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arXiv:1410.3826 · Physical Review A

Perfect Zeno-like effect through imperfect measurements at a finite frequency

David Layden, Eduardo Martín-Martínez, Achim Kempf

Published Feb 10, 2015• 20 citations
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Abstract

The quantum Zeno effect is usually thought to require infinitely frequent and perfect projective measurements to freeze the dynamics of quantum states. We show that perfect freezing of quantum states can also be achieved by more realistic nonprojective measurements performed at a finite frequency.
DOI
10.1103/physreva.91.022106

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The quantum Zeno effect, in its original form, uses frequent projective measurements to freeze the evolution of a quantum system that is initially governed by a fixed Hamiltonian. We generalize this effect simultaneously in three directions by allowing open system dynamics, time-dependent evolution equations, and general quantum operations in place of projective measurements. More precisely, we study Markovian master equations with bounded generators whose time dependence is Lipschitz continuous. Under a spectral gap condition on the quantum operation, we show how frequent measurements again freeze the evolution outside an invariant subspace. Inside this space, the evolution is described by a modified master equation.
Measurements in quantum mechanics can not only effectively freeze the quantum system (the quantum Zeno effect) but also accelerate the time evolution of the system (the quantum anti-Zeno effect). In studies of these effects, a quantum state is prepared repeatedly by projecting the quantum state onto the initial state. In this paper, we repeatedly prepare the initial quantum state in a different manner. Instead of only performing projective measurements, we allow unitary operations to be performed, on a very short time-scale, after each measurement. We can then repeatedly prepare the initial state by performing some projective measurement and then, after each measurement, we perform a suitable unitary operation to end up with the same initial state as before. Our objective is to find the projective measurements that minimize the effective decay rate of the quantum state. We find such optimal measurements and the corresponding decay rates for a variety of system-environment models such as the pure dephasing model and the spin-boson model. We find that there can be considerable differences between this optimized effective decay rate and the usual decay rate obtained by repeatedly projecting onto the initial state. In particular, the Zeno and anti-Zeno regimes can be considerably modified.
A purely dynamical explanation for the quantum Zeno effect is proposed. It is argued that a quantum system undergoes a quantum-Zeno-type dynamics as a consequence of a particular type of evolution involving a series of frequent spectral decompositions. The role of quantum measurements and of the ``collapse of the wave function'' is investigated and it is clarified that, provided a final observation is performed, the dynamical quantum Zeno effect can be obtained without making use of von Neumann's projection postulate. The meaning of infinitely frequent measurements is critically discussed and it is argued that it should be regarded as a mathematical idealization, impossible to realize from a physical point of view.
The quantum Zeno effect typically refers to freezing the dynamics of a quantum system through frequent observations. In general, quantum Zeno dynamics is obtained with an error of order O(1/N)\mathcal{O}(1/N)O(1/N), where NNN is the number of projective measurements performed within a fixed evolution time. In this work, we develop higher-order Zeno sequences that achieve faster convergence to Zeno dynamics, yielding an improved error scaling of O(1/N2k)\mathcal{O}(1/N^{2k})O(1/N2k), where kkk describes the order of the Zeno sequence. This is achieved by relating higher-order Zeno sequences to higher-order Trotter formulas that achieve similar convergence behavior. We leverage this relation to develop higher-order Zeno sequences for different manifestations of the quantum Zeno effect, including frequent projective measurements and unitary kicks. We go on to discuss achieving quantum Zeno dynamics through periodic control fields of high frequency. We explicitly develop control fields that yield a second-order type improvement in the Zeno error scaling and present shorter Zeno sequences. Finally, we discuss the connection to randomized and Uhrig dynamical decoupling to develop more efficient implementations in the weak coupling regime.
If frequent measurements ascertain whether a quantum system is still in its initial state, transitions to other states are hindered and the quantum Zeno effect takes place. However, in its broader formulation, the quantum Zeno effect does not necessarily freeze everything. On the contrary, for frequent projections onto a multidimensional subspace, the system can evolve away from its initial state, although it remains in the subspace defined by the measurement. The continuing time evolution within the projected "quantum Zeno subspace" is called "quantum Zeno dynamics:" for instance, if the measurements ascertain whether a quantum particle is in a given spatial region, the evolution is unitary and the generator of the Zeno dynamics is the Hamiltonian with hard-wall (Dirichlet) boundary conditions. We discuss the physical and mathematical aspects of this evolution, highlighting the open mathematical problems. We then analyze some alternative strategies to obtain a Zeno dynamics and show that they are physically equivalent.
Abstract. If frequent measurements ascertain whether a quantum system is still in its initial state, transitions to other states are hindered and the quantum Zeno effect takes place. However, in its broader formulation, the quantum Zeno effect does not necessarily freeze everything. On the contrary, for frequent projections onto a multidimensional subspace, the system can evolve away from its initial state, although it remains in the subspace defined by the measurement. The continuing time evolution within the projected “quantum Zeno subspace ” is called “quantum Zeno dynamics: ” for instance, if the measurements ascertain whether a quantum particle is in a given spatial region, the evolution is unitary and the generator of the Zeno dynamics is the Hamiltonian with hard-wall (Dirichlet) boundary conditions. We discuss the physical and mathematical aspects of this evolution, highlighting the open mathematical problems. We then analyze some alternative strategies to obtain a Zeno dynamics and show that they are physically equivalent.