Quantum state sensitivity to initial conditions

The different time-dependent distances of two arbitrarily close quantum or classical-statistical states to a third fixed state are shown to imply an experimentally relevant notion of state sensitivity to initial conditions. A quantitative classification scheme of quantum states by their sensitivity and instability in state space is given that reduces to the one performed by classicalmechanical Lyapunov exponents in the classical limit. PACS number(s): 03.65.Bz, 05.45.+b Typeset using REVTEX ∗Electronic mail: [email protected] 1 Stimulated by the pioneering work of Peres [1,2] on state sensitivity to small changes in the Hamiltonian, several new developments have taken the study of quantum state sensitivity to the experimental realm. Schack and Caves [3] have analyzed the sensitivity to perturbations as the amount of information about a perturbing environment that is needed to keep entropy from increasing. Their approach links the study of quantum sensitivity to statistical mechanics and to quantum information theory. Recently, Gardiner et al. [4] have proposed a detailed experimental set-up of an ion trap to study quantum state sensitivity to changes in the Hamiltonian. A different approach has been taken by Ballentine and Zibin [5] that study the emergence of classical state sensitivity from quantum theory in computational time reversal. The difficulty of directly tackling the problem of quantum state sensitivity to initial conditions stems from the following reason. Classical sensitivity to initial conditions is made precise when measured by Lyapunov characteristic exponents [6] λc(x0) = lim dc(0)→0 lim t→∞ 1 t ln ( dc (xt,yt) dc (x0,y0) ) (1) with dc (xt,yt) the Euclidean distance [7] between two phase space vectors xt and yt. There is classical sensitivity to initial conditions when λc(x0) > 0 and the trajectory starting at x0 is said to be unstable. A direct attempt to define the quantum analog of the classical Lyapunov characteristic exponent in (1) substitutes the phase space distance between two classical trajectories by the Hilbert-space H distance between two close quantum vectors giving λH [|ψ(0)〉] = lim dH(0)→0 lim t→∞ 1 t ln ( dH (|ψ(t)〉 , |φ(t)〉) dH (|ψ(0)〉 , |φ(0)〉) ) = 0, (2) a zero quantum Lyapunov exponent for all states as any two states do not separate at all with dH (|ψ(t)〉 , |φ(t)〉) ≡ ‖|ψ(t)〉 − |φ(t)〉‖ = ‖|ψ(0)〉 − |φ(0)〉‖ and ‖·‖ = √ 〈· |·〉. According to (2) all possible quantum states are stable and no sensitivity of quantum states to initial conditions can be found. However, there are three strong objections against the use of λH as a measure of state sensitivity to initial conditions and instability. Firstly, the quantum mechanical state 2 space is the complex projective Hilbert space CPn that has a curvature not present in the complex Hilbert space H C . It is a nontrivial Kähler manifold with a symplectic form and an associated Riemannian metric [8]. Secondly, equation (2) is obtained by substituting the classical phase space trajectory distance in (1) by a quantum distance. This is not a mere technical error but a conceptual one as the closest classical object to a quantum state is a Liouville density, not a single trajectory [2,3,5]. Recall for example that in the classical limit the dynamical equation for the Wigner function or for the coherent state representation of the quantum density reduce to the Liouville equation [2]. A definition analogous to (2) for Liouville densities gives λL [ρ(0)] = 0 with dL (ρ1(t), ρ2(t)) ≡ √∫ dqdpρ 1 (q, p, t)ρ 1/2 2 (q, p, t) = dL (ρ1(0), ρ2(0)) by Koopman ′s theorem. The fact that λL(ρ(t)) = 0 for all possible Liouville densities tells us that we cannot know from (2) if quantum states have sensitivity as this type of measure for classical Liouville densities is insensitive to classical-mechanical instability. Thirdly, unlike the classical trajectory distance, dH is a bounded metric, thus precluding the exponential divergence. In this paper we propose to overcome the above three objections in three steps to provide a working scheme to classify quantum and classical-statistical states by their sensitivity and instability in projective space Pn. This sensitivity and instability are shown to be measurable by the different transition probabilities of two nearby states to a third fixed state. Step 1. A classical density ρ(x) can be mapped into a real Hilbert spaceH R by taking its square root, ψc(x) ≡ √ ρ(x). The expectation of a classical operator F is 〈F 〉 = 〈Fψ2 c 〉 / 〈ψ2 c 〉 , which means that the physical space is not H R but the space of equivalence classes obtained by identifying ψc ∼ λψc for any λ ∈ R − {0}. This physical space is the real projective space RPn. Similarly, in quantum mechanics the true physical space is not the complex Hilbert space H C but the space of equivalent classes obtained by identification of vectors ψQ ∼ λψQ for any λ ∈ C−{0} , the complex projective space CPn. Classical and quantum states in Pn will be written as ψ̃. The distance between two points in Pn (real or complex) can be defined as