Stochastic decoherence of qubits

Stochastic decoherence of qubits

Krzysztof Wodkiewicz

Institute of Theoretical Physics, Warsaw UniversityWarsaw 00-681, Poland

andDepartment of Physics and Astronomy, University of New Mexico

Albuquerque NM 87131, [email protected], [email protected]

Abstract: We study the stochastic decoherence of qubits using theBloch equations and the Bloch sphere description of a two-level atom.We show that it is possible to describe a general decoherence processof a qubit by a stochastic map that is dependent on 12 independentparameters. Such a stochastic map is constructed with the help of thedamping basis associated with a Master equation that describes thedecoherence process of a qubit.

c© 2001 Optical Society of AmericaOCIS codes: (270.2500) Fluctuations, relaxations, and noise

References and links1. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1987).2. G. S. Agarwal, Quantum Statistical Theories of Spontaneous Emission and Their Relation to

Other Approaches (Springer, Berlin, Heidelberg, 1974), Vol. 70.3. C. W. Gardiner, Handbook of Stochastic Processes (Springer, Berlin, Heidelberg, 1984).4. M. B. Plenio and P. L. Knight, “Realistic lower bounds for the factorisation time of large numbers

on a quantum computer,” Phys. Rev. A 53, 2986-2990 (1996).5. C. H. Benett and P. W. Shore, “Quantum Information Theory,” IEEE Trans. Info. Theory 44,

2724-2748 (1998).6. M. B. Ruskai, S. Szarek and E. Werner, “A Characterisation of Completely-Positive Trace Pre-

serving Maps on M2,” preprint quantum-ph/0005004, http://xxx.lanl.gov/

7. K. Wodkiewicz and J. H. Eberly, “Random telegraph theory of effective Bloch equations withapplications to free induction decay,” Phys. Rev. A 32, 992-1001 (1985).

8. K. Kraus, States, Effects and Operations: Fundamental Notions of Quantum Theory (Springer-Verlag, Berlin Heidelberg, 1983).

9. C. W. Gardiner, Quantum Noise (Springer, Berlin, Heidelberg,1991).10. H. J. Briegel and B. -G. Englert, “Quantum optical master equations: The use of damping bases,”

Phys. Rev. A 47, 3311-3328 (1993).11. C. King and M. B. Ruskai, “Minimal Entropy of States Emerging from Noisy Channels,” preprint

quantum-phy/9911079, http://xxx.lanl.gov/

1 Introduction

For many years the two level atom provided the simplest and the most natural frameworkfor the investigation of resonance phenomena in the presence of damping. The naturaldescription of the two-level atom in terms of the Bloch vector~b = (u, v, w) gives a simple,yet powerful, geometrical description of the coherent and the incoherent dynamics ofa two-level atom. The classical by now textbook of Allen and Eberly [1] provides adetailed description of the two-level dynamics. In this description the coherent dynamicsis represented by a rotation of the Bloch vector on a Bloch sphere defined as u2 + v2 +w2 = const, while the incoherent dynamics leads to a spiraling of the Bloch vectorinto a steady state that is not necessarily on the Bloch sphere. The physical source

(C) 2001 OSA 15 January 2001 / Vol. 8, No. 2 / OPTICS EXPRESS 145#29382 - $15.00 US Received November 16, 2000; Revised January 02, 2000

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http://xxx.lanl.gov/

of damping characterizing the incoherent part of the dynamics has been attributedto quantum or stochastic fluctuations of various properties of the reservoir coupled tothe two-level atom. Perhaps the best known source of decoherence is the spontaneousemission damping due to quantum vacuum fluctuations of the electromagnetic field[2]. Other possible sources of noise can involve stochastic phase,frequency or amplitudefluctuations. There is vast literature devoted to stochastic models of collisions, phasediffusion or frequency fluctuations [3]. In the framework of a theory involving a driventwo-level atom by a multiplicative Gaussian white noise, the Bloch vector descriptionis still valid, with damping rates given by the diffusion coefficients of the correspondingfluctuations.

A fundamental new milestone has been achieved, when it was recognized that twoinput bits in a channel can be replaced by a two-level atom forming a quantum bitof information called: qubit. In realistic setups the qubit is usually exposed to a noisychannel leading to decoherence of quantum information. Due to decoherence there is nosuch thing as a perfect qubit or a perfect quantum logical gate [4]. If a qubit is exposedto a noisy channel one can observe a loss of coherence and it is still an open questionwhether there is an intrinsic lower limit to the decoherence rate for an arbitrary qubit.The quantum or the stochastic noise is an undesirable but an unavoidable property ofquantum logical gates.

The issue that stochastic noise of a qubit can dramatically change logical opera-tions has become one of the central themes of error correction and noise stabilizationin quantum computational schemes. The control of decoherence of a qubit is a centralproblem of various models of quantum computation, where an incoming qubit is trans-mitted through a noisy channel. Due to the noise the statistical properties of such aqubit can be changed. For example, pure states become mixed states. For a class of suchnoisy channels, one could ask the question of a possible optimalization or minimaliza-tion of the decoherence effect of a qubit. One fundamental goal is to seek those noisesthat minimize a fidelity or an entropy of such a channel. In order to perform such atask a general understanding of a wide class of noises acting on the qubit is required.This problem has attracted a lot of attention and has been studied in the framework ofquantum information theory [5].

It is the purpose of this work to address the problem of stochastic decoherence ofa qubit in the framework of a standard quantum optics description, involving suchproperties like the Bloch sphere, the optical Bloch equations, and the Master equation.

2 Decoherence of two level atoms

The most widely used description of a two-level atom with a coherent and incoherentdynamics is given by the Bloch equations:

u = − 1T2

u − ∆v

v = − 1T2

v + ∆u + χw

w = − 1T1

(w − weq) − χv . (1)

The meaning of all the parameters is standard and is described in the classical textbookon this subject [1]. The incoherent dynamics is described by two lifetimes. The transverselifetime T2 gives a decay rate of the atomic dipole moment characterized by u and v,while the longitudinal lifetime T1 gives the decay rate of the atomic inversion w into anequilibrium state weq. The Bloch equations have been applied as the most successful


tool used in quantum optics to describe and to study the impact of incoherent effectson the two-level atom dynamics.

Using the Bloch equations one can obtain a simple geometrical picture of the two-level atom decoherence. In this picture, the Bloch vector and the Bloch sphere are theessential geometrical ingredients of the description.

In order to make contact with the language of quantum information theory, we shalldenote the decoherence effects as a stochastic noise operation acting on the Bloch vector.We shall denote this stochastic noise by the following map:

Φ : b 7→ b′ (2)

and call it for short: stochastic map. In the following Sections the structure and theproperties of this stochastic map acting on a qubit will be investigated in the frameworkof the two-level Bloch theory based on Eqs. (1).

3 Decoherence of qubits

In order to investigate the impact of the stochastic map given by Eq. (2) on a qubitwe shall use the matrix representation of the Bloch vector in the form of the followinglinear combination of Pauli matrices:

B = ~b · ~σ =[

w u − ivu + iv −w

]. (3)

Under the coherent evolution (no noise) the Bloch vector stays on the Bloch spheredefined by the property

detB = −(u2 + v2 + w2) . (4)

This means that the coherent (unitary) evolution of the qubit leads to an orthogonaltransformation of the Bloch vector, leaving Eq. (4) invariant.

Because of the stochastic noise (2), the B matrix is transformed as follows:

Φ : B 7→ B′ . (5)

In general, the stochastic map (2) can transform the Bloch vector to a vector inside theBloch ball which consists of the interior of the Bloch sphere. In order to be inside ofthe Bloch ball such a stochastic transformation has to be a contraction, i.e., it has tosatisfy the condition:

|detB′| ≤ |detB| . (6)

The stochastic map given by Eq. (2), if applied to an arbitrary density operator of aqubit, leads to the following transformation:

Φ : ρ 7→ Φ(ρ) . (7)

Such a stochastic map of the density operator has to be trace preserving

Tr {Φ(ρ)} = Tr {ρ} = 1 (8)

and positivity preserving Φ(ρ) ≥ 0. Such positive maps play a central role in quantuminformation theory [6].

In the following we present two examples of such stochastic maps given by Eq. (2)or by Eq. (5). These two examples provide a hint about the general nature of suchtransformations. We shall see that such transformations have a simple relation to theBloch equations (1). In the next sections of this paper we provide an explicit constructionof the stochastic map Φ(ρ) directly from the Bloch equations.


Stochastic map for spontaneous emission noise

It is well known that the spontaneous emission noise is described by 1T1

= 2T2

= Aand weq = −1. The damping terms are characterized by the Einstein A coefficient ofspontaneous emission. In geometrical terms, the dynamics of a coherently driven qubitwith spontaneous noise can be seen as the following three operations on the Blochsphere. Due to coherent interaction, the first operation amounts to a rotation of theBloch vector. The spontaneous emission noise leads to two additional operations: adamping and a linear shift of the Bloch vector. In this case we have ΦA : B 7→ B′ with

ΦA : ~b 7→ ~b′ = M~b +~b0 (9)

where the matrix M = RNA has been written as a product of a coherent rotation R andthe spontaneous emission noise damping NA. This damping noise matrix is diagonal

NA =

Λ1 0 0

0 Λ2 00 0 Λ3

(10)

with the following damping eigenvalues

Λ1 = Λ2 = exp(−At

2) , Λ3 = exp(−At) . (11)

The important property of the spontaneous emission noise is that it leads to an addi-tional linear translation of the Bloch vector by:

~b0 = (0, 0, Λ3 − 1) . (12)

Stochastic map for frequency diffusion noise

As a different model of a stochastic map (5) we take the case of frequency (detuning)white-noise fluctuations [7]. In this case weq = 0, i.e., the steady inversion becomesthermal. The damping rate 1

T1= 0 and only 1

T2= Γ is responsible for the frequency

diffusion. For such stochastic fluctuations the stochastic map ΦΓ leads to no translationof the Bloch vector, i.e., ~b0 = 0 and as in the case of the spontaneous emission noiseM = RNΓ with a diagonal damping matrix NΓ with eigenvalues

Λ1 = Λ2 = exp(−Γt) , Λ3 = 1 . (13)

The spontaneous emission noise or the frequency diffusion noise results from thecoupling of the qubit to a quantum or stochastic reservoir of noise fluctuations. Sucha coupling can be described by a Master equation [2]. With these two examples weare now in a position to write the general stochastic noise transformation of the Blochvector based on the corresponding Master equation.

4 Arbitrary stochastic noise

A general stochastic map of the form given by Eq. (2) of a qubit can be written in theform

Φ : ~b 7→ ~b′ = M~b +~b0 (14)

of an affine transformation of the Bloch vector. From the fact that the Bloch vector isreal it follows that such a stochastic map is fully characterized by a real 3x3 matrixM and by 3 real shift parameters ~b0. This means that an arbitrary linear stochastic


map is fully characterized by an affine transformation labeled by 12 real independentparameters.

As we have already noted the coherent evolution is given by a rotation matrix, andas a result we can write that, in general, M = RN, where the damping matrix Nis symmetric and is characterized by 6 real parameters. The damping matrix N canbe diagonalized by another orthogonal transformation T, leading to a diagonal matrixΛ with eigenvalues Λ1, Λ2, Λ3. Each of the two orthogonal transformations R and Tcorresponds geometrically to a rotation (with inversion) of the Bloch vector on the Blochsphere. The resulting matrix Λ describes the three fundamental decoherence dampingeigenvalues of the qubit in some selected principal axes of the Bloch sphere.

The density operator of the qubit has the familiar form:

ρ =12(I +~b · ~σ) =

12(I + B) . (15)

For the two-level atom, pure states are on the Bloch sphere ~b ·~b = 1, while mixed statescorrespond to ~b ·~b ≤ 1. Under the stochastic map (5) the density operator becomes

Φ : ρ 7→ Φ(ρ) =12(I + B′) (16)

with the property that the affine shift leads to

Φ(I) = I +~b0~σ . (17)

It is well known that orthogonal transformations are represented by unitary operatorsacting on the density operators. Based on this property, we conjecture that a generalstochastic map of the density operator can be written in the following formal way:

Φ(ρ) = e−ihR SL e−ihT ρ eihT SReihR (18)

where the hermitian operators hR and hT correspond to the two orthogonal transfor-mations (rotations) of the Bloch vector and SL,R are some superoperators acting onthe density operator. The form of these superoperators remains to be established. Inthe next section we provide an explicit construction of such a transformation using aMaster equation.

From general properties of positive maps it is known that an arbitrary positivestochastic map can be written in Kraus form [8]:

Φ(ρ) =∑

λ

A†λρAλ (19)

with the condition ∑λ

Aλ A†λ = I . (20)

From the explicit form of the superoperators Eq. (18), one can derive the Kraus operators(19).

5 General stochastic transformation

We shall start the construction of the stochastic map (7) assuming that the densityoperator for the qubit satisfies a Master equation in the form:

dρ

dt=

1ih

[H, ρ] + Lρ . (21)


The first term of the time evolution describes the standard coherent two-level dynamicsand the last term accounts for the gain and the damping mechanisms. This term hasthe form of the Liouville superoperator which can be written in the Lindblad form [9]:

Lρ =∑

i

[F †

i Fiρ + ρF †i Fi − 2FiρF †

i

](22)

where Fi and F †i form a collection of generalized atomic creation and annihilation oper-

ators characteristic for a particular problem. The Lindblad form of the Master equationguarantees that the interaction with the damping reservoir preserves the positivity ofthe density operator.

To study the impact of the noise on a qubit we shall use the theory of the dampingbasis for the Liouville operator developed in [10]. In this theory one looks for eigenoper-ators of the Liouville superoperator. Such an operator has right and left eigenoperators

LRλ = λRλ LλL = λLλ . (23)

The right and the left eigenoperators form an orthogonal basis with respect to theHilbert-Schmidt scalar product:

Tr {RλLλ′} = δλ,λ′ . (24)

The right and the left eigenstates of the Liouville operator form a damping basis for thecorresponding Master equation given by Eq. (22). We shall use this basis to describethe time evolution of an arbitrary initial density operator ρ(0). Such a time evolutionis given by

ρ(t) =∑

λ

Λλ rλ(0)Lλ =∑

λ

Λλ lλ(0)Rλ (25)

where the eigenvalues of a damping matrix are in the form of Λλ = eλt and give thedynamical evolution of the density operator. From the properties of the damping basis(24) we obtain

rλ(0) = Tr {Rλ ρ(0)} and lλ(0) = Tr {Lλ ρ(0)} . (26)

With the help of these properties one can construct an explicit form of the stochasticmap generated by the superoperators (18) using the relation:

SL ρSR =∑

λ

Λλ Tr {Rλ ρ} Lλ =∑

λ

Λλ Rλ Tr {ρ Lλ} . (27)

This is the central result of this paper. In the following section we show how an explicitcalculation of Eq. (27) can be performed for the Bloch equations.

6 Stochastic noise of bloch equations

In this section we give an explicit construction of the stochastic map (7) for a Masterequation associated with the Bloch equations. The form of the corresponding Lindbladoperators (22) for the two-level atom is well known and has the form [2]:

Lρ = − 14T1

(1 − weq)[σ†σρ + ρσ†σ − 2σρσ†]

− 14T1

(1 + weq)[σσ†ρ + ρσσ† − 2σ†ρσ

]

− (1

2T2− 1

4T1) [ρ − σ3ρσ3] . (28)


The right

R0 = 1√2

(I + weqσ3) , R1 = 1√2

(σ + σ†) , R2 = 1√

2

(σ − σ†) , R3 = 1√

2σ3 (29)

and the left

L0 = 1√2I, L1 = 1√

2

(σ† + σ

), L2 = 1√

2

(σ† − σ

)L3 = 1√

2(σ3 − weqI) , (30)

eigenoperators of the Liouville operator have been calculated in [10]. These eigenoper-ators correspond to the following four eigenvalues of the damping matrix:

Λ0 = 1 , Λ1 = Λ2 = exp(− t

T2) , Λ3 = exp(− t

T1) . (31)

With these explicit formulas for the damping basis and the damping eigenvalues we cancalculate the transformation (27).

If we assume that the initial density operator of the qubit is in the form of a matrix:

ρ =[

a dd∗ c

](32)

a simple calculation shows that the stochastic map (7) of this matrix is:

Φ(ρ) =[

A DD∗ C

](33)

where

A =12(a + c)(1 + weq) +

12Λ3(a − c − weq(a + c)) ,

C =12(a + c)(1 − weq) − 1

2Λ3(a − c − weq(a + c)) ,

D = Λ2d , D∗ = Λ2d∗ . (34)

From the resulting formula (33) we see that the stochastic map is a contraction if theBloch sphere u2 + v2 + w2 = 1 is mapped into the interior of the Bloch ball. This isobtained if:

Λ22u

2 + Λ22v

2 + (Λ3w + (1 − Λ3)weq)2 ≤ 1 . (35)

This condition leads to various inequalities between the parameters of the Bloch dy-namics that guarantee the positivity of the stochastic map. Such relations have beeninvestigated in [6] and [11]. It is worthwhile to point out that in the approach presentedin this paper the positivity of the stochastic map is guaranteed by the Lindblad formof the Master equation (21).

From the formula (33) one can calculate the image of the set of pure state den-sity matrices generated by the stochastic map. Such an image is given by a family ofellipsoids:

(u

Λ1

)2

+(

v

Λ1

)2

+(

w

Λ3+ weq(1 − 1

Λ3))2

= 1 . (36)

With the help of such an equation one can obtain a geometric visualization of thestochastic map of a set of pure states using the Bloch equations.


7 Conclusions

It has been the purpose of this paper to study the decoherence of a qubit with the helpof a stochastic map that can be derived from the quantum optical Bloch equations. Itis clear that despite the fact that we have used a specific Master equation, the generalapproach based on the damping basis is general and flexible to handle more complicatedcases. It is worth pointing out that the use of a Master equation in the Lindblad formguarantees that the noise map of the qubit preserves the complete positivity of thedensity operator.

Acknowledgments

I would like to acknowledge discussions with J. H. Eberly, K. Banaszek, A. Ekert, C.Caves, B. -G. Englert and S. Daffer. This paper has been written to honor the 65birthday of Prof. Eberly whose seminal contribution to the two-level dynamics hasallowed for a natural extension of the Bloch theory to a general decoherence problemof an arbitrary qubit. This work has been partially supported by the Polish KBN grantNo. 2 P03B 089 16.


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Stochastic decoherence of qubits