Whether a quantum system has a potential capacity to be accelerated is an important question in a quantum process. A concept which may weigh this question is 1 where τ_QSL is a quantum speed limit (QSL) time^[1–6] and τ is an actual evolution time. Equation (1) gives a percentage of the potential acceleration of a quantum system, and its physical meaning will be revealed in the next section: it may play a decisive role in quantum computation,^[7,8] quantum communication,^[9,10] quantum optimal control,^[11–14] and quantum metrology.^[15–17]

Using Eq. (1), it is obvious that the QSL time which has been studied in recent years^[18–25] needs to be calculated to obtain the speed-up capacity. A unified lower bound of a QSL time involving Mandelstam–Tamm (MT)^[26] and Margolus–Levitin (ML)^[27] types in open systems has been derived by Deffner and Lutz.^[22] It has also been confirmed that the ML type bound based on the operator norm provides the sharpest bound of the QSL time in an open system and non-Markovianity leads to a smaller QSL time. The QSL time of a multi-qubit open system has now caught increasing attention from researchers. Since it was found that the QSL time can be reduced into a special class of multi-qubit states in an amplitude-damping channel even in a memoryless environment,^[28] three interesting questions have arisen. 1) What is a common case for the speed-up capacity of multi-qubit open systems in different types of quantum channels? 2) Does the memory effect play the same role as the speed-up capacity in different channels? 3) What will happen when the memory degree is changed? In this paper we consider these queries in a two-qubit system and show that even different quantum channels lead to different speed-up capacities that exist in most cases. We demonstrate that the capacity will benefit from a memory environment. Moreover, we find two key factors which may be helpful in controlling the speed-up capacity of a quantum process in experiments.

The rest of this paper is organized as follows. In Section 2 we explain the physical meaning of the speed-up capacity and deal with the dynamics of multi-qubit open systems by Kraus operators. Evolutions of the speed-up capacity of typical two-qubit states in different quantum channels without the memory effect are explored and explained in Section 3. The memory and memory degree effect on the speed-up capacity are discussed in Section 4. Finally, the results obtained in this work are summarized in Section 5.

With the form of Eq. (1), it can be seen that the speed-up capacity is just a ratio of difference between τ and τ_QSL to τ. Since τ_QSL and τ are the minimal evolution time and the actual evolution time respectively, the difference between them represents a time length which may be potentially reduced in evolution. Therefore, greater difference causes longer potential time, so that the evolution process may be speeded up. In this sense, equation (1) gives a maximal percentage of the evolution of a system, by which the process can be accelerated in theory within a given actual evolution time. This is the exact physical meaning of Eq. (1), serving as the definition of the quantum speed-up capacity. To obtain the capacity, the QSL time which can be derived by combining the results of MT and ML bounds^[3,4,14,29] need to be calculated. A definition of the QSL time in an open system is^{[22,24,28,30,31]} 2 where denotes the Bures angle between an initial state ρ₀ = |Ψ₀ ⟩ ⟨ Ψ₀| and its target state ρ_τ, with L_t being a superoperator which satisfies d ρ_t/dt = L_t (ρ_t). Here ||A||_op = α₁, and are the operator norm, trace norm and the Hilbert–Schmidt norm, respectively, α_i being the singular value of A.^[32]

A popular and convenient description which indicates the dynamics of a state in a quantum channel is the Kraus representation.^[33] With this description, the evolution of a state ρ can be written in the form of 3 where the operators K_μ are the so-called Kraus operators and satisfy for all t’s. When the system is composed of N subsystems with independent environments respectively, equation (3) is replaced by^[34] 4

By using Eq. (4), the evolution of a multi-qubit system can be evaluated.

Now we focus on the evolutions of the speed-up capacity of two-qubit states in different quantum channels where the N in Eq. (4) equals 2. The operator norm which has been proved to provide the sharpest bound^[22] is used here. Two classes of typical Bell-type initial states, |Ψ₁⟩ and |Ψ₂⟩ with coefficient a ∈ [0,1], are considered as the initial states respectively. The evolving state ρ (t) is used as a target state to show the dynamics of the capacity.

3.1. Amplitude-damping channel

This channel represents the dissipative interaction between a qubit and its environment. The Hamiltonian model for the process can be written as^[35] 5 where σ_± are the raising and lowering operator with ω₀ being the transition frequency of the qubit, ω_k denotes different field modes of the reservoir where ) is the annihilation (creation) operator and g_k is the coupling constant. A damped Jaynes–Cummings model is considered with 6 where λ defines the spectral width and γ₀ quantifies the coupling strength. The decoherence function of the model is 7 where . The environment is Markovian (memoryless) when γ₀ < λ/2, otherwise a non-Markovian (memory effect) environment is caused.^[36–40] The Kraus operators of this model are given as 8 where the damping parameter p(t) equals G²(t). The evolution can be easily expanded to two-qubit systems by using Eq. (4).

The speed-up capacities of |Ψ₁⟩ and |Ψ₂⟩ each as a function of the scaled time λt and coefficient a without memory effect are shown in Fig. 1. It can be found that |Ψ₁⟩ always has no speed-up capacity no matter how long it evolves. Yet |Ψ₂⟩ has a nonzero speed-up capacity at the initial time (except a = 1) and then it increases to an invariant value within a short time of evolving. The capacity is in inverse proportion to coefficient a. These phenomena mean that |Ψ₁⟩ reaches the best accelerated performance in this channel, while |Ψ₂⟩ can obtain a further acceleration even when the environment is memoryless. It is an important character for state selecting in an experiment.

Fig. 1.

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	Fig. 1. Evolutions of the speed-up capacity of (a) |Ψ1⟩ and (b) |Ψ2⟩ in an amplitude-damping channel as a function of scaled time λt and coefficient a with λ = 50.

3.2. Phase-damping channel

The phase damping process describes a pure dephasing type of interaction between a qubit and a bosonic reservoir. The Hamiltonian is written as follows:^[35] 9 A spectral density of an Ohmic-like form is considered here: 10 where s is the Ohmic parameter and ω_c is the cutoff frequency of the environment. By changing the relationships for s and constant 1, we obtain different Ohmic spectra which correspond to sub-Ohmic environment (s < 1), Ohmic environment (s = 1), and super-Ohmic environment (s > 1), respectively. Besides, s > 2 may cause a memory effect with zero T.^[41] Kraus operators of this channel are given as 11 where p(t) is a dephasing parameter and can be calculated from 12 with γ(t) being the dephasing rate, and it can be expressed as follows: 13 with Γ(s) representing the Euler function. It can also be easily expanded to two-qubit systems by using Eq. (4).

Evolutions of the speed-up capacities of |Ψ₁⟩ and |Ψ₂⟩ in this channel are very different from those in an amplitude-damping channel. A biggest distinction is that neither |Ψ₁⟩ or |Ψ₂⟩ has a different speed-up capacity any more. As shown in Fig. 2, the two classes of states have an identical invariant speed-up capacity when coefficient a is the same in a memoryless environment. There is a non-monotonic relationship between the capacity and the coefficient a. This relationship can also be used to select an appropriate state for experiment aims.

Fig. 2.

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	Fig. 2. (color online) Evolutions of the speed-up capacities of (a) |Ψ1⟩ and (b) |Ψ2⟩ in a phase-damping channel as a function of scaled time ωct and coefficient a with s = 1, ωc = 1.

3.3. Bit flip, phase flip, and bit-phase flip channels

These three channels are all under the Markov approximation in which the memory effect does not exist. Their unified Lindblad operator in a single-qubit system is expressed as follows: 14 where γ is the time-independent dephasing rate and σ_i is the Pauli matrix with i = x, y, z denoting bit flip, phase-bit flip, and phase flip channels, respectively. The set of Kraus operators for each one of these channels is given as^[34] 15 where p(t) = 1 −exp [−γt].

In these three channels the speed-up capacities of |Ψ₁⟩ and |Ψ₂⟩ are the same. The evolution in a bit flip channel is shown in Fig. 3. It can be seen that the capacity in this channel rises from zero to a very small value within a short time. The relationship between the final value and coefficient a is still non-monotonic but it is different from that in a phase-damping channel. The dynamics in a phase flip channel is similar to that in a phase-damping channel as shown in Fig. 4. This phenomenon is easy to be understood since it has been known that a phase-damping channel and a phase flip channel have exactly the same quantum operation.^[42] In Fig. 5 it is found that the speed-up capacity in a bit-phase flip channel is the same as that in a bit flip channel. This phenomenon can be easily confirmed mathematically by using Eqs. (2), (4), and (15).

Fig. 3.

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	Fig. 3. (color online) Evolutions of the speed-up capacities of (a) |Ψ1⟩ and (b) |Ψ2⟩ in a bit flip channel as a function of scaled time γt and coefficient a with γ = 10.

Fig. 4.

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	Fig. 4. (color online) Evolutions of the speed-up capacities of (a) |Ψ1⟩ and (b) |Ψ2⟩ in a phase flip channel as a function of scaled time γt and coefficient a with γ = 10.

Fig. 5.

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	Fig. 5. (color online) Evolutions of the speed-up capacities of (a) |Ψ1⟩ and (b) |Ψ2⟩ in a bit-phase flip channel as a function of scaled time γt and coefficient a with γ = 10.

In this section we explore the scenario in which the memory effect appears. How to distinguish whether the environment is memory or memoryless has been described in the previous section. In an amplitude-damping channel we see that things become a little different from previous ones when the system is affected by memory. As shown in Fig. 6, |Ψ₁⟩ still has no speed-up capacity at the beginning, but suddenly rises to an invariant value which is unrelated to a. For |Ψ₂⟩ the circumstance is similar to what happens in a memoryless environment, however it takes a shorter time to reach the invariant value which is higher than that in a memoryless environment.

Fig. 6.

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	Fig. 6. (color online) Evolutions of the speed-up capacities of (a) |Ψ1⟩ and (b) |Ψ2⟩ in an amplitude-damping channel as a function of scaled time λt and coefficient a with λ = 50.

Dynamics of the speed-up capacity in a phase-damping channel with memory effect is shown in Fig. 7. Although it seems to be the same as that in a memoryless environment, it is found that the capacity slightly rises to a higher value within a short time of evolving when the environment is a memory one as shown in Fig. 8.

Fig. 7.

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	Fig. 7. (color online) Evolutions of the speed-up capacities of (a) |Ψ1⟩ and (b) |Ψ2⟩ in a phase-damping channel as a function of scaled time ωct and coefficient a with s = 5 and ωc = 1.

Fig. 8.

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	Fig. 8. (color online) Comparison of speed-up capacity between (a) |Ψ1⟩ and (b) |Ψ2⟩ in a memory and a memoryless environment in a phase-damping channel with a = 0.5.

Overall, though the dynamics of the capacity is different in different channels, it can be concluded that the memory effect causes a more potentially accelerated ability in two-qubit open systems, which in fact is a powerful supplement to the result that non-Markovianity may reduce the QSL time in a Jaynes–Cummings model of a single qubit open system.^[22] It also testifies to the generalization of the relationship between the non-Markovianity and the QSL time. Since the essential reason why quantum speedup connects directly with the non-Markovianity in a single qubit open system has been revealed in Ref. [46], our result regarding the relationship between the quantum speed-up capacity and the non-Markovianity in a two-qubit open system may also be explained by using the conclusion of Ref. [46].

Finally, we deal with the question over what will happen if the memory degree is changed. We know that the memory effect comes from the non-Markovianity of the environment. So if the degree of non-Markovianity is higher, the memory degree is higher. It has been proved that the degree of non-Markovianity is in proportion to the coupling strength γ₀ in a damped Jaynes–Cummings model.^[37] Therefore the memory degree is also in proportion to the γ₀ in a damped Jaynes–Cummings model. The speed-up capacities of |Ψ₁⟩ and |Ψ₂⟩ as a function of scaled time λt and γ₀ are shown in Fig. 9. It is found that the higher the memory degree, the stronger the speed-up capacity is. Namely, the stronger the memory effect, the greater the potential acceleration is. It is remarkable that when γ₀ is high enough, there is a short fluctuation for |Ψ₁⟩ before it reaches an invariant speed-up capacity. Some methods which may control the evolution speed of a single qubit system in theory and experiment have been presented,^[46–57] so our results may make some contribution to experiments in further work.

Fig. 9.

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	Fig. 9. (color online) Evolutions of the speed-up capacities of (a) |Ψ1⟩ and (b) |Ψ2⟩ in an amplitude-damping channel as a function of scaled time λt and the coupling strength γ0 (in proportion to memory degree) with λ = 50 and a = 0.5.

In this work, we have presented a formula to describe the speed-up capacity of a quantum system. We find that the capacity has different dynamic behaviors in different channels and exists in most situations of two-qubit open systems. We explain the characters of the capacity and demonstrate that the memory effect can always improve the capacity. We also find that the coefficient a of the initial state and the memory degree are two key factors which may be useful in controlling the capacity experimentally.