Quantum speed limit for arbitrary initial states

Quantum speed limit for arbitrary initial states

Abstract

We investigate the generic bound on the minimal evolution time of the open dynamical quantum system. This quantum speed limit time is applicable to both mixed and pure initial states. We then apply this result to the damped Jaynes-Cummings model and the Ohimc-like dephasing model starting from a general time-evolution state. The bound of this time-dependent state at any point in time can be found. For the damped Jaynes-Cummings model, the corresponding bound first decreases and then increases in the Markovian dynamics. While in the non-Markovian regime, the speed limit time shows an interesting periodic oscillatory behavior. For the case of Ohimc-like dephasing model, this bound would be gradually trapped to a fixed value. In addition, the roles of the relativistic effects on the speed limit time for the observer in non-inertial frames are discussed.

pacs:
03.65.Yz, 03.67.Lx, 42.50-p

Introduction.—Quantum mechanics acting as a fundamental law of nature imposes limit to the evolution speed of quantum systems. The utility of these limits is shown in different scenarios, including quantum communication Bekenstein (), the identification of precision bounds in quantum metrology Giovanetti (), the formulation of computational limits of physical systems Lloyd (), as well as the development of quantum optimal control algorithms Caneva (). The minimal time a system needs to evolve from an initial state to its one orthogonal state is defined as the quantum speed limit time (QSLT). The study of it has been focused on both closed and open quantum systems. For closed system with unitary evolution, a unified lower bound of QSLT is obtained by Mandelstam-Tamm (MT) type bound and Margolus-Levitin (ML) type bound Mandelstam (); Fleming (); Anandan (); Vaidman (); Margolus (); Levitin (). The extensions of the MT and ML bounds to nonorthogonal states and to driven systems have been investigated in Refs. 11 (); 12 (); 13 (); 14 (); 15 (); 16 (). The QSLT for nonunitary evolution of open systems is also studied 17 (); 18 (); 19 (). It is shown that a unified bound of QSLT including both MT and ML types for non-Markovian dynamics can be formulated 19 (). However, while this unified bound is applicable for a given driving time for the pure initial states, it is not feasible for mixed initial states. As we all know that decoherence and inaccurate operations are indispensable which may result in mixed initial states.

In this Letter, we shall derive a QSLT for mixed initial states by introducing relative purity as the distance measure, which can characterize successfully the speed of evolution starting from an arbitrary time-evolution state in the generic nonunitary open dynamics. Let us consider the states of a driven system in the damped Jaynes-Cummings model starting from a certain pure state which corresponds to a special case of our result, one may observe that the QSLT is equal to the driving time in the Markovian regime 19 (). While by calculating the QSLT starting from the time-evolution state at any point in time which is in general a mixed state, it is interesting to find that the QSLT first begins to decrease from the driving time and then gradually increases to this driving time in the Markovian dynamics. So the speed of evolution in the whole dynamical process exhibits an acceleration first and then deceleration process. Additionally in the case of the non-Markovian regime, the memory effect of the environment leads to a periodical oscillatory behavior of the QSLT. We can also focus on the widely used Ohmic-like reservoir spectra to investigate the QSLT for the time-dependent states of the purely dephasing dynamics process. We demonstrate that the QSLT will be reduced with the starting point in time for Ohmic and sub-Ohmic dephasing model, that is to say, the open system executes a speeded-up dynamics evolution process. While for the super-Ohmic environments, due to the occurrence of coherence trapping 35 (), we specifically point out that this QSLT would be gradually trapped to a fixed value, and therefore leads to a uniform evolution speed for the open system. We remark that the findings of those phenomena rely on our general result of QSLT for arbitrary initial states. Finally, we also investigate the influence of the relativistic effect on the QSLT for the observer in non-inertial frames in the above two quantum decoherence models.

Quantum speed limit time for mixed initial states.—In the following, we shall consider a driven open quantum system and look for the minimal time that is necessary for it evolve from a mixed state to its final state . Under the general nonunitary quantum evolutions of open system, the final state will be generally mixed. One general choice of distance measure between two mixed states and is fidelity . In this case of the initially mixed state should be treated by purification in a sufficiently enlarged Hilbert space. And the fidelity can be written , where the maximization is over all on a larger Hilbert space that are purifications of the mixed states on the smaller system . But performing the optimization over all possible purifications is a challenging task that will be very hard to perform in the general case.

Here we follow the relative purity as a distance measure to derive lower bound on the QSLT for open quantum systems. The so-called relative purity between initial and final states of the quantum system is defined as 29 () . To evaluate the QSLT, let us now characterize the derivative of the relative purity, . The rate of change of will serve as the starting point for our derivation to ML and MT type bounds on the minimal evolution time of an initially mixed state , using, respectively, the von Neumann trace inequality and the Cauchy-Schwarz inequality.

By using the von Neumann trace inequality, we begin to provide a derivation of ML type bound to arbitrary time-dependent nonunitary equation of the form Let such a map govern the evolution and consider

 ˙f(t)=tr[ρτLt(ρt)]tr(ρ2τ)=tr[Lt(ρt)ρτ]tr(ρ2τ). (1)

Then, we introduce the von Neumann trace inequality for operators which reads von Neumann (); 30 (), , where the above inequality holds for any complex matrices and with descending singular values, and . The singular values of an operator are defined as the eigenvalues of 30 (). In the case of Hermitian operator, they are given by the absolute value of the eigenvalues of . We thus find , with are the singular values of and those of the initial mixed state . Since the singular values of satisfy , the trace norm of would satisfy , so

 |˙f(t)|≤∑ni=1σiϱitr(ρ2τ)≤∑ni=1σitr(ρ2τ). (2)

Integrating Eq. (2) over time from to , we arrive at the ineqality

 τ≥max{1¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯∑ni=1σiϱi,1¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯∑ni=1σi}|f(τ+τD)−1|tr(ρ2τ), (3)

where . For unitary processes, is equal to the time-averaged energy , so the ML bound for closed systems can be expressed .

By noting the following inequality holds , then . So we can therefore simplify Eq. (3) as

 τ≥|f(τ+τD)−1|tr(ρ2τ)¯∑ni=1σiϱi. (4)

Regarding to general nonunitary open system dynamics, Eq. (4) expresses a ML type bound on the speed of quantum evolution valid for mixed initial states.

Next we want to derive a unified bound on the QSLT for the open systems. According to Ref. 18 (), the rate of change of relative purity can be bounded with the help of the Cauchy-Schwarz inequality for operators, . Then , since is a mixed state, , and we obtain

 |˙f(t)|≤√tr[Lt(ρt)†Lt(ρt)]tr(ρ2τ)=∥Lt(ρt)∥tr(ρ2τ), (5)

where is the Hilbert-Schmidt norm. Integrating Eq. (5) over time leads to the following MT type bound for nonunitary dynamics process,

 τ≥|f(τ+τD)−1|tr(ρ2τ)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯√∑ni=1σ2i, (6)

where means the time-averaged variance of the energy.

Here, combining Eqs. (4) and (6), we obtain a unified expression for the QSLT of arbitrary initially mixed states in open systems, as following

 τQSL=max{1¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯∑ni=1σiϱi,1¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯√∑ni=1σ2i}|f(τ+τD)−1|tr(ρ2τ).

Interestingly, for a pure initial state , the singular value , then . So expression (LABEL:10) thus reduces to the unified bound for the QSLT has been given in Ref. 19 () based on fidelity, since relative purity and fidelity are identical for pure initial states. That is to say, in expression (LABEL:10) can also be defined as the minimal time a system needs to evolve from a pure initial state to its final state.

The speed of evolution in the exactly solvable open system dynamics.—In order to clear which bound on the speed limit time can be attained and tight, we must compare and . In case , the ML type bound provides the tighter bound on the QSLT. The unified expression (LABEL:10) of the speed limit time for mixed initial states presented above is one of the results in this Letter. Next we shall illustrate its use for the quantum evolution speed of a qubit system in two decoherence channels. An generally mixed state of a qubit can be written in terms of Pauli matrices, whose coefficients define the so-called Bloch vector , where is the identity operator of the qubit, is the Pauli operator, and .

We firstly consider the exactly solvable damped Jaynes-Cummings model for a two-level system resonantly coupled to a leaky single mode cavity. The environment is supposed to be initially in a vacuum state. The nonunitary generator of the reduced dynamics of the system is , where are the Pauli operators and the time-dependent decay rate. In the case of only one excitation in the whole qubit-cavity system, the environment can be described by an effective Lorentzian spectral density of the form , where is the width of the distribution, denots the frequency of the two-level system, and the coupling strength. Typically, weak-coupling regime (), where the behavior of the qubit-cavity system is Markovian and irreversible decay occurs, and strong-coupling regime (), where non-Markovian dynamics occurs accompanied by an oscillatory reversible decay. The time-dependent decay rate is then explicitly given by , with . The reduced density opertor of the system at time reads

 ρt=12(2−(1−vz)pt(vx−ivy)√pt(vx+ivy)√pt(1−vz)pt), (8)

where .

For the generally mixed state of a qubit, is always less than , so we reach the result that the ML type bound on the QSLT is tight for the open system. The unified expression (LABEL:10) proposed for the mixed initial states in this Letter, can demonstrate the speed of the dynamics evolution form an arbitrary time-dependent mixed state to another by a driving time . We examine the whole dynamics process where the system starts in the excited state, and . Figs. and show the QSLT for a time-dependent mixed state as a function of in the Markovian and non-Markovian dynamics process, respectively, in the case . The QSLT can initially reduce to a minimum, and gradually reach to the driving time in the Markovian regime. While for the non-Markovian regime, the speed limit time first decreases to a minimum in the beginning of the evolution, then occurs a periodical oscillatory of the time . That is to say, in the Markovian regime, the evolution of the qubit first exhibits a speeded-up process for and then shows gradual deceleration process for . However, the speed of evolution for the qubit in the non-Markovian dynamics process complies with an interesting periodical oscillatory behavior.

The above behavior can be explained by evaluating the QSLT for the qubit to evolve from to ,

 τQSL=|(pτ−pτ+τD)(1−2pτ)|1τD∫τ+τDτ|˙pt|dt. (9)

For the Markovian regime , the value of can be given by , then the speed limit time is simplified as . So the appearance seen in Fig. depends only on the decay of the excited population for the time-dependent state , and the critical time . Furthermore, the oscillatory behavior shown by Fig. in the the non-Markovian regime, appears as a consequence of the oscillatory time dependence of the decay rate .

In what follows, we consider a spin-boson-type Hamiltonian that describes a pure dephasing type of interaction between a qubit and a bosonic environment. It is worth stressing that this qubit-plus-environment model admits an exact solution 31 (); 32 (). There exists no correlations between the system and the environment at ; furthermore, the environment is initially in its vacuum state at zero temperature. The nonunitary generator of the reduced dynamics of the system is By considering the bosonic environment operator is simply a sum of linear couplings to the coordinates of a continuum of harmonic oscillators described by a spectral function 33 (); 34 (), then . Here, we suppose that the spectral density of the environmental modes is Ohmic-like , with being the cutoff frequency and a dimensionless coupling constant. By changing the -parameter one goes from sub-Ohmic reservoirs () to Ohmic () and super-Ohmic () reservoirs, respectively. For zero temperature, and , the dephasing rate can be obtained , where is the Euler Gamma function. Taking the limit carefully, one also finds . The time evolution of the reduced density matrix of the qubit satisfies

 ρt=12(1+vz(vx−ivy)qt(vx+ivy)qt(1−vz)), (10)

where .

In this Ohmic-like dephasing model, the QSLT of a qubit can also be given by ML type bound. In the dephasing evolution, by considering an arbitrary mixed state to another under a driving time , the QSLT can be calculated

 τQSL=C(ρ0)1/2|qτqτ+τD−q2τ|1τD∫τ+τDτ|˙qt|dt, (11)

with means the coherence of . With this, it is easy to show that is independent of , and not only relate to the dephasing rate of the Ohmic-like environment but also to the coherence of the initial state under a given driving time . Fig. 2 presents the results of our analysis for in the Ohmic-like dephasing process with different . We observe that, for the same driving time , the lager coherence of the initial state can decrease the speed of evolution of a quantum system, and thus demand the longer QSLT. By choosing the -parameters satisfied Markovian regime PRARP (), the speed limit time can be rewritten as . Hence, for a given initial state , the speed of evolution in the dynamics process is determined by the decay rate of the coherence for the mixed state . Due to the specific form of the spectral density for Ohmic-like dephasing model 35 (), in the case of zero temperature, the qubit dephasing will predict vanishing coherences in the long time limit for . On the other hand, for the qubit dephasing will stop after a finite time, therefore leading to coherence trapping, as shown by the inset of Fig. . So the other notable observation about Fig. is shown: The open system executes a speeded-up dynamics evolution process in the Ohmic and sub-Ohmic dephasing models. But for the super-Ohmic dephasing model the qubit firstly exhibits a speeded-up dynamics process before a finite time, and then complies with an uniform evolution speed after this finite time.

Quantum speed limit time in non-inertial frames.—If the observer for a quantum system in a uniformly accelerated frame with acceleration , the relativistic effect should be taken into account 20 (); 21 (); 22 (); 25 (); 26 (); 27 (); 28 (). So here we shall investigate the influence of the relativistic effect on the QSLT. Owing to the relativistic effect, the coherence of the changed initial state turns into , and becomes much less than that of . so the relativistic effect can increase the speed of evolution of a quantum system in the purely Ohmic-like dephasing channels. The parameter above is defined by , the speed of light in the vacuum, and the central frequency of the fermion wave packet. But for the damped Jaynes-Cummings model, in spite of the weaker coherence of the initial state brought by the relativistic effect, the larger excited population in the changed initial state can also be acquired. As well as the QSLT mainly depends on the population of excited state under a given driving time in the amplitude-damping channels 36 (), so the relativistic effect would slow down the quantum evolution of the qubit in the damped Jaynes-Cummings model, therefore leads to a smaller QSLT.

Conclusions.—We have derived a QSLT for arbitrary initial states to characterize the speed of evolution for open systems. In particular, considering the damped Jaynes-Cummings model, we have obtained that the speed of evolution in the Markovian regime exhibits an acceleration first and then deceleration process, and shows a peculiar periodical oscillatory behavior in non-Markovian regime. Moreover, in the case of the purely dephasing environments, the QSLT would be gradually reduce to a fixed value for the super-Ohmic dephasing model, and hence leads to a uniform evolution speed for the open system. Our results may be of both theoretical and experimental interests in exploring the speed of quantum computation and information processing in the presence of noise.

This work was supported by ¡°973¡± program under grant No. 2010CB922904, the National Natural Science Foundation of China under grant Nos. 11175248, 61178012, 11204156, 11304179 and 11247240.

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