Quantum Central Limit Theorems, Emergence of Classicality and Time-Dependent Differential Entropy

We derive some quantum central limit theorems for the expectation values of macroscopically coarse-grained observables, which are functions of coarse-grained Hermitian operators consisting of non-commuting variables. Thanks to the Hermiticity constraints, we obtain positive-definite distributions for the expectation values of observables. These probability distributions open some pathway for the emergence of classical behaviours in the limit of an infinitely large number of identical and non-interacting quantum constituents. This is in contradistinction to other mechanisms of classicality emergence due to environmental decoherence and consistent histories. The probability distributions thus derived also enable us to evaluate the non-trivial time-dependence of certain differential entropies.


I. OPENING REMARKS
The Central Limit Theorem (CLT) [1] for sums of independent identically distributed (iid) random variables is one of the most fundamental pillars of classical probability theory.
It and various generalisations [2] have found numerous applications in diverse fields including mathematics, physics, information theory, economics, finance and psychology.
In this paper we derive a quantum version of the CLT for expectation values of hermitian operators only, and not of general operators.This hermicity constraint for observables results in positive-definite probability distributions -in contradistinction to the Wigner function, which is a quasiprobability distribution and although is real-valued but not positive-definite in general.Our probability distributions below, (8) and (18), are also unique and independent of the operators being considered.We employ in our derivation a renormalisation blocking approach to obtain explicit expressions for the probability distributions.These are presented in the next three sections.
Note that, on the other hand, previous versions of quantum CLT consider general operators, including non-hermitian ones, and thus do not explicitly express the resulting quasidisributions but only implicitly through expectation values with gaussian states.The exact forms of those quasi-distributions, as such, may also be dependent on the operators being considered.
With the explicit forms of our so-derived probability distributions, there affords a pathway for emergence of classical behaviours from quantum mechanics of a system having non-interacting constituents when the number of constituents is taken to infinity.This is discussed in section V. Our pathway for an emergence of classicality is quite different from that afforded by decoherence and/or consistent histories.
We then use our explicit probability distributions for the evaluation of particular form of differential entropy for some simple quantum systems in section VII.In the literature for both open and closed quantum systems, different information-theoretical entropy measures have been discussed [12][13][14][15][16].The measure of differential entropy we employ is a special case of relative entropy, argued for based on the considerations by Jaynes [17].
The last section of the paper contains our concluding remarks.

II. QUANTUM CENTRAL LIMIT THEOREM AND HEURISTIC RENORMAL-ISATION BLOCKING
Renormalisation group blocking plays a central role in understanding emerging bulk behaviours and collective phenomena.Heuristically, one could start with a path integral or partition function in some set of fundamental microscopic variables / operators.As an illustration, let us take the path integral expression for a quantum system having the action In deriving coarse-graining behaviours from the system, we introduce the coarse-grained field variable Ξ as a function of the fundamental variables ξ in some chosen blocking scheme Ξ j = h(ξ), as in an averaging scheme, for example, where V j is a measure of the "volume" of each block.We have to choose the blocking function h in such a way that the coarse-grained variables are not growing indefinitely in magnitude when we keep on coarse-graining the coarse-grained variables successively to the next level -hence the volume denominator in our example above.
The expectation value of a quantum operator F (Ξ) of the coarse-grained variables could then be expressed as To convert the last path integral in Dξ to that in DΞ, we insert the resolution of unity into (3) then interchange the order of integration to obtain where Successive repeating of the last expression defines a renormalisation group flow.
We will phrase the quantum central limit theorems in this paper as a restricted renormalisation blocking in the sense that we consider only the expectation values of hermitean operators and not the full path integral / partition function for arbitrary operators.
Our restricted consideration results in positive-definite measures which can be interpreted as probability measures, from which the fixed-point distributions of the renormalisation blocking emerge.

III. CENTRAL LIMIT THEOREM FOR SINGLE HERMITIAN VARIABLE
The centre of mass, or intensive variables in general, of a composite systems of N components can be expressed as We now consider a system with identical and non-interacting components (such as the case of an ideal gas) where |φ i = |φ , for all i.
With some general function f , we obtain the following result for where The derivation of the above is given in Appendix A.
In particular, we can derive, as a special case from the above, the probability density for finding X around some X which is a gaussian distribution.
We could estimate from the derivation of the above that the size of the system should satisfy the condition N ≫ | x 3 / x 2 | for the approximation.
We can also easily generalise the result to the case when the initial state is a mixed state instead of being pure.
Note also that the above result can be readily generalised to the case when where m is integer and g() is some arbitrary function.
In the limit of N → ∞ the gaussian distribution in ( 8) converges to a delta distribution, We thus have from (8), for arbitrarily finite integer m, This is an indication of an emergence of classical behaviours for macroscopically blocked variable X, as the right hand side of the last expression contains x m rather than x m .
In order to verify such emergence we will need to further consider quantum mechanically non-commuting variables in the next section.

IV. A CENTRAL LIMIT THEOREM FOR NON-COMMUTING VARIABLES
We additionally consider the momentum operators pi , the non-commuting conjugate variables of the position operators.and introduce the blocked variable P While X of the last section is the centre of mass, this blocked variable P corresponds to a measure of the velocity of the centre of mass.
Even for system of interacting components, we have we have, because of the approximate commutativity above, We now consider a hermitian combination of some finite sum of products of X and P , which can be expressed in general as, by constraint of hermiticity, where c mn are c-numbers.For expectation values of general observables, we can indeed further restrict the above to real values of c mn .
For N ≫ 1, we obtain the following result, of which the derivation is presented in Ap- where the probability distribution for the real parts ℜ(c mn ) is while the probability distribution for the imaginary parts ℑ(c mn ) is In the above, N 1 and N 2 are normalising factors, and and also It is noted that the probability distribution for the imaginary parts, P im (X, P ), explicitly contains the commutator of x and p in the quantities θ − and ∆ − .In fact, were x and p commutative then θ − = 0 and For the probability distribution for the real parts, P re (X, P ), we have a product of gaussian distributions mixing combinations of the two generally non-commuting variables X and P .However, were xp c = 0 then we would have a factorisation into two gaussian distributions in X and P separately.
As a special case, upon the substitution in (18), the probability distribution P im (X, P ) vanishes and the remaining distribution P re (X, P ) reduces to a product of distributions of single variable in (8).Alternatively, we could get these same results as with ( 8) by letting n = 0 in (18).

V. EMERGENCE OF CLASSICALITY
From the results of the last section, we can readily derive the following expectation values Similarly, and Furthermore, it can be shown that the correlation between the coarse-grained/renormalisation block variables X and P 1 2 Φ X P + P X Φ N →∞ ∼ dXdP XP P re (X, P ), indicating that, in this limit, the coarse-grained/renormalisation block variables are uncorrelated and behaving as classically independent variables.
For the expectation value of the hermitian commutator, i Φ X P − P X Φ , we integrate (18) with the distribution P im (X, P ) for the imaginary part (20) to obtain It thus follows also that were xp − px = 0 then so would be Φ X P − P X Φ = 0, identically for any value of N.
We further observe that, in the limit of infinitely many identical and non-interacting quantum subsystems, N → ∞, P re (X, P ) And The right hand side above now involves only x i and p j (with some integers i and j), and contains neither x i nor p j , nor the quantum correlations x i p j .Implied also in this last expression, which does not include the imaginary parts ℑ(c mn ), is that the expectation value of the commutator of the coarse-grained/renormalisation block variables X and P is vanishingly small with sufficiently large N, in agreement with (34).
In general, any classical observable can be expressed indeed as a restricted form of the left hand side of (37) with real c mn -thus removing the need to consider the distribution for the imaginary part P im (X, P ).
As a consequence, a regime of classicality could be emerging due to the fact that quantum correlations and all traces of quantum behaviours are now suppressed, except those inherent in the quantum expectation values x and p .

VI. DIFFERENTIAL ENTROPIES
A direct generalisation of information Shannon entropy for discrete probabilities p d [18] to the case of continuous probability distributions might be DEnt 1 = −k B P r(X, P ) ln P r(X, P ) dXdP.
(39) This is normally called the differential entropy.
This definition of differential entropy, however, does not share all properties of discrete entropy.For example, the differential entropy above can be negative; more importantly, it is not invariant under continuous coordinate transformations.In fact, Jaynes [17] showed that the expression above is not the correct limit of the expression for a finite set of probabilities.
He introduced a modification of differential entropy to address defects in the initial definition of differential entropy by adding an invariant measure factor to correct this [17].
In information theory, this is the limiting density of discrete points in an adjustment to the formula of Shannon for differential entropy.
In the phase space volume ∆X i ∆P i , the transition from discrete probability to continuous probability density should be If this passage to the limit is sufficiently well behaved, we would have lim where N X is the number of points in the X dimension, and m(X i ) is the density in this dimension.As a result, the differences ∆X i in the neighbourhood of any particular value of X i will have to be lim We have, on the other hand, for the probability density m(X i ) = P r(X i , P )dP.
(43) Thus, lim Similarly, lim component (19) (which suffices for classical observables) into our adopted entropy (47), we arrive at We see from this explicit expression that the non vanishing of xp c in general, due to quantum correlations, that enables some non-trivial time dependence for the differential entropy.

A. Free particles
For free particles in one dimension, we have for the individual constituent, in the Heisenberg picture, The time-dependent variance of the centre of mass, with finite initial variances σ 2 x (0) and σ 2 p (0), assumes the following temporal behaviours: and It then follows that the coarse-grained entropy (48), for a sizable collection of N free and independent particles and for sufficiently large time, is behaving as which is increasing irreversibly with time (unless the individual subsystem is initially in a momentum eigenstate, whereby σ 2 p (0) = 0 = x(0)p(0) c ).Such entropy is increasing with time although invariant with time-reversal, t → −t and p → −p -as is the symmetry of the underlying dynamics of an individual constituent particle.

B. Uniform and constant force
For a system under an uniform and constant external force, we have in the Heisenberg picture Ĥ = p2 2m − ax, p(t) = p(0) + at.
From which follow the time dependence and Upon which, the coarse-grained entropy is, for large time, also increasing irreversibly, unless σ 2 p (0) = 0 and x(0)p(0) c = 0, that is, when the individual subsystem is in a momentum eigenstate initially.Initial position eigenstate is also not applicable here because that would imply an unbounded variance of the momentum due to quantum uncertainty relation.

C. Oscillatory particles
On the other hand, an example in which the differential entropy is not monotonic in time is that of the quantum simple harmonic oscillator, p(t) = p(0) cos(ωt) − mω x(0) sin(ωt).
From which, and In this case, the differential entropy (48) is not, even for large time, a monotonic function of the time.

VIII. SUMMARY AND CONCLUDING REMARKS
We derive some quantum mechanical versions of Central Limit Theorems for expectation values of coarse-grained observables, which are functions of coarse-grained hermitean operators.In the above, the coarse-grained variables considered correspond to the center of mass and its classical velocity.
Our derivation methodology could also be rephrased explicitly as a restricted form of renormalisation blocking applied only for observables, and not for non-hermitean operators.
Even though incomplete in that sense, the restricted renormalisation is important and useful enough for consideration of all the bulk behaviours that are observable and measurable.
From such hermicity constraints, we obtain for the expectation values positive-definite distributions, which also are the fixed points of the restricted renormalisation group flows.
Our probability distributions are also unique and independent of the operators being considered.Those are the results in (8) for functions of single macroscopically coarse-grained variables and that in (18) for functions of macroscopically coarse-grained non-commutative quantum variables.In the latter case, we have two separate distributions for the real and imaginary parts (19) and (20), respectively -even though we need only consider the real part for observables.
Furthermore, our results herein could be applied also to systems of interacting constituents when approximations whereby the many-body problem could be essentially reduced to a one-body problem, like the mean field Hartree method, are applicable.
Our probability distributions enable a path way for emergence of classical coarse-graining behaviours, as far as observable and measurable, in the limit of an infinitely large number of identical and non-interacting quantum constituents (having finite variances for relevant variables).This is the result of the fact that quantum correlations and all traces of quantum behaviours are now suppressed as shown in (37), except those inherent in x and p of the constituents.
It should be emphasised that this particular mechanism for such emergence is entirely due to coarse graining in the macroscopic limit, and neither because of environmental decoherence nor due to some kinds of interactions among the constituents.
It is important to note that, because the wave functions are time-dependent in general, in the derivation of the results above we have had to work with a same time instant for all the microscopic constituent wave functions, as demonstrated by (A3).That is, expectation values of the different components xi and pi in the block variables X and P must be evaluated at the same time.This situation is in stark contrast to the classical Central Limit Theorems, which, when dealing with time-independent iid components, can be employed for averaging measurement results over different moments in time.This distinction is important in our context to recognise that an emergence of classicality would be applicable only for macroscopically block variables -and not for microscopic variables repeatedly measured and averaged over time.The double-slit experiments could illustrate our point here.Single electron one by one going through the apparatus still exhibits interference after averaging over many such identical and independent electrons, but macroscopic particles (a macroscopic bunch of many electrons at the same moment of time) may not.
The probability distributions of the quantum Central Limit Theorem further allow us to evaluate some differential entropies for composites of macroscopically coarsed-grained systems.Those entropies are symmetric with respect to time reversal (t → −t, p → −p and φ → φ * ), as is the underlying quantum dynamics.Nevertheless, they could have some interesting and non-trivial temporal dependence.It is noted that it is the quantum origin of the non-factorisation of P r(X, P ) (18) into product of component distributions of X and P that gives rise to some interesting and non-trivial temporal behaviours of the entropies.
In fact, in some instances, they could also increase with time approximately monotonically -as functions of the absolute value of the time, for sufficiently large time.
As with the case of classical Central Limit Theorems which have been generalised to cover some less stringent constraints on the behaviours of the constituent components [2], we expect that further quantum Central Limit Theorems may also be similarly generalised.
ACKNOWLEDGEMENT I want to thank Peter Hannaford for some input for this paper.This is the probability distribution for the real part P re (X, P ) of (19).
Similar to the derivation above, it can also be shown that the distribution for the imagi-nary part P im (X, P ) of ( 20) is, for some integers m and n, We could recover the results from ( 8) by either putting n = 0 or replacing P by 1 in (18).