How fast a quantum state can evolve has attracted considerable attention in connection with quantum measurement and information processing. A lower bound on the orthogonalization time, based on the energy spread DeltaE, was found by Mandelstam and Tamm. Another bound, based on the average energy E, was established by Margolus and Levitin. The bounds coincide and can be attained by certain initial states if DeltaE=E. Yet, the problem remained open when DeltaE not equal E. We consider the unified bound that involves both DeltaE and E. We prove that there exist no initial states that saturate the bound if DeltaE not equal E. However, the bound remains tight: for any values of DeltaE and E, there exists a one-parameter family of initial states that can approach the bound arbitrarily close when the parameter approaches its limit. These results establish the fundamental limit of the operation rate of any information processing system.
How fast a quantum state can evolve has attracted considerable attention in connection with quantum measurement and information processing. A lower bound on the orthogonalization time, based on the energy spread DeltaE, was found by Mandelstam and Tamm. Another bound, based on the average energy E, was established by Margolus and Levitin. The bounds coincide and can be attained by certain initial states if DeltaE=E. Yet, the problem remained open when DeltaE not equal E. We consider the unified bound that involves both DeltaE and E. We prove that there exist no initial states that saturate the bound if DeltaE not equal E. However, the bound remains tight: for any values of DeltaE and E, there exists a one-parameter family of initial states that can approach the bound arbitrarily close when the parameter approaches its limit. These results establish the fundamental limit of the operation rate of any information processing system.
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.
Quantum speed limits set the maximal pace of state evolution. Two well-known limits exist for a unitary time-independent Hamiltonian: the Mandelstam-Tamm and Margolus-Levitin bounds. The former restricts the rate according to the state energy uncertainty, while the latter depends on the mean energy relative to the ground state. Here we report on an additional bound that exists for states with a bounded energy spectrum. This bound is dual to the Margolus-Levitin one in the sense that it depends on the difference between the state's mean energy and the energy of the highest occupied eigenstate. Each of the three bounds can become the most restrictive one, depending on the spread and mean of the energy, forming three dynamical regimes which are accessible in a multilevel system. The new bound is relevant for quantum information applications, since in most of them, information is stored and manipulated in a Hilbert space with a bounded energy spectrum.
One version of the energy-time uncertainty principle states that the minimum time ${T}_{\ensuremath{\perp}}$ for a quantum system to evolve from a given state to any orthogonal state is $h/(4\mathrm{\ensuremath{\Delta}}E)$, where $\mathrm{\ensuremath{\Delta}}E$ is the energy uncertainty. A related bound called the Margolus-Levitin theorem states that ${T}_{\ensuremath{\perp}}\ensuremath{\ge}h/(2\ensuremath{\langle}E\ensuremath{\rangle})$, where $\ensuremath{\langle}E\ensuremath{\rangle}$ is the expectation value of energy and the ground energy is taken to be zero. Many subsequent works have interpreted ${T}_{\ensuremath{\perp}}$ as defining a minimal time for an elementary computational operation and correspondingly a fundamental limit on clock speed determined by a system's energy. Here we present local time-independent Hamiltonians in which computational clock speed becomes arbitrarily large relative to $\ensuremath{\langle}E\ensuremath{\rangle}$ and $\mathrm{\ensuremath{\Delta}}E$ as the number of computational steps goes to infinity. We argue that energy considerations alone are not sufficient to obtain an upper bound on computational speed, and that additional physical assumptions such as limits to information density and information transmission speed are necessary to obtain such a bound.
The Mandelstam-Tamm and Margolus-Levitin quantum speed limits bound how fast a state evolves, using the energy variance and the mean energy above the ground state. Speed limits on observables -- the change of an expectation value <A(t)> -- have so far used the variance (the Mandelstam-Tamm / quantum-Fisher-information lineage); the mean energy has not been brought to bear on the change of <A> in a fixed state. We close this branch. First, a no-go theorem: there is no state-independent linear mean-energy bound on the time to change an observable's expectation by Delta = |<A(T)> - <A(0)>|; the optimal state-independent exponent of Delta is exactly two. Second, the corresponding sharp quadratic bound, T (<H> - E_0) >= C_* Delta^2 / sigma_A^2, for every time-independent Hamiltonian H (ground energy E_0), every bounded observable A with spectral spread sigma_A = (lambda_max - lambda_min)/2, and every pure or mixed state, with the dimension-independent constant C_* = 1/(8 sin x_*) = 0.172506267461..., where x_* is the smallest positive root of tan(x/2) = x. The bound is tight, approached but not attained by a near-ground two-level family. We then give the exact energy-time/swing trade-off curve of which C_* is the small-swing slope -- tight at every swing, the observable analog of the Giovannetti-Lloyd-Maccone curve for states -- show the constant survives for mixed states via joint convexity of the trace distance, sharpen it for bandwidth-limited generators and several observables at once, and show the quadratic law degrades to a linear one when the initial state is an eigenvector of the observable. It completes the (mean-energy x observable) corner of the speed-limit landscape and, being quadratic, is most constraining where the linear variance bounds are weakest. Its cleanest physical home is the autonomous quantum clock, where it gives a coherent mean-energy resolution floor.
Abstract Quantum speed limit (QSL) is the study of fundamental limits on the evolution time of quantum systems. For instance, under the action of a time-independent Hamiltonian, the evolution time between an initial and a final quantum state obeys various mutually complementary lower bounds. They include the Mandelstam–Tamm (MT) bound, the Margolus–Levitin (ML) bound, the Luo–Zhang bound, the Lee–Chau (LC) bound together with the dual ML bound introduced by Ness and coworkers. Here we show that the MT bound can be obtained by optimizing the LC bound over a certain parameter. More importantly, we report a QSL that includes all the above bounds as special cases before optimizing over the physically meaningless reference energy level of a quantum system. This unifying bound depends on a certain parameter p. For any fixed p, we find all pairs of time-independent Hamiltonian and initial pure quantum state that saturate this unifying bound. More importantly, these pairs allow us to compute this bound accurately and efficiently using an oracle that returns certain pth moments related to the absolute value of energy of the quantum state. Moreover, this oracle can be simulated by a computationally efficient and accurate algorithm for finite-dimensional quantum systems as well as for certain infinite-dimensional quantum states with bounded and continuous energy spectra. This makes our computational method feasible in a lot of practical situations. We further compare the performance of this bound for the case of a fixed p as well as the case of optimizing over p with existing QSLs. We find that if the dimension of the underlying Hilbert space is ≲ 2000 , our unifying bound optimized over p can be computed accurately in a few minutes using Mathematica code with just-in-time compilation in a typical desktop. Besides, this optimized unifying QSL is at least as good as all the existing ones combined and can occasionally be a few percent to a few times better.
Quantum speed limits (QSLs) impose fundamental constraints on the evolution speed of quantum systems. Traditionally, the Mandelstam-Tamm (MT) and Margolus-Levitin (ML) bounds have been widely employed, relying on the standard deviation and mean of energy distribution to define the QSLs. However, these universal bounds only offer loose restrictions on the quantum evolution. Here we introduce the generalized ML bounds, which prove to be more stringent in constraining dynamic evolution, by utilizing moments of energy spectra of arbitrary orders, even noninteger orders. To validate our findings, we conduct experiments in a superconducting circuit, where we have the capability to prepare a wide range of quantum photonic states and rigorously test these bounds by measuring the evolution of the system and its photon statistics using quantum state tomography. While, in general, the MT bound is effective for short-time evolution, we identify specific parameter regimes where either the MT or the generalized ML bounds suffice to constrain the entire evolution. Our findings not only establish new criteria for estimating QSLs but also substantially enhance our comprehension of the dynamic evolution of quantum systems.