quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

, 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 an ...

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 disc ...

of a density matrix 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 Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...

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 which satisfies a stronger, more robust condition. Definition Let A and B be C*-algebras. A linea ...

, 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, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...

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