Tight, robust, and feasible quantum speed limits for open dynamics

Campaioli, Francesco; Pollock, Felix A.; Modi, Kavan

doi:10.22331/q-2019-08-05-168

Cited by 81 publications

(75 citation statements)

References 74 publications

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“…The results of [1,2] were extended to include cases where the evolved state is not orthogonal to the initial state in [3]. Moreover, in addition to the previous definitions valid for closed quantum systems, several authors proposed different generalizations to open quantum systems applicable for both Markovian and non-Markovian dynamics [4][5][6][7][8][9]. Nowadays, QSLs are investigated in connection to a number of topics, from quantum metrology to quantum computation, from quantum control to quantum thermodynamics, as reviewed, e.g.…”

Section: Introductionmentioning

confidence: 99%

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

2019

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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...

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Section: Introductionmentioning

confidence: 99%

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

2019

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“…It is therefore of particular importance to find the pervasive bound for QSLT of open quantum systems. To this end, much work has been done to define the proper QSLT for open quantum systems [4][5][6][7][8][9][10][11][12][13][14][15][16]. Here we use the geometric approaches based on relative purity for driving the QSLT for open quantum systems [7,8].…”

Section: Introductionmentioning

confidence: 99%

The Effect of Homodyne-Based Feedback Control on Quantum Speed Limit Time

Haseli

2020

Int J Theor Phys

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The minimal time a system requires to transform from an initial state to target state is defined as the quantum speed limit time. quantum speed limit time can be applied to quantify the maximum speed of the evolution of a quantum system. Quantum speed limit time is inversely related to the speed of evolution of a quantum system. That is, shorter quantum speed limit time means higher speed of quantum evolution. In this work, we study the quantum speed limit time of a two-level atom under Homodyne-based feedback control. The results show that the quantum speed limit time is decreased by increasing feedback coefficient. So, Homodyne-based feedback control can induce speedup the evolution of quantum system. PACS numbers:

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“…The contribution to the squared speed from the unitary evolution (the first term in (11)) has recently been identified in [26] as the speed of the evolution of the state, when the distance is measured with respect to the Euclidean angular separation. This term resembles, but is different from, the Wigner-Yanase skew information I = tr(Ĥ 2ρ ) − tr(Ĥ √ρĤ√ρ ).…”

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confidence: 99%

“…The apparent discrepancy between this result and that obtained in [5] is that here we measure the speed with respect to the Euclidean norm in R n 2 −1 , whereas in [5] the evolution speed of the state is obtained using the Hilbert space norm. While the latter is more useful in the context of state estimation (because the Fisher-Rao metric for unitary evolution is given by the skew information), as remarked in [26] for the analysis of the evolution speed and time, the use of the Euclidean metric is computationally more effective for it does not involve taking the square-root of the density matrix. We shall refer to S(X) = tr(X †Xρ2 ) − tr(XρX †ρ ) as the 'modified skew information' for the operatorX.…”

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confidence: 99%

“…We examined example systems that show that the speed of evolution is typically decreasing in time, but this need not be the Note added. While completing this work we came across a closely-related work [26], in which the Euclidean norm is used to investigate bounds on the evolution time for general open systems. Various merits in the use of the Euclidean norm are also discussed therein.…”

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Evolution speed of open quantum dynamics

Brody

Longstaff

2019

Phys. Rev. Research

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The space of density matrices is embedded in a Euclidean space to deduce the dynamical equation satisfied by the state of an open quantum system. The Euclidean norm is used to obtain an explicit expression for the speed of the evolution of the state. The unitary contribution to the evolution speed is given by the modified skew information of the Hamiltonian, while the radial component of the evolution speed, connected to the rate at which the purity of the state changes, is shown to be determined by the modified skew information of the Lindblad operators. An open-system analogue of the quantum navigation problem is posed, and a perturbative analysis is presented to identify the amount of change on the speed. Properties of the evolution speed are examined further through example systems, showing that the evolution speed need not be a decreasing function of time.

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Tight, robust, and feasible quantum speed limits for open dynamics

Cited by 81 publications

References 74 publications

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

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

The Effect of Homodyne-Based Feedback Control on Quantum Speed Limit Time

Evolution speed of open quantum dynamics

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