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
Outline
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 nonoverlapping probability distributions, is permitted.
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)].
The Quantum No-cloning Theorem
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
Wootters and Zurek, Nature (1982).
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 timedependent probability distribution P(x ; t) evolving according to
Liouville equation,
Hamiltonian Systems
In the case of a Hamiltonian system with n degrees of freedom we
have N = 2n, and
where the qi and the pi stand for generalized coordinates and
momenta, respectively.
Hamiltonian dynamics exhibits the important feature of being
divergence-free
Conserved Entropic Distances
Let us now consider a functional depending on two time dependent
solutions of Liouville equation, P1 and P2 ,
where g[…] denotes an arbitrary function (we assume that the integral
in (6) converges).
The functional G[P1 ; P2] is preserved by the Liouville dynamics,
Particular cases of the functional G
I) Kullback-Leibler distance,
II) Overlap,
Classical Cloning
We study the distances G[P1 ; P2] 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 Qj , would verify,
Classical No-Cloning Theorem
The conservation of G[P1 ; P2] 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.
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
Classical No Deleting Theorem
Solo una cosa no hay. Es el olvido.
J.L. Borges: Everness
For classical deleting processes, the conservation of
G[P1 ; P2 ] implies that the information deleted from the
target system is entirely transferred into the final state of
the deleting machine.
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 E2 for the (unbiased)
estimation of  obeys the Cramer-Rao bound
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)].
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.
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.