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arXiv:1506.03199 · Physics Letters A

Quantum coherence sets the quantum speed limit for mixed states

Debasis Mondal, Chandan Datta, Sk Sazim

Published Dec 19, 2015• 113 citations
arXivHTMLPDF
DOI
10.1016/j.physleta.2015.12.015

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Abstract The question of how fast a quantum state can evolve is considered. Using the definition of squared speed based on the Euclidean distance given in (Brody and Longstaff 2019 Phys. Rev. Res. 2 033127), we present a systematic framework to obtain the optimal speed of a d-dimensional system evolved unitarily under a time-independent Hamiltonian. Among the set of mixed quantum states having the same purity, the optimal state is obtained in terms of its purity parameter. We show that for an arbitrary d, the optimal state is represented by a X-state with an additional property of being symmetric with respect to the secondary diagonal. For sufficiently low purities for which the purity exceeds the purity of maximally mixed state I / d by at most 2 / d 2 , the only nonzero off-diagonal entry of the optimal state is ϱ 1 d , corresponding to the transition amplitude between two energy eigenstates with minimum and maximum eigenvalues, respectively. For larger purities, however, whether or not the other secondary diameter entries ϱ i , d − i + 1 take nonzero values depends on their relative energy gaps | E d − i + 1 − E i | . The effects of coherence and entanglement, with respect to the energy basis, are also examined and found that for optimal states both resources are monotonic functions of purity, so they can cause speed up quantum evolution leading to a smaller quantum speed limit. Our results show that although the coherence of the states is responsible for the speed of evolution, only the coherence caused by some off-diagonal entries located on the secondary diagonal play a role in the fastest states.
Abstract The minimum evolution time between multi-qubit quantum states is estimated for non-Markovian quantum channels. We consider the maximally coherent pure and mixed states as well as multi-qubit X states as initial states and discuss the impact of initial coherence and the behaviour of coherence on their speed of evolution for both dephasing and dissipative processes. The role of the non-zero value of initial coherence under information backflow conditions for the non-unital dissipative process is revealed by the flow of quantum speed limit time ( τ QSL ). The trade-off between mixedness and coherence on the speed limit time reveals the nature of the quantum process the states undergo. The complementarity effect between mixedness and coherence is more prominent in the quantum non-unital dissipation process. The parametric trajectory of speed limit time vividly depicts the difference in the evolution of pure and mixed initial states, and this could be used to distinguish between the unital and non-unital channels studied in this work. Our investigation of quantum speed limit time on multi-qubit entangled X states reveals that τ QSL can be identified as a potential dynamical witness to distinguish multi-qubit states in the course of evolution.
A unified bound on the quantum speed limit is obtained for open quantum systems with a mixed initial state by utilizing the function of relative purity proposed by F. Campaioli et al. [Phys. Rev. Lett. 120, 060409 (2018)]. As applications, it is found that the quantum speed limit bound for the damped Jaynes-Cummings model is determined by the competition among the non-Markovianity, the population of the initial excited state, and the initial-state coherence, which shows that the population of the initial excited state and the coherence of the initial state can make the quantum speed limit bound tighter even though the non-Markovian effects can accelerate the evolution of open quantum system. For the dephasing model, a simple factorization law with the initial-state coherence shows that the quantum speed limit is governed by only the competition between the non-Markovianity and the coherence of the initial state.