There is no general connection between the quantum speed limit and non-Markovianity

Abstract

The quantum speed limit (QSL) sets a bound on the minimum time required for a quantum system to evolve between two states. For open quantum systems this quantity depends on the dynamical map describing the time evolution in presence of the environment, on the evolution time τ, and on the initial state of the system. We consider a general single qubit open dynamics and show that there is no simple relationship between memory effects and the tightness of the QSL bound. We prove that only for specific classes of dynamical evolutions and initial states, there exists a link between non-Markovianity and the QSL. Our results shed light on the connection between information back-flow between system and environment and the speed of quantum evolution. than 1 when it is non-Markovian. Their result suggests that in the Markovian case the dynamics saturates the bound, giving the most efficient evolution, whereas in the non-Markovian case the actual limit can still be lower than the evolution time. The explicitly derived dependency between QSL and non-Markovianity has proven useful in several applications [10,[15][16][17][18][19][20][21][22][23][24][25][26][27][28].Our main goal is to tackle the question of the connection between non-Markovianity and the QSL not starting from a specific model but in full generality, looking in detail at the role played by the dynamical map, the evolution time τ, and the initial state, in the achievement of the QSL bound. We show that, for the most general cases, there is no simple connection between the Markovian to non-Markovian crossover and the QSL. Under certain more restrictive assumptions, however, we can characterize families of one-qubit dynamical maps for which the QSL speed-up coincides with the onset of non-Markovianity, as indicated by the Breuer-Laine-Piilo (BLP) non-Markovianity measure [13]. For these families we derive analytical formulas for the QSL as a function of the BLP measure. Our results also show that, for a given open quantum system model, both the evolution time τ and the initial state play a key role and cannot be overlooked when making claims on the QSL. As an example, we generalize results in [4] to a broader set of pure initial states, and show that the QSL bound is saturated only for very few initial states even in the fully Markovian case.The paper is structured as follows. In section 2 we briefly present the formalism of open quantum systems and recall the common mathematical definitions of QSL. In section 3 we present the Jaynes-Cummings (JC) model used in [4] and discuss briefly their results concerning non-Markovianity and quantum speed-up. In section 4 we study how the actual evolution time affects the QSL for the same JC system. In section 5, we calculate the general conditions for the QSL optimal dynamics, and study the connection between BLP non-Markovianity and QSL. In section 6 we study the initial state dependence of the QSL for the Markovian dynamics arising from Pauli and phase-covariant master equations. In section 7 we study the effects of Markov...

Quantum Speed Limit and Divisibility of the Dynamical Map

“…This connection was later studied in more detail, and it was found that the speed-up is not always dependent on the information backflow and can sometimes be present without any non-Markovian effects [10]. In this case, the presence of information backflow coincides with the loss of P-divisibility.…”

“…In [ 4 ], it was shown that, for an amplitude damping system, as given by master Equation ( 2 ) with and , the QSL is directly dependent on the information backflow as where and is the Breuer–Laine–Piilo (BLP) non-Markovianity measure, as given by This connection was later studied in more detail, and it was found that the speed-up is not always dependent on the information backflow and can sometimes be present without any non-Markovian effects [ 10 ]. In this case, the presence of information backflow coincides with the loss of P-divisibility.…”

“…This result was then experimentally confirmed in [ 9 ]. A more thorough analysis on the role of non-Markovianity was performed in [ 10 ], where it was shown that its connection with QSL is not as straightforward and the speed-up can be present, even when the dynamics is Markovian.…”

The effect of quantum memory on quantum speed limit time

“…Quantum speed limit as a measure of non-Markovianity, and its connection with the hierarchy of quantum correlations of composite quantum system [28,29] are discussed in [30,31]. Even though quantum speed limit time seems to identify the information backflow due to the non-Markovianity, in [30] it is shown that there exists no simple connection between non-Markovianity and speed limit time. This raises the question of whether there is a quantum concept of memory that is better indicated by the quantum speed limit.…”

Non-uniform magnetic field as a booster for quantum speed: faster quantum information processing

“…The definition of QS in the context of open quantum systems [6] was developed in the last decade [7][8][9]. The concept of QS has been used to shed light into various facets of quantum information [10,11], open systems [12][13][14][15], control of quantum systems [16] and quantum thermodynamics [17,18]. Further, using causality and thermodynamics, the important Bremermann-Bekenstein bound [19,20] relates the energy cost per bit of information to the QSL time.…”