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Quantum speed limit for a thermal state after a quench

Oleg Valentinovich Lychkovskiy

Published May 13, 2020• 0 citations
OpenAlex

Abstract

Quantum speed limits are rigorous estimates on how fast a state of a quantum system can depart from the initial state in the course of quantum evolution. We derive a quantum speed limit for a system initially prepared in a thermal state and evolving after a quench of its Hamiltonian. This quantum speed limit exploits the structure of the thermal state and, in particular, explicitly depends on the temperature. As a consequence, it can be dramatically tighter than general quantum speed limits applied to thermal states.

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Quantum speed limits are rigorous estimates on how fast a state of a quantum system can depart from the initial state in the course of quantum evolution. We derive a quantum speed limit for a system initially prepared in a thermal state and evolving after a quench of its Hamiltonian. This quantum speed limit exploits the structure of the thermal state and, in particular, explicitly depends on the temperature. As a consequence, it can be dramatically tighter than general quantum speed limits applied to thermal states.
Quantum speed limits are rigorous estimates on how fast a state of a quantum system can depart from the initial state in the course of quantum evolution. Most known quantum speed limits, including the celebrated Mandelstam-Tamm and Margolus-Levitin ones, are general bounds applicable to arbitrary initial states. However, when applied to mixed states of many-body systems, they, as a rule, dramatically overestimate the speed of quantum evolution and fail to provide meaningful bounds in the thermodynamic limit. Here we derive a quantum speed limit for a closed system initially prepared in a thermal state and evolving under a time-dependent Hamiltonian. This quantum speed limit exploits the structure of the thermal state and, in particular, explicitly depends on the temperature. In a broad class of many-body setups it proves to be drastically stronger than general quantum speed limits.
In the Schr\"odinger picture, the state of a quantum system evolves in time and the quantum speed limit describes how fast the state of a quantum system evolves from an initial state to a final state. However, in the Heisenberg picture the observable evolves in time instead of the state vector. Therefore, it is natural to ask how fast an observable evolves in time. This can impose a fundamental bound on the evolution time of the expectation value of quantum-mechanical observables. We obtain the quantum speed limit time bound for observable for closed systems, open quantum systems, and arbitrary dynamics. Furthermore, we discuss various applications of these bounds. Our results can have several applications ranging from setting the speed limit for operator growth, correlation growth, quantum thermal machines, quantum control, and many-body physics.
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.
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.
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.