Editing Quantum decoherence (section)

===Operator-sum representation===
Consider a system ''S'' and environment (bath) ''B'', which are closed and can be treated quantum-mechanically. Let <math>\mathcal H_S</math> and <math>\mathcal H_B</math> be the system's and bath's Hilbert spaces respectively. Then the Hamiltonian for the combined system is

: <math>\hat{H} = \hat H_S \otimes \hat I_B + \hat I_S \otimes \hat H_B + \hat H_I,</math>

where <math>\hat H_S, \hat H_B</math> are the system and bath Hamiltonians respectively, <math>\hat H_I</math> is the interaction Hamiltonian between the system and bath, and <math>\hat I_S, \hat I_B</math> are the identity operators on the system and bath Hilbert spaces respectively. The time-evolution of the [[density operator]] of this closed system is unitary and, as such, is given by

: <math>\rho_{SB}(t) = \hat U(t) \rho_{SB}(0) \hat U^\dagger(t),</math>

where the unitary operator is <math>\hat U = e^{-i\hat{H}t/\hbar}</math>. If the system and bath are not [[Quantum entanglement|entangled]] initially, then we can write <math>\rho_{SB} = \rho_S \otimes \rho_B</math>. Therefore, the evolution of the system becomes

: <math>\rho_{SB}(t) = \hat U (t)[\rho_S(0) \otimes \rho_B(0)] \hat U^\dagger(t).</math>

The system–bath interaction Hamiltonian can be written in a general form as

: <math>\hat H_I = \sum_i \hat S_i \otimes \hat B_i,</math>

where <math>\hat S_i \otimes \hat B_i</math> is the operator acting on the combined system–bath Hilbert space, and <math>\hat S_i, \hat B_i</math> are the operators that act on the system and bath respectively. This coupling of the system and bath is the cause of decoherence in the system alone. To see this, a partial trace is performed over the bath to give a description of the system alone:

: <math>\rho_S(t) = \operatorname{Tr}_B\big[\hat U(t)[\rho_S(0) \otimes \rho_B(0)] \hat U^\dagger(t)\big].</math>

<math>\rho_S(t)</math> is called the ''reduced density matrix'' and gives information about the system only. If the bath is written in terms of its set of orthogonal basis kets, that is, if it has been initially diagonalized, then <math>\textstyle\rho_B(0) = \sum_j a_j |j\rangle \langle j|</math>. Computing the partial trace with respect to this (computational) basis gives

: <math>\rho_S(t) = \sum_l \hat A_l \rho_S(0) \hat A^\dagger_l,</math>

where <math>\hat A_l, \hat A^\dagger_l</math> are defined as the ''Kraus operators'' and are represented as (the index <math>l</math> combines indices <math>k</math> and <math>j</math>):

: <math>\hat A_l = \sqrt{a_j} \langle k| \hat U |j\rangle.</math>

This is known as the ''[[Decoherence-free subspaces#Operator-sum representation formulation|operator-sum representation]]'' (OSR). A condition on the Kraus operators can be obtained by using the fact that <math>\operatorname{Tr}[\rho_S(t)] = 1</math>; this then gives

: <math>\sum_l \hat A^\dagger_l \hat A_l = \hat I_S.</math>

This restriction determines whether decoherence will occur or not in the OSR. In particular, when there is more than one term present in the sum for <math>\rho_S(t)</math>, then the dynamics of the system will be non-unitary, and hence decoherence will take place.