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Register a new userQuantum speed of evolution in a Markovian bosonic environment
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We present explicit evaluations of quantum speed limit times pertinent to the Markovian dynamics of an open continuous-variable system. Specifically, we consider the standard setting of a cavity mode of the quantum radiation field weakly coupled to a thermal bosonic reservoir. The evolution of the field state is ruled by the quantum optical master equation, which is known to have an exact analytic solution. Starting from a pure input state, we employ two indicators of how different the initial and evolved states are, namely, the fidelity of evolution and the Hilbert-Schmidt distance of evolution. The former was introduced by del Campo {em et al.} who derived a time-independent speed limit for the evolution of a Markovian open system. We evaluate it for this field-reservoir setting, with an arbitrary input pure state of the field mode. The resultant formula is then specialized to the coherent and Fock states. On the other hand, we exploit an alternative approach that employs both indicators of evolution mentioned above. Their rates of change have the same upper bound, and consequently provide a unique time-dependent quantum speed limit. It turns out that the associate quantum speed limit time built with the Hilbert-Schmidt metric is tighter than the fidelity-based one. As apposite applications, we investigate the damping of the coherent and Fock states by using the characteristic functions of the corresponding evolved states. General expressions of both the fidelity and the Hilbert-Schmidt distance of evolution are obtained and analyzed for these two classes of input states. In the case of a coherent state, we derive accurate formulas for their common speed limit and the pair of associate limit times.
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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.
One of the fundamental physical limits on the speed of time evolution of a quantum state is known in the form of the celebrated Mandelshtam-Tamm inequality. This inequality gives an answer to the question on how fast an isolated quantum system can evolve from its initial state to an orthogonal one. In its turn, the Fleming bound is an extension of the Mandelshtam-Tamm inequality that gives an optimal speed bound for the evolution between non-orthogonal initial and final states. In the present work, we are concerned not with a single state but with a whole (possibly infinite-dimensional) subspace of the system states that are subject to the Schroedinger evolution. By using the concept of maximal angle between subspaces we derive an optimal estimate on the speed of such a subspace evolution that may be viewed as a natural generalization of the Fleming bound.
The dynamics of an open quantum system with balanced gain and loss is not described by a PT-symmetric Hamiltonian but rather by Lindblad operators. Nevertheless the phenomenon of PT-symmetry breaking and the impact of exceptional points can be observed in the Lindbladean dynamics. Here we briefly review the development of PT symmetry in quantum mechanics, and the characterisation of PT-symmetry breaking in open quantum systems in terms of the behaviour of the speed of evolution of the state.
Continuous-time Markovian evolution appears to be manifestly different in classical and quantum worlds. We consider ensembles of random generators of $N$-dimensional Markovian evolution, quantum and classical ones, and evaluate their universal spectral properties. We then show how the two types of generators can be related by superdecoherence. In analogy with the mechanism of decoherence, which transforms a quantum state into a classical one, superdecoherence can be used to transform a Lindblad operator (generator of quantum evolution) into a Kolmogorov operator (generator of classical evolution). We inspect spectra of random Lindblad operators undergoing superdecoherence and demonstrate that, in the limit of complete superdecoherence, the resulting operators exhibit spectral density typical to random Kolmogorov operators. By gradually increasing strength of superdecoherence, we observe a sharp quantum-to-classical transition. Furthermore, we define an inverse procedure of supercoherification that is a generalization of the scheme used to construct a quantum state out of a classical one. Finally, we study microscopic correlation between neighbouring eigenvalues through the complex spacing ratios and observe the horse-shoe distribution, emblematic of the Ginibre universality class, for both types of random generators. Remarkably, it survives superdecoherence and supercoherification.
We study how useful random states are for quantum metrology, i.e., surpass the classical limits imposed on precision in the canonical phase estimation scenario. First, we prove that random pure states drawn from the Hilbert space of distinguishable particles typically do not lead to super-classical scaling of precision even when allowing for local unitary optimization. Conversely, we show that random states from the symmetric subspace typically achieve the optimal Heisenberg scaling without the need for local unitary optimization. Surprisingly, the Heisenberg scaling is observed for states of arbitrarily low purity and preserved under finite particle losses. Moreover, we prove that for such states a standard photon-counting interferometric measurement suffices to typically achieve the Heisenberg scaling of precision for all possible values of the phase at the same time. Finally, we demonstrate that metrologically useful states can be prepared with short random optical circuits generated from three types of beam-splitters and a non-linear (Kerr-like) transformation.
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