Let us have a Hamiltonian H and 2 states which can time evolve into each other via this Hamiltonian. In this particular situation, say one of the states evolves into the other in time t. Now let us fix these two states ; then we may have infinite Hamiltonians (other than H, and one of them is say H') that can time evolve one of them into the other. But the time taken for this evolution varies from t and is say t' (that is, t' is the time Hamiltonian H' takes to time evolve one of the fixed states into the other). From now on, we shall consider only those Hamiltonians that can time evolve one of the states into the other. The question is given any small positive number a, will there exist a Hamiltonian H such that t' = a? Answer: Yes , the Hamiltonian Ht/a (obtained by rescaling the original Hamiltonian) will work. But, there are bounds that show that as long as the average energy and its fluctuations are bounded (thus, no rescaling allowed), you cannot go arbitrarily fast from one state to another orthogonal state. This discussion falls under the study of quantum speed limits. This document looks into this field of active research. We shall study with care how the question of a speed limit arises: from varying the Hamiltonians between two fixed states ; and hence produce an accurate speed limit between any two fixed states. The process and results shall develop as concretely and smoothly as the problem which is produced.
Let us have a Hamiltonian H and 2 states which can time evolve into each other via this Hamiltonian. In this particular situation, say one of the states evolves into the other in time t. Now let us fix these two states ; then we may have infinite Hamiltonians (other than H, and one of them is say H') that can time evolve one of them into the other. But the time taken for this evolution varies from t and is say t' (that is, t' is the time Hamiltonian H' takes to time evolve one of the fixed states into the other). From now on, we shall consider only those Hamiltonians that can time evolve one of the states into the other. The question is given any small positive number a, will there exist a Hamiltonian H such that t' = a? Answer: Yes , the Hamiltonian Ht/a (obtained by rescaling the original Hamiltonian) will work. But, there are bounds that show that as long as the average energy and its fluctuations are bounded (thus, no rescaling allowed), you cannot go arbitrarily fast from one state to another orthogonal state. This discussion falls under the study of quantum speed limits. This document looks into this field of active research. We shall study with care how the question of a speed limit arises: from varying the Hamiltonians between two fixed states ; and hence produce an accurate speed limit between any two fixed states. The process and results shall develop as concretely and smoothly as the problem which is produced.
The time needed by a quantum system to reach a state fully distinguishable from the original one provides a natural way of determining how fast the corresponding dynamical evolution is. This orthogonality time admits a lower bound, expressible in terms of the energy's expectation value and the energy's standard deviation, that yields a 'quantum speed limit'. Composite quantum systems need entanglement in order to achieve this limit. So far, most studies on the connection between entanglement and the quantum speed limit have focused on the case of non-interacting systems. The connection between quantum speed and entanglement is systematically investigated here for the case of a system of two interacting qubits, taking into consideration all possible initial states that evolve into an orthogonal one.
The traditional quantum speed limits are not attainable for many physical processes, as they tend to be loose and fail to determine the exact time taken by quantum systems to evolve. To address this we derive exact quantum speed limits for the unitary dynamics of pure-state quantum systems that outperform the existing quantum speed limits. Using these exact quantum speed limits, we can precisely estimate the evolution time for two- and higher-dimensional quantum systems. Additionally, for both finite- and infinite-dimensional quantum systems, we derive an improved Mandelstam-Tamm bound for pure states and show that this bound always saturates for any unitary generated by self-inverse Hamiltonians. Furthermore, we show that our speed limits establish an upper bound on the quantum computational circuit complexity and yield a tighter quantum Cramér-Rao bound. These results will have a significant impact on our understanding of quantum physics as well as rapidly developing quantum technologies, such as quantum computing, quantum control, and quantum batteries.
The Margolus-Levitin (ML) bound says that for any time-independent Hamiltonian, the time needed to evolve from one quantum state to another is at least $\ensuremath{\pi}\ensuremath{\alpha}(\ensuremath{\epsilon})/2\ensuremath{\langle}E\ensuremath{-}{E}_{0}\ensuremath{\rangle}$, where $\ensuremath{\langle}E\ensuremath{-}{E}_{0}\ensuremath{\rangle}$ is the expected energy of the system relative to the ground state of the Hamiltonian and $\ensuremath{\alpha}(\ensuremath{\epsilon})$ is a function of the fidelity $\ensuremath{\epsilon}$ between the two states. For a long time, only an upper bound ${\ensuremath{\alpha}}_{>}(\ensuremath{\epsilon})$ and a lower bound ${\ensuremath{\alpha}}_{<}(\ensuremath{\epsilon})$ are known, although they agree up to at least seven significant figures. Recently, H\"ornedal and S\"onnerborn [arXiv:2301.10063] proved an analytical expression for $\ensuremath{\alpha}(\ensuremath{\epsilon})$, a fully classified system whose evolution time saturates the ML bound, and gave this bound a symplectic-geometric interpretation. Here I solve the same problem through an elementary proof of the ML bound. By explicitly finding all the states that saturate the ML bound, I show that ${\ensuremath{\alpha}}_{>}(\ensuremath{\epsilon})$ is indeed equal to ${\ensuremath{\alpha}}_{<}(\ensuremath{\epsilon})$. More importantly, I point out a numerical stability issue in computing ${\ensuremath{\alpha}}_{>}(\ensuremath{\epsilon})$ and report a simple way to evaluate it efficiently and accurately.
Quantum speed limits are relations yielding lower bounds on the evolution time of quantum systems. These results have been generalized in some ways, in particular by including evolutions to non-orthogonal states. However, there was a gap in the literature on this area, for only unitary evolutions -- closed quantum systems -- had been considered. On this Ph.D. thesis, such limitation is overcome: our main result is a bound for quantum-system evolutions in general, whether unitary or not, and correctly recovers the known bounds in the unitary case. Applications of this bound to several concrete cases of interest are herein presented. This bound is also used to extend to the non-unitary case the discussion of the role of entanglement in fast evolutions, leading to nontrivial results. For the derivation of the results, a geometric approach has been employed, which allows a clear interpretation of the bounds and a discussion of the criteria for their saturation. No previous knowledge of quantum-state geometry by the reader has been assumed.
The minimal time a system needs to evolve from an initial state to its one orthogonal state is defined as the quantum speed limit time, which can be used to characterize the maximal speed of evolution of a quantum system. This is a fundamental question of quantum physics. We investigate the generic bound on the minimal evolution time of the open dynamical quantum system. This quantum speed limit time is applicable to both mixed and pure initial states. We then apply this result to the damped Jaynes-Cummings model and the Ohimc-like dephasing model starting from a general time-evolution state. The bound of this time-dependent state at any point in time can be found. For the damped Jaynes-Cummings model, when the system starts from the excited state, the corresponding bound first decreases and then increases in the Markovian dynamics. While in the non-Markovian regime, the speed limit time shows an interesting periodic oscillatory behavior. For the case of Ohimc-like dephasing model, this bound would be gradually trapped to a fixed value. In addition, the roles of the relativistic effects on the speed limit time for the observer in non-inertial frames are discussed.
As one of the fundamental questions, quantum speed limit (QSL) is of vital role in the community of quantum physics, which sets the ultimate, maximal speed with which a quantum system can evolve. QSL originated from the understanding of the Heisenberg uncertainty relation. In 1945, Mandelstam and Tamm found the time Δt in the celebrated relation ΔEΔt≥ℏ could be thought as the minimum time that a quantum system needs to evolve from an initial state to the orthogonal state for unitary evolution. QSL sets the lower bound of the evolution. They established the relationship between the minimum evolution time ( τQSL ) and the energy uncertainty of the system (ΔE), namely τQSL=πℏ/(2ΔE) , the Mandelstam-Tamm (MT) bound. Later, Margolus and Levitin derived another different bound for a closed system which connected the τQSL with the average energy of the system ( 〈E〉 ), i.e., τQSL=πℏ/(2〈E〉) , the Margolus-Levitin (ML) bound. Combining the MT and ML bounds together, we can have a unified bound for a unitary system and orthogonal states. In the case of open system dynamics, because there is energy and/or coherence exchange between the system and the environment, the orthogonal final state of the initial state can hardly be reached any longer. An investigation of the QSL on non-unitary quantum evolution needs to be considered. The generalized QSLs are established by employing the distance between the distinguishable states. Different metrics define different distances, leading to different QSL bounds. We show a few typical generalized QSL bounds based on different metrics. And the unified QSL bound for non-unitary evolution is obtained by combining the generalized MT and ML bounds under the same metric. The generations and developments of QSL have successfully generalized the MT bounds. Even the ML bounds have been attempted to generalize, the current studies of ML bound are thought as the significantly harder task, and keep as the open questions. The influences of external environment and quantum system dynamics which include the role of entanglement in QSL for open dynamics and many-body systems, the relation between the maximum interaction speed and QSL in quantum spin systems, the non-Markovianity effect of the environment on accelerating the speed of evolution have been investigated. Furthermore, by employing the gauge invariant distance in the Riemannian metric, we establish a distinct QSL bound, which relates the quantum speed limit time to the geometry phase, and presents the geometric and gauge invariant properties of it. The bound shows the impacts of the transmission modes of the state vectors on the evolution path in the manifold. Different from the established MT and ML bounds, our bound establishes the relation between the QSL and the Pancharatnam connection 1-form, and our results show that the QSL can be controlled via manipulating the generalized Pahcharatnam connection. Also, our results of QSL are holonomy via the generalized Pahcharatnam connection, which show us a possible technique to speed up the holonomic quantum computation. The bound is also interesting to the metrology and non-equilibrium thermodynamics, etc. QSL has also been studied in the quantum representation of Wigner function. In the future research of quantum speed limit, there are many issues that need to be considered, such as the generalized ML bound, the unified QSL bound under different metrics, the application of maximum rate of quantum information derived from the QSL, and the relation between the quantum speed limit and the quantum heat engine, etc.