Microscopic derivation of the one qubit Kraus operators for amplitude and phase damping

This article presents microscopic derivation of the Kraus operators for (the generalized) amplitude and phase damping process. Derivation is based on the recently developed method [Andersson et al, J. Mod.Opt. 54, 1695 (2007)] which concerns finite dimensional systems (e.g. qubit). The form of these operators is usually estimated without insight into the microscopic details of the dynamics. The behavior of the qubit dynamics is simulated and depicted via Bloch sphere change.


INTRODUCTION
Quantum information processing substantially depends on the mathematical details of the environmental influence exerted on the qubit-registers [1].
Nevertheless, to the best of our knowledge, theoretical origin of the widely used Kraus operators for the one-qubit quantum noise-channels [1][2][3][4][5][6][7] has not been investigated yet. In this paper we perform a thorough analysis of the microscopic models for the standard one-qubit amplitude damping and phase damping quantum processes. To this end we use a recently formulated method [8] for derivation of the Kraus operators from a microscopic master-equation description of the processes. We find the unitary equivalence of the here derived Kraus operators with those widely used in the literature, thus presenting the same quantum noise process. The converse conclusion regarding the one-qubit depolarizing process will be presented elsewhere [9].
In Section 2, we overview the method of Andersson et al [8]. In Section 3 we derive the Kraus operators for the generalized amplitude damping process, while in Section 4 we derive the Kraus operators for the phase damping process. Section 5 is the conclusion.

OVERVIEW OF THE METHOD
In the paper [8], the authors developed a general procedure for deriving a Kraus decomposition from the known master equation and vice versa, regarding the finite-dimensional quantum systems. The only assumption is that the master equation is local in time.
The so-called Nakajima-Zwanzig projection method [11,13] gives the following master equation for the system's density operatorρ S (t), ( = 1): whereĤ represents the system's self-Hamiltonian (that includes the so-called Lamb-shift term) and K t,s is the memory kernel which accounts for the nonunitary effects due to the environment.
Certain processes can be written in a local-in-time form [11,13]: where Λ t is a linear map which preserves hermiticity, positivity and unit trace ofρ S (t) and has the property: Alternatively, dynamics can be presented in a non-differential, "integral" form [8,11,13]:ρ where φ t is a completely positive and trace preserving linear map.
It can be shown [8] that linear maps Λ t and φ t are connected via the matrix differential equation:Ḟ where the matrix elements of L are given by: In eq.(6), {G k } is any orthonormal basis of the Hermitian operators acting on the system's Hilbert space. For the time independent Λ t , i.e. L, eq. (5) has the unique solution: Complete positivity of the map φ t (and hence of the matrix F ) is equivalent to the positivity of the, so called, Choi matrix, S [8,10], whose elements are defined as [8]: With the use of equation (8), eq.(4) takes the form: which, after diagonalization of the S matrix: gives rise to a Kraus decomposition. The eigenvalues d i and the eigenvectors of the S matrix constitute the diagonal matrix D and the unitary matrix U = (u ij ) respectively; columns of the unitary U operator are the normalized eigenvectors of the S matrix. Then the Kraus operators: yield the Kraus decomposition of the dynamical map φ t : Therefore, the chain of the construction is established: from a master equation to calculate L, then via relation (7) to obtain the matrix F and, due to eq.(8) and diagonalization eq.(10) of the Choi matrix to calculate the Kraus operators eq.(11).

THE GENERALIZED AMPLITUDE DAMPING CHANNEL
The standard master equation for the amplitude damping process, at absolute zero, T = 0K, reads [2]: while the corresponding standard Kraus operators: and the Pauli operatorsσ . To describe the amplitude damping process for all temperatures, the following Hamiltonian is often regarded [11]: The first term on the right side of eq.(15) denotes the system's self-Hamiltonian, the second denotes the self-Hamiltonian of the environment (a thermal bath of linear non-interacting harmonic oscillators andâ ω representing the bosonic "annihilation" operator for the frequency ω) while the last term represents the interaction with the coupling-coefficients h(ω). ω max is the 'cutoff frequency' for the bath's oscillators; one may take the limit ω max → ∞ providing that the h(ω) sufficiently quickly decreases.
As distinct from eq.(13), the microscopic Markovian master equation, in interaction picture, derived from Hamiltonian (15) reads [11]: Lamb-like shift and the Stark-like shift contributions from the vacuum and the thermal field, respectively, and curly brackets stand for anti-commutator.
P.V. stands for the Cauchy principal value of the integral. J(ω) represents the spectral density of the bath.
The master equation (16) reduces to the standard AD master equation (13) for T = 0K, and is therefore often called generalized amplitude damping (GAD) channel. Below, due to the procedure described in Section 2, from eq.(16) we derive the GAD Kraus operators, which will turn out to be unitary equivalent with the known GAD Kraus operators [12]: where To ease the calculation, we introduce the following notation: Now, from (18) and using eq.(6) from the body text, theσ z -representation of the L matrix takes the form: In order to facilitate the calculation of the exponential F matrix, we multiply the L matrix by 2 (y+z) that allows introduction of new variables: Then follows: and which obtains the form: From eq.(22) we obtain the corresponding Choi matrix (Section 2), whose diagonalization gives the following set of eigenvalues: and the respective non-normalized eigenvectors: Hence we obtain the first two Kraus matrices for GAD, eq.(4): By introducing: B ± = e −4τ 2 + 2e 2τ ± Ω 2 + e 2τ (16 + (−2 + e 2τ ) Ω 2 ) , and another pair of Kraus matrices, E 3 and E 4 , can be written as: and It is straightforward yet tedious task to confirm the completeness relation and and (37).
Also we study the GAD channel for various (high) temperatures via investigating temporal behavior of the Bloch sphere volume.   [13]. On the other hand, Fig.1(left) reveals a faster change of the "excited" state |0 for higher temperatures.
Time dependence of the Bloch-sphere volume is: equivalently V (t) = 4π 3 e −2(z+y)t . The relative change of the Bloch sphere volume, κ(t) = 1 V 0 dV (t) dt : Eq.(42) regarding GAD process is presented in Fig.2; to take the following forms in the asymptotic limit , τ → ∞:

THE PHASE DAMPING CHANNEL
The phase damping (PD) quantum channel models pure decoherence without loss of energy for a single-qubit system. The Hamiltonian for the total (closed) system is given by [11]: Notation and the meaning of the terms in eq.(47) are the same as in eq.(15).
The model eq.(47) gives rise to the following microscopic Markovian master equation in the interaction picture [11]: whereby the decay rate r

CONCLUSION
Detailed microscopic analysis of the differential form of the amplitude damping and phase damping processes on a single qubit gives rise to the Kraus operators that describe exactly the same process as the standard Kraus operators widely used for these processess.