Enhancing the Precision and Accuracy in Quantum Metrology = 양자 계측의 정밀도 및 정확도 향상

다국어 초록 (Multilingual Abstract)

Enhancing the Precision and Accuracy in Quantum Metrology Hyukgun Kwon Department of Physics and Astronomy The Graduate School Seoul National University Precise and accurate parameter estimation plays a crucial role in the development of science and technology. The main purpose of the dissertation is two-fold: to investigate the role of general quantum networks in terms of enhancing the estimation precision and to study the efficacy of error mitigation on quantum metrology in terms of reducing systematic error (bias) which enhances the estimation accuracy. In the first part, we investigate the quantum metrological power of typical continuous-variable (CV) quantum networks. Particularly, we show that most CV quantum networks provide an entanglement to quantum states in distant nodes that enables one to achieve the Heisenberg scaling in the number of modes for distributed quantum displacement sensing, which cannot be attained using an unentangled probe. Notably, our scheme only requires local operations and measurements after generating an entangled probe using the quantum network. In addition, we find a tolerable photon-loss rate that maintains the quantum enhancement. Finally, we numerically demonstrate that even when CV quantum networks are composed of local beam splitters, the quantum enhancement can be attained when the depth is sufficiently large. In the second part, we study the efficacy of virtual purification on quantum metrology. Noise in quantum systems poses a major obstacle to fully utilizing quantum advantages. Especially in quantum parameter estimation, unknown noise entails systematic error (bias). Recently, virtual purification- based error mitigation (VPEM) was introduced to address this issue in quantum metrology but its general applicability remained unclear yet. To shed light on the general applicability of VPEM on quantum metrology, we identifiy factors influencing the efficacy of VPEM: 1) Closeness between the dominant eigenvector of a noisy state and the noise-free quantum probe. The closeness determines reducible bias by VPEM. 2) A reference point of the target parameter. Inappropriate reference point can increase bias even if one applies VPEM. Finally, we analyze VPEM in phase estimation under various noise conditions, predicting and verifying the effectiveness of VPEM in bias reduction. Keywords : Quantum Metrology, Quantum Entanglement, Quantum Error Mitigation Student Number : 2018-24362

국문 초록 (Abstract)

정밀하고 정확한 매개변수 추정은 과학과 기술의 발전에 중요한 역할을 한다. 이와 관련하여, 본 학위 논문에서는 크게 두 가지 주제를 다룬다.
첫째로, 대부분의 양자 네트워크는 고전적으로는 달성할 수 없었던 더 높은 정밀도를 달성 가능하게 해준다는 것을 증명하였다. 기존의 연구 결과에 의하면 양자 네트워크를 사용하지 않고 변위 추정을 할 경우, 계측 정밀도가 양자 상태의 모드의 개수 M의 역수 1/M에 비례하는 반면, 아주 특별한 구조의 양자 네트워크를 이용하면 계측 정밀도를 1/M2에 비례하게 낮출 수 있다는 사실이 알려져 있었다. 우리는 더 나아가 일반적인 양자 네트워크를 사용하더라도 이러한 향상된 정밀도를 얻을 수 있다는 사실을 밝혀냈다. 본 연구는 최초로 일반적인 양자 네트워크의 유용성을 증명했을 뿐만 아니라, 현재 기술로 구현 가능한 선형 광학 시스템과 조임 상태만을 이용했다는 점, 그리고 아주 특별한 구조의 양자 네트워크를 만들어야 한다는 제약을 없앴다는 점 등에서 관련 실험연구에도 많은 영향을 미칠 것으로 기대된다.
둘째로, Virtual Purification (VP) 이라는 양자 오류 완화 방법을 양자 계측에 적용하였을때의 효용성에 관하여 연구하였다. 환경과의 상호작용은 양자 계측의 정확도를 낮추게 되고 이는 양자 이득을 얻는데에 큰 걸림돌이 된다. 이를 해결하기 위해 최근, VP를 양자 계측에 적용한 연구가 있었지만, 일반적인 양자 계측 상황에서 VP가 항상 효과적으로 정확도를 복원해줄지에 대해서는 알려진바가 없었다. 본 연구에서 우리는, VP가 효과적으로 정확도를 복원할 수 있는지의 여부에 영향을 미치는 두 가지 요소를 밝혀내었다. 첫번째 요소는 잡음이 발생한 양자 상태의 가장 우세한 고유 벡터와 잡음이 없을때의 이상적인 양자 상태의 가까운 정도이다. 두 상태의 가까운 정도가 VP를 이용해 복원할 수 있는 정확도의 준위를 결정한다. 두번째 요소는 우리가 추정 하고자 하는 매개변수의 기준점이다. 우리의 연구 결과에 의하면, 잘못된 기준점을 잡는다면 VP를 적용하더라도 오히려 정확도가 낮아질 수 있다. 마지막으로 앞선 분석들을 여러 종류의 잡음들이 존재하는 간섭계에서의 위상 매개 변수 추정에 적용하여보았다.

목차 (Table of Contents)

• Abstract i
• I. Introduction 1
• II. Preliminary 5
• 2.1 Continuous variable system 5
• 2.1.1 Quadrature operators 5
• 2.1.2 Beam splitter operation . 6
• 2.1.3 Displacement operator . 7
• 2.1.4 Squeezed state 8
• 2.2 Quantum estimation theory 9
• III. Distributed Quantum Sensing 15
• 3.1 Introduction 15
• 3.2 Distributed quantum disaplacement sensing 17
• 3.2.1 Schematic of distributed displacement sensing 17
• 3.3 QFI of the sensing scheme 19
• 3.3.1 QFI 19
• 3.4 Main Results . 21
• 3.4.1 Lemma 1 21
• 3.4.2 Theorem 1 24
• 3.4.3 Corollary 1 28
• 3.4.4 Theorem 2 29
• iii
• 3.4.5 Optimality of homodyne measurement 34
• 3.4.6 Effect of Loss 36
• 3.4.7 Local beam splitter network . 39
• 3.5 Appendix 42
• 3.5.1 Optimal state for distributed displacement sensing . 42
• 3.5.2 Upper bound of QFI 45
• 3.5.3 Without an Entanglement 47
• 3.5.4 Local optimized QFI 47
• 3.5.5 Haar random unitary representation 49
• 3.5.6 Concentration of measure inequality 50
• 3.6 Conclusion 51
• IV. Efficacy of virtual purification on quantum metrology . 53
• 4.1 Introduction 53
• 4.2 MSE with bias . 56
• 4.2.1 Ideal Case 57
• 4.2.2 Error case 58
• 4.2.3 Error Mitigated Quantum Metrology 60
• 4.3 Crucial factors for reducing bias via VPEM 64
• 4.3.1 Expectation value of  over the dominant eigenvector
• . 65
• 4.3.2 Reference point of the parameter 68
• 4.4 Numerical simulation of VPEM for phase estimation scheme 71
• 4.4.1 N00N state . 73
• 4.4.2 Product of coherent and squeezed states . 78
• iv
• 4.5 Appendix 82
• 4.5.1 Calculations of the average and variance of sample
• mean of mitigation case 82
• 4.5.2 Phase estimation exploiting N00N state . 87
• 4.5.3 N00N state in the presence of the phase diffusion 92
• 4.5.4 N00N state in the presence of the photon loss . 94
• 4.5.5 Phase estimation exploiting product of coherent and
• squeezed states 98
• 4.5.6 Product of coherent and squeezed states in the presence
• of the photon loss . 100
• 4.5.7 Product of coherent and squeezed states in the presence
• of additive Gaussian noise 104
• 4.6 Conclusion 105
• V. Conclusion . 115
• Bibliography 117
• Abstract in Korean 127
• v

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