1. Liouville Dynamics and Classical Analogues of Information-Related Quantum Impossible Processes A.R. Plastino University of Pretoria, South Africa A.Daffertshofer Vrije Universiteit, Amsterdam A. Plastino Universidad de la Plata, Argentina

2. The conservation of generalized Kullbak measures (entropic distances) under Liouville dynamics yields classical analogues of information-related, quantum mechanical impossible operations, such as universal quantum cloning. Universal cloning or deleting of classical states (described by probability distributions associated with classical statitical ensembles) are forbidden. However, the cloning or deleting of certain states, described by non- overlapping probability distributions, is permitted. Outline

3. Classical Analogues of Quantum Phenomena 1) Classical Analogue of Entanglement [Collins and Popescu, Phys.Rev. A (2002)]. 2) Classical Analogue of Quantum Search Algorithms [Grover and Sengupta, Phys.Rev. A (2002)]. 3) Classical dynamical settings leading to non-Bolean logics [Westmoreland and Schumacher, Phys.Rev. A (1993)]. 4) Classical No-Cloning and No-Deleting Theorems [Daffertshofer, A.R. Plastino, and A. Plastino, Phys. Rev. Lett. (2002) ; A.R. Plastino and A. Daffertshofer, Phys. Rev. Lett. (2004)].

4. The Quantum No-cloning Theorem Wootters and Zurek, Nature (1982). A hallmark feature of quantum information is that it cannot be cloned: An unknown quantum state of a given (source) system cannot be perfectly duplicated while leaving the state of the source system unperturbed. No unitary (quantum mechanical) transformation exists that can perform the process : For arbitrary states

5. Liouville Equation We are going to consider general classical deterministic dynamical systems governed by equations of motion of the form: where x denotes a point in the concomitant N-dimensional phase space. Statistical ensembles of such systems are described by a time- dependent probability distribution P(x ; t) evolving according to Liouville equation,

6. Hamiltonian Systems In the case of a Hamiltonian system with n degrees of freedom we have N = 2n, and where the q i and the p i stand for generalized coordinates and momenta, respectively. Hamiltonian dynamics exhibits the important feature of being divergence-free

7. Conserved Entropic Distances Let us now consider a functional depending on two time dependent solutions of Liouville equation, P 1 and P 2, where g[…] denotes an arbitrary function (we assume that the integral in (6) converges). The functional G[P 1 ; P 2 ] is preserved by the Liouville dynamics,

8. Particular cases of the functional G I) Kullback-Leibler distance, II) Overlap,

9. Classical Cloning We study the distances G[P 1 ; P 2 ] between two solutions j = 1; 2 of the Liouville equation of a tripartite system composed of a copy machine (m), a source system (s), and a target system (t). The initial states (probability distributions) read The corresponding final distributions of the cloning process, denoted by Q j, would verify,

10. Classical No-Cloning Theorem The conservation of G[P 1 ; P 2 ] implies that it is not possible to implement a universal cloning process on the basis of Liouville dynamics [Daffertshofer, A.R. Plastino, and A. Plastino, Phys. Rev. Lett. (2002)]. Even if Liouville dynamics forbids universal cloning of ensemble distributions, the cloning or deleting of some particular distributions are not necessarily forbidden. This is the case of non-overlapping states. Entirely known classical states described by  -distributions are special instances of this “non-overlapping” situation.

11. Classical Deleting The aim of the process is to delete information of the target system against that of the source system. The initial and final distributions, respectively, are

12. Classical No Deleting Theorem For classical deleting processes, the conservation of G[P 1 ; P 2 ] implies that the information deleted from the target system is entirely transferred into the final state of the deleting machine. Solo una cosa no hay. Es el olvido. J.L. Borges: Everness

13. Fisher's Information Measure Fisher's information is a non-negative quantity that plays a key role in statistical estimation theory. When inferring the parameter from one sample x chosen from the distribution P, the mean squared error E 2 for the (unbiased) estimation of obeys the Cramer-Rao bound

14. Classical No Cloning Theorem from Fisher's Information The conservation of Fisher's information under Liouville dynamics implies that the “distinguishability” of phase space ensembles does not change under Liouvillian evolution. On the contrary, final states generated by a universal cloning machine would be more “distinguishable” than the concomitant initial states. [A.R. Plastino and A. Daffertshofer, Phys. Rev. Lett. (2004)].

15. Conclusions The conservation of information distances between time dependent solutions of the Liouville equation allows for the identification of classical analogues of information-related, quantum mechanical impossible operations, such as universal quantum cloning and universal quantum deleting. The physical impossibility of universal cloning or deleting is a basic feature of classical probabilistic settings arising from an incomplete knowledge of the system's state.

16. Conclusions However, complete knowledge of classical states is possible, at least in principle, and cloning and deleting are not forbidden in such cases (they are possible even in the case of non-overlapping probability distributions). In this regard, the quantum mechanical situation is more strict since universal cloning or deleting are impossible even for the set of completely determined states, that is, for pure states.