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Classical Simulation of Quantum Noise

Daniel Crow and Robert Joynt Physics Department, University of Wisconsin-Madison, Madison, WI, US

(Dated: September 26, 2013)

Dephasing decoherence induced by interaction with a quantum bath can be simulated classicallyby random unitary evolution without the need for a bath and this random unitary evolution is

equivalent to the quantum case. For a general dephasing model and a single qubit system, weexplicitly construct the noise functional and completely specify the random unitary evolution. Todemonstrate the technique, we applied our results to three paradigmatic models: spin-boson, centralspin, and quantum impurity. We also give examples of 2-qubit models that can be simulatedclassically, and show that the the depolarizing channel can be simulated classically for an arbitrarynumber of qubits.

I. INTRODUCTION

The notion of an open quantum system has receivedintense scrutiny in recent years, motivated initially byquestions about the location and nature of the bound-ary (if any) between the quantum and classical worlds[1]. Further impetus has been given by the desire tominimize decoherence in quantum information process-ing [2, 3]. Open quantum systems have been conceivedin two quite distinct ways: a system and bath consideredas a single quantum entity with overall unitary dynam-ics, or a quantum system acted on by random classicalforces. The rst (system-bath or SB) approach is usu-ally thought of as being more general since it includesthe transformation of information and the idea that de-coherence is connected to the entanglement of a systemwith its environment [4]. In the random classical (RC)force approach these effects are absent. Furthermore, inthe question of the quantum-classical divide, it is often

seen as important that the bath be macroscopic since thisprecludes the complete measurement of the bath degreesof freedom.

It is very clear from the Kraus operator representationthat the SB approach is more general than the RC one,since the operators need not be unitary in SB. However,this does not illuminate what precisely is the role of en-tanglement in separating the SB and RC methods. Infact, it has been pointed out that SB models whose cou-pling is of dephasing type should always be able to berewritten as classical models [5]. This is based on gen-eral results concerning existence of certain positive maps[6]. A specic example of a single qubit bath has been

given [7]. Since SB entanglement is present in manydephasing models, this implies that this entanglement isnot the decisive feature of SB models that distinguishesthem from RC models.

In order to understand this particular aspect of thedifference between quantum theory and classical physics,

Electronic address: decrow@wisc.edu Electronic address: rjjoynt@wisc.edu

it would be useful to have a general explicit constructionof a method to classically simulate quantum dephasingSB models. The possibility of such simulation of coursedoes not imply that system-bath entanglement is not im-portant for quantum decoherence, but it does imply thatthe distinction between quantum and classical noise is

more subtle than has been previously recognized.In this paper, we give the explicit construction of theclassical model that corresponds to any given quantumdephasing model. As examples, we show explicitly howthree paradigmatic system-macroscopic dephasing bathmodels can be simulated by subjecting the quantum 2-level system to a classical random force without everneeding the bath. We do this by describing completelythe classical noise source that perfectly mimics the effectof entanglement of the qubit with the system. The threemodels are the popular spin-boson model, a specic cen-tral spin model, and the quantum impurity model. Thelast model is particularly interesting in this context since

it has a classical and a quantum phase. [8]. Wealso construct the classical simulations of certain 2-qubitmodels and show that pure depolarizing models can al-ways be simulated classically for an arbitrary number of qubits.

II. CLASSICAL SIMULATION OF QUANTUMDEPHASING MODELS

A. Quantum Models

The class of quantum models considered here consistsof a single qubit subject to dephasing by an arbitraryquantum system. Thus the total Hamiltonian of theopen quantum system is

H = H S [i ] + H B [ i ] + H SB [i , i ] (1)

where 0,x,y,z are the identity matrix and the Pauli ma-trices that act on the qubit and the matrices j aresome complete set of Hermitian operators for the bath,taken for simplicity to be nite-dimensional. H B [ i ] +H SB [i , i ] are linear functions of the i . 0 is propor-tional to the identity, and Tr B i = 0 for i > 0. For an

a r X i v : 1 3 0 9 . 6 3 8 3 v 1 [ q u a n t - p h ] 2 5 S e p 2 0 1 3

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N -level bath the i for i > 0 could be chosen as propor-tional to the N 2 1 generators of SU (N ) and 0 = I/N.Tr B is the trace over bath variables. We shall normalizethe i by the condition that Tr B [ i j ] = ij for i, j > 0.H B [ i ] is the bath Hamiltonian. Innite-dimensionalbaths can be treated by the same method at the cost of some additional notational complexity.

The total density matrix can be expanded as

=ij

N ij i j (2)

and we can invert this equation :

N ij =12

Tr [( i j ) ] .

The reduced density matrix of the system is

S = Tr B =12

i

n i i

and n i = (1 , n x , n y , n z ) is the expanded Bloch vector of the system. For quantum noise we have

(t) = U (0) U

with U = exp ( iHt ) where we have set = 1. In termsof components this isN ij (t) =

12

kl

Tr (i j ) U (k l ) U N kl (0) .

Let us write the initial in a product form (0) =S (0) B (0) , which in components says that N ij (0) =12 n i (0) m j , where

m j = Tr B i B (0) .

Then the dynamics of the system can be rewritten as

N ij (t) =12

kl

Tr (i j ) U (k l (0)) U n k (0) m l

and the Bloch vector of the qubit at time t is given by

n i (t) = 2 N i 0 (t)

=kl

Tr (i 0) U (k l (0)) U nk (0) m l

Finally we have a linear relation between the initial andnal expanded Bloch vectors:

n i (t) =k

T (Q )ik nk (0)

where the quantum transfer matrix is given by

T (Q )ik (t) =l

Tr (i 0) U (k l ) U m l . (3)

One should note that this represents a linear relationbetween the initial and nal expanded Bloch vectors.However, if construed as a relation between the usual 3-dimensional Bloch vectors, it is an affine relation, since if n i (t = 0) = (1 , 0, 0, 0) , then the initial state has zero or-dinary Bloch vector, but the nal state has n i (t) = T

(Q )i 0 ,

which is not zero. Expanding the Bloch vector to fourcomponents is just a convenient way of including theaffine part.

The whole dynamics of the qubit is contained in thematrix T (Q )ik (t) .

B. Classical Noise

A 2-level system acted on by dephasing classical noisehas the 2 2 Hamiltonian

H Cl = 12

B z +12

h (t) z . (4)

h (t) is a random classical (c-number) function of timethat represents an external source of noise. To com-pletely specify the model we need a probability functional

P [h] on the noise histories. Then the system dynamicsis given by(Cl ) (t) = D[h (t )]P [h (t )]

U Cl [h (t )] (Cl ) (0) U Cl [h (t )] .(5)

This is a functional integral over all real functions h (t )dened on the interval 0 t t. Each h (t ) is assigneda probability P [h (t )] and

D[h (t )]P [h (t )] = 1.Here the time-ordered exponential U Cl [h (t)] =

T exp i t

0 H Cl (t ) dt is a 22 matrix, unlike U Q (t) ,which is innite-dimensional. In the dephasing casetwo Hamiltonians taken at different times commute:[H Cl (t) , H Cl (t )] = 0 and the time-ordering can bedropped. Note the simplicity of the classical problemrelative to the quantum one.

We now follow the steps of the discussion of the quan-tum models with the appropriate modications. Restat-ing the results in terms of the 3

3 transfer matrix T (Cl ) ,

we have, in exact analogy to the section above,

S =12

3

i=0n i i ,

n i =12

Tr i S

n i (t) =3

j =0

T (Cl )ij (t) n j (0)

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(It is interesting to note that this decomposition of thevector ( c, s) is not unique.) Then the classical evolutionsubmatrix is written as

T (Cl )sub =12

cos1 sin1

sin1 cos1+

12

cos2 sin2

sin2 cos2.

T (Cl )c is the convex sum of rotations through the twoangles

1 (t) =tan 1s cc + s

and

2 (t) =tan 1s + cc s

.

(13)

Hence, by comparison to Eq. ( 7), it denes a classicalmodel. Dene the elds h1 (t) and h2 (t) by

h1 = 1t

+ B

and

h2 = 2t

+ B.

(14)

Then the equivalent classical model is given by

H Cl = 12

B z +12

h (t) z

and the probability distribution for h is: h (t) = h1 (t)with probability 1 / 2 and h (t) = h2 (t) with probabil-ity 1/ 2. More formally, P [h (t)] = (1 / 2) [h h1] +(1/ 2) [h h2] . Writing i (t) =

t0 (B + h i (t )) dt and

following Eq. (6) we nd

T (Cl )ij = 12Tr i e

i2 z 1 ( t ) j e

i2 z 1 ( t )

+12

Tr i ei2 z 2 ( t ) j e

i2 z 2 ( t )

The contents of this section can be summarized by thefollowing theorem and constructive proof.Theorem. The dynamics of the open quantum systemgiven by Eq. ( 8) can be simulated by the classical noisemodel given by Eq. ( 4): the density matrix of the qubitis the same for the two models at all times.

Proof. The quantum Hamiltonian H of Eq. (8) deter-mines the noise functions u(t) and v (t) in Eqs. (9) andhence the matrix T (Q ) in Eqs. (10) and ( 11) uniquely.The classical elds h1 (t) and h2 (t) are given explicitlyin terms of the matrix elements of T (Q ) by Eqs. ( 12)through ( 14). h1 (t) and h2 (t) in turn dene the classi-cal Hamiltonian H Cl and noise probability functional P given in Eqs. (4) and (5). The evolution of the qubitdensity matrix according to H Q with the usual partialtrace is precisely the same as the qubit evolution accord-ing to the classical Hamiltonian H Cl with the averagingover noise histories.

III. CLASSICAL SIMULATION OF DEPHASINGMODELS

A. Spin-Boson Model

The spin-boson Hamiltonian is

H SB = 12B z + z k gk bk + gk bk + k k bk bk .

The initial density matrix is (t = 0) = S (t = 0)B (t = 0) , where S (t = 0) is the initial state of the 2-level system and S (t = 0) is the thermal state of thebath at temperature 1 / . We are interested in the caseof a macroscopic bath, and it is then conventional to de-ne the coupling function J () = 4 k ( k ) |gk |

2 .Using U Q (t) = exp( iH Q t) , the system dynamics isgiven by (Q )S (t) = Tr B U Q (t) (t = 0) U Q (t). The ini-tially unentangled system and bath are entangled byU Q (t). This model has been well-studied and is ex-actly solvable [ 9], with the result that the off-diagonalelements of S (t) are proportional to e( t ) , where ( t) =

0 d J ()coth( / 2)(1 cos t) /2 < 0.

More explicitly,

(Q ) = 00 01 e( t )+ iBt

10 e( t )iBt 11

=00 + 11

20 +

00 112

z

+12

e( t ) 01 eiBt + 10 eiBt x

+12

e( t ) 10 eiBt 01 eiBt y .ij are the (time-independent) initial values for the ele-ments of . In terms of the Bloch vector we have

nx (t) = e( t ) (nx (0)cos Bt + ny (0)sin Bt )

ny (t) = e( t ) (nx (0)sin Bt + n y (0)cos Bt )We are free to consider the polarization vector along thez direction by change of basis and we can also begin witha pure state since total evolution is simply proportionalto the initial polarization vector. Written in terms of elements of T (Q ) ,

csb = e( t )

cos Btssb = e( t ) sin Bt sb =

1 e2( t )e( t )

.

According to the prescription in the previous section,therefore,

H ( sb )Cl = 12

B z +12

h ( sb ) (t) z

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and the noise source h( sb ) (t) is

h( sb )1 (t) = t

tan 1 s sb sb csbcsb + sb ssb

+ B ; p1 =12

h( sb )2 (t) = t

tan 1 s sb + sb csbcsb sb ssb

+ B ; p2 =12

(15)

where pi is the probability of h i .

Eq. 15 gives the result for a general coupling func-tion J () . A common choice for J is the ohmic bath:J () = Ae/ , important in quantum optics. In thiscase there is [9] the exact result

(t) = 12

ln 1 + 2 t2 lnsinh( t/ )

t/ .

where is the cutoff frequency and = 1T is the thermalcorrelation time where we have taken kB = 1. Thisexpression for ( t) allows an exact calculation of h1 (t)and h2 (t). The results of this calculation are plotted inFig. 1 for the case = 20.

FIG. 1: Fields h 1 (t ) and h 2 (t ) for the spin-boson model.

The classical elds have a complicated form, with aninitial quadratic decay crossing over to exponential atlonger times. The behavior for t is due to thermaldecay while the behavior for t is determined by thecutoff and is due to uctuations of the eld [ 9].

B. Central Spin Model

The central spin Hamiltonian describes a qubit coupledto a bath of nuclear spins. In the case of free inductiondecay, the Hamiltonian is

H CS = 12

B z + H int + H hf

where

H int =i

i J zi +i = j

bij J +i J j 2J zi J zj

and

H hf = z12 i

Ai J zi +i = j

Ai Aj4B

J +i J j .

i are the nuclear Zeeman splittings, bij contain the dipo-lar interaction, and Ai are the hyperne couplings. Weassume that the initial state is a product state of thequbit and the equilibrium bath state. For realistic con-ditions on i , bij , and Ai , the qubit dynamics can becalculated approximately at experimentally relevant timescales [1012]. The off-diagonal component of the qubitdensity matrix is given by 10 (t) = 10 (0) Dcs (t) where

D cs (t) =ei arctan( t )iBt

1 + 2 t2where is a complicated function of the parameters of the model. The explicit dependence is found in [ 10] and determines the relevant time scales since the model isvalid only for t 1. In reference [10], 1/ 20 s forGaAs dot.The qubit gains an additional phase from the bath in-teraction. In terms of T (Q ) , we have

ccs =1

1 + 2 t2 cos(arctan ( t ) Bt )scs =

1 1 + 2 t2 sin(arctan ( t ) Bt )

cs =t

1 + 2 t2 .Then we can write

1 = Bt + arctan ( t ) + arcsint

1 + 2 t22 = Bt + arctan ( t ) arcsin

t 1 + 2 t2

Noting that arctan( t ) = arcsin t 1+ 2 t 2 and applyingEqs. (14) we get

h (cs )1 (t) =2

1 + 2 t2and

h (cs )2 (t) =0 .

This model shows a Lorentzian fall-off of one of thetwo possible elds but the other one vanishes. Unlikethe spin-boson model, these elds have a non-zero timeaverage which indicates that the central spin noise in-duces an additional relative phase between the twosystem states. Furthermore, if we write the decoher-ence function as re i , for r, R so that 10 (t) =10 (0)eiBt r (t)ei ( t ) , then the central spin model

gives a simple relation between r and . Namely,r = cos for all values of t .

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C. Quantum Impurity Model

The quantum impurity Hamiltonian is

H QI = 12

B z +12

v dd z + H B

where

H B = 0dd +k

tk ck d + H.c. +k

k ck ck .

ck creates a reservoir electron and d creates an impu-rity electron. v represents the qubit-impurity couplingstrength and tk are the tunneling amplitudes betweenthe impurity and level k. Dene the tunneling rate = 2 k |tk |

2 ( k 0). Assuming an initial productstate and an equilibrium bath, this model can be solvedvia numerically exact techniques [ 13]. The off-diagonalcomponents of the qubit density matrix satisfy

10 (t) = 10 (0) eiBt Dqi (t).

Using the numerical methods outlined in [13], we com-pute D (t) = r (t)ei (t ) for r (t) > 0. Applying the methodoutlined earlier yields

i = Bt + (t) cos1(r (t))and the classical noise source h(qi ) (t) can be calculatedusing Eqs. ( 14). Numerical results are shown in Fig. 2.

FIG. 2: Fields h 1 ( t ) and h 2 (t ) for the quantum impuritymodel. The solid line shows coupling v = 3 (quantum phase).The dotted line shows v = 2 (crossover). The dashed lineshows v = 0 . 6 (classical phase). Note the oscillatory behaviorof the stronger coupling, indicative of coherence oscillation.

This model is of particular interest because it shows aclassical phase for v and a quantum phase forv with a crossover region in between. The quan-tum phase is characterized by oscillation in the coher-ence measures such as the visibility. These oscillations

are clearly caused by the oscillations in the noise eldsas shown in Fig. 2. There is, however, nothing quantumabout the elds at any value of v: they are classical noisesources.

IV. TWO-QUBIT DEPHASING

We consider a classical two-qubit dephasing modelwith a Hamiltonian of the form

H = H S + H B + H SB

such that all diagonal two-qubit Pauli operators commutewith the total Hamiltonian. That is,

[H, i j ] = 0

for i, j {0, 3}. Note that these diagonal Pauli operatorsalso commute with each other. We assume that the sys-tem and bath are initially in a product state so that thetwo-qubit system is in the initial state S . The diagonalelements of the two-qubit reduced density matrix will re-main constant under time evolution. For a classical noisesource to simulate the system-bath Hamiltonian and pre-serve the diagonal elements, each noise source must com-mute with each diagonal Pauli. Thus, the time evolutionof the density matrix is given by ij (t) = r ij (t)ij (0) forr ij (t) = r ji (t) , |r ij (t)| 1, and r ii (t) = r ij (0) = 1.Consider a classical noise source H with probabilitydistribution p() for R . The corresponding evolu-tion of the two-qubit density matrix is given by

(t) = p( )U (t)(0)U (t)d.In order to preserve the diagonal entries, we must havethat each H is a linear combination of diagonal two-qubit Paulis. Note that the diagonal Paulis span theentire set of Hermitian diagonal matrices so we can con-sider each H as a diagonal matrix with diagonal en-tries d i (t) where we have chosen to scale each H by the parameter . Then, time evolution is givenby diagonal unitary operators U with diagonal entriesexp(i i (t)) for

i (t) = t

0di (t )dt .

Then the action U U gives

ij ei ij ( t ) ijwhere ij (t) = i (t)j (t). We now make a simplifyingassumption that H is simply proportional to so that ij (t) = ij (t). Applying this result and averaging theresult over the probability distribution gives

ij (t) = p()ij (0)ei ij ( t ) d= ij (0) p ( ij (t)) (16)

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where p(t) = p()eit d . That is, for the model

specied above,

r ij (t) = p( ij (t)) . (17)

This model gives a highly restrictive set conditions of r ij (t). However, note that we can again average over avariety of such models to obtain more complicated be-havior.

V. N-DIMENSIONAL DEPOLARIZATION

Given a quantum system in N dimensions with initialdensity matrix 0 , the depolarization channel is given by

(t) = (1 p(t))0 + p(t)1N

I

where p(t) [0, 1] for all t and p(0) = 0.We now present a classical model for depolarization

in arbitrary dimension. Consider the unitary groupSU (N ). For each U SU (N ), let H U be dened by

eiH U = U

where the eigenvalues of H U are all in [, ]. Note thateach H U can be expanded asH U =

rhr r

where h r are real and r are the generators of SU (N ).Then consider the Hamiltonian dened by the set {H U }with the probability distribution equal to the uniformdistribution over the unitary group. For convenience,choose

SU (N ) dU = 1where the integral is taken with respect to the Haar mea-sure. Time evolution is then given by

(t) = SU (N ) eiH U t (0)eiH U t dU. (18)We need to show that this time evolution is purely

depolarizing: the polarization vector remains parallel tothe initial polarization vector at all values of t and (1)is proportional to the identity giving the totally mixedstate. Let 0 be given by polarization vector v so that

(0) =1N

I +r

vr r .

Then consider the unitary operator

V = exp ir

vr r .

where is an arbitrary non-zero real number. It is clearthat

V 0V = 0and we can write that

(t) = SU (N ) eiH U t V 0V eiH U t dU from which we can easily obtain

V (t)V = SU (N ) V eiH U t V 0V eiH U t V dU.Since the above integral runs over the entire unitarygroup SU (N ), conjugation by V simply amounts to areparameterization of the integral. Thus, we can write

V (t)V = SU (N ) eiH U t 0eiH U t dU = (t). (19)Hence, any rotation leaving the initial polarization vectorxed must also leave (t) xed, the polarization vector

of (t) must be parallel to that of 0 .Now, consider the case t = 1. From our initial deni-tion of H U , we see that

(1) = SU (N ) U0U dU.Further consider an arbitrary unitary operator W . Thenconjugation gives

W(1)W = SU (N ) W U0U W dU =

SU (N )

U0U dU

since if U runs over the entire unitary group than so doesWU . Thus, we see that

W (1)W = (1) .

Since this holds for arbitrary unitary operator W , wemust have that (1) is proportional to the identity. Thus,(1) is the totally mixed state and we see that the modelpresented is indeed a depolarizing channel.

For the single qubit case, the calculation can be evalu-ated explicitly. Without loss of generality, assume thatthe qubit state is specied by

(0) =12 (I + z ) .

Then

(t) =12

(I + n z (t)z ) .

The integration in Eq. 18 reduces to an integral over the3-sphere S 3 . It can be evaluated explicitly and one nds

n z (t) =13

+sin(2t )

3(t t3). (20)

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This expression indeed satises n z (0) = 1 and n z (1) = 0.Somewhat surprisingly, this function has a root near t =0.77 and has a limiting value of 1 / 3 as t . Becauseof the root, proper depolarization behavior ends beforet = 1, however a simple reparameterization of time allowsfor more general depolarizing behavior. The graph of n z (t) is shown in Fig. 3.

FIG. 3: The z component of the Bloch vector as a functionof time for single qubit classical depolarization. Note thatproper depolarization ends at the rst root ( t 0. 77).

VI. CONCLUSION

We have shown that decoherence arising from the in-teractions between a qubit and an external bath can besimulated classically via random unitary evolution forthe pure dephasing case. We demonstrated that thenoise functional takes on the particularly simple form of two uniformly distributed random unitary evolutions andgave explicit construction of the corresponding Hamilto-nian elds. We offered exact results of the calculationof these elds for the spin-boson model, approximate re-sults for the central spin model, and numerically exactresults for the quantum impurity model.

Classical simulation of quantum noise acting on multi-ple qubits is also possible in certain cases. For two inter-acting qubits, we showed that the simulation is possible if the interaction is sufficiently simple. Lastly, we demon-strated that the depolarization channel can be simulatedclassically for an arbitrary number of qubits.

Acknowledgments

We acknowledge with pleasure illuminating discussionswith W. Strunz and B.L. Hu. This project was fundedby DARPA-QuEst Grant No. MSN118850.

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