The quantum speed limit is important in determining the minimum evolution time of a quantum system, and thus is essential for quantum community. In this Letter, we derive a novel unified quantum speed limit bound for Hermitian and non-Hermitian quantum systems. The bound is quantified by the changing rate of phase of the quantum system, which represents the transmission mode of the quantum states over their evolution. The bound leads to further insights beyond the previous bounds on concrete evolution modes of the quantum system, such as horizontal or parallel transition or horizontal joining of the two quantum states in Hilbert space. The bound is linked to the feasibility of the evolutions of the state vectors, and provides a tighter upper bound. In addition, the generalized Margolus-Levitin bound is discussed.
The quantum speed limit is important in determining the minimum evolution time of a quantum system, and thus is essential for quantum community. In this Letter, we derive a novel unified quantum speed limit bound for Hermitian and non-Hermitian quantum systems. The bound is quantified by the changing rate of phase of the quantum system, which represents the transmission mode of the quantum states over their evolution. The bound leads to further insights beyond the previous bounds on concrete evolution modes of the quantum system, such as horizontal or parallel transition or horizontal joining of the two quantum states in Hilbert space. The bound is linked to the feasibility of the evolutions of the state vectors, and provides a tighter upper bound. In addition, the generalized Margolus-Levitin bound is discussed.
We derive a distinct bound of the quantum speed limit for a non-Hermitian quantum system by employing the gauge invariant and geometric natures of quantum mechanics. The bound is of geometric properties since it relates to the geometric phase of the quantum system, and it is tighter than the Mandelstam-Tamm and Margolus-Levitin bounds in some cases. Also, by making the geodesic assumption, the analog of the Margolus-Levitin bound is derived for the time-dependent (non-)Hermitian quantum system. These two bounds reflect the impacts of the transmission modes of the state vectors on the evolution path in the manifold.
The quantum speed limit (QSL) refers to the maximum speed of a quantum system to evolve from an initial state to its orthogonal states. The bound on the QSL for Hermitian systems, for example the Mandelstam-Tamm (MT) and Margolus-Levitin (ML) as well as Sun-Zheng(SZ) bound, was studied respectively from the perspectives of average value and variance of the system Hamiltonian as well as the geometry of the system. While the compactness of the MT-type, ML-type and SZ-type bounds has been examined well for Hermitian systems, a compact QSL for non-Hermitian systems has not been well studied. In this work, based on the biorthogonal basis theory we derive two distinct and tighter bounds on the QSL for non-Hermitian systems, which correspond to the MT and ML bounds for Hermitian systems. We show that the shortest evolution time corresponding to the two bounds of the non-Hermitian system can be attained by certain initial states, showing the compactness and tightness of our bounds. These initial states dubbed fastest initial states(FIS) are different from that in Hermitian systems. A bound close to QSL for non-FIS is presented and comparison of our bound with others in literature is performed. To illustrate our results, we present a minimal non-Hermitian system to show QSL, and the condition for the shortest evolution time is derived analytically using the present theory.
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 describe how fast a system can evolve from one state to another. There are several well-known quantum speed limits such as the Mandelstam–Tamm and Margolus–Levitin bounds which were originally derived for pure states and are not always tight for mixed states. In this thesis a stronger bound is derived for mixed qubit states, which has been shown to be tight, by using additional information about the system. We chose to focus on a single qubit and use the Bloch representation to describe its evolution geometrically. Due to both the energy and the purity of the system being conserved under unitary time evolution, the Bloch vector moves in a circle. By expressing the fidelity between the initial and final state in terms of Bloch vectors, we can derive a stricter quantum speed limit for mixed qubit states. This new lower bound depends on both fidelity and purity and is tighter than the Margolus–Levitin bound. The result shows that taking conserved quantities like purity into account can lead to improved quantum speed limits and similar ideas might be useful for more complex systems as well.
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.
Non-Hermitian Hamiltonians play a crucial role in describing open quantum systems and nonequilibrium dynamics. In this paper, we derive trade-off relations for systems governed by non-Hermitian Hamiltonians, focusing on the Margolus-Levitin-type and Mandelstam-Tamm-type bounds, which are originally derived as quantum speed limits in isolated quantum dynamics. While the quantum speed limit for the Mandelstam-Tamm bound in general non-Hermitian systems was derived in the literature, we obtain a Mandelstam-Tamm quantum speed limit for continuous measurement using the continuous matrix product state formalism. Moreover, we derive a Margolus-Levitin quantum speed limit in the non-Hermitian setting. We derive additional bounds on the ratio of the standard deviation to the mean of an observable, which take the same form as the thermodynamic uncertainty relation. As an example, we apply these bounds to the continuous measurement formalism in open quantum dynamics, where the dynamics is described by discontinuous jumps and smooth evolution induced by the non-Hermitian Hamiltonian. Our work provides a unified perspective on the quantum speed limit and thermodynamic uncertainty relations in open quantum dynamics from the viewpoint of the non-Hermitian Hamiltonian, extending the results of previous studies.