Parallel Session BThe speed of any quantum process is limited by quantum mechanics via time-energy uncertainty relations and they imply that time and energy are tradeoff with each other. As such, we propose to measure the time-energy as a single unit for quantum channels. We consider a time-energy measure for quantum channels and compute lower and upper bounds of it using the channel Kraus operators. For a special class of channels (which includes the depolarizing channel), we obtain the exact value of the time-energy measure. Our result can be used to compare the time-energy resources of similar quantum processes. In particular, we show that erasing quantum information requires √(n + 1)/n) times more time-energy resource than erasing classical information, where n is the system dimension. This work is published in [1].postprin

Chau, HF

Fung, FCH

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Title Measuring time-energy resources for quantum processesAuthor(s) Fung, FCH; Chau, HFCitation The 13th Asian Conference on Quantum Information Science(AQIS'13), Chennai, India, 25-30 August 2013.Issued Date 2013URL http://hdl.handle.net/10722/190222Rights Creative Commons: Attribution 3.0 Hong Kong LicenseMeasuring Time-Energy Resources for Quantum ProcessesChi-Hang Fred Fung1 ∗ H. F. Chau11 Department of Physics and Center of Theoretical and Computational Physics, University of Hong Kong, PokfulamRoad, Hong KongAbstract. The speed of any quantum process is limited by quantum mechanics via time-energy uncer-tainty relations and they imply that time and energy are tradeoff with each other. As such, we propose tomeasure the time-energy as a single unit for quantum channels. We consider a time-energy measure forquantum channels and compute lower and upper bounds of it using the channel Kraus operators. For aspecial class of channels (which includes the depolarizing channel), we obtain the exact value of the time-energy measure. Our result can be used to compare the time-energy resources of similar quantum processes.In particular, we show that erasing quantum information requires√(n+ 1)/n times more time-energy re-source than erasing classical information, where n is the system dimension. This work is published in [1].Keywords: Quantum process, time-energy measure, time-energy uncertainty relationQuantum processes in nature and quantum computa-tion processes designed by human all require time andenergy to evolve. The evoluation speed of a physical de-vice is governed by physical laws and is limited by theenergy of the device. Under the constraints of quantummechanics, time-energy uncertainty relations (TEURs)set limits on system evolutions [2]. The investigation ofTEURs has a long history. Mandelstam and Tamm [3]proved the first major result of a TEUR. This was fol-lowed by subsequent work on isolated systems (e.g., [4, 5])and composite systems with entanglement (e.g., [6]) Re-cently, TEURs for general quantum processes have alsobeen proved [7, 8]. The general form of TEURs is aninequality that sets a lower limit on the product of thesystem energy (or a function of the energies) and the timeit takes to evolve an initial state to a final state (e.g., anorthogonal state). One implication of the form of TEURsis that time and energy are tradeoff against each other.Thus, we propose to regard time-energy as a single prop-erty of a quantum process. The intuition is that the morecomputation or work a quantum process performs, themore time-energy it requires. And it is up to the systemdesigner (or nature) to perform it with more time butless energy, or vice versa. Thus, our goal in this paper isto investigate the time-energy requirements of quantumprocesses by using a time-energy measure. Research inthis direction has been carried out before. Chau [9] pro-posed a time-energy measure for unitary transformationsthat is based on a TEUR proved earlier [10]. In this pa-per, we extend this measure to quantum processes. TheTEUR due to Chau [10] is tight in the sense that it canbe saturated by some states and Hamiltonians, and thusit serves to motivate a good definition for a time-energymeasure. To see this, let’s start with this TEUR. Givena time-independent Hamiltonian H of a system, the timet needed to to evolve a state |Φ〉 under the action of H toa state whose fidelity 1 is less than or equal to  satisfies∗chffung@hku.hk1We adopt the fidelity definition F (ρ, σ) =(Tr√ρ1/2σρ1/2)2for two quantum states ρ and σ.the TEURt ≥ (1−√)~A∑j |αj |2|Ej |(1)where Ej ’s are the eigenvalues of H with the corre-sponding normalized energy eigenvectors |Ej〉’s, |Φ〉 =∑j αj |Ej〉, and A ≈ 0.725 is a universal constant. Essen-tially, after time t, the state transforms unitarily accord-ing to U = e−iHt/~. The same U could be implementedwith either a high energy H run for a shorter time or alow energy H run for a longer time. Based on Eq. (1),a weighted sum of |tEj |’s serves as an indicator of thetime-energy resource needed to perform U , and as suchthe following time-energy measure on unitary matriceswas proposed by Chau [9]:‖U‖~µ =r∑j=1µj |θj |↓where U has eigenvalues exp(−iEjt/~) ≡ exp(θj), and|θj |↓ denotes |θj | ordered non-increasingly |θ1|↓ ≥ |θ2|↓ ≥· · · ≥ |θr|↓. Also, ~µ = [µ1, µ2, . . . , µr] 6= ~0 with µ1 ≥ µ2 ≥· · · ≥ µr ≥ 0. Note that ‖U‖~µ satisfies the multiplicativetriangle inequality ‖UV ‖~µ ≤ ‖U‖~µ+‖V ‖~µ [9]. In essence,a large value of ‖U‖~µ suggests that a long time may beneeded to run a Hamiltonian that implements U for afixed energy, and vice versa.In this paper, we are interested in an analogous mea-sure for quantum channels which include unitary trans-formations as special cases. We are given a quantumchannel F(ρ) acting on system A that maps n × n den-sity matrix ρ to another one with the same dimension.There exist unitary extensions UBA in a larger Hilbertspace with an ancillary system B such that F(ρ) =TrB [UBA(|0〉B〈0| ⊗ ρA)U†BA]. Each UBA could have adifferent time-energy spectrum and we want to select theone requiring the least resource for F . We extend theresource indicator for U to quantum channel F by defin-ing‖F‖~µ ≡ minU‖U‖~µs.t.F(ρ) = TrB [UBA(|0〉B〈0| ⊗ ρA)U†BA] ∀ρ.This gives a U that consumes the least time-energy re-source. Thus, ‖F‖~µ is an indicator of the resource neededto perform F .There are some interesting consequences by using thistime-energy measure. In particular, we can compare thetime-energy resources needed to erase quantum informa-tion and classical information. We show that√(n+ 1)/ntimes more resource is required in the quantum settingthan in the classical setting. Also, we study the time-energy scaling of consecutive runs of the depolarizingchannel. It turns out that the amount of time-energyresource needed for k runs of the depolarizing channelscales as√k when the noise is small.In this work, we focus on two special cases of the time-energy measure:• Sum time-energy: ‖U‖sum ≡r∑j=1|θj | .• Max time-energy: ‖U‖max ≡ max1≤j≤r|θj | = |θ1|↓.Note that the subscript “sum” is short for ~µ = [1, 1, . . . , 1]and “max” for ~µ = [1, 0, . . . , 0].For these two cases, we derive lower and upper boundson the time-energy resource measure ‖F‖max and ‖F‖sumfor any quantum channel F given its Kraus operators.The derivation is based on analyzing a few intermediateoptimization problems. It turns out that the lower andupper bounds are all dependent on the eigenvalues ofsome Kraus operator of F . Specifically, we prove that‖F‖max ≥ minv: ‖v‖≤1max1≤i≤ncos−1[Re(λi(d∑j=1vjFj))](2)‖F‖max ≤ minv: ‖v‖≤1n∑i=1cos−1[Re(λi(d∑j=1vjFj))](3)‖F‖sum ≥ minv: ‖v‖=1max1≤i≤n2 cos−1 |λi(d∑j=1vjFj)| (4)‖F‖sum ≤ minv: ‖v‖≤1n∑i=12 cos−1[Re(λi(d∑j=1vjFj))](5)where ‖v‖ =√∑dj=1 |vj |2, Fj ∈ Cn×n, j = 1, . . . , d arethe Kraus operators of F , and λi(·) denotes the ith eigen-value of its argument. Note that F(ρ) =∑dj=1 FjρF †j .For a class of channels (which includes the depolariz-ing channel), we obtain the exact value for ‖F‖max. Inparticular, when F is a depolarizing channel with proba-bility q that the input state is unchanged, its time-energyrequirement is ‖F‖max = cos−1√q + (1− q)/n2.We summarize the approach used to establish the lowerand upper bounds. We cast the original problem of find-ing the most time-energy efficient U that implements Fas the problem of finding a U that transforms some giveninitial vectors to some given final vectors. The time-energy of this U is lower bounded by that of any U ′ thattransforms a subset of the vectors. This is essence ofhow we obtain a lower bound for ‖F‖max and ‖F‖sum, bysearching for U ′ that transforms only one vector. To de-rive the upper bound, we construct a sequence of single-vector transformations such that their product gives theoriginal U . Since ‖U‖~µ satisfies the multiplicative trian-gle inequality ‖UV ‖~µ ≤ ‖U‖~µ + ‖V ‖~µ [9], we obtain theupper bound. Thus, both the derivations of the lower andupper bounds rely on the solution to the single-vector-transformation problem.We make a few other remarks. A related concept abouterasure and energy is the Landauer’s principle [11] whichputs lower limits on the energy dissipated to the envi-ronment in erasing (qu)bits. There is a difference be-tween the erasure considered here and the erasure of theLandauer’s principle. For future investigation, it is in-structive to obtain the time-energy for various quantumprocesses such as some standard gates or algorithms, toconsider this time-energy measure in the thermodynamicsetting, and to explore deeper operational meaning aboutthis measure.References[1] C.-H. F. Fung and H. F. Chau. Time-energy measurefor quantum processes. Phys. Rev. A, 88:012307, Jul2013.[2] S. Lloyd. Ultimate physical limits to computation.Nature, 406:1047–1054, 2000.[3] L. Mandelstam and I. Tamm. The uncertainty re-lation between energy and time in nonrelativisticquantum mechanics. J. Phys. USSR, 9:249–254,1945.[4] K. Bhattacharyya. Quantum decay and themandelstam-tamm-energy inequality. J. Phys. A,16(13):2993, 1983.[5] J. Anandan and Y. Aharonov. Geometry of quantumevolution. Phys. Rev. Lett., 65:1697–1700, Oct 1990.[6] V. Giovannetti, S. Lloyd, and L. Maccone. The roleof entanglement in dynamical evolution. Europhys.Lett., 62(5):615, 2003.[7] M. M. Taddei, B. M. Escher, L. Davidovich, andR. L. de Matos Filho. Quantum speed limit forphysical processes. Phys. Rev. Lett., 110:050402, Jan2013.[8] A. del Campo, I. L. Egusquiza, M. B. Plenio, andS. F. Huelga. Quantum speed limits in open systemdynamics. Phys. Rev. Lett., 110:050403, Jan 2013.[9] H. F. Chau. Metrics on unitary matrices andtheir application to quantifying the degree of non-commutativity between unitary matrices. Quant.Inf. Compu., 11:0721–0740, 2011.[10] H. F. Chau. Tight upper bound of the maximumspeed of evolution of a quantum state. Phys. Rev.A, 81:062133, Jun 2010.[11] R. Landauer. Irreversibility and heat generation inthe computing process. IBM J. Res. Dev., 5(3):183–191, 1961.

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