ==> www.Ebook777.com Springer Theses Recognizing Outstanding Ph.D Research For further volumes: http://www.springer.com/series/8790 www.Ebook777.com Aims and Scope The series ‘‘Springer Theses’’ brings together a selection of the very best Ph.D theses from around the world and across the physical sciences Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field As a whole, the series will 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foreword by the supervisor outlining the significance of its content • The theses should have a clearly defined structure including an introduction accessible to scientists not expert in that particular field Yu Watanabe Formulation of Uncertainty Relation Between Error and Disturbance in Quantum Measurement by Using Quantum Estimation Theory Doctoral Thesis accepted by The University of Tokyo, Tokyo, Japan 123 Free ebooks ==> www.Ebook777.com Supervisor Prof Masahito Ueda The University of Tokyo Tokyo Japan Author (Current Address) Dr Yu Watanabe Kyoto University Kyoto Japan ISSN 2190-5053 ISBN 978-4-431-54492-0 DOI 10.1007/978-4-431-54493-7 ISSN 2190-5061 (electronic) ISBN 978-4-431-54493-7 (eBook) Springer Tokyo Heidelberg New York Dordrecht London Library of Congress Control Number: 2013947354 Ó Springer Japan 2014 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of 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trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made The publisher makes no warranty, express or implied, with respect to the material contained herein Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) www.Ebook777.com Parts of this thesis have been published in the following journal articles: (i) Y Watanabe, T Sagawa, and M Ueda, Optimal Measurement on Noisy Quantum Systems, Phys Rev Lett 104, 020401 (2010) (ii) Y Watanabe, T Sagawa, and M Ueda, Uncertainty Relation Revisited from Quantum Estimation Theory, Phys Rev A 84, 042121 (2011) (iii) Y Watanabe, M Ueda, Quantum Estimation Theory of Error and Disturbance in Quantum Measurement, arXiv:1106.2526 (2011) Supervisor’s Foreword In this thesis, Dr Yu Watanabe applies quantum estimation theory to investigate uncertainty relations between error and disturbance in quantum measurement In his seminal work, Heisenberg discussed a thought experiment concerning the position measurement of a particle by using a gamma-ray microscope, and discovered a trade-off relation between the error of the measured position and the disturbance on the quantum-mechanically conjugate momentum caused by the measurement process This trade-off relation epitomizes the complementarity in quantum measurements: we cannot perform a measurement of an observable without causing disturbance in its canonically conjugate observable However, Heisenberg’s argument was rather qualitative, and the quantitative understanding of the trade-off relationship was elusive because in his era, quantum measurement theory had not been established Meanwhile, Kennard and Robertson discussed a different type of inequality concerning inherent fluctuations of observables This version of Heisenberg’s uncertainty relation is commonly described in quantum mechanics textbooks and often erroneously interpreted as a mathematical formulation of the complementarity From the modern point of view, Heisenberg’s uncertainty relation is the trade-off relation between the information gain about an observable and the concomitant information loss about its conjugate observable In this thesis, Dr Watanabe argues that the best solution to this problem is to apply the estimation theory to the outcomes of the measurement for quantifying the error and disturbance in quantum measurement He has successfully formulated the error and disturbance in terms of the Fisher information content, which gives the upper bound of the accuracy of the estimation Moreover, Dr Watanabe has derived the attainable bound of the error and disturbance in quantum measurement The obtained bound is determined by the quantum fluctuations and correlation functions of the observables, which characterize the non-classical fluctuation of the observables Notably, this bound is stronger than the conventional one set by the commutation relation of the observables I believe that this thesis provides a vii viii Supervisor’s Foreword groundbreaking work that establishes the fundamental bound on the accuracy of one measured observable and the disturbance on the conjugate observable in the original spirit of Heisenberg, and I expect that the method developed here will be applied to a broad class of problems related to quantum measurement Tokyo, 31 March 2012 Prof Masahito Ueda Acknowledgments I would like to thank my supervisor, Prof Masahito Ueda for providing helpful comments and suggestions I would like to thank Takahiro Sagawa for a work on error in quantum measurement and uncertainty relations; Yuji Kurotani for guiding me to uncertainty relations and quantum measurement theory when I was an undergraduate student; Prof Masahito Hayashi for fruitful discussions I express my appreciation to Prof Mio Murao, Prof Akira Shimizu, Prof Kimio Tsubono, Prof Masato Koashi, and Prof Makoto Gonokami for refereeing my thesis and for valuable discussions Finally, I am grateful to the numerous researchers who have provided me with opportunities for many helpful discussions ix Free ebooks ==> www.Ebook777.com Contents Introduction References Reviews of Uncertainty Relations 2.1 Heisenberg’s Gamma-Ray Microscope 2.2 Von Neumann’s Doppler Speed Meter 2.3 Kennard-Robertson’s Inequality and Schrödinger’s Inequality 2.4 Arthurs-Goodman’s Inequality 2.5 Ozawa’s Inequality References 7 11 12 14 17 19 19 23 28 30 36 37 37 38 39 42 44 Expansion of Linear Operators by Generators of Lie Algebra su(d) 5.1 Generators of Lie Algebra suðdÞ 45 45 Classical Estimation Theory 3.1 Parameter Estimation of Probability Distributions 3.2 Cramér-Rao Inequality and Fisher Information 3.3 Monotonicity of the Fisher Information ˇ encov’s Theorem and C 3.4 Maximum Likelihood Estimator References Quantum Estimation Theory 4.1 Parameter Estimation of Quantum States 4.2 Monotonicity of the Fisher Information in Quantum Measurement 4.3 Quantum Cramér-Rao Inequality and Quantum Fisher Information 4.4 Adaptive Measurement References xi www.Ebook777.com 106 Uncertainty Relations Between Measurement Errors of Two Observables The quantum fluctuations and correlation function satisfy the following lemma Lemma (Generalized Schrödinger inequality) ˆ α Q ( B) ˆ − CQS ( A, ˆ B) ˆ 2∃ α Q ( A) ˆ B] ˆ ≥[ A, (8.29) Proof Let Pˆi be the projection operator onto Hi From the positivity of the RLD Fisher information and (6.113), ˆ CQ ( A, ˆ B) ˆ α Q ( A) ˆ ˆ ˆ CQ ( B, A) α Q ( B)2 ∃ (8.30) Thus, det ˆ CQ ( A, ˆ B) ˆ α Q ( A) ˆ B) ˆ α Q ( B) ˆ CQ ( A, ˆ α Q ( B) ˆ − CQ ( A, ˆ B)C ˆ Q ( B, ˆ A) ˆ = α Q ( A) ˆ α Q ( B) ˆ − CQS ( A, ˆ B) ˆ + ≥[ A, ˆ B] ˆ = α Q ( A) ˆ B) ˆ − ≥[ A, ˆ B] ˆ CQS ( A, ˆ B] ˆ ˆ α Q ( B) ˆ − CQS ( A, ˆ B)C ˆ QS ( B, ˆ A) ˆ − ≥[ A, = α Q ( A) ∃ (8.31) ∗ ∈ Therefore, (8.29) is proved ˆ we In terms of the quantum fluctuations and correlation function of Aˆ and B, obtain the following theorem ˆ Bˆ ⊂ Lh (H), and Theorem 12 For all quantum states γˆ ⊂ S(H), observables A, ˆ B), ˆ the following inequality is all probabilistic PVM measurements M ⊂ M( A, satisfied: ˆ α Q ( B) ˆ − CQS ( A, ˆ B) ˆ ˆ M)Δ( B; ˆ M) ∃ α Q ( A) (8.32) Δ( A; The equality of this inequality is attainable, that is there exist POVMs that satisfy the equality of (8.32) Proof First, we show (8.32) Let a, b, c and d be coefficients such that Aˆ = a0 Iˆ + λˆ · a, (8.33a) Bˆ = b0 Iˆ + λˆ · b, (8.33b) Cˆ = c0 Iˆ + λˆ · c, (8.33c) Dˆ = d0 Iˆ + λˆ · d (8.33d) 8.3 Attainable Bound of the Product of the Measurement Errors Note that Cˆ Dˆ Aˆ Bˆ =X cT dT ⊗ =X 107 aT bT (8.34) ˆ D, ˆ q) ⊂ M( A, ˆ B), ˆ the inverse of the From (6.34), for an arbitrary POVM M(C, ˜ reduced Fisher information J (M) is calculated to be J˜(M)−1 = aT bT = X −1 J (M)−1 a b ˆ CS (C, ˆ D) ˆ − CQS (C, ˆ D) ˆ ˆ + 1−q α Q (C) α (C) q q ˆ D) ˆ − CQS (C, ˆ D) ˆ α ( D) ˆ 2+ ˆ CS (C, 1−q α Q ( D) (X T )−1 (8.35) From (6.72), the inverse of the reduced SLD Fisher information J˜S is calculated to be J˜S−1 = = aT bT JS−1 a b X −1 ˆ B) ˆ ˆ CS ( A, α ( A) ˆ B) ˆ α ( B) ˆ CS ( A, = X −1 ˆ D) ˆ ˆ CS (C, α (C) (X T )−1 ˆ ˆ ˆ CS (C, D) α ( D) (8.36) We define the reduced error matrix Δ˜ (M) ⊂ R2×2 as Δ˜ (M) := J˜(M)−1 − J˜S−1 = X −1 1−q ˆ q α Q (C) ˆ D) ˆ −CQS (C, ˆ D) ˆ −CQS (C, q ˆ 1−q α Q ( D) (X T )−1 (8.37) From the quantum Cramér-Rao inequality, Δ˜ (M) is positive Note that the measurement errors of Aˆ and Bˆ are ˆ M) = Δ˜ (M)11 , Δ( A; ˆ M) = Δ˜ (M)22 Δ( B; (8.38) From (8.37), we obtain ˆ M)Δ( B; ˆ M) ∃ det[˜Δ (M)] Δ( A; = det X −1 = det ˆ CQS (C, ˆ D) ˆ α Q (C) (X T )−1 ˆ ˆ ˆ CQS (C, D) α Q ( D) ˆ CQS ( A, ˆ B) ˆ α Q ( A) ˆ B) ˆ α Q ( B) ˆ CQS ( A, ˆ α Q ( B) ˆ − CQS ( A, ˆ B) ˆ = α Q ( A) (8.39) 108 Uncertainty Relations Between Measurement Errors of Two Observables Thus, the inequality (8.32) is proved Next, we show that the equality of (8.32) is attainable From the derivation (8.39), we see that the equality is attained if the off-diagonal elements of Δ˜ (M) vanish Noting that Δ˜ (M) = 1−q X X −1 −q q(1 − q) ˆ CQS ( A, ˆ B) ˆ α Q ( A) 1−q XT ˆ ˆ ˆ −q CQS ( A, B) α Q ( B) (X T )−1 , (8.40) the condition of the equality in (8.32) being attained is written as Δ˜ (M)12 = ⇔ e1 · X −1 1−q 0 −q XT X ˆ CQS ( A, ˆ B) ˆ α Q ( A) ˆ ˆ ˆ CQS ( A, B) α Q ( B) 1−q 0 −q (X T )−1 e2 = (8.41) For proving the existence of the POVM that attains (8.32), it is sufficient to show that ˆ there exists a set of X and q that satisfies (8.41) If we scale Cˆ ∼ k1 Cˆ and Dˆ ∼ k2 D, ˆ ˆ the POVM M(C, D, q) is conserved Therefore, without loss of generality, we can write X as cos φ sin φ X= (8.42) cos τ sin τ If q = 1/2, we have XT 1/2 (X T )−1 = −1/2 sin(φ + τ) − cos φ cos τ sin φ sin τ − sin(φ + τ) (8.43) Thus, by choosing φ = τ + 2/δ , we have XT XT 1/2 (X T )−1 e1 = −1/2 1/2 (X T )−1 e2 = −1/2 cos 2τ sin 2τ sin 2τ − cos 2τ =: x1 , =: x2 (8.44a) (8.44b) Equation (8.44a) and (8.44b) indicate that the vectors x1 and x2 are orthogonal to each other Then, the condition (8.41) becomes x1 · ˆ CQS ( A, ˆ B) ˆ α Q ( A) x2 = ˆ ˆ ˆ CQS ( A, B) α Q ( B) (8.45) 8.3 Attainable Bound of the Product of the Measurement Errors 109 The matrix in the above equation is symmetric, and thus the two eigenvectors of the matrix are orthogonal to each other By choosing τ so that x1 and x2 become the eigenvectors, (8.45) is satisfied and the equality in (8.32) is attained ∗ ∈ From the Schrödinger inequality (8.29) for the quantum fluctuations and correlation function, the bound set by (8.32) is tighter than that by (8.14) Because there always exist POVM measurements that attain the equality in (8.32), the inequality is the tightest The inequality (8.32) reduces to the trade-off relation found in Ref [1] for dim H = and γˆ = Iˆ/2 For arbitrary linear unbiased estimators Aest and B est for ≥ Aˆ and ≥ Bˆ , respectively, that is ni (8.46a) φi , Aest = n i B est = τi i where φi and τi satisfy Aˆ = the following inequality i ni , n φi Mˆ i and Bˆ = (8.46b) i τi Mˆ i , Nagaoka [2, 3] proved ˆ + α ( B) ˆ + Tr |γˆ 1/2 [ A, ˆ B] ˆ γˆ 1/2 | nVar[Aest ] + nVar[B est ] ∃ α ( A) (8.47) Because the estimators are restricted to the linear estimators, ˆ ∃ Δ( A; ˆ M) nVar[Aest ] − α ( A) (8.48) ˆ as is satisfied By defining ΔNagaoka ( A) ˆ = nVar[Aest ] − α ( A) ˆ 2, ΔNagaoka ( A) (8.49) Nagaoka’s inequality can be written as ˆ + ΔNagaoka ( B) ˆ ∃ Tr |γˆ 1/2 [ A, ˆ B] ˆ γˆ 1/2 | ΔNagaoka ( A) (8.50) As shown in Fig 8.1, (8.32) is stronger than (8.50) A simplest but not optimal way to estimate ≥ Aˆ and ≥ Bˆ is to perform the PVM measurement P Aˆ on n Aˆ samples and another PVM measurement P Bˆ on n Bˆ := n−n Aˆ samples This measurement scheme is asymptotically equivalent to the POVM ˆ B, ˆ q) with q = n ˆ /n The reduced error matrix Δ˜ [M( A, ˆ B, ˆ q)] measurement M( A, A satisfies 1−q ˆ ˆ ˆ q α Q ( A) −CQS ( A, B) ˆ B, ˆ q)] = Δ˜ [M( A, (8.51) q ˆ ˆ ˆ α Q ( B) −CQS ( B, A) 1−q 110 Uncertainty Relations Between Measurement Errors of Two Observables ˆ Fig 8.1 Bounds of the ⇒ measurement errors for dim H = (S = 3/2), γˆ = { I /(2S+1)+|S ≥S|}/2, Aˆ = Sˆ x , and Bˆ = ( Sˆ x + Sˆ y )/2, where Sˆi is the spin operator in the x-direction (i = x, y, z) ˆ B), ˆ and the curves that are closer to the origin have Each axis is [Δ( Xˆ ; M)/α ( Xˆ )2 + 1]−1 ( Xˆ = A, tighter bounds The red dash-dotted, black solid, green dotted, and blue dashed curves show the bounds set by (8.14), (8.32), (8.50), and (8.52), respectively Therefore, errors in this measurement scheme satisfy ˆ M)Δ( B; ˆ M) = α Q (A)2 α Q (B)2 Δ( A; (8.52) However, this measurement scheme does not exploit possible correlations between ˆ To utilize them, it is sufficient to perform PVM the observables Aˆ and B measurements of two observables Cˆ and Dˆ with probabilities q and − q, respecˆ tively, where Cˆ and Dˆ are linear combinations of Aˆ and B Although, the inequality (8.32) is shown for the probabilistic PVM measurements, for the 2-dimensional Hilbert space, we will show that the inequality is satisfied for every POVM in Theorem 13 Before showing the theorem, we prove the following two lemmas Lemma For all γˆ ⊂ S(H), Tr J (M)JS−1 ≤ d − (8.53) is satisfied for every POVM measurement M This lemma was also proved by Gill and Massar [4] Proof From the monotonicity of the Fisher information under the Markov mapping, it is sufficient to consider the case in which all the elements of M are rank Then, every element of M satisfies (8.54) Mˆ i2 = ki Mˆ i , where ki is the non-zero eigenvalue of Mˆ i We obtain 8.3 Attainable Bound of the Product of the Measurement Errors i vi · JS−1 vi = pi i ki pi − pi2 = pi Tr J (M)JS−1 = = 111 i α ( Mˆ i )2 pi ki − (8.55) i From the completeness of the POVM, we have Tr Mˆ i = d ki = i (8.56) i ∗ ∈ Therefore, the inequality (8.53) is proved Lemma Let K be a × matrix such that −1/2 ˜ −1/2 K := J˜S J (M) J˜S (8.57) For the 2-dimensional Hilbert space, the following inequality is satisfied for every M: det[K −1 − I ] ∃ (8.58) Proof For an arbitrary × matrix K , det[K −1 − I ] = det[K −1 ] det[I − K ] = − Tr K + det[K ] det[K ] (8.59) is satisfied Therefore, det[K −1 − I ] ∃ ⇔ Tr K ≤ (8.60) By denoting X := ηϕ/ηθ , we obtain Tr K = Tr J˜(M) J˜S−1 = Tr [X J (M)−1 X T ]−1 X JS−1 X T = Tr P J (M)1/2 JS−1 J (M)1/2 ≤ Tr J (M)1/2 JS−1 J (M)1/2 ≤ d − 1, where P := J (M)−1/2 X T [X J (M)−1 X T ]−1 X J (M)−1/2 (8.61) (8.62) is a projection matrix, which satisfies the idempotency condition P = P Therefore, for d = 2, the inequality (8.58) is proved ∗ ∈ Theorem 13 Let H be a 2-dimensional Hilbert space For all γˆ ⊂ S(H) and observˆ Bˆ ⊂ Lh (H), (8.32) is satisfied for every POVM measurement M ables A, 112 Uncertainty Relations Between Measurement Errors of Two Observables Proof For 2-dimensional Hilbert space, it is not necessary to consider the blockˆ The quantum fluctuation and the symmetrized quantum diagonalization of Aˆ and B correlation function become ˆ = α Q ( A) ˆ α ( A) ˆ B) ˆ = CQS ( A, ˆ B] ˆ = 0); ([ A, (otherwise), ˆ B) ˆ CS ( A, ˆ B] ˆ = 0); ([ A, (otherwise) (8.63) (8.64) ˆ M) and If Aˆ and Bˆ are commutable, the bound set by (8.32) vanishes, and Δ( A; ˆ Δ( B; M) can vanish by performing the P measurement corresponding to the spectral ˆ Hence, in the following, we consider the case in which decomposition of Aˆ and B ˆ ˆ A and B are non-commuting From Lemmas and 9, we obtain ˆ M)Δ( B; ˆ M) ∃ det[˜Δ (M)] = det[ J˜−1 ] det[K −1 − I ] Δ( A; S ∃ det ˆ B) ˆ ˆ CS ( A, α ( A) ˆ A) ˆ α ( B) ˆ CS ( B, ˆ α ( B) ˆ − CS ( A, ˆ B) ˆ = α ( A) Therefore, (8.32) is proved for the 2-dimensional Hilbert space (8.65) ∗ ∈ Fig 8.2 Plots of measurement errors of 109 randomly ⇒ chosen POVMs for dim H = (S = 3/2), γˆ = { Iˆ/(2S + 1) + |S ≥S|}/2, Aˆ = Sˆ x , and Bˆ = ( Sˆ x + Sˆ y )/2, where Sˆi is the spin operator in ˆ B), ˆ and the curves that the x-direction (i = x, y, z) Each axis is [Δ( Xˆ ; M)/α ( Xˆ )2 + 1]−1 ( Xˆ = A, are closer to the origin have tighter bounds The red dash-dotted, black solid, green dotted, and blue dashed curves show the bounds set by (8.14), (8.32), (8.50), and (8.52), respectively Numerically calculated 109 errors lie within the area set by the black solid curve, and there are no points between the black solid and the red dash-dotted curves 8.3 Attainable Bound of the Product of the Measurement Errors 113 In Theorem 12, (8.32) is proved only for the probabilistic PVM measurements The following theorem states that every POVM measurement satisfies (8.32) for a set of specific quantum states The inequality (8.32) is rigorously proved for every POVM measurement for ˆ B) ˆ for dim H ∃ For higher dimensional dim H = and for every M ⊂ M( A, Hilbert spaces from dim H = to 7, we numerically calculate the measurement errors of 109 randomly chosen POVMs for randomly chosen 10 pairs of quantum ˆ B) ˆ We find that the calculated measurement errors states and two observables (γ, ˆ A, satisfy (8.32) A typical example of the numerical calculation is shown in Fig 8.2 The area within the bound is blacked out by 109 data points with no point found outside of the bound The range dim H = to includes prime numbers (dim H = 3, 5, 7), a power of a prime (dim H = 4), and a composite number that is not a power of a prime (dim H = 6) Therefore, we propose the following conjecture ˆ Bˆ ⊂ Lh (H), Conjecture For all quantum state γˆ ⊂ S(H) and observables A, inequality (8.32) is satisfied for every POVM measurement M References T Sagawa, M Ueda, Phys Rev A 77, 012313 (2008) H Nagaoka, Trans Jap Soc Ind Appl Math 1, 43 (1991) M Hayashi, Asymptotic Theory of Quantum Statistical Inference: Selected Papers (World Scientific, Singapore, 2005) R.D Gill, S Massar, Phys Rev A 61, 042312 (2000) Chapter Uncertainty Relations Between Error and Disturbance in Quantum Measurements Heisenberg originally discussed the trade-off relation between error and disturbance in quantum measurement In this chapter, we prove several trade-off relations between error and disturbance As shown in Chap 7, the error and disturbance in measurement are quantified in terms of Fisher information contents By expanding relevant operators in terms of the generator λˆ of the Lie algebra, we calculate the error and disturbance, and prove the trade-off relations 9.1 Heisenberg’s Uncertainty Relation in Terms of Fisher Information Contents In this section, we show that the error and disturbance is bounded by the commutation relation of the observables Before showing the bound, we prove the following lemma about the error and disturbance ˆ Bˆ ≥ Lh (H), and Lemma 10 For all quantum state ρˆ ≥ S(H), observables A, measurement operators K, there exist POVMs M that satisfy ˆ K)η( B; ˆ K) ≥ ε( A; ˆ M)ε( B; ˆ M) ε( A; (9.1) Note that M is not the POVM derived from K in general Proof Let M = { Mˆ i }i be the POVM corresponding the measurement operators K = { Kˆ i, j }i j : Mˆ i = (9.2) Kˆ i,† j Kˆ i, j j ˆ K) is finite, there exists a PVM For every measurement operators K for which η( B; measurement P = { Pˆi }i such that Y Watanabe, Formulation of Uncertainty Relation Between Error and Disturbance in Quantum Measurement by Using Quantum Estimation Theory, Springer Theses, DOI: 10.1007/978-4-431-54493-7_9, © Springer Japan 2014 115 116 Uncertainty Relations Between Error and Disturbance ˆ K) = b · [J⊂ (P)−1 − J−1 ]b η( B; S ∼ b · (JS⊂ )−1 b = b · J⊂ (P)−1 b, (9.3) where J ⊂ (P) and JS⊂ are the classical and SLD Fisher information contents of the post-measurement state ρˆ ⊂ := i j Kˆ i, j ρˆ Kˆ i,† j , in other words, they are based on the quantum-statistical model defined in (6.55) Let M⊂ = { Mˆ ⊂ }i be the POVM defined by i Kˆ †j,k Pˆi Kˆ j,k Mˆ i⊂ = (9.4) jk The probability distributions for the situations, in which P is performed on ρˆ ⊂ and ˆ coincide: M⊂ is performed on ρ, Tr[ρˆ Kˆ †j,k Pˆi Kˆ j,k ] = Tr[ρˆ Mˆ i⊂ ] pi = Tr[ρˆ ⊂ Pˆi ] = (9.5) jk Therefore, the Fisher information J ⊂ (P) satisfies J ⊂ (P) = J (M⊂ ) (9.6) From the monotonicity of the Fisher information under the Markov mapping, we find that (9.7) J (M⊂ ) ≤ J (M⊂⊂ ) is satisfied, where M⊂⊂ = { Mˆ i,⊂⊂ j }i, j is the POVM defined by † ˆ ˆ Kˆ i,k P j K i,k Mˆ i,⊂⊂ j = (9.8) k Therefore, we obtain ˆ K) = ε( B; ˆ M⊂ ) ≥ ε( B; ˆ M⊂⊂ ) η( B; (9.9) Because M can be written as † ˆ Kˆ i,k K i,k = Mˆ i = k † ˆ ˆ Kˆ i,k P j K i,k = jk Mˆ ⊂⊂j , (9.10) j the following inequality is satisfied: J (K) = J (M) ≤ J (M⊂⊂ ) (9.11) ˆ K) = ε( A; ˆ M) ≥ ε( A; ˆ M⊂⊂ ) ε( A; (9.12) Therefore, we obtain 9.1 Heisenberg’s Uncertainty Relation in Terms of Fisher Information Contents 117 From (9.9) and (9.12), we obtain ˆ M⊂⊂ ) ˆ K)η( B; ˆ K) ≥ ε( A; ˆ M⊂⊂ )ε( B; ε( A; (9.13) ∃ ˆ the following Theorem 14 For all quantum states ρ, ˆ and observables Aˆ and B, inequality is satisfied for every set of measurement operators K: ˆ K)η( B; ˆ K) ≥ ε( A; ˆ B]∗ ˆ ∈[ A, (9.14) Proof From Theorem 11 and Lemma 10, we obtain ˆ K)η( B; ˆ K) ≥ ε( A; ˆ M)ε( B; ˆ M) ≥ ε( A; ˆ B]∗ ˆ ∈[ A, (9.15) ∃ This theorem shows that the error and disturbance satisfies Heisenberg’s uncertainty relation and that the product of the error and disturbance is bounded by the commutation relation of the observables However, because the equality of (8.14) is not attainable, the equality of (9.14) is not attainable, either 9.2 Attainable Bound of the Product of Error and Disturbance In the previous section, we obtain Heisenberg’s uncertainty relation about the error and disturbance in quantum measurement However, the equality of the trade-off relation is not attainable in general In this section, we obtain the attainable bound of the product of error and disturbance ˆ Bˆ ≥ Lh (H), if Theorem 15 For all quantum state ρˆ ≥ S(H) and observables A, ˆ B), ˆ the following a set of measurement operators K induces POVM M ≥ M( A, inequality is satisfied: ˆ σ Q ( B) ˆ − CQS ( A, ˆ B) ˆ ˆ K)η( B; ˆ K) ≥ σ Q ( A) ε( A; (9.16) The equality of this inequality is attainable, that is, there exist some measurement operators that satisfy the equality of (9.16) Proof First, we show (9.16) Let M be a POVM whose elements are all rank 1: Mˆ i = xi |ψi ∗∈ψi | (9.17) 118 Uncertainty Relations Between Error and Disturbance Any set K of measurement operators that induces M satisfies Kˆ i, j = yi z i, j |ϕ j ∗∈ψi |, (9.18) where yi , z i, j ≥ C satisfy |yi |2 = xi and j |z i, j | = For such measurement operators K, the post-measurement state ρˆ ⊂ becomes Kˆ i, j ρˆ Kˆ i,† j = ρˆ ⊂ = i, j |z i, j |2 |ϕ j ∗∈ϕ j | pi (9.19) i, j The probability distribution qi of measurement outcomes when a POVM measurement M⊂ is performed on the post-measurement state is |z j,k |2 ∈ϕk | Mˆ i⊂ |ϕk ∗ p j qi = Tr[ρˆ ⊂ Mˆ i⊂ ] = (9.20) j,k Therefore, the mapping p ⊗ q is a Markovian From the monotonicity of the Fisher information, we obtain J (M) ≥ J ⊂ (M⊂ ), ∼ b · J(M)−1 b ≤ b · J⊂ (M⊂ )−1 b (9.21) Since M⊂ is arbitrary, we obtain Therefore, we obtain b · J(M)−1 b ≤ b · JS⊂−1 b (9.22) ˆ M) ≤ η( B; ˆ M) ε( B; (9.23) ˆ B), ˆ we obtain From Theorem 12, if M ≥ M( A, ˆ σ Q ( B) ˆ − CQS ( A, ˆ B) ˆ ˆ K)η( B; ˆ K) ≥ ε( A; ˆ M)ε( B; ˆ M) ≥ σ Q ( A) ε( A; (9.24) Next, we show that the equality of (9.16) is attainable For proving the attainability of (9.16), it is sufficient to show that there exist a set of measurement operators K that ˆ B) ˆ be a POVM that attain the equality attain the equality Let M = { Mˆ i }i ≥ M( A, of (8.32) Because every element of M is rank 1, every element of M can be written in the following form: (9.25) Mˆ i = xi |ψi ∗∈ψi | Let K = { Kˆ i }i be measurement operators that satisfy √ Kˆ i = xi |ϕi ∗∈ψi | (9.26) 9.2 Attainable Bound of the Product of Error and Disturbance 119 This measurement operators K induces M: Kˆ i† Kˆ i = Mˆ i The Markov mapping p ⊗ q (9.20) becomes qi = ∈ϕ j | Mˆ i⊂ |ϕ j ∗ p j (9.27) j If ∈ϕi |ϕ j ∗ = δi j and M⊂ is a PVM P whose arbitrary element takes the form Pˆi = |ϕi ∗∈ϕi |, the mapping (9.27) becomes invertible Then, we obtain and J (M) = J ⊂ (M), (9.28) ˆ M) = η( B; ˆ K) ε( B; (9.29) Since M attains the equality of (8.32), K also attains the equality of (9.16): ˆ σ Q ( B) ˆ − CQS ( A, ˆ B) ˆ ˆ K)η( B; ˆ K) = ε( A; ˆ M)ε( B; ˆ M) = σ Q ( A) ε( A; (9.30) ∃ Note that to prove Theorem 15 it is impossible to use Lemma 10, because even ˆ B), ˆ the POVM that satisfies (9.1), in general, K induces the POVM M ≥ M( A; ˆ ˆ ˆ B) ˆ Lemma 10 can be utilized if does not belong to M( A, B), i.e., M ⇒≥ M( A, an equality is proved for every POVM measurement M Therefore, we obtain the following theorem from Theorem 13 Theorem 16 Let H be a 2-dimensional Hilbert space For all ρˆ ≥ S(H) and observˆ Bˆ ≥ Lh (H), (9.16) is satisfies for every set of measurement operators K ables A, If Conjecture is proved, the following conjecture can also be proved by using Lemma 10 ˆ Bˆ ≥ Lh (H), Conjecture For all quantum state ρˆ ≥ S(H) and observables A, inequality (9.16) is satisfied for every set of measurement operators K Chapter 10 Summary and Discussion We have studied quantum estimation theory of error and disturbance in quantum measurement, and derived several uncertainty relations between the measurement errors of two observables, and the uncertainty relations between the error and disturbance In the following we will summarize these results and discuss some outstanding issues In Chap 7, we have formulated error and disturbance in quantum measurement by using quantum estimation theory We have clarified the importance of the estimation process, which is implicitly involved in Arthurs and Goodman’s inequality, and found that the unbiasedness condition is necessary not for measurements, but for the estimation We have shown that the estimated values from measurement outcomes consist of three types of error: inherent fluctuation of an observable, error in the measurement, and estimation error caused by the unoptimality of the estimator We have extracted the measurement error from the variance of the estimator in terms of the Fisher information, which gives the upper bound of the accuracy of the estimator We have also shown that the disturbance caused by the backaction of the measurement is quantified in terms of the loss of the Fisher information contents during the measurement process Our study is based on finite-dimensional Hilbert spaces For infinite-dimensional Hilbert spaces, the Fisher information cannot be defined in general because the number of the parameters that determine the quantum state becomes infinite [1] Therefore, it is an outstanding issue to define error and disturbance for the infinitedimensional Hilbert spaces Our analysis about the error and disturbance is based on the premise that we want to know the expectation values of observables on an unknown quantum state However, it is possible to consider whether the expectation value of the observables are larger or smaller than thresholds Such a situation is called testing [2] Because of the non-commutativity of the observables, it is expected other trade-off relations can be derived based on quantum testing theory In Chap 8, we have rigorously proved the Heisenberg-type uncertainty relation between the measurement errors of two observables: the product of the errors is Y Watanabe, Formulation of Uncertainty Relation Between Error and Disturbance in Quantum Measurement by Using Quantum Estimation Theory, Springer Theses, DOI: 10.1007/978-4-431-54493-7_10, © Springer Japan 2014 121 Free ebooks ==> www.Ebook777.com 122 10 Summary and Discussion bounded by the commutation relation of the observables We have shown that the bound cannot be attained in general, and found the attainable bound of the product of the errors The attainable bound is given by the quantum fluctuation and correlation function of the observables The quantum fluctuation and correlation function characterize the non-classical fluctuation and correlation of the observables, respectively We have found the attainable bound for an arbitrary measurement for the 2-dimensional Hilbert space, and for a class of measurements for higher dimensions We have shown numerical evidences for the validity of the bound for an arbitrary measurement for higher-dimensional Hilbert spaces It is an outstanding issue to prove the attainable bound rigorously In Chap 9, we have proved Heisenberg’s uncertainty relation between the error and disturbance: the product of the error and disturbance is also bounded by the commutation relation of the observables We have also shown that the bound cannot be attained, and found the attainable bound of the product of the errors The attainable bound is rigorously proved for the 2-dimensional Hilbert space, and for a class of measurements for the higher dimensions If the attainable bound for the product of the errors is proved, the bound for the error and disturbance can be proved The measurements that attain the bound, which we have found until now, require an ancilla system that has, at least, a 2-dimensional Hilbert space If we cannot use the ancilla, the bound may not be attainable Therefore, we will address the question of what the attainable bound is for the case without the ancilla This remains yet another outstanding issue References S Amari, H Nagaoka, Methods of Information Geometry (American Mathematical Society, Providence, 2007), pp 1–206 M Hayashi, Quantum Information: An Introduction (Springer Verlag, Berlin, 2006), pp 1–429 www.Ebook777.com ... analyze error and disturbance in quantum measurement, and define the error and disturbance in terms of Fisher information contents In Chap 8, we derive uncertainty relations of the measurement errors... position x of the particle by the following relation: L1 ≥ x= x, (2.2) L2 Y Watanabe, Formulation of Uncertainty Relation Between Error and Disturbance in Quantum Measurement by Using Quantum Estimation... is, by choosing X appropriately, we assume that p(x; θ ) > for all x ∈ X and θ ∈ ε Y Watanabe, Formulation of Uncertainty Relation Between Error and Disturbance in Quantum Measurement by Using Quantum