Time Optimal Quantum Operartion for Mixed States(Micro-Macro Duality in Quantum Analysis)

(1)

Time Optimal Quantum Operartion for Mixed

States

Alberto Carlini,1,* Akio Hosoya,1,\dagger Tatsuhiko

Koike,2,\ddagger

and Yosuke Okudaira1,\S

1_Department

of

Physics, Tokyo Institute

_of

Technology, Tokyo, Japan

2Department

_of

Physics, Keio University, Yokohama, Japan

(Dated: January26, 2007)

We formulate a variational principle for finding the time.optimal quantum evolution ofmixed

states subject to amaster equation, when theHamiltonian$H$ and the Lindbladoperators $L_{f}$ are

subject to certain constraints. Weshow that the problem can be reduced to solving first a

fun-damentalequation (the “quanbum brachistochrone“) for $H(t)$, whichcanbe writtendown once_the

$\infty nstraints$are specified, and then solving the constraints and the master equation for the$L_{f}(t)s$

and the densityoperator $\rho(t)$

.

As anapplication ofour fomalism, we _{$an\Phi ticM$}solve asimple

one$qu$,bit model where theoptimalLindbladoperatorscorrespondeither toameasurementortoa

decoherence processbytheenvironment.

PACSnumbers:

I. INTRODUCTION

Quantumcontrol theoryofpurestates has been

stud-iedbymanypeople(foranexcellent review of the subject,

see,e.g., [1]). Forthemixedstate case, the master$\Re ua-$

tion in theLindbladform has been used in [10]. Anatural

problem to investigate is

the

optimal quantum control of

such systems. Around twenty years ago, Peirce, Dahleh

and Rabitz [2] considered

a

variational method to

man-ufacture a wave packet as close

as

possible to

a

target

wave

packet startingfrom

a

giveninitial

wave

packet. In

our

previous workwe haveestablished

a

general theory

based

on

the variational principle to find

a

(time)

opti-mal Hamiltonian which transforms agiven initial state

to

a

targetstate [26], and to find the (time) optimal

uni-tary operation for arbitrary initial states [27] which is

more

relevant for quantum computation, where the

in-put may be unknown. Recently, many works related to

time optimal quantumcomputationhave appearedinthe

literature[15-18,20-25] (fora reviewsee,e.g.,[27]). The

minimization

of physical timetoachieve

a

given unitary

transformation

provides

a

more

physical description of the complexity ofquantum algorithms.

Here we

extend

our

previousworks$[26, 27]$

on

thetime

optimal unitary evolution for pure quantum states and

we

formulate

a

variational principle for the time optimal

quantum control ofopensystems where the dynamics is

driven by amaster equation inLindblad $[12, 13]$ form: $\frac{d\rho}{dt}$

$:= \mathcal{L}(\rho)=-i[H,\rho]+\sum_{j}(L_{j}\rho L_{j}^{\dagger}-\frac{1}{2}\{L_{j}^{\dagger}L_{j)}\rho\}),$ $(1)$

forthe density operator$\rho(t)$, where $H(t)$ is the

Hamilto-nian, $L_{j}(t)(j=1, \ldots N^{2}-1)$

are

theLindbladoperators

’Electronicaddress: $c*rlinllth$

.

phys. titech.ac.jp

\dagger_Electronic_address: $*ho\epsilon oy*\partial tb$

.

phys.$t$it$\cdot$ch.ac.jp $*Electronic$ address: $koik\cdot 0phy*$

.

keio.ac.jp

\S_{Electronic address:} $okudaira9tb$

.

phys.$t$itech.ac.jp

and$N$is thedimension of the Hilbert spaceof thesystern.

The Hamiltonian represents the unitary evolution part

while the Lindblad operators express generalized $me*$

surements

or

decoherence processesdue tothe coupling

ofthe system with

an

environment. Note that $H(t)$ and

$L_{j}(t)$

are

consideredhere

as

dynamical variables

evolv-ing in time, besides the usual time dependent $\rho(t)$

.

The masterequationis

a

Markovian,i.e.

zero

memory

evolu-tion equaevolu-tion that defines aquantum mechanical semi-group, andit

can

be physically realized if the interaction

between the main physical systemand withthe

environ-mentis weak and theinteraction timeissmall compared

with the typical time scale of the physical system. The

Hamiltonian and the

Lindblad

operators

are

constrained

by

some

conditions due to physical laws

or

the

experi-mental set-up. E.g., a normalization $\infty nstraint$ for the

Hamiltonian is necessary because

one

can

afford only

a

finite amount of energy in experiments. The condition

on

the Lindbladoperators is necessary because at least

oneshould know the worst noise (i.e. decoherence rate)

toperform any sensibleexperiment.

Note that the authors of [14] also considered the prob-lem of control in dissipative quantum dynamics in

or-der to achieve optimal purification of

a

quantum state,

but they worked within the standard framework of

a

set

of constant Lindblad operators. Furthermore, although

thereshouldbenoconceptual difficulty in extending

our

work to the problem of optimal quantum control via

quantum feedback by introducing a stochastic tem in

the master equation [5, 7, 8],

we

will not discuss this

problem here.

The paper is organiaed

as

follows. In Section II

we

introducethe problem by defining

an

action principlefor

the time optimal unravelling of

an

open system under

the condition that the evolution is driven by

a

master

equation inLindblad form and of the existence of

a

set

ofconstraints for the available Hamiltonians and

Lind-blad operators, and

we

derive the

fundamental

equations of motion. In Section

III we

explicitly show how

our

the

oryvia the exampleof

a

one-qubit systemand

we

derive

thetimeoptimal Hamiltonian, whichgenerates the

uni-数理解析研究所講究録

(2)

tary evolution part of the density operator, and thetime

optimal Lindblad operators, which

can

represent either

a

measurement or a

decoherence

process

by the

environ-ment. Finally, Section IV is devoted to thesummary and

discussion of

our

results.

II. AVARIATIONAL PRINCIPLE

Let

us

consider the problem of making the transition

from

a

given initialstateto atarget state in the shortest

time by controllingacertain physical system. We

assume

that the mixedstateis governed by the master equation

(1) with the traceless Hamiltonian $H$ and the traceless

Lindbladoperators$\{L_{j}\}$

.

Mathematically this is a time

optimality problem for the evolution of the density $ma_{r}$.

trix$\rho(t)$ accordingto (1)and bycontrolling the Hamilto-nian

and the Lindblad

operators. We

assume

thatat least

the ‘magnitude’ of theHamiltonian andof the Lindblad

operators isbounded. Physically this corresponds to the

fact that

one

can

affordonly

a

finite energyinthe

exper-iment, and that

a

maximum level of noise is tolerated.

Besides this $no_{\mathfrak{R}^{alizati_{0}n}}$ constraint, the available

op-erations may be subject also toother constraints,which

can

represent eitherexperimental requirements (e.g., the

specifications of the apparatusinuse)

or

theoretical

con-ditions (e.g., allowing no operations involvin$g$ three or

more

qubits). The mixed state is repraeented by

an

N-dimensional positive definite matrix, $\rho\in \mathcal{M}_{N}$

,

whose

trace is preserved through the evolution by the master

equation. We then define the following action for the

dynamicalvariables $\rho(t),$ $H(t)$ and $\{L_{j}(t)\}$

.

$S= \int dt\{c(\rho, H, L_{j})+b[\sigma(\dot{\rho}-\mathcal{L}(\rho))]$

$+ \frac{1}{2}\lambda_{0}(hH^{2}-Nw^{2})+\sum_{j}\frac{1}{2}\lambda_{j}(hL_{j}^{\dagger}L_{j}-N\gamma_{j}^{2})\gamma_{2})$

where the first term gives the time duration

as

the cost

when

we

choose $c=1$, thesecondterm guarantees that

the quantum evolution is governed by the master

equa-tion through the Lagrange multiplier $\sigma$, while thethird

and fourth terms constrain the amplitude of the

Hamilto-nian$H$ and of theLindbladoperators $\{L_{j}\}$ through the

Lagrange multipliers $\{\gamma_{j}\}$

.

The operator $\sigma$ is traceless

because the master equation does not contain thetrace

part. Therefore, .Qaking variations ofthe action with

re-spectto $G$and $th^{\dot{j}}e$traceless part of

$\rho$

, we

obtain:

$\dot{\rho}=\mathcal{L}(\rho)$ (3)

$\dot{\sigma}=i[\sigma)H]-P[\sum_{j}(L_{j}^{\dagger}\sigma L_{j}+\tau 1\{\sigma, L_{j}^{\dagger}L_{j}\})]$

,

(4)

where$P(X)=x_{-\pi}^{I}$CE $X$isaprojection from$X\in \mathcal{M}_{N}$

to the traceless part of$X$

.

Iiurthermore, variations with

respect to $H$ and $L_{j}^{\dagger}$ give

$-i[\rho, \sigma]=\lambda_{0}H$ (5)

$P(\sigma L_{j}\rho-\frac{1}{2}L_{j}\{\rho, \sigma\})=\lambda_{j}L_{j}$

.

(6)

III. ONE QUBITEXAMPLE

For

a

onequbit system, the equations above

can

be

decomposed into three dimensional vector equations by

using the Pauli baeis $\{\sigma_{x}, \sigma_{y}, \sigma_{z}\}$

.

Namely,

we

have_$\rho=$

$\frac{1}{2}+r\cdot\sigma,$ $\sigma=s\cdot\sigma,$ $H=h\cdot\sigma$ and $L_{j}=l_{j}\cdot\sigma$ where$r,$$s$

and $h\in R^{3}$ and$l_{j}\in \mathbb{C}^{3}$

.

Accordingto thisnotation, the

set of equations (1), (4), (5) and (6)

can

be rewritten as

follows:

$?=2h \cross r+\sum_{l}(2R\epsilon((l\cdot r)l^{*})-2|l|^{2}r+ilxl\cdot)(7)$

$\dot{\epsilon}=2[hxs-\sum_{l}({\rm Re}((l\cdot\epsilon)l^{*})-|l|^{2}\epsilon)]$ (8)

$rx\epsilon=\lambda_{0}h$ (9)

$(s\cdot l_{j})r+(r\cdot l_{j})\epsilon+i\epsilon xl_{j}=\nu_{j}l_{j}$, (10)

where$v_{j};=r\cdot s+\lambda_{j}$

.

When $r$ and $\epsilon$

are

not parallel,

the components of the Hamiltonian $h$

are

given by $h= \pm w\frac{r\cross\epsilon}{|rx\epsilon|}$ (11)

becauseof the constraint Tr$H^{2}=2\omega^{2}$

.

Using the

master

equation (7), (8) and the eigenvalue equation (10),

one

can

see

that

$\frac{d}{dt}(rx\epsilon)=2hx(\sigma\cdot x\epsilon)$ (12)

which, together with (9), guarantees the conservation of

the vector $rxs$

.

Components of theLindbladoperators $\{l_{j}\}$ are

deter-mined as eigenvectors of the eigenvalue equation (10)

with constraints $|l_{j}|=\gamma_{j}$

.

In

some

instant,

we

can

parametrize$r$ and$s$

as

$r=r( \cos\frac{\theta}{2}e_{x}+\sin\frac{\theta}{2}e_{y})$ (13)

$\epsilon=s(coe\frac{\theta}{2}eae-\sin_{5}^{\theta}e_{y})$ (14)

where $r\cdot\epsilon=rs$

cos

$\theta$ with therange

$r \in[0, \int]$ and $\theta\in$

$[0, \pi]$, and rewrIte (10)

as

$[0-2rs \sin_{\theta}^{2\theta}iscoe_{E}0\mathfrak{T}-is\cos-is\sin\frac{\theta}{\int_{f}}0]l_{j}=\lambda_{j}l_{j}$

.

(15)

If the initial conditions satisfy $r(O)xs(O)=0$

,

then

we

also have $r(t)\cross\epsilon(t)=0$ because of (12), and the

componentsofthe Lindbladoperators

are

given by (15)

as

thefollowingconstants:

$\iota_{\pm}=\frac{\gamma\pm}{\sqrt{2}}(e_{l}\pm ie_{y})$ (16)

$l_{0}=\gamma_{0}e_{z}$, (17)

where$e_{\iota}$is theunitvectorparallelto$r$

.

Sincethe

Hamil-toniancannot bezero,exceptfor the$ca\epsilon e\cdot of\omega=0$, from

(11)

we see

that $\lambda_{0}=0$ and $h$ is arbitrary. To get rid

of the effect of the Hamiltonian,

we

move

to the

inter-action picture by the transformation $\rho’=U_{0}\rho U_{0}^{\dagger}$ with

(3)

$U_{0}(t)= \mathcal{T}\exp(-i\int H(t)dt)$

.

In the

new

frame, the

mas-ter equation reads

$\dot{r}=\sum_{l}[2R(l\cdot r)l^{*}-2|l|^{2}r+ilxl^{*}]$

$(i18))$

,

and forthe initialconditions

as

above we have

$\dot{r}=-2(\gamma_{+}^{2}+\gamma_{-}^{2})r+(\gamma_{+}^{2}-\gamma_{-}^{2})\epsilon_{z}$ (19)

whichguarantees$\dot{e}_{z}=0$

.

Therefore

we

obtainthe

follow-ingsolution for the componentsofthe density operator $r(t)= \{\frac{(\gamma_{+}^{2}-\gamma_{-}^{2})}{2(\gamma_{+}^{2}+\gamma_{-}^{2})}+c_{e^{-2(\gamma_{*}^{2}+\gamma_{-}^{l})t}}\}e_{z}$

.

(20)

Choosing magnitudesof the Lindbladoperators

as

$\gamma+=$

$\gamma_{-}$

,

the state willjustlose thecoherence, but the

$\infty her-$

ence

can

be

recovered

when magnitudes ofthe

Lindblad

operators

are

different

(see fig.1).

FIG.1: Analytlcal, timeoptimalevolutionof$\rho(t)$inthe Bloch

sphere for the case of$rxs=0$with aconstantHamiltonian

$H=h\cdot\sigma,$$\gamma+\neq 0,$ $\gamma_{-}=\gamma 0=0$and$\rho(0)=(1+\sigma_{l})/2$

.

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IV. SUMMARYAND DISCUSSION

We have developd aframework for finding the time optimalquantumoperationtotransform

a

givenpure

or

mixedstate to another, when thephysicalsystemobeysa Markovianmasterequationin Lindbladform. The

equa-tions forthe

Hamiltonian

$H$ and the

Lindblad

operator

$L_{a}$

can

bewritten down

once

the constraints for $H$ and $L_{a}$

are

specified according to theproblem. One then

ob-tains the time optimal operation $(H(t), L_{a}(t))$ and the

optimal time duration $T$ by solving the equations and

imposing the initial and final conditions $\rho(0)=\rho$

:

and

$\rho(T)=\rho’$

.

We have praeented

a

single qubit model to

demon-strate

our

variational formalismof thequantum

bruhis-tochrone for the mixed state

case.

The optimal

Hamilto-nian

is

obtained

by$solv\dot{\bm{o}}g$

the

quantumbrachistochrone equationand then

a

set ofordinary

differential

equations givethe time evolution of thedensity operator together

with theLindbladoperatorswhichrepresenttheoptimal

non-unitary operations corresponding to

a

meesurement

or

todecoherence. In aparticularcase,

an

analytical

so-lution

was

given. The fun version of this work

can

be

seen

in [28]

ACKNOWLEDGEMENTS

This research

was

partially supported by the

MEXT

of Japan, under grant

No. 09640341

(A.H. andT.K.), by the JSPS with grant L05710 (A.C.), by the COE21

project

on

‘Nanometer-scaleQuantum Physics’ at Tokyo

Institute

of Technology and by Macquarie University (A.C.).

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