quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...

, especially in the study of

open quantum system In physics, an open quantum system is a quantum-mechanical system that interacts with an external quantum system, which is known as the ''environment'' or a ''bath''. In general, these interactions significantly change the dynamics of the system a ...

s, reduced dynamics refers to the

time evolution Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state (also called ''stateful systems''). In this formulation, ''time'' is not required to be a continuous parameter, but may be discr ...

of a

density matrix In quantum mechanics, a density matrix (or density operator) is a matrix used in calculating the probabilities of the outcomes of measurements performed on physical systems. It is a generalization of the state vectors or wavefunctions: while th ...

for a system coupled to an environment. Consider a system and environment initially in the state

\rho_ (0) \,

(which in general may be entangled) and undergoing unitary evolution given by

U_t \,

. Then the reduced dynamics of the system alone is simply :

\rho_S (t) = \mathrm_E_t \rho_ (0) U_t^\dagger

If we assume that the mapping

\rho_S(0) \mapsto \rho_S(t)

linear In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x) ...

and

completely positive In mathematics a positive map is a map between C*-algebras that sends positive elements to positive elements. A completely positive map is one that satisfies a stronger, more robust condition. Definition Let A and B be C*-algebras. A linear m ...

, then the reduced dynamics can be represented by a

quantum operation In quantum mechanics, a quantum operation (also known as quantum dynamical map or quantum process) is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. This was first discusse ...

. This mean we can express it in the operator-sum form :

\rho_S = \sum_i F_i \rho_S (0) F_i^\dagger

where the

F_i \,

are operators on the

Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...

of the system alone, and no reference is made to the environment. In particular, if the system and environment are initially in a product state

\rho_ (0) = \rho_S (0) \otimes \rho_E (0)

, it can be shown that the reduced dynamics are completely positive. However, the most general possible reduced dynamics are ''not'' completely positive.

Notes

References

* Nielsen, Michael A. and Isaac L. Chuang (2000). ''Quantum Computation and Quantum Information'', Cambridge University Press, Quantum information science {{quantum-stub