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arXiv:2010.04121 · Annales Henri Poincaré

Quantum Zeno Effect in Open Quantum Systems

Simon Becker, Nilanjana Datta, Robert Salzmann

Published Jun 5, 2021• 14 citations
arXivHTMLPDF

Abstract

Abstract We prove the quantum Zeno effect in open quantum systems whose evolution, governed by quantum dynamical semigroups, is repeatedly and frequently interrupted by the action of a quantum operation. For the case of a quantum dynamical semigroup with a bounded generator, our analysis leads to a refinement of existing results and extends them to a larger class of quantum operations. We also prove the existence of a novel strong quantum Zeno limit for quantum operations for which a certain spectral gap assumption, which all previous results relied on, is lifted. The quantum operations are instead required to satisfy a weaker property of strong power-convergence. In addition, we establish, for the first time, the existence of a quantum Zeno limit for open quantum systems in the case of unbounded generators. We also provide a variety of physically interesting examples of quantum operations to which our results apply.

DOI
10.1007/s00023-021-01075-8

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Zeno dynamics in quantum open systems

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Quantum Zeno effect shows that frequent observations can slow down or even stop the unitary time evolution of an unstable quantum system. This effect can also be regarded as a physical consequence of the statistical indistinguishability of neighboring quantum states. The accessibility of quantum Zeno dynamics under unitary time evolution can be quantitatively estimated by quantum Zeno time in terms of Fisher information. In this work, we investigate the accessibility of quantum Zeno dynamics in quantum open systems by calculating noisy Fisher information when a trace preserving and completely positive map is assumed. We firstly study the consequences of non-Markovian noise on quantum Zeno effect and give the exact forms of the dissipative Fisher information and the quantum Zeno time. Then, for the operator-sum representation, an achievable upper bound of the quantum Zeno time is given with the help of the results in noisy quantum metrology. It is of significance that the noise reducing the accuracy in the entanglement-enhanced parameter estimation can conversely be favorable for the accessibility of quantum Zeno dynamics of entangled states.

arXiv:1306.2145•Scientific Reports•Jun 23, 2015•29 citations
Abstract We prove the quantum Zeno effect in open quantum systems whose evolution, governed by quantum dynamical semigroups, is repeatedly and frequently interrupted by the action of a quantum operation. For the case of a quantum dynamical semigroup with a bounded generator, our analysis leads to a refinement of existing results and extends them to a larger class of quantum operations. We also prove the existence of a novel strong quantum Zeno limit for quantum operations for which a certain spectral gap assumption, which all previous results relied on, is lifted. The quantum operations are instead required to satisfy a weaker property of strong power-convergence. In addition, we establish, for the first time, the existence of a quantum Zeno limit for open quantum systems in the case of unbounded generators. We also provide a variety of physically interesting examples of quantum operations to which our results apply.
Abstract In open quantum systems, the quantum Zeno effect consists in frequent applications of a given quantum operation, e.g., a measurement, used to restrict the time evolution (due, for example, to decoherence) to states that are invariant under the quantum operation. In an abstract setting, the Zeno sequence is an alternating concatenation of a contraction operator (quantum operation) and a C0C_0C0​ C 0 -contraction semigroup (time evolution) on a Banach space. In this paper, we prove the optimal convergence rate O(1n)\mathcal {O}(\tfrac{1}{n})O(n1​) O ( 1 n ) of the Zeno sequence by proving explicit error bounds. For that, we derive a new Chernoff-type n\sqrt{n}n​ n -Lemma, which we believe to be of independent interest. Moreover, we generalize the convergence result for the Zeno effect in two directions: We weaken the assumptions on the generator, inducing the Zeno dynamics generated by an unbounded generator, and we improve the convergence to the uniform topology. Finally, we provide a large class of examples arising from our assumptions.
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.
Frequent applications of a mixing quantum operation to a quantum system slow down its time evolution and eventually drive it into the invariant subspace of the named operation. We prove this phenomenon, the quantum Zeno effect, and its continuous variant, strong damping, in a unified way for infinite-dimensional open quantum systems, while merely demanding that the respective mixing convergence holds pointwise for all states. Both results are quantitative in the following sense: Given the speed of convergence for the mixing limits, we can derive bounds on the convergence speed for the corresponding quantum Zeno and strong damping limits. We apply our results to prove quantum Zeno and strong damping limits for the photon loss channel with an explicit bound on the convergence speed.
If frequent measurements ascertain whether a quantum system is still in a given subspace, it remains in that subspace and a quantum Zeno effect takes place. The limiting time evolution within the projected subspace is called quantum Zeno dynamics. This phenomenon is related to the limit of a product formula obtained by intertwining the time evolution group with an orthogonal projection. By introducing a novel product formula, we will give a characterization of the quantum Zeno effect for finite-rank projections in terms of a spectral decay property of the Hamiltonian in the range of the projections. Moreover, we will also characterize its limiting quantum Zeno dynamics and exhibit its (not necessarily bounded from below) generator as a generalized mean value Hamiltonian.
We provide a self‑contained mathematical analysis of the quantum Zeno limit for a dissipative quantum Markov semigroup with a GKSL generator. Frequent projective measurements modelled by a projection PP lead to an effective generator GG on the invariant subspace PHPH. The generator splits into a natural GKSL part and a leakage term −12∑kγk{Dk†Dk,ρ}−21∑kγk{Dk†Dk,ρ} with Dk=P⊥LkPDk=P⊥LkP. Perfect confinement (trace preservation) holds iff P⊥LkP=0P⊥LkP=0 for all kk. Two contrasting examples are worked out: a two‑qubit system with internal relaxation and state‑dependent leakage, and a free particle with a translation jump giving spatially structured leakage. For unbounded Hamiltonians we prove an unconditional Zeno limit under the sole hypothesis that the unitary Zeno limit defines a self‑adjoint Zeno Hamiltonian (a condition known to be broadly satisfied, e.g.\ for Dirichlet obstacles), using the Exner–Ichinose strong convergence, a standard core property of derivation generators, and a Duhamel perturbation argument; this removes any additional Banach‑space core condition. The trace of the limit state decays according to the leakage term, and the limit semigroup is a strongly continuous contraction semigroup. Beyond the qualitative limit, we establish three further quantitative results. First, for bounded generators we prove an explicit O(1/n)O(1/n) Zeno convergence rate ∥Vn(t)−etG∥≤t2κ2etκ/n∥Vn(t)−etG∥≤t2κ2etκ/n, with κκ the operator norm of the GKSL generator, sharp in its order. Second, writing Γ=∑kγkDk†DkΓ=∑kγkDk†Dk for the leakage operator, we prove a two‑sided exponential sandwich e^{-\gamma_{\max}t}\tr\rho\le\tr(e^{tG}\rho)\le e^{-\gamma_{\min}t}\tr\rho governed by spec⁡(Γ)spec(Γ), together with a matching numerical‑range localization of the eigenvalues of the associated effective non‑Hermitian Hamiltonian. Third, we show that iterated confinement through a family of pairwise commuting projections is order‑independent and agrees exactly with direct confinement through their product, exhibiting P↦GPP↦GP as a meet‑homomorphism on the associated Boolean algebra of Zeno subspaces. We also present a new algebraic identity linking the leakage operator to the projected jump intensity, yielding a compact form of the confined generator and a Gram‑matrix reduction for the leakage norm. Applications to quantum error correction, quantum thermodynamics, and the emergence of Markovian dissipation as a Zeno effect on the environment are discussed. We close with a set of genuinely open problems and a cautionary de Sitter example showing the limits of a naive horizon‑leakage analogy.
Quantum Zeno effect shows that frequent observations can slow down or even stop the unitary time evolution of an unstable quantum system. This effect can also be regarded as a physical consequence of the statistical indistinguishability of neighboring quantum states. The accessibility of quantum Zeno dynamics under unitary time evolution can be quantitatively estimated by quantum Zeno time in terms of Fisher information. In this work, we investigate the accessibility of quantum Zeno dynamics in quantum open systems by calculating noisy Fisher information when a trace preserving and completely positive map is assumed. We firstly study the consequences of non-Markovian noise on quantum Zeno effect and give the exact forms of the dissipative Fisher information and the quantum Zeno time. Then, for the operator-sum representation, an achievable upper bound of the quantum Zeno time is given with the help of the results in noisy quantum metrology. It is of significance that the noise reducing the accuracy in the entanglement-enhanced parameter estimation can conversely be favorable for the accessibility of quantum Zeno dynamics of entangled states.