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arXiv:2212.12352 · New Journal of Physics

Quantum speed limits for change of basis

Moein Naseri, Chiara Macchiavello, Dagmar Bruß, Paweł Horodecki, Alexander Streltsov

Published Feb 1, 2024• 4 citations
arXivHTMLPDF

Abstract

Abstract Quantum speed limits provide ultimate bounds on the time required to transform one quantum state into another. Here, we introduce a novel notion of quantum speed limits for collections of quantum states, investigating the time for converting a basis of states into an unbiased one as well as basis permutation. Establishing an unbiased basis, we provide tight bounds for the systems of dimension smaller than 5, and general bounds for multi-qubit systems and the Hilbert space dimension d. For two-qubit systems, we show that the fastest transformation implements two Hadamards and a swap of the qubits simultaneously. We further prove that for qutrit systems the evolution time depends on the particular type of the unbiased basis. Permuting a basis, we obtain the exact expression for the Hilbert space of dimension d. We also investigate speed limits for coherence generation, providing the minimal time to establish a certain amount of coherence with a unitary evolution.

DOI
10.1088/1367-2630/ad25a5

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Abstract Quantum speed limits provide ultimate bounds on the time required to transform one quantum state into another. Here, we introduce a novel notion of quantum speed limits for collections of quantum states, investigating the time for converting a basis of states into an unbiased one as well as basis permutation. Establishing an unbiased basis, we provide tight bounds for the systems of dimension smaller than 5, and general bounds for multi-qubit systems and the Hilbert space dimension d. For two-qubit systems, we show that the fastest transformation implements two Hadamards and a swap of the qubits simultaneously. We further prove that for qutrit systems the evolution time depends on the particular type of the unbiased basis. Permuting a basis, we obtain the exact expression for the Hilbert space of dimension d. We also investigate speed limits for coherence generation, providing the minimal time to establish a certain amount of coherence with a unitary evolution.
Quantum speed limits are the bounds that define how quickly one quantum state can transform into another. Instead of focusing on the transformation between pairs of states, we provide bounds on the speed limit of quantum evolution by unitary operators in arbitrary dimensions. These do not depend on the initial and final state but depend only on the trace of the unitary operator that is to be implemented and the gross characteristics (average and variance) of the energy spectrum of the Hamiltonian which generates this unitary evolution. The bounds that we find can be thought of as the generalization of the Mandelstam-Tamm (MT) and the Margolus-Levitin (ML) bound for state transformations to implementations of unitary operators. We will discuss the application of these bounds in several classes of transformations that are of interest in quantum information processing.
Abstract In quantum control, quantum speed limits provide fundamental lower bounds on the time that is needed to implement certain unitary transformations. Using Lie algebraic methods, we link these speed limits to symmetries of the control Hamiltonians and provide quantitative bounds that can be calculated without solving the controlled system dynamics. In particular we focus on two scenarios: On one hand, we provide bounds on the time that is needed for a control system to implement a given target unitary U and on the other hand we bound the time to implement the most difficult to reach evolution e − i θ H s of a target Hamiltonian H s . We apply our abstract bounds on physically relevant systems like coupled qubits, spin chains, globally controlled Rydberg atoms and NMR molecules and compare our results to the existing literature. We hope that our bounds can aid experimentalists to identify bottlenecks and design faster quantum control systems.
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 laws of quantum physics place a limit on the speed of computation, in particular the evolution time of a system cannot be arbitrarily fast. Bounds on the speed of evolution for unitary dynamics have been long studied. A few bounds on the speed of evolution for noisy dynamics have also been obtained recently, these bounds, however, are in general not tight. In this article we present a new framework for quantum speed limit of noisy dynamics. With this framework we obtain the exact maximal angle that a noisy dynamics can achieve at any given time, this then provides tight bounds on the evolution time for noisy dynamics. The obtained bound reveals that noisy dynamics are generically different from unitary dynamics, in particular we show that the 'orthogonalization' time, which is the minimum time needed to evolve any state to its orthogonal states, is in general not applicable to noisy dynamics.
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.
The speed limit on entanglement is defined as the maximal rate at which entanglement can be generated or degraded in a physical process. We derive the speed limits on entanglement, using the relative entropy of entanglement and trace-distance entanglement, for unitary and arbitrary quantum dynamics, where we assume that the dynamics of the closest separable state can be approximately described by the closest separable dynamics of the actual dynamics of the system. For the unitary dynamics of isolated bipartite systems which are described by pure states, the rate of entanglement production is bounded by the product of fluctuations of the system's driving Hamiltonian and the surprisal operator, with an additional term reflecting the time-dependent nature of the closest separable state. Removing restrictions on the purity of the input and on the unitarity of the evolution, the two terms in the bound get suitably altered. Furthermore, we find a lower bound on the time required to generate or degrade a certain amount of entanglement by arbitrary quantum dynamics. We demonstrate the tightness of our speed limits on entanglement by considering quantum processes of practical interest.