I report a tight upper bound of the maximum speed of evolution from one
quantum state $\rho$ to another $\rho'$ with fidelity $F(\rho,\rho')$ less than
or equal to an arbitrary but fixed value under the action of a time-independent
Hamiltonian. Since the bound is directly proportional to the average absolute
deviation from the median of the energy of the state ${\mathscr D}E$, one may
interpret ${\mathscr D}E$ as a meaningful measure of the maximum information
processing capability of a system.Comment: 4 pages, 1 figure, minor changes with an additional reference added,
  to appear in PR

H. F. Chau

N. Margolus

English

arXiv

I report a tight upper bound of the maximum speed of evolution from one quantum state ρ to another ρ′ with fidelity F(ρ,ρ′) less than or equal to an arbitrary but fixed value under the action of a time-independent Hamiltonian. Since the bound is directly proportional to the average absolute deviation from the median of the energy of the state DE, one may interpret DE as a meaningful measure of the maximum information-processing capability of a system. © 2010 The American Physical Society.published_or_final_versio

Chau, HF

HKU Scholars Hub

Title Tight upper bound of the maximum speed of evolution of aquantum stateAuthor(s) Chau, HFCitation Physical Review A - Atomic, Molecular, And Optical Physics,2010, v. 81 n. 6Issued Date 2010URL http://hdl.handle.net/10722/123858Rights Creative Commons: Attribution 3.0 Hong Kong LicensePHYSICAL REVIEW A 81, 062133 (2010)Tight upper bound of the maximum speed of evolution of a quantum stateH. F. Chau*Department of Physics and Center of Computational and Theoretical Physics, University of Hong Kong, Pokfulam Road, Hong Kong(Received 7 March 2010; published 30 June 2010)I report a tight upper bound of the maximum speed of evolution from one quantum state ρ to another ρ ′with fidelity F (ρ,ρ ′) less than or equal to an arbitrary but fixed value under the action of a time-independentHamiltonian. Since the bound is directly proportional to the average absolute deviation from the median of theenergy of the stateDE, one may interpretDE as a meaningful measure of the maximum information-processingcapability of a system.DOI: 10.1103/PhysRevA.81.062133 PACS number(s): 03.65.Xp, 03.67.−a, 89.70.EgI. MOTIVATION AND PRIOR WORKIt is impossible to build an arbitrarily fast and powerfulcomputer, quantum or classical, because several fundamentalphysical limits bound the maximum speed of logical operationsand the size of memory space [1]. In particular, Bhattacharyya[2], Uhlmann [3], and Pfeifer [4] found that the time τ neededto evolve a (mixed) state ρ to another state ρ ′ under the actionof a time-independent HamiltonianH is tightly lower-boundedbyτ  τTEUR ≡ h¯ cos−1(√)E≡ gTEUR()πh¯2E, (1)where  = F (ρ,ρ ′) ≡ [Tr(√√ρρ ′√ρ)]2 is the fidelity be-tween the two states andE =√Tr(H 2ρ) − E2 ≡√Tr(H 2ρ) − [Tr(Hρ)]2 (2)is the standard deviation of the energy of the system. [Actually,the authors of Refs. [3] and [4] considered the more generalsituation of a time-dependent Hamiltonian whose results canbe reduced to Eq. (1) in the time-independent case.] Becauseof the form of Eq. (1), it is sometimes called the time-energyuncertainty relation (TEUR) bound. Bounds of this type areinteresting for they depend only on a modest description ofthe system. Later on, Margolus and Levitin [5,6] discoveredanother tight lower bound on the time required to evolve a(pure) state to another state in its orthogonal subspace underthe action of a time-independent Hamiltonian H . Their boundis inversely proportional to the average energy of the systemabove the ground state, E − E0. Giovannetti et al. [7] extendedthe Margolus-Levitin (ML) theorem by showing that the timeτ required to evolve between two (mixed) states with fidelityless than or equal to a fixed  ∈ [0,1] under the action of atime-independent Hamiltonian is tightly lower-bounded byτ  τML ≡ gML()πh¯2(E − E0) (3)for some smooth function gML. Although no closed-formexpression is known for gML, it can be approximated to withina few percent of error by [7]gML() ≈ [gTEUR()]2 . (4)*hfchau@hkusua.hku.hkMore importantly, Giovannetti et al. found examples inwhich the ML bound in Eq. (3) is better than the TEURbound [7]. Note that the smaller the τ , the faster the systemcan be used for quantum-information processing. In thisrespect, the tight bounds in Eqs. (1) and (3) show thatE and E − E0 are reasonable measures of the maximumpossible quantum-information-processing rate of a system[1,5–7].Here I report another tight lower bound on the time neededto evolve from one (mixed) state to another under the action of atime-independent Hamiltonian such that the fidelity is less thanor equal to a fixed value  ∈ [0,1]. Recall from the discussionsof Margolus and Levitin in Refs. [5,6] that the faster the timeτ for a quantum system to evolve between two orthogonalstates, the more powerfully the system can process quantuminformation. In this respect, a lower bound of the time τ posesa so-called quantum speed limit on the maximum quantum-information-processing rate of the system. This notion of aquantum speed limit was then generalized by Giovannetti et al.to the study of evolution between two nonorthogonal states [7].Since the tight evolution-time bound reported here is inverselyproportional to the so-called average absolute deviation fromthe median (AADM) of the energy of the state DE of thesystem, I conclude that DE is also a reasonable measure ofthe maximum possible quantum-information-processing rateof a system. Finally, I compare this bound with the TEURbound [2–4] and the ML bound [5–7].II. THE EVOLUTION-TIME BOUNDA. An auxiliary inequalityI begin by considering an inequality with a simple geometricmeaning. The first quadrant of Fig. 1 depicts the (unique) linewith the greatest slope that passes through the origin and meetsthe curve y = 1 − cos x at two distinct points (namely, x = 0and x = xm). Clearly, this line is the tangent to the curve atx = xm; and its slope A is given byA = max{1 − cos xx: x > 0}. (5)Numerically, I find thatA ≈ 0.724 611 (6a)andxm ≈ 2.331 12. (6b)1050-2947/2010/81(6)/062133(4) 062133-1 ©2010 The American Physical SocietyH. F. CHAU PHYSICAL REVIEW A 81, 062133 (2010)10 5 0 5 10x0.51.01.52.02.53.0yFIG. 1. The curve y = 1 − cos x and the broken line y = A|x|defined in the text.By considering the mirror image of this line with respectedto the y axis, it is obvious thatcos x  1 − A|x| (7)for all x ∈ R. (The geometric meaning of this inequality isapparent from Fig. 1.)B. The pure-state caseNow, I may use Margolus and Levitin’s argument inRefs. [5,6] to obtain the required bound for the case of purestates. Suppose |(0)〉 =∑j αj |Ej 〉, where the |Ej 〉’s arethe normalized energy eigenvectors of the time-independentHamiltonian H and∑j |αj |2 = 1. Then, under the actionof H ,〈(0)|(t)〉 ≡ 〈(0)|e−iH t/h¯|(0)〉 =∑j|αj |2e−iEj t/h¯. (8)In other words, at the time when the system evolves to a statewhose fidelity is less than or equal to  from |(0)〉, the realpart of Eq. (8) obeys∣∣∣∣∣∣∑j|αj |2 cos(−Ej th¯)∣∣∣∣∣∣ √. (9)Applying the inequality in Eq. (7) to Eq. (9), I get1 − Ath¯∑j|αj |2|Ej | √. (10)Therefore, the earliest time τ at which |(0)〉 evolves to a statewhose fidelity is less than or equal to  from |(0)〉 satisfiesthe inequalityτ  (1 −√) h¯A∑j |αj |2|Ej |. (11)By means of the fact that the reference energy level of asystem has no physical meaning, I can further strengthen thebound in Eq. (11) as follows: Recall that the function f (x) =∑j |αj |2|Ej − x| attains its minimum when x equals M , themedian of the Ej ’s with relative frequency of occurrence ofEj equal to |αj |2. More precisely, consider the cumulativedistribution functionC(x) =∑j :Ejx|αj |2. (12)ThenM = 12(limy→0.5−C−1(y) + limy→0.5+C−1(y)). (13)(The above assertion can be proven by checking whendf/dx = 0.) In statistics, the quantityDE ≡∑j|αj |2|Ej − M|= Tr[√(H − M)†(H − M)|(0)〉〈(0)|] (14)is known as the AADM of the energy. Thus, I conclude thatτ  τC ≡ (1 −√) h¯A∑j |αj |2|Ej − M|≡ (1 −√) h¯A DE≡ gC() h¯A DE.(15)C. The mixed-state caseTo extend the above bound to cover the case of mixed states,one simply needs to repeat the argument used by Giovannettiet al. in Ref. [7]: One can always purify the initial and finalmixed states. And one may consider a particular choice of thepurified states such that the two sets of orthonormal state ketsof the ancillary systems used in the purification are identical.Clearly, for other choice of the purified states, the evolutiontime τ can never be shorter than the above choice. In addition,the fidelity between this pair of particularly chosen purifiedstates does not exceed the fidelity between the original pairof mixed states. By applying the time bound in Eq. (15) tothis particular choice of purified states, one concludes that thebound is also applicable to mixed states [7].D. Tightness of the boundThe time bound in Eq. (15) is certainly tight when  = 1.Hence, to show that this bound is tight for all , I need only toconsider the case of  < 1. Let me consider the state|ϕ(0)〉 = √1 − α|0〉 +√α2| − E〉 +√α2|E〉, (16)whereα = 1 −√Axm≈ 0.592 011(1 − √) ∈ [0,1]. (17)Furthermore, |0〉, |E〉, and | − E〉 are normalized energyeigenkets with energies 0, E , and −E , respectively. Notethat 〈ϕ(0)|ϕ(t)〉 = 1 − α + α cos(E t/h¯) is a real-valued sinu-soidal function of t . Besides, it starts to decrease at t = 0until t = πh¯/E . Therefore, the earliest time τ at whichF (|(0)〉,|(τ )〉) = |〈(0)|(τ )〉|2   obeys√ = 1 − α + α cos(Eτh¯)= 1 − α + α cos(τ DEαh¯).(18)062133-2TIGHT UPPER BOUND OF THE MAXIMUM SPEED OF . . . PHYSICAL REVIEW A 81, 062133 (2010)TABLE I. Comparison between the three lower bounds on τ for  = 0.|(0)〉 τ τTEUR τML τC1√2 (| − E〉 + |E〉) πh¯2E πh¯2E = τ πh¯2E = τ h¯AE ≈ 0.879τ|ϕ(0)〉 as defined in Eq. (16) h¯AαEπh¯2√αE ≈ 0.876τ πh¯2E ≈ 0.674τ h¯AαE = τ|ϕ(0)〉 as defined in Eq. (16) but 7πh¯12E πh¯√4−√2+√6Eπh¯2E4h¯AEwith α = 44−√2+√6 ≈ 0.794 ≈0.764τ ≈0.857τ ≈0.598τ1√3 (|0〉 + | − E〉 + |E〉) 2πh¯3E πh¯E√38 ≈ 0.919τ πh¯2E = 0.75τ 3h¯2AE ≈ 0.988τ1√2n∑n−1k=0[| − (k + 12 )E〉 + |(k + 12 )E〉] πh¯nE (n√34n2−1 )τ ( n2n−1 )τ ( 2Aπ )τ≈0.866τ for large n ≈0.5τ for large n ≈0.879τ1√2n+1∑nk=−n |kE〉 2πh¯(2n+1)E ( 2n+14√3n(n+1) )τ ( 2n+14n )τ ( (2n+1)22n(n+1)πA )τ≈0.866τ for large n ≈0.5τ for large n ≈0.879τ for large nFrom Eq. (17), I arrive at(1 − √) cos(τAxmDE(1 − √) h¯)= (1 − √)(1 − Axm)= (1 − √) cos xm. (19)Note that I have used the fact that the line y = Ax intersectswith the curve y = 1 − cos x at x = xm to arrive at the lastline of the above equation. Since  < 1, the general solutionof Eq. (19) isτADE(1 − √)h¯ = 1 +2nπxm(20)for all n ∈ Z. From Eq. (6b), I know that 2π/xm > 1.Therefore, the earliest time τ at which |〈ϕ(0)|ϕ(τ )〉|2 = obeys τADE/[(1 − √)h¯] = 1. Thus, the bound stated inEq. (15) is tight. After all the discussions above, it is clearthat the maximum speed of evolution of a quantum stateunder the action of a time-independent Hamiltonian H istightly upper-bounded by ADE/[(1 − √)h¯]. And since thereciprocal of the speed of evolution of a quantum systemsignifies its quantum-information-processing rate [1,5–7], theAADM of the energy DE is also a reasonable measure of themaximum possible quantum-information-processing rate of asystem.III. COMPARISON WITH EXISTINGMINIMUM-EVOLUTION-TIME BOUNDSNow, I start to compare the performance of the three boundsbased on E, E − E0, and DE for fixed values of . Table Ishows the values of these three bounds when  = 0 for a fewcases in which the τ ’s are known. Clearly, the three boundscomplement each other. Moreover, τC is the best wheneverDE/E and DE/(E − E0) are small. This finding is easy tounderstand. From Eqs. (1), (3), and (15), it is clear that for afixed value of , the performances of these three bounds aredetermined by the ratio E:E − E0:DE. And the τC boundworks best when DE 	 min(E,E − E0).Observe that E − E0 is the average absolute deviation fromthe ground-state energy E0. (Consequently, the three boundsare in fact based on three different statistical dispersion mea-sures of the eigenvalues of H whose frequencies of occurrenceare given by the |αj |2’s.) So, from our earlier discussions onAADM, DE  E − E0. Furthermore, by a straight-forwardapplication of the Cauchy-Schwarz inequality, one can showthat DE  E. Note, however, that even though DE  Eand E − E0, it is still possible for the other two bounds to out-perform Eq. (15) because the ratio gTEUR():gML():2gC()/πalso plays a role in determining which bound is better. But inany case, if the distribution formed by the eigenvalues of Hwhose frequencies of occurrence are given by the |αj |2’s hasa small kurtosis (whose value depends on , of course), thenthe time bound due to DE is better than the other two. Asan illustration, I consider the special case in which  = 0 andthe eigenvalues of H are drawn uniformly from an interval[a,b]. The expected values of E, E − E0, and DE are(b − a)√3/6, (b − a)/2, and (b − a)/4, respectively. Thus,as the Hilbert space dimension of the state ket increases, theratio τTEUR:τML:τC approaches√3:1:4/(πA) ≈ 1.732:1:1.757for a typical state ket |(0)〉. So, as a rule of thumb, τC has agood chance of giving a better time bound for τ when  ≈ 0provided that the kurtosis of the distribution of eigenvaluesof H is greater than or equal to the kurtosis of a uniformdistribution, namely, −6/5.Finally, I study the effect of  on the performance ofthe three bounds. Note from Eqs. (1), (4), and (15) thatgTEUR(0) = gML(0) = gC(0) = 1. Moreover, by differentiat-ing gC()/gTEUR() and gC()/gML() with respect to ,I conclude that gC()/gTEUR() and gC()/gML() are de-creasing and increasing functions of , respectively. In fact,lim→0+ gC()/gTEUR() = 0. In other words, for a sufficientlysmall value of , it is likely that τTEUR  τC  τML.IV. CONCLUSIONSTo summarize, I presented a tight lower bound τC for thetime required to evolve between two states with fidelity lessthan or equal to  under the action of a time-independentHamiltonian. This time bound τC works best when the fidelitybetween the two states, , is small and the kurtosis of the062133-3H. F. CHAU PHYSICAL REVIEW A 81, 062133 (2010)distribution of eigenvalues of H is >∼−6/5. My result alsoimplies that the AADM of the energy DE is a reasonablemeasure of the maximum quantum-information-processingrate of a system.ACKNOWLEDGMENTSI thank C.-H. F. Fung, K. Y. Lee, and H.-K. Lo fortheir discussions. This work is supported by the RGC GrantNo. HKU 700709P of the HKSAR Government.[1] S. Lloyd, Nature (London) 406, 1047 (2000).[2] K. Bhattacharyya, J. Phys. A 16, 2993 (1983).[3] A. Uhlmann, Phys. Lett. A 161, 329 (1992).[4] P. Pfeifer, Phys. Rev. Lett. 70, 3365 (1993); 71, 306(E) (1993).[5] N. Margolus and L. B. Levitin, in Proceedings of the FourthWorkshop on Physics and Computation, edited by T. Toffoli,B. Biafore, and J. Lea˜o (New England Complex SystemsInstitute, Boston, 1996), pp. 208–211 .[6] N. Margolus and L. B. Levitin, Physica D 120, 188(1998).[7] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. A 67,052109 (2003).062133-4

Tight Upper Bound Of The Maximum Speed Of Evolution Of A Quantum State

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