Fundamental Speed Limits to the Generation of Quantumness

Quantum physics dictates fundamental speed limits during time evolution. We present a quantum speed limit governing the generation of nonclassicality and the mutual incompatibility of two states connected by time evolution. This result is used to characterize the timescale required to generate a given amount of quantumness under an arbitrary physical process. The bound is shown to be tight under pure dephasing dynamics. More generally, our analysis reveals the dependence on the initial state and non-Markovian effects.

Scientific RepoRts | 6:38149 | DOI: 10.1038/srep38149 quantumness, that quantifies the degree to which the time-evolving state is mutually incompatible with the initial state under arbitrary dynamics. The new bound allows one to classify different dynamics according to their power to generate nonclassicality and is shown to be saturated under pure dephasing dynamics, whether it is induced by a Markovian or a non-Markovian environment.

Quantum speed limit to the dynamics of quantumness
The nonclassicality of quantum systems can be conveniently quantified using the Hilbert-Schmidt norm of the commutator of two states, which is proposed to witness the "state incompatibility" between any two admissible states ρ a and ρ b 39,40 . The "quantumness" is then defined as where the pre-factor is required for normalization and ≡ † A AA Tr( ) 2 is the Hilbert-Schmidt norm of A. As a quantumness witness, and Q(ρ a , ρ b ) = 0 iff [ρ a , ρ b ] = 0 39,40 . Choosing ρ a = ρ 0 and ρ b = ρ t , Q (ρ 0 , ρ t ) allows one to quantify the capacity of an arbitrary physical process to generate or sustain quantumness in case of [ρ 0 , ρ t ] ≠ 0. Clearly, if ρ 0 is a diagonal density matrix in a given basis and time evolution just alters the weight distribution without generating coherences, the quantumness between the initial and time-evolving states Q(ρ 0 , ρ t ) remains zero. Consequently we can generally expect a QSL different in nature from those previously derived for the fidelity decay, which would remain valid as weaker lower bounds. As shown in the section Methods, we obtain the lower-bound of evolution time associated with quantumness, , and  is the Liouville super-operator describing the time derivative of density matrix:

Results
The lower bound obtained in Eq. (3) constitutes the main result of this work. In what follows, this bound is analyzed in a series of relevant scenarios, that will be used to identify the salient physical principles governing the generation of quantumness. After a discussion of its dynamics in isolated systems we consider a system embedded in an environment, exhibiting possible non-Markovian effects, and discuss the limits of pure dephasing and dissipative processes.
Unitary quantum dynamics. Consider a general two-parameter unitary transformation for a two-level system (setting ħ ≡ 1 from now on) x y where θ and α are arbitrary real functions of time and σ is the Pauli operator 41 (We apply the standard conventions that σ x = |1〉 〈 0| + |0〉 〈 1| and σ y = i|1〉 〈 0| − i|0〉 〈 1|). When the system is prepared in an initially pure states In this case, the system Hamiltonian is found to be Specially in the case that both θ  and α are constant numbers, θ θ =  X 2 cos 2 2 2 . According to Eq. (5), the dynamics is generated by θ σ α σ α = − +  H ( cos sin ) x y . Then due to Eq. (6), the exact evolution time saturates the lower bound τ = τ Q in the regime that 0 < θ < π/4. It is worth pointing out that in this case, the independence of the result on the angle α is coincident with the result in the Bloch-vector formalism set up by the angles θ and α. When the final state reaches θ = π/4, the lower bound τ Q attains its maximal value and starts to decline when θ goes over this point. After that, Eq. (6) remains valid while loosing the tightness in the regime 0 < θ < π/4. In another special case with both θ and α ≠  0 constant during the evolution, one can find that (sin sin cos cos sin cos s in ) by a nonvanishing α, τ Q (θ = π/4) = τ/|α| by Eq. (6). Therefore, the QSL ruling the evolution of quantumness exhibits a pronounced dependence on the initial and final states.
A similar analysis can be extended to higher-dimensional systems. Consider the stimulated Raman adiabatic passage (STIRAP) 42 in a three-level atomic system, under the Hamiltonian as The system can be transferred from ρ 0 to ρ τ = |ψ τ 〉 〈 ψ τ |, where ψ θτ θτ = − + τ sin ( ) 2 cos ( ) 0 without disturbing the quasistable state |1〉 . The QSL bound also becomes tight and matches the exact time of evolution τ Q = τ when θ < π/4 and θ  and α are time-independent.
Nonunitary process. In this part, we consider the scenario of open quantum systems. We will use the quantum-state-diffusion (QSD) equation 43,44 as a general framework to derive the exact master equation before discussing the relevant QSL. In doing so, we treat both Markovian and non-Markovian environments in a unified way. In particular, we consider an Ornstein-Uhlenbeck process for the environmental noise. The correlation function reads where Γ implies the system-environment coupling strength and γ is inversely proportional to the memory time of the environment. Here, 0 < γ < ∞ and the lower and upper limits of γ correspond to the strongly non-Markovian and Markov environments, respectively. For a single two-level system, the system-environment Hamiltonian is z tot where ω and ω λ are the frequencies of the system and the λ-th environmental mode, respectively, and g λ is their coupling strength. L is the coupling operator and a λ ( λ † a ) is the annihilation (creation) operator for the λ-th environmental mode.
When L = σ z , QSD equation describes a pure dephasing process. In the rotating frame with respect to the system bare Hamiltonian, the exact super-operator  is found to be Q where β τ ≡ 2Γ τ. Remarkably, the bound is tight and reachable under pure-dephasing dynamics, when τ = τ Q as shown in Fig. 1. Equation (12) also applies to the non-Markovian case as long as β τ in Eq. (11) is modified into Qualitatively, QSL timescale depends on the choice of initial state parameter θ, specifically, the initial population distribution determined by θ cos (2 ). QSL is therefore symmetric as a function of θ with respect to θ = π/4. In Fig. 1, for different initial states, we compare the new QSL timescale τ Q obtained in Eq. (3) and that τ B based on the fidelity evolving with time (see ref. 30) in the presence of a Markovian dephasing environment. Specifically, it was then shown that for an initially pure state, the minimum time for the (squared) fidelity or relative purity F(t) = Tr[ρ 0, ρ t ] to decay to a given value F(τ) is lower bounded by . Figure 1 illustrates that τ Q not only provides a tighter bound than τ B for the generation of quantumness, but also it actually captures the real evolution pattern. Furthermore, it is known that as the system progressively goes to a steady state, which depends on the initial coherence between the up and down states, the dephasing rate should be asymptotically slowed down. This pattern has not been captured by τ B . When the quantumness approaches a final value determined by θ, τ Q increases rapidly while the rate of τ B is nearly invariant.
Next we consider the effect of the environmental memory, which is parameterized by γ, on the QSL timescale. In Fig. 2, τ Q is evaluated for a fixed initial state (with θ = π/5) and the other parameters except γ and Q. The dependence of τ Q on the quantumness Q of system and environment is monotonic. The environmental memory timescale is inversely proportional to γ. As an upshot, in the presence of a strongly non-Markovian environment the evolution speed is greatly suppressed, resulting in larger values of τ Q . Yet it is found that at the end of the dephasing process, the quantum speed limit timescale quickly approaches the same asymptotical value. The difference between the QSL timescale of the system in the extremely non-Markovian environment [τ Q (γ/Γ = 0.1)] and that in a nearly Markov environment [τ Q (γ/Γ = 2.0)] is increased with increasing Q before the system goes to the steady state.
For an n-qubit system in a common dephasing environment, we can rigorously discuss the scaling behavior of QSL for certain states. By a treatment in the Kraus representation 45 , a general GHZ state ψ θ θ = + ⊗ ⊗ cos 1 sin 0 n n 0 evolves into ρ t = C(t)○ ρ 0 . Here ○ denotes the entry-wise product and effectively C(t) (as well as ρ t ) can be expressed in a 2 × 2 matrix expanded by |0 ⊗n 〉 and |1 ⊗n 〉 , where the off-diagonal terms are r n 2 with ( ) 0 and the diagonal terms are unity. By Eqs (11) and (12), we can find that when ∫ τ dtf t ( ) 0 is sufficiently small (e.g., with a Markov environment, r = e −2Γt is sufficiently close to unity in the short time limit), both the quantumness Q and QSL time τ Q scale with the number of qubits n as n 2 .
When L = σ − , the total Hamiltonian describes a dissipation (energy relaxation) model, whose exact super-operation is found to be is a complex function of time. In the Markov limit, P(t) = Γ /2 and then ξ = Γ τ/2. In the non-Markovian situation, P(t) satisfies ∂ t P(t) = Γ γ/2 − γP(t) + P 2 (t) with P(0) = 0. Consequently, according to Eq. (3), it is found that . Note here θ sin 2 is not allowed to be zero, otherwise, Q(ρ 0 , ρ τ ) will vanish according to its definition in Eq. (1). Equations (14) and (15) indicate that in the dissipation model, it is hard to find a closed analytical expression for τ Q , and one has to resort to the numerical evaluation.
In Fig. 3, we demonstrate the dependence of the QSL timescale on the environmental memory parameter γ, measured in units of the system-environment coupling strength Γ , for a fixed initial state. From the numerically exact dynamics, we find that τ Q monotonically decreases with increasing γ. With a nearly Markovian environment (see e.g., the dot-dashed line for γ/Γ = 2.0), τ Q approaches a steady value. As expected in an environment with short memory time, the energy dissipated into the environment from the system has nearly no chance to come back to the system. The dissipation process becomes therefore irreversible. This favors the evolution of the system towards a final incompatible state. As a result, two different regimes are observed. For nearly memoryless dynamics, γ/Γ ≥ 1, the QSL timescale is found to rapidly increase as the system approaches the steady state through a roughly exponential decay. Regarding the spectral function G(t, s), a smaller γ then yields a lesser damping rate of the system. In the strong non-Markovian regime 0.1 ≤ γ/Γ < 1, the pattern becomes complex and the QSL timescale appears to be greatly enhanced by decreasing γ. In this regime, it is difficult for the time-evolving state to become classically incompatible with the initial state.

Discussion
We have studied the generation of nonclassicality via the quantumness witness defined as the Hilbert-Schmidt norm of the commutator of the initial and the final quantum states, resulting from time evolution. For arbitrary physical processes we have derived a quantum speed limit that sets the minimum timescale τ Q for the generation of a given amount of quantumness. This novel QSL has been computed and analyzed in a variety of relevant scenarios including unitary evolution, pure dephasing (of both single-and multiple-qubit system), and energy dissipation. In addition, we have discussed the generation of quantumness in non-unitary evolutions, by employing the exact quantum-state-diffusion equations.
While standard quantum speed limits characterizing the fidelity decay become too conservative and even fail to capture the correct dependence of this timescale on the parameters of the system, the new bound is tight and is saturated under pure dephasing dynamics, whether induced by a Markovian or non-Markovian environment.

Method
We consider the time-evolution of the initial density matrix to be described by a master equation of the form