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.In the harmonic-oscillator case, approximateproviding that half-widths of the coherent states in x andreversibility is effectively guaranteed, since the actionp decrease to zero as ’!0.This would be assured when,associated with the 1- contour of the Gaussian statefor instance, in Eqs.(5.41) and (5.42),increases with time at the rate (Zurek, Habib, and Paz,2.(5.55)1993)Thus individual coherent-state Gaussians approachkBT0.(5.58)phase-space points.This behavior indicates that in amacroscopic open system nothing but probability distri-Action I is a measure of the lack of information aboutbutions over localized phase-space points can survive inphase-space location.Hence its rate of increase is a mea-the ’!0 limit for any time of dynamical or predictivesure of the rate of predictability loss.The trajectory is asignificance.[Coherence between immediately adjacentlimit of the tube swept through phase space by thepoints separated only by , Eq.(5.55), can last longer.moving contour representing the instantaneous uncer-This is no threat to the classical limit.Small-scale coher-tainty of the observer about the state of the system.Evo-ence is a part of a quantum halo of the classical pointerlution is approximately deterministic when the area ofstates (Anglin and Zurek, 1996).]this contour is nearly constant.In accord with Eqs.The mathematical classical limit implemented by let-ting ’!0 becomes possible in the presence of decoher- (5.56) and (5.57) 0 tends to zero in the reversible mac-roscopic limit:ence.It is tempting to carry this strategy to its logicalconclusion and represent every probability density in0 O 1/R.(5.59)phase space in the point(er) basis of narrowing coherentstates.Such a program is beyond the scope of this re- The existence of an approximately reversible trajectory-view, but the reader should by now be convinced that it like thin tube provides an assurance that, having local-is possible.Indeed, Perelomov (1986) showed that a ized the system within a regular phase-space volume atgeneral quantum state could be represented in a sparse t 0, we can expect to find it later inside the Liouville-basis of coherent states that occupy the sites of a regular transported contour of nearly the same measure.Similarlattice, providing that the volume per coherent state conclusions follow for integrable systems.Rev.Mod.Phys., Vol.75, No.3, July 2003 745Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical3.Ignorance inspires confidence in classicalitynent, while Vx /Vxxx characterizes the dominantscale of nonlinearities in the potential V(x), and pDynamical reversibility can be achieved with einselec-gives the coherence scale in the initial wave packet.Thetion in the macroscopic limit.Moreover, 0/I or otherabove estimate, Eq.(3.5), depends on the initial condi-measures of predictability loss decrease with the in-tions.It is smaller than, but typically close to, trcrease of I.This is especially dramatic when quantified1ln I/ , Eq.(3.6), where I is the characteristic ac-in terms of the von Neumann entropy, that, for Gaussiantion of the system.By contrast, phase-space patches ofstates, increases at the rate (Zurek, Habib, and Paz,regular systems undergo stretching with a power of time.1993)Consequently, loss of correspondence occurs only over aI 1 much longer tr (I/ ) , which depends polynomially on" 0 ln.(5.60a).I 1The resulting " is infinite for pure coherent states (I1.Restoration of correspondence1), but quickly decreases with increasing I.Similarly,the rate of purity loss for Gaussians isExponential instability spreads the wave packet to aparadoxical extent at the rate given by the positiveÙ0/I2.(5.60b)(i)Lyapunov exponents.Einselection attempts to en-Again, it tapers off for more mixed states.force localization in phase space by tapering off interfer-This behavior is reassuring.It leads us to concludeence terms at a rate given by the inverse of the decoher-that irreversibility quantified through, say, von Neumann1ence time scale, ( / x)2.The two processesD Tentropy production, Eq.(5.60a), will approach " reach status quo when the coherence length of thec20/I, vanishing in the limit of large I.When, in thewave packet makes their rates comparable, that is,spirit of the macroscopic limit, we do not insist on the1.(5.61)Dmaximal resolution allowed by quantum indeterminacy,the subsequent predictability losses measured by the in-This yields an equation for the steady-state coherencecrease of entropy or through the loss of purity will di-length and for the corresponding momentum dispersion:minish.Illusions of reversibility, determinism, and exact/2 ; (5.62)c Tclassical predictability become easier to maintain in thepresence of ignorance about the initial state!/ 2D/.(5.63)c cTo think about phase-space points one may not evenAbove, we have quoted results (Zurek and Paz, 1994)need to invoke a specific quantum state.Rather, a pointcan be regarded as the limit of an abstract recursive pro- that follow from a more rigorous derivation of the co-cedure in which the phase-space coordinates of the sys- herence length than the rough and ready approachcthat led to Eq.(5.61).They embody the same physicaltem are determined better and better in a succession ofargument, but seek asymptotic behavior of the Wignerincreasingly accurate measurements.One may betempted to extrapolate this limiting process ad infinit- function that evolves according to the equationessimum, which would lead beyond Heisenberg s inde-2n n2n 1 2n 1terminacy principle and to a false conclusion that ideal- † H,W V Wx p22n 2n 1 !n 1ized points and trajectories exist objectively, and that theinsider view of Sec.II can always be justified.While in 2D W.(5.64)pour quantum universe this conclusion is wrong, and theThe classical Liouville evolution generated by the Pois-extrapolation described above illegal, the presence,son bracket ceases to be a good approximation of thewithin Hilbert space, of localized wave packets near thedecohering quantum evolution when the leading quan-minimum-uncertainty end of such imagined sequencestum correction becomes comparable to the classicalof measurements is reassuring.Ultimately, the ability torepresent motion in terms of points and their time- force:ordered sequences (trajectories) is the essence of classi-2 2Vx Wpcal mechanics.VxxxWppp.(5.65)24 24 2 2cThe term VxWp represents the classical force in theE.Decoherence, chaos, and the second law Poisson bracket.Quantum corrections are small when.(5.66)cThe breakdown of correspondence in this chaotic set-ting was described in Sec.III.It is anticipated to occur in Equivalently, the Moyal bracket generates approxi-all nonlinear systems, since stretching of the wave mately Liouville flow when the coherence length satis-packet by the dynamics is a generic feature, absent only fiesfrom a harmonic oscillator.However, the exponential in-.(5.67)cstability of chaotic dynamics implies rapid loss ofquantum-classical correspondence after the Ehrenfest This last inequality has an obvious interpretation: it is a1time, t ln p/.Here is the Lyapunov expo- condition for localization to within a region that iscRev.Mod.Phys., Vol.75, No
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