Non-Hermitian Type Uncertainty Relation and its Application

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1 9.. The 37th Syposiu on Inoration Theory and its Applications SITA) Unazuki, Toyaa, Japan, Dec. 9, Non-Heritian Type Uncertainty Relation and its Application Kenjiro Yanai Abstract In quantu echanics it is well known that Heisenber/Schrödiner uncertainty relations hold or two non-coutative observables and density operator. These are soe kinds o trace inequalities. Recently Dou and Du [5, 6] obtained several uncertainty relations or two noncoutative non-heritian observables and density operator. In this paper we show that their results can be iven as corollaries o our non-heritian type uncertainty relations or eneralized etric adjusted skew inorations or eneralized etric adjusted correlation easures. Keywords Trace inequality, etric adjusted skew inoration, etric adjusted correlation easure Introduction M n C) n n coplex atrices, M n,sa C) n n sel-adjoint atrices, M n,+ C) M n C) strictly positive eleents, M n,+, C) strictly positive density atrices, M n,+, C) { M n C) T r[], >. aithul states > M n C) Hilbert-Schidt A, B T r[a B] Winer-Yanase skew inoration) [] I H) [ [ ]) ] T r i /, H T r[h ] T r[ / H / H]. M n,+, C) H M n,sa C) coutator [X, Y ] XY Y X Dyson Winer-Yanase-Dyson skew inoration I,α H) T r[i[α, H])i[ α, H])] T r[h ] T r[ α H α H], α [, ] I,α H) E.H.Lieb [7] Winer-Yanase, , Graduate School o Science and Enineerin, Yaauchi University, -6-, Tokiwadai, Ube , Japan, E-ail:yanai@yaauchi-u.ac.jp skew inoration uncertainty relation [9]. Winer-Yanase-Dyson skew inoration uncertainty relation [5, 3] [3, ], uncertainty relation [5] two paraeter uncertainty relation [7] Dou-Du Uncertainty Relations Dou-Du [5, 6] Heisenber/Schrödiner uncertainty relations. A, B M n C), M n,+, C) ) [A, B] [A, B] + [A, B ]), [A, B] AB BA. ) {A, B {A, B + {A, B ), {A, B AB + BA. 3) V ar A) T r[a A ], A A T r[a]i. ) V ar A) V ar A) + V ar A )).. A, B M n C), M n,+, C) uncertainty relations ) V ar A) V ar B) T r[[a, B]]. ) V ar A) V ar B) T r[{a, B ]. 3) V ar A) V ar B) T r[[a, B] ] + T r[{a, B ]. ) U A) U B) T r[[a, B] ], U A) V ar A)) V ar A) I A)), I A) T r[i[/, A ])i[ /, A])]. 67

2 3 :, + ) R n N A B A, B M n,+ C) A) B) operator onotone) operator onotone unction x) xx ) syetric ) norarized 3. F op :, + ), + ). ),. tt ) t), 3. operator onotone. 3. F op RLD x) x x +, W Y x) ) x +, BKM x) x lo x, SLDx) x +, x ) W Y D x) α α) x α )x α, α, ). ) Reark 3. F op onotone etricsquantu Fisher inorations ) A, B, T ra L, R ) B)), L A) A, R A) A. A, B M n,+, C) tanent vectors [], [] ).. []), Fop r k > x ) x) k x) x ) x) x) k F op.) x). A, B M n C), M n,+, C) Corr s,) A, B) k i[, A], i[, B],, x x + x) x +, x >. F op ) li x x) reular non-reular F r op { F op ), F n op { F op ) F op F r op F n op. 3. Fop r x) [ x + ) x ) ) ], x >. x) 3. [8], [], [6]) F r op F n op Generalized Quasi-Metric Adjusted Skew Inoration and Generalized Quasi- Metric Adjusted correlation Measure ean) operator onotone unction) A, B M n,sa C) I,) U,) A) I,) A) Corr s,) A, A), C A, B) T r[a L, R )B], C A) C A, A), C A) + C A))CA) C A)), A), Corr s,) A, B) eneralized quasietric adjusted skew inoration, eneralized quasietric adjusted correlation easure. A, B M n C), M n,+, C) A A T r[a]i, B B T r[b]i.. I,). J,) 3. U,) A) A) I,) A ) CA ) C A ), A) CA ) + C A ), A) J,) A). I,,). Corr s,) A, B) Corr s,) A, B ).. F r op I,) A) I,) B) Corr s,) A, B), A, B) A / A / BA / )A /. A, B M n C), M n,+, C). 675

3 .. X, Y M n C) Corr s,) X, Y ) k i[, X], i[, Y ],. Corr s,) X, Y ) kt ri[, X]) L, R ) i[, Y ]) kt ril R )X) L, R ) il R )Y ) T rx L, R )Y ) T rx L, R )Y ), Corr s,) X, Y ) M n C) Schwarz inequality. Fop r, l > x) + x) lx).) U,) A) U,) B) kl T r[a, B]),.3) A, B M n C), M n,+, C)....).) x, y) x, y) klx y).. :.),.) x y) x, y) x, y) k x, y)..) x, y) + x, y) l x, y),.5) A, B M n C), M n,+, C) A A T r[a]i, B B T r[b]i I,) A) { λ j, λ k ) λ j, λ k ) a jk, J,) A) { λ j, λ k ) + λ j, λ k ) a jk,. :.3) T r[a, B]) T r[a, B]). λ j λ k )a jk b kj, kl T r[a, B]) kl λj λ k a jk b kj λ j λ k a jk b kj. λ j, λ k ) λ j, λ k ) ) / ajk b kj ) λ j, λ k ) λ j u, λ k ) a jk ) λ j, λ k ) + λ j, λ k ) b kj I,) A)J,) B). I,) B)J,) A) cd T r[a, B])..),.5) x, y) x, y) { { x, y) x, y) x, y) + x, y) )x y) x, y) l x, y) klx y)., I,). U,) A) A), J,) A),. { ϕ, ϕ,, ϕ n, {λ, λ,, λ n a jk ϕ j A ϕ k, b jk ϕ j B ϕ k,.3) 5 Dou-Du x), x), k, l x) x +, x ) x) α α) x α )x α, < α < ), ) k ), l. x ) x) x) k x) xα + x α ). 676

4 . x) + x) lx) x α )x α ) {x α )x α) ) α α)x ) α / I,) A) I,) A ) T r[a A ] + T r[a A ] T r[ / A / A ]. 5. Dou-Du )) A, B M n C) M n,+, C) U A) U B) T r[[a, B]] I T r[[a, B]] T r[[a, B]] T r[[a, B]] T r[[a, B]] + T r[[a, B ]] [ T r [A, B] + ] [A, B ] T r[[a, B] ]. 5. Dou-Du ),)) A, B M n C) M n,+, C) ) V A) V B) U A) U B) T r[[a, B]]. ) V A) V B) T r[{a, B ]. ) A, B M n C) x) x+, M n C) A, B T r[a L, R )B ]. Schwarz s inequality A, B A, B A, A B, B. ] [A T r L + R B T r[a B ] + T r[a B ] T r[b A ] + T r[a B ] T r[{a, B ]. A, A T r [ ] A L + R A T r[a A ] + T r[a A ] T r[a A ] + T r[a A ] V ar A) T r[{a, B ] V ar A) V ar B). A A T r[{a, B ] V ar A ) V ar B). V ar A) V ar A ) 6 Reark Dou-Du 3) V ar A) V ar B) 6.) T r[[a, B] ] + T r[{a, B ]. 5. V ar A) V ar B) 6.) U A) U B) T r[[a, B]]. ) ) ) 3 i, A, B. i 6.) < 6.). 3 ) ) ) i, A, B. i 6.) > 6.). Acknowledeent The author was partially supported by JSPS KAK- ENHI Grant Nuber 69. [] K.Audenaert, L.Cai and F.Hansen, Inequalities or quantu skew inoration, Lett.Math.Phys., vol.858), pp [] J.C.Bourin, Soe inequalities or nors on atrices and operators, Linear Alebra and its Applications, vol.9999), pp

5 [3] L.Cai and S.Luo, On convexity o eneralized Winer-Yanase-Dyson inoration, Lett.Math.Phys., vol.838), pp [] P.Chen and S.Luo, Direct approach to quantu extensions o Fisher inoration, Front.Math.China, vol.7), pp [5] Y.N.Dou and H.K.Du, Generalizations o the Heisenber and Schrödiner uncertainty relations, J.Math.Phys., vol.53), pp [6] Y.N.Dou and H.K.Du, Note on the Winer-Yanase- Dyson skew inoration, Int.J.Theor.Phys., vol.53), pp [7] S.Furuichi and K.Yanai, Schrödiner uncertainty relation, Winer-Yanase-Dyson skew inoratio and Metric adjusted correlation easure, J.Math.Anal.Appl., vol.388), pp7-56. [8] P.Gibilisco, D.Iparato and T.Isola, Uncertainty principle and quantu Fisher inoration, II, J.Math.Phys., vol.87), pp [9] P.Gibilisco, D.Iparato and T.Isola, A Robertsontype uncertainty principle and quantu Fisher inoration, Linear Alebra and its Applications, vol.88), pp [] Gibilisco, P., Hansen, F., Isola, T.: On a correspondence between reular and non-reular operator onotone unctions, Linear Alebra and its Applications, vol.39), pp.5-3. [] P.Gibilisco, F.Hiai and D.Petz, Quantu covariance, quantu Fisher inoration, and the uncertainty relations, IEEE Trans.Inoration Theory, vol.559), pp [] P.Gibilisco and T.Isola, On a reineent o Heisenber uncertainty relation by eans o quantu Fisher inoration, J.Math.Anal.Appl., vol.375), pp [3] F.Hansen, Metric adjusted skew inoration, Proc.Nat.Acad.Sci., vol.58), pp [] W.Heisenber, Über den anschaulichen Inhat der quantuechanischen Kineatik und Mechanik, Zeitschrit ür Physik, vol.397), pp [5] H.Kosaki, Matrix trace inequality related to uncertainty principle, Internatonal Journal o Matheatics, vol.65), pp [6] Kubo, F., Ando, T.: Means o positive linear operators, Math.Ann., vol.698), pp.5-. [7] E.H.Lieb, Convex trace unctions and the Winer-Yanase-Dyson conjecture, Adv.Math., vol.973), pp [8] S.Luo, Heisenber uncertainty relation or ixed states, Phys.Rev.A, vol.75), p.. [9] S.Luo and Q.Zhan, On skew inoration, IEEE Trans.Inoration Theory, vol.5), pp , and Correction to On skew inoration, IEEE Trans.Inoration Theory, vol.55), p.3. [] Petz, D.: Monotone etrics on atrix spaces, Linear Alebra and its Applications, vol.996), pp [] E.Schrödiner, About Heisenber uncertainty relation, Proc.Prussian Acad.Sci., Phys.Math., vol.xix93), p.93, Section. [] E.P.Winer and M.M.Yanase, Inoration content o distribution, Proc.Nat.Acad.Sci. U,S,A., vol.9963), pp [3] K.Yanai, S.Furuichi and K.Kuriyaa, A eneralized skew inoration and uncertainty relation, IEEE Trans.Inoration Theory, vol.55), pp.-. [] K.Yanai, Uncertainty relation on Winer-Yanase-Dyson skew inoration, J.Math.Anal.Appl., vol.365), pp.-8. [5] K.Yanai, Uncertainty relation on eneralized Winer-Yanase-Dyson skew inoration, Linear Alebra and its Applications, vol.33), pp [6] K.Yanai, Metric adjusted skew inoration and uncertainty relation, J.Math.Anal.Appl., vol.38), pp [7] K.Yanai, S.Furuichi and K.Kuriyaa, Uncertainty relations or eneralized etric adjusted skew inoration and eneralized etric adjusted correlation easure, J.Uncertainty Anal.Appl., vol.3), pp-. [8] K.Yanai, Generalized eric adjusted skew inoration and uncertainty relation, Proc. ISBFS, vol.iv), pp

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