Quantum Zeno effect

Figure 4.4 Frequency-domain representation of the dynamically controlled decoherence rate in various limits (Section 4.4). (a) Golden-Rule limit, (b) Anti-Zeno effect (AZE) limit (c) Quantum Zeno effect (QZE) limit. Here, F,( ) and G(w) are the modulation and bath spectra, respectively and F are the interval of change and width of G( ), respectively and is the interruption rate.

This limit is that of the quantum Zeno effect (QZE), namely, the suppression of relaxation as the interval between interruptions decreases [64-66]. In this limit, the system-bath exchange is reversible and the system coherence is fully maintained (Figure 4.4c). Namely, the essence of the QZE is that sufficiently rapid interventions prevent the excitation escape to the continuum, by reversing the exchange with the bath. 

The prevailing view until recently has been that successive frequent measurements (interruptions of the evolution) known as the quantum Zeno effect must slow down the decay of any unstable system. A few years ago, Kofman et al. [Kofman 2000 Kofman 2001 (a)] showed that, in fact, the opposite is commonly true for decay into open-space continua the anti-Zeno effect (AZE), i.e., decay acceleration by frequent measurements1, is far more ubiquitous than the QZE [Milonni 2000 Seife 2000]. How can this conclusion be understood and what was missing in standard treatments that claimed the QZE universality The last paper of this part, by G. Kurizki et al. shows that ... 

Abstract The protection of the coherence of open quantum systems against the influence of their environment is a very topical issue. The main features of quantum error-correction are reviewed here. Moreover, an original scheme is proposed which protects a general quantum system from the action of a set of arbitrary uncontrolled unitary evolutions. This method draws its inspiration from ideas of standard error-correction (ancilla adding, coding and decoding) and the Quantum Zeno Effect. A pedagogical demonstration of our method on a simple atomic system, namely a Rubidium isotope, is proposed. 

The phenomenon known as the quantum Zeno effect takes place in a system which is subject to frequent measurements projecting it onto its (necessarily known) initial state if the time interval between two projections is small enough the evolution of the system is nearly "frozen". This effect, and its inverse (the anti-Zeno effect), have been widely investigated theoretically [Khalhn 1957-58 Winter 1961 Misra 1977 Fonda 1978 Kofman 1996 Kof-man 2000 Lewenstein 2000 Kofman 2001 (a) Schmidt 2003 / 2004] as well as experimentally [Cook 1988 Itano 1990 Wilkinson 1997 Fischer 2001], Generalizations have been proposed which employ incomplete measurements [Facchi 2002] in this setting, the Hilbert space is split into "Zeno subspaces" (degenerate multidimensional eigenspaces of the measured observable), and the state vector of the system is compelled by frequent measurements of the physical observable to remain in its initial Zeno subspace. The dynamics of the system in the Zeno subspaces has also been studied in different specific situations [Facchi 2001 (b)]. 

The paper is organized as follows. In Sec. 1, we introduce the main features of quantum error-correction, and, particularly, we present the already well-developed theory of quantum error-correcting codes. In Sec. 2, we present a multidimensional generalization of the quantum Zeno effect and its application to the protection of the information contained in compound systems. Moreover, we suggest a universal physical implementation of the coding and decoding steps through the non-holonomic control. Finally, in Sec. 3, we focus on the application of our method to a rubidium isotope. 

As we can see at the end of this short review of the main practical error-correction methods, it seems that all the ways of explicitly building quantum codes, or more generally that all the existing protection schemes require special features from the errors they combat. In the following, we address the problem of unitary errors and show that information can be protected from their action through a generalization of the quantum Zeno effect, without making any symmetry assumption about them. 

The section starts by the presentation of a multidimensional generalization of the quantum Zeno effect, which we then employ to protect information in compound systems. Finally, we suggest the non-holonomic control technique as a physical way to implement the coding / decoding steps of our scheme. 

The standard quantum Zeno effect [Khalhn 1957-58 Winter 1961 Misra 1977 Fonda 1978 Kofman 1996 Kofman 2000 Lewenstein 2000 Kofman 2001 (a) Schmidt 2003 / 2004] implies that we can nearly "freeze" the evolution of the system by measuring it frequently enough in one of its predetermined initial states in other words, this effect allows us to protect the onedimensional subspace spanned by the initial state of the system from the influence of the error-inducing Hamiltonian (11). In what follows, we generalize this effect so as to protect an arbitrary multidimensional subspace C from H(t), under certain conditions that we will explicit in the following. 

Let us assume that we are physically able to perform a measurement-induced projection onto C in the system S (see the following subsection for a discussion of such projections in compound systems comprising an information subsystem and an ancilla). If we just follow the standard quantum Zeno effect procedure and merely project the state vector I tpe) (resulting from the infinitesimal evolution of the initial state ip)) onto C, we obtain a vector ipv), which (a priori) differs from ip) (see fig. 1 a). This occurs due to the fact that usually the... 

Figure 1. Multidimensional quantum Zeno effect a) a simple projection fails to recover the initial vector, b) the sequence coding-decoding-projection protects the initial vector.

The multidimensional generalization of the quantum Zeno effect we have just described allows us to protect an arbitrary subspace C of a Hilbert space H... 

Information protection through the multidimensional quantum Zeno effect... 

The loss of coherence for an individual quantum system is always due to an entanglement with some other system and this holds even for a thermal reservoir. This can be seen from the following consideration assume that the quantum system studied is coupled to the reservoir with a coupling strength a and the reservoir relaxation rate is given by k. Standard adiabatic elimination of the reservoir then yields an atomic decay rate 7 = a2/k which tends to zero for an arbitrarily fast reservoir. This is due to the fact that for such a reservoir the combined system would truly remain in a product state and no entanglement would ever be built up. This may also be taken as a variant of quantum Zeno effect, in which the decay into an infinitely fast reservoir corresponds to very rapidly repeated measurements that prevent the spontaneous decay. 

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