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 「Mathematical Studies on Quantum Systems and Locally Quantum Systems」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。

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

Mathematical Studies on Quantum Systems and Locally Quantum Systems

吉田, 裕哉 名古屋大学

2022.01.05

概要

一般確率論(GPTs)と呼ばれる理論モデルがあり,そこでは状態,測定,測定値を得る確率が扱える.たとえば,古典確率論と量子論は一般確率論であり,それぞれ状態として,確率ベクトルと密度行列を持つ.しばしば一般確率論では,有限個のGPT A!,...,Anをまとめ,1つのGPTΑ=Α1···Αηとして扱う.この場合,を部分系,义を全体系と呼ぶ.部分系が全て量子系であるものは局所量子系と呼ばれ,量子系も局所量子系の1つである.しかし,量子系でない局所量子系はいくらでも存在する.本論文では,量子系や局所量子系に関する3つの話題を調べる.

 1つ目は,容量である.各GPTに対して,同時かつ完全に識別可能な状態の最大の個数を容量と呼ぶ.容量はあまりGPTに依存しないことが知られている.本論文では,特殊な局所量子系の容量を導出するための主張Sを提示し,それより弱い主張'WSを示す.主張WSは複素数体上のテンソル積空間に関する主張であり,複素数体を一般の無限体に置き換えても同じ証明が機能する.有限体の場合には部分的な結果を示す.

 2つ目は,差分プライバシーである.差分プライバシーは,個人情報を保護しつつ活用するための研究(プライバシー保護データマイニング)から生まれた.そのため,古典確率論に基づいた研究がほとんどであり,その量子拡張はほとんど研究されていない.本論文では,差分プライバシーの量子拡張の1つ(古典量子差分プライバシー)を定め,その数学的側面を調べる.古典量子差分プライバシーを満たすπ個の量子状態の組全体の集合CQnと,本質的に古典的であるCQnの部分集合ECnを定める.πが2ならECn=CQnであるが,ηが3以上ならECn≠CQnであることを示す.

 3つ目は,2状態完全識別である.量子論では,直交性が2状態を完全に識別するための必要十分条件である(この同値性をEと記す).しかし,一般のGPTに対して同値性Eが成り立つとは限らない.本論文では,局所量子系からなる比較的自然な連続1パラメーター族を構成し,そこでの2状態完全識別を調べる.その連続1パラメーター族は量子系を含むため,量子系にいくらでも近い局所量子系を含んでいる.それにも関わらず,量子系以外では同値性Eが成り立たないことを示す.

参考文献

[1] S. Aaronson and G. N. Rothblum. Gentle measurement of quantum states and differential privacy. In STOC’19—Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 322–333. ACM, New York, 2019.

[2] N. Alon and L. Lov´asz. Unextendible product bases. J. Combin. Theory Ser. A, 95(1):169–179, 2001.

[3] H. Arai, Y. Yoshida, and M. Hayashi. Perfect discrimination of non-orthogonal separable pure states on bipartite system in general probabilistic theory. J. Phys. A, 52(46):465304, 14, 2019.

[4] G. Aubrun and S. J. Szarek. Tensor products of convex sets and the volume of separable states on n qudits. Phys. Rev. A, 73(2):022109, 10, 2006.

[5] P. Bag, S. Dey, M. Nagisa, and H. Osaka. The order-n minors of certain (n + k) × n matrices. Linear Algebra Appl., 603:368–389, 2020.

[6] H. Barnum and J. Hilgert. Strongly symmetric spectral convex bodies are jor- dan algebra state spaces. preprint, available at https://arxiv.org/abs/1904. 03753, 2019.

[7] C. H. Bennett, D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal. Unextendible product bases and bound entanglement. Phys. Rev. Lett., 82(26, part 1):5385–5388, 1999.

[8] S. Boyd and L. Vandenberghe. Convex optimization. Cambridge University Press, Cambridge, 2004.

[9] A. Chefles, R. Jozsa, and A. Winter. On the existence of physical transfor- mations between sets of quantum states. Int. J. Quantum Inf., 2(01):11–21, 2004.

[10] J. Chen and N. Johnston. The minimum size of unextendible product bases in the bipartite case (and some multipartite cases). Comm. Math. Phys., 333(1):351–365, 2015.

[11] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt. Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett., 23:880–884, 1969.

[12] T. Cubitt, A. Montanaro, and A. Winter. On the dimension of subspaces with bounded Schmidt rank. J. Math. Phys., 49(2):022107, 6, 2008.

[13] D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal. Unex- tendible product bases, uncompletable product bases and bound entanglement. Comm. Math. Phys., 238(3):379–410, 2003.

[14] A. C. Doherty, P. A. Parrilo, and F. M. Spedalieri. Complete family of separa- bility criteria. Phys. Rev. A, 69(2):022308, 20, 2004.

[15] Y. Du, M.-H. Hsieh, T. Liu, D. Tao, and N. Liu. Quantum noise protects quantum classifiers against adversaries. Phys. Rev. Research, 3(2):023153, 18, 2021.

[16] Y. Du, M.-H. Hsieh, T. Liu, S. You, and D. Tao. Quantum differentially pri- vate sparse regression learning. preprint, available at https://arxiv.org/abs/ 2007.11921, 2020.

[17] J. C. Duchi, M. I. Jordan, and M. J. Wainwright. Local privacy and statis- tical minimax rates. In 2013 IEEE 54th Annual Symposium on Foundations of Computer Science—FOCS 2013, pages 429–438. IEEE Computer Soc., Los Alamitos, CA, 2013.

[18] C. Dwork. Differential privacy. In Automata, languages and programming, Part II, volume 4052 of Lecture Notes in Comput. Sci., pages 1–12. Springer, Berlin, 2006.

[19] C. Dwork, F. McSherry, K. Nissim, and A. Smith. Calibrating noise to sensitivity in private data analysis. In Theory of cryptography, volume 3876 of Lecture Notes in Comput. Sci., pages 265–284. Springer, Berlin, 2006.

[20] I. Ekeland and R. T´emam. Convex analysis and variational problems, volume 28 of Classics in Applied Mathematics. Society for Industrial and Applied Math- ematics (SIAM), Philadelphia, PA, english edition, 1999. Translated from the French.

[21] K. Feng. Unextendible product bases and 1-factorization of complete graphs. Discrete Appl. Math., 154(6):942–949, 2006.

[22] Q. Geng, P. Kairouz, S. Oh, and P. Viswanath. The staircase mechanism in differential privacy. IEEE J. Sel. Topics Signal Process., 9(7):1176–1184, 2015.

[23] Q. Geng and P. Viswanath. The optimal noise-adding mechanism in differential privacy. IEEE Trans. Inform. Theory, 62(2):925–951, 2016.

[24] Q. Geng and P. Viswanath. Optimal noise adding mechanisms for approximate differential privacy. IEEE Trans. Inform. Theory, 62(2):952–969, 2016.

[25] L. Gurvits and H. Barnum. Largest separable balls around the maximally mixed bipartite quantum state. Phys. Rev. A, 66(6):062311, 7, 2002.

[26] M. Hayashi. Quantum Information Theory: Mathematical Foundation, Second Edition. Springer, Berlin, Heidelberg, 2017.

[27] M. Hayashi, S. Ishizaka, A. Kawachi, G. Kimura, and T. Ogawa. Introduction to Quantum Information Science. Springer, Berlin, Heidelberg, 2015.

[28] R. Hildebrand. Entangled states close to the maximally mixed state. Phys. Rev. A, 75(6):062330, 10, 2007.

[29] N. Holohan, D. J. Leith, and O. Mason. Extreme points of the local differential privacy polytope. Linear Algebra Appl., 534:78–96, 2017.

[30] N. Holohan, D. J. Leith, and O. Mason. Optimal differentially private mecha- nisms for randomised response. IEEE Trans. Inf. Forensics Secur., 12(11):2726– 2735, 2017.

[31] M. Horodecki, P. Horodecki, and R. Horodecki. Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A, 223(1-2):1–8, 1996.

[32] P. Horodecki. Separability criterion and inseparable mixed states with positive partial transposition. Phys. Lett. A, 232(5):333–339, 1997.

[33] Z. Huang, C.-K. Li, E. Poon, and N.-S. Sze. Physical transformations between quantum states. J. Math. Phys., 53(10):102209, 12, 2012.

[34] N. Jacobson. Basic algebra. I. W. H. Freeman and Company, New York, second edition, 1985.

[35] P. Janotta. Generalizations of boxworld. In Proc. 8th International Workshop on Quantum Physics and Logic, volume 95, pages 183–192, 2012.

[36] P. Janotta and H. Hinrichsen. Generalized probability theories: what determines the structure of quantum theory? J. Phys. A, 47(32):323001, 32, 2014.

[37] P. Jordan, J. von Neumann, and E. Wigner. On an algebraic generalization of the quantum mechanical formalism. Ann. of Math. (2), 35(1):29–64, 1934.

[38] P. Kairouz, S. Oh, and P. Viswanath. Extremal mechanisms for local differential privacy. J. Mach. Learn. Res., 17:Paper No. 17, 51, 2016.

[39] M. Koecher. Positivit¨atsbereiche im Rn. Amer. J. Math., 79:575–596, 1957.

[40] L. Lami, C. Palazuelos, and A. Winter. Ultimate data hiding in quantum me- chanics and beyond. Comm. Math. Phys., 361(2):661–708, 2018.

[41] L. Masanes and M. P. Mu¨ller. A derivation of quantum theory from physical requirements. New J. Phys., 13(6):063001, 29, 2011.

[42] K. Matsumoto and G. Kimura. On additivity of strong converse bound of noiseless channels in locally quantum systems —in relation to the radius of the separable ball—. In Proc. of The 37th Quantum Information Technology Symposium (QIT37), pages 13–16, 2017. https://www.ieice.org/ken/paper/ 20171116Z1AP/eng/.

[43] M. P. Mu¨ller, O. C. O. Dahlsten, and V. Vedral. Unifying typical entanglement and coin tossing: on randomization in probabilistic theories. Comm. Math. Phys., 316(2):441–487, 2012.

[44] M. P Mu¨ller and C. Ududec. Structure of reversible computation determines the self-duality of quantum theory. Phys. Rev. Lett., 108:130401, 5, 2012.

[45] K. R. Parthasarathy. On the maximal dimension of a completely entangled subspace for finite level quantum systems. Proc. Indian Acad. Sci. Math. Sci., 114(4):365–374, 2004.

[46] M. Pl´avala and M. Ziman. Popescu-Rohrlich box implementation in general probabilistic theory of processes. Phys. Lett. A, 384(16):126323, 6, 2020.

[47] S. Popescu and D. Rohrlich. Quantum nonlocality as an axiom. Found. Phys., 24(3):379–385, 1994.

[48] A. J. Short and S. Wehner. Entropy in general physical theories. New J. Phys., 12(March):033023, 34, 2010.

[49] A. Uhlmann. The transition probability for states of ∗-algebras. Ann. Physik (7), 42(4-6):524–532, 1985.

[50] E`. B. Vinberg. Homogeneous cones. Soviet Math. Dokl., 1:787–790, 1960.

[51] N. R. Wallach. An unentangled Gleason’s theorem. In Quantum computation and information (Washington, DC, 2000), volume 305 of Contemp. Math., pages 291–298. Amer. Math. Soc., Providence, RI, 2002.

[52] S. L. Warner. Randomized response: a survey technique for eliminating evasive answer bias. J. Amer. Statist. Assoc., 60(309):63–69, 1965.

[53] Y. Yoshida, H. Arai, and M. Hayashi. Perfect discrimination in approx- imate quantum theory of general probabilistic theories. Phys. Rev. Lett., 125(15):150402, 5, 2020.

[54] Y. Yoshida and M. Hayashi. Asymptotic properties for Markovian dynamics in quantum theory and general probabilistic theories. J. Phys. A, 53(21):215303, 43, 2020.

[55] Y. Yoshida, M.-H. Yung, and M. Hayashi. Optimal mechanism for random- ized responses under universally composable security measure. In 2019 IEEE International Symposium on Information Theory (ISIT), pages 547–551. IEEE, Paris, France, 2019.

[56] L. Zhou and M. Ying. Differential privacy in quantum computation. In 2017 IEEE 30th Computer Security Foundations Symposium (CSF), pages 249–262. IEEE Computer Soc., Santa Barbara, CA, 2017.

参考文献をもっと見る

全国の大学の
卒論・修論・学位論文

一発検索!

この論文の関連論文を見る