Gaussian ensembles distributions from mixing quantum systems

In the context of the mixing dynamical systems we present a derivation of the Gaussian ensembles distributions from mixing quantum systems having a classical analog that is mixing. We find that mixing factorization property is satisfied for the mixing quantum systems expressed as a factorization of quantum mean values. For the case of the kicked rotator and in its fully chaotic regime, the factorization property links decoherence by dephasing with Gaussian ensembles in terms of the weak limit, interpreted as a decohered state. Moreover, a discussion about the connection between random matrix theory and quantum chaotic systems, based on some attempts made in previous works and from the viewpoint of the mixing quantum systems, is presented.


I INTRODUCTION
Gaussian ensemble theory emerged from the study of the complex nuclei and the long lived resonance states in 1950's by Wigner, and later by Dyson. They realized that many statistical features of energy levels of these systems can be adequately described by a simple statistical model of random Hermitian matrices. Wigner's central idea was that for quantum systems with many degrees of freedom like a heavy nucleus, we can assume that the matrix elements of the hamiltonian in a typical basis can be treated as independent Gaussian random numbers. The main prediction of this approach is that the statistical distribution of spacings between adjacent energy levels, denoted by S n = E n + 1 − E n , normalized such that the average spacing S equals one, obeys universal distributions which only depend on certain symmetry properties. These universal distributions define the Gaussian Orthogonal Ensemble (GOE), the Gaussian Unitary Ensemble (GUE) and the Gaussian Symplectic Ensemble (GSE) corresponding to the distribution of the hamiltonian matrix elements invariant under orthogonal, unitary and symplectic transformations, respectively.
On the other hand, in [12] we studied the Quantum Ergodic Hierarchy (QEH) [11,12] which expresses the cancelation of the correlations between states and observables for large times t → ∞. The cancelation of these quantum correlations occurs when any stateρ reaches the relaxation in an equilibrium stateρ * (weak limit). The relaxation is understood in the sense of the mean values (i.e. the decoherence of a set of observables) and it can be expressed in terms of a relevant subalgebra O R of the quantum algebra A which are the only observables accessible experimentally [13,14,15,16,20]. In paper [12] we also interpreted the chaotic behavior of the Casati-Prosen model [5,6] and the kicked rotator [1,2] in terms of the quantum mixing level. Given an initial stateρ, an observableÔ ∈ O R and the weak limitρ * ofρ, the QEH correlation C(ρ(t),Ô) betweenρ andÔ at time t is (ρ(t)|Ô) − (ρ * |Ô), where (f |ĝ) denotes the mean value of the observableĝ in the statef , i.e. tr(f .ĝ) = (f |ĝ). Thus the quantum mixing level is given by the condition The interpretation of the Eq. (1) is that the weak limitρ * has the meaning of a "coarse graining" ofρ over the set of observables O R where the mean values (ρ(t)|Ô) tend to the constant value (ρ * |Ô) for large times. Considering that the mean values (ρ(t)|Ô) contain information about the quantum correlations ofρ at time t then the mean values of the form (ρ * |Ô) represent the correlations ofρ in the asymptotic limit t → ∞. In this sense we say that the weak limitρ * is such as an "equilibrium state" since all the correlations ofρ in the asymptotic limit are contained inρ * . This property is one of the keys to obtain the main results of the paper. Furthermore, if S is a quantum system whose classical limit S cl has N + 1 constants of the motion in involution locally 1 we can summarize the role of the weak limitρ * in the steps involved in the classical limit [paper limite clasico, 1 I.e. the classical Hamiltonian O0(φ) = H(φ) is a constant of motion defined over the phase space Γ and there exist a set of N constants of motion {Oi(φ)}i=1,...,N satisfying on a maximal domain of integration D φ i ⊆ Γ around any point φi ∈ Γ for all i = 1, ..., N .

paper olimpia mario]:
Quantum mechanics =⇒ Classical statistical mechanics =⇒ Classical mechanics The steps of the Eq. (3) can be considered as a formalism of the classical limit that allows study several phenomena as decoherence (in closed and open systems [citar papers mario fune olimpia]), quantum chaos [citar papers nacho mario]. In the first step, the non-diagonal part of (ρ(t)|Ô) denoted by (t) is responsible for the purely quantum effects (quantum interference, entanglement, etc) and contains quantum correlations 2 ofρ(t) as it evolves in time. Instead, the diagonal part (ρ * |Ô) admits an interpretation by ignorance, i.e. it has a structure of classical statistical mean value. The second step concers the classical statistical limit ( → 0, t → ∞) within the time intervals of the quantum chaos timescales t τ = τ ( ) in order to be compatible with the correspondence principle and fundamental graininess [citar paper limite clasico]. The weak limit ofρ(t) is taken from the descomposition (ρ(t)|Ô) = (ρ * |Ô) + (t) where (t) → 0 for t → ∞, i.e. the quantum correlations vanish in the classical limit (see for details). The equilibrium density is obtained from ρ * (φ) = symbρ * that is the time independent part of the Wigner function ρ( where f (φ, t) manifests the quantum effects in the phase space 3 . Finally, in the third step the equilibrium density ρ * (φ) is decomposed in classical trajectories H(φ) = ω, P iI (φ) = p iI , τ (φ) = τ 0 + ωt, θ φi (φ) = θ φi0 + p φi t corresponding to the constants of motion (globally or locally resp.) of the classical limit (integrable or non-integrable resp.) of the quantum system in question [paper limite clasico]. Therefore, the classical trajectories are obtained and classical mechanics is recovered. But the manner in which the correlations of the equilibrium stateρ * are strongly affected by the quantum mixing level are given by the property: LetÔ 1 ,Ô 2 ,Ô 3 , ...,Ô n ∈ O R be n observables and consider thatρ * is a pure state (i.e. ρ 2 * =ρ * ). Then we have (see appendix) The factorization of the right hand of Eq. (4) expresses a kind of "uncorrelation" between the observablesÔ 1 ,Ô 2 , ...,Ô n in the classical limit → 0. This property is the key to obtain the uncorrelation of the Hamiltonian matrix elements of a quantum mixing system. The two conditions for the probability P (H 11 , ..., H N N ) of the Hamiltonian matrix elements H ij which define the Gaussian ensembles are (see [1] pag. 73, 74) whereĤ ′ is obtained fromĤ by an orthogonal, unitary or symplectic transformation according to the GOE, GUE or GSE ensembles. The Eq. (5) expresses the invariance of the probability P (H 11 , H 12 , ..., H N N ) against an orthogonal, unitary or simplectic transformation. The Eq. (6) says that in the fully chaotic regime of a classically chaotic quantum system the details of the interactions are not relevant so we can replace the Hamiltonian by a matrix whose elements are uncorrelated between them, i.e. randomly chosen.

II GAUSSIAN ENSEMBLES FROM THE QUANTUM ERGODIC HIERARCHY (QEH)
In this section we establish the link between the Random Matrix Theory and the quantum mixing level of the QEH. For this we consider a quantum mixing system 4 , denoted by S, whose HamiltonianĤ of dimension N × N is unknown. Since we are looking at the asymptotic properties (t → ∞) and by what we said in the introduction then we assume that S is represented by an equilibrium stateρ * which is the weak limit of some initial stateρ. Now we are able to show that the conditions of the Eqns. (5) and (6) can be obtained from Eq. (4). In order to accomplish this, our main assumption is as follows: * "The probabilities P (H ij ) of the Eq. (6) can be computed on the equilibrium stateρ * by the Born rule as P (H ij ) = (ρ * |π ij ) whereπ ij are projectors for all i, j = 1, ..., N ".
This assumption is motivated by the following reasons, phenomenological in nature 5 . It is well known that genuine quantum chaos is possible only within its characteristic timescales t τ (with τ ∝ −α in the regular case and τ ∝ − log in the chaotic case) where typical phenomena with a semiclassical description are possible such as relaxation, exponential sensitivity, etc. These phenomena constitute the dynamical aspects of quantum chaos in the time domain. However, in this time domain the universality of level fluctuations laws predicted by the Gaussian ensembles of the Random Matrix Theory (RTM) are also manifested. This fact shows the two closely connected viewpoints of quantum chaos, the stationary aspects in the energy domain and the dynamical aspects in the time domain. More precisely, the time interval of application of the RTM is found from the Ehrenfest time (τ E ∝ − log ) including the Heisenberg time (τ H = ∆ ∝ −α , ∆ =mean level spacing). With this in mind, it is reasonable to consider that the probabilities P (H ij ) can be thought in the classical limit t → ∞, → 0 with t τ and then here comes the connection with the quantum mixing level of the Quantum Ergodic Hierarchy. By what we said in the introduction the representative state of S in the classical limit isρ * then in order to be compatible with the Born rule we consider that P (H ij ) = (ρ * |π ij ) for certain projectorsπ ij for all i, j = 1, ..., N . We point out that the explicit form of these projectors is not necessary to know for the deduction of the results of this paper but if its mere existence. The next step is to express P (H 11 , H 12 , ..., H N N ) in terms of the stateρ * and the operatorsπ ij . Since theπ ij are projectors then from the quantum logic viewpoint (see [citar libro svozil]) they represent properties A ij of the states of S, in particular aboutρ * . Moreover, given that P (H ij ) = (ρ * |π ij ) we reasonably assume that each A ij is associated with the proposition "the ij-component of the Hamiltonian is equal to H ij " about our unknown HamiltonianĤ. Then we can make the "logical operation" and of the N × N projectorsπ ij that is identified with (for noncommuting operators, see pag. 11 of [svozil])π Now, by the laws of the quantum logic we know that the projectorπ represent the conjunction 6 of all A ij with i, j = 1, ..., N and since each A ij is associated with the proposition "the ij-component of the Hamiltonian is equal to H ij " through the formula P (H ij ) = (ρ * |π ij ) then we conclude that The Eq. (8) is the expression of the Gaussian ensembles correlations in the quantum mixing language. Beyond that we are interested in the behavior of P (H 11 , H 12 , ..., H N N ) in the classical limit → 0, let us first see what information gives the formula (ρ * |lim n N 2 −1 →∞ (...lim n2→∞ (lim n1→∞ (π 11π12 ) n1π 13 ) n2 ...π N N ) n N 2 −1 ) up to terms of order O( ). For this purpose we use the following properties of the Wigner transform denoted by symb : A → A q where A and A q are the quantum algebra of operators and the quasiclassical algebra of distribution functions defined over the phase space Γ respectively 7 : 5 Furthermore, it is not hard to think a projector that mathematically satisfies these requirements at least in the classical limit → 0.
Then in the classical limit → 0 we have symb(π 2 ij −πij ) ≈ 0 and recalling that symb is a bijective linear map we conclude that when → 0 it resultsπ 2 ij =πij , i.e.πij is a projector. 6 More, precisely, Aij is a closed subspace of the Hilbert space of S soπ represents the property A = A11 ∩ A12 ∩ ... ∩ ANN where ∩ denotes the intersection of subspaces. 7 Here we are considering that the dimension of Γ is 2(M + 1), i.e. φ a = (q1, ..., qM+1, p1, ..., pM+1). and where is the symplectic form of Γ. Since we are only interested in terms up to order O( ) then we can approximate the Eq. (10) as where C is the operator defined by ∂pi∂qi . The Eq. (12) says us that the discontinuities in the second-order mixed derivatives of g(φ) are precisely those terms that provide corrections of order O( ). Using that C(f g) = f C(g) + gC(f ) and from successive iterations of the Eq. (12) is not difficult to show that where f j (φ) = symbf j for all j = 1, ..., K and β l (j) In the particular case whenf 1 ,f 2 , ...,f K are projectors (i.e.f 2 j =f j ) we have symbf 2 j = symbf j then by the Eq. (10) it results Therefore, neglecting terms of order O( 2 ) we have f j (φ) 2 = f j (φ). This means that iff 1 ,f 2 , ...,f K are projectors then the Eq. (14) becomes From the Eqns. (13) and (16) we have where Now we can see what is the effect of the terms of order O( ) in the quantum mixing correlations given by the Eq. (8).
where we have used that when → 0 isπ =π 11π12 ...π N N andπ ′ =π ′ 11π ′ 12 ...π ′ N N . Thus we have obtained the invariance condition of equation (5). Summing up, assuming the quantum system is quantum mixing and that P (H ij ) = (ρ * |π ij ) where:π ij are projectors for all i, j = 1, ..., N andρ * the weak limit, we have demonstrated the formulas (5) and (6) From Eq. (32) we see that in the quantum mixing framework the corrections of order O( ) to the statistical distributions of GE are precisely given in terms of P (H 11 , H 12 , ..., H N N |H ij ) and the integrals M+1 ∂pa∂qa . Roughly speaking, in the semiclassical approximation (i.e. q = S ≪ 1) the probabilities P (H 11 , H 12 we have The Eq. (34) says us that P (H 11 , H 12 , ..., H N N |H ij ) can be interpreted as a (N 2 − 1)− point correlation function since this is the probability density to find H ij at a given value, irrespective of all other Hamiltonian matrix elements.
On the other hand, the integrals I ij = M+1 a=1 contain information about the discontinuities of the mixed second partial derivatives of π ij (φ) and noting thatπ ij are projectors then we can reasonably assume that π ij (φ) are characteristic functions. In such case, if π ij (φ) = 1 Aij (φ) and C ij is the frontier of A ij then we have

III GAUSSIAN ENSEMBLES FOR MIXTURES OF PURE STATES
In the previous section we assumed that ρ * was a pure state. This is the typical situation in quantum chaos from the wavefunctions viewpoint. That is, if |ψ N is the eigenfunction of a quantum system S with a chaotic classical description for a large number N ≫ 1 then the reasoning of the previous section can be applied to the pure state ρ ψN = |ψ N ψ N | which is an stationary state. In particular ρ ψN is an equilibrium state. However, a natural question arises for a generic state |ψ = j α j |ψ j which is superposition of eigenfunctions |ψ j : What is the description of |ψ in terms of the quantum mixing level? In such case we have that ρ ψ = |ψ ψ| = j α j α * k |ψ j ψ k | has the weak limit (under certain assumptions, see) ρ * = j |α j | 2 |ψ j ψ j |. Moreover, the quantum ergodic hierarchy says that the description of |ψ is given by the quantum mixing characteristics of its equilibrium state ρ * = j |α j | 2 |ψ j ψ j |. As we saw in section 2, in the classical limit → 0 the quantum mixing level implies the Gaussian ensembles for quantum chaotic systems which has a description given by an equilibrium state ρ * that is a pure state. In the case of an equilibrium state ρ * = j |α j | 2 |ψ j ψ j | we have that each |ψ j ψ j | satisfy the Gaussian ensemble conditions (see equations (5) and (6)) for the probability P (H 11 , ..., H N N ) of the hamiltonian matrix elements H ij (see equations (5) and (6)). Therefore, if we apply the equations (??) and (??) to the state ρ * = j |α j | 2 |ψ j ψ j | we have where P j (H kl ) = tr(|ψ j ψ j |π A kl ) = (|ψ j ψ j ||π A kl ) is the probability that the kl-component of H is H kl in the state |ψ j ψ j | and we have used that P j (H 11 , H 12 , ..., H N N ) = P j (H 11 )P j (H 12 )...P j (H N N ) because |ψ j ψ j | are pure eigenstates for all j. Since the Gaussian ensembles are determined univocally by the equations (5), (6) where we have used that ψ|ψ = j |α j | = 1 (normalization condition). Therefore, from (36) we have generalized the Gaussian ensembles for hamiltonians such that the equilibrium state ρ * of any state ρ is a mixture of pure eigenstates |ψ j ψ j |. More precisely, If ρ * = j |α j | 2 |ψ j ψ j | is the equilibrium state of an initial state ρ where the second equality of (37) (invariance condition) is obtained using the equations (29), (30), (31) and the above reasoning. where O R are the observables that are only accessible experimentally. Therefore, it can be considered as a extension of the Gaussian ensembles to the functional approach proposed by the Brussels school (led by Ilya Prigogine) in [25]. Moreover, the quantum ergodic hierarchy formalism contains the Gaussian ensembles as a particular case of the mixing spectral correlations for large times t → ∞. Where the correlated probability P (H 11 , H 12 , ..., H N N ) is interpreted, by the Born rule, as the probability of the property A = {the ij − component of H is H ij ∀ i, j = 1, ..., N } calculated in the weak limit ρ * . As we saw above, since the quantum ergodic hierarchy formalism contains randomly correlations in the general form of the factorization (ρ * |O 1 O 2 ...O n ) = (ρ * |O 1 )(ρ * |O 2 )...(ρ * |O n ) then the QEH does not need the random matrix assumption of the randomly distribution of the hamiltonian matrix elements to obtain the Gaussian ensembles. For comparison, in the next table we list the characteristics of the random matrix theory and the mixing level of the quantum ergodic hierarchy: O R is a relevant subalgebra of observables • ρ * is interpreted as the decoherence which are the only that are accessible state with a relaxation time t R = ∞. experimentally, O R depends on the quantum system.
• Exponential localization of the kicked rotator and interference destruction of the Casati-Prosen model as consequences of the quantum mixing level (see [12]).
In this table we see that the Gaussian ensembles are a particular case of the factorization property (equation (4)) satisfied by the weak limit ρ * in the classical limit → 0. Moreover, we also see that typical quantum chaos phenomena like the exponential localization of the kicked rotator and the destruction interference of the Casati-Prosen model can be interpreted in terms of the mixing level, because this contains the appropriate spectral correlations given by the Gaussian ensembles. On the other hand, since the weak limit ρ * represents the decoherence of any initial state ρ in the sense of the mean values for large times (i.e. the relaxation time is t R = ∞) then the mixing level connects two theories in a simple way: decoherence and random matrix theory. The following scheme illustrates this relationship.
where (ρ|O) is the mean value of O in the state ρ at time t = 0, the sums D , N D are a short notation of the diagonal and non diagonal terms of (ρ|O) and ρ * is the weak limit of ρ. The limit (ρ|O) = D + N D −→ (ρ * |O) = D when t → ∞ is the decoherence of the observable O in the functional approach. Therefore, we arrive to our main conclusion: "The universal spectral distributions GOE, GUE and GSE are a natural consequence of the no-correlations (ρ * |O 1 O 2 ...O n ) between any observables O 1 , O 2 , ..., O n in the classical limit → 0. Where ρ * is interpreted as the equilibrium (decoherence) state of any initial state ρ and the relaxation time is t R = ∞." From all these facts we conclude that the quantum ergodic hierarchy can be considered an alternative approach enough to obtain at least global properties of the quantum chaos like the universal spectral distributions GOE, GUE and GSE.