quantum markov channels for qubits

, USA

PHYSICAL REVIEW A 67, 062312 ~2003!

Quantum Markov channels for qubits

Sonja Daffer,1 Krzysztof Wodkiewicz,1,2 and John K. McIver11Department of Physics and Astronomy, University of New Mexico, 800 Yale Boulevard NE, Albuquerque, New Mexico 87131

2Instytut Fizyki Teoretycznej, Uniwersytet Warszawski, Hoz˙a 69, Warszawa 00-681, Poland~Received 31 October 2002; published 26 June 2003!

We examine stochastic maps in the context of quantum optics. Making use of the master equation, thedamping basis, and the Bloch picture we calculate a nonunital, completely positive, trace-preserving map withunequal damping eigenvalues. This results in what we call the squeezed vacuum channel. A geometrical pictureof the effect of stochastic noise on the set of pure state qubit density operators is provided. Finally, we studythe capacity of the squeezed vacuum channel to transmit quantum information and to distribute Einstein-Podolsky-Rosen states.

DOI: 10.1103/PhysRevA.67.062312 PACS number~s!: 03.67.Hk, 42.50.Lc, 03.65.Yz

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I. INTRODUCTION

One of the aims of quantum information theory isachieve the storage or transmission of information encoin quantum states in a fast and reliable way@1#. It is unreal-istic to consider a physical system, in which informationstored, as being isolated. It is well known that, whensystem of interest interacts with its environment, irreversidecoherence occurs, which is, in most cases, both undable and unavoidable@2#. This interaction causes pure statto become mixed states. This process describes the influof noise on quantum states, which results in information pcessing errors.

The question of how to reliably transmit information bgan with communication systems. Shannon’s noisy chancoding theorem is the fundamental theorem of informattheory @3,4#. It states that information can be transmittwith arbitrarily good reliability over a noisy channel provided the transmission rate is less than the channel capand that a code exists which achieves this. There has brecent interest in studying quantum channels for sendquantum information and defining quantum channel capties @5–11#.

As in classical information theory, a quantum channelpacity is characterized by the type of noise present inchannel. There exists a set of input states or alphabet wthe sender transmits through the channel. The noise inchannel generally degrades the states. The receiver trierecover the message that was sent from the output stThis process of induced errors may be described by a sysinteracting with a reservoir. For a classical communicatchannel, the channel is completely characterized by its tsition probability matrix which determines the errors that coccur. In constrast, a quantum channel is characterizedcompletely positive, trace-preserving or stochastic mwhich takes the input state to an output state. This charaizes the type of noise present in the channel.

In this paper, we use a special basis of left and ridamping eigenoperators for a Lindblad superoperator toculate explicitly the image of a stochastic map for a wiclass of Markov quantum channels. We use this methodderive a noisy quantum channel for qubits, which we callsqueezed vacuum channel. This channel is nonunital w

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unequal damping eigenvalues, which makes it different frpreviously introduced channels@12#. We use this channel togive geometrical insight into the Holevo channel capacity

We begin this paper by defining a noisy quantum chanin Sec. II in terms of stochastic maps and the Kraus decoposition. A special case of the Markov channel is discussIn Sec. III, a general Lindblad equation for a finitedimensional Hilbert space is introduced. In Sec. IV, tdamping basis is introduced as an alternative way to calate the stochastic map explicitly without using a Kraus dcompostion. Stochastic maps in the context of quantumtics described by a set of Bloch equations are discusseSec. V. Section VI reviews some known quantum channand presents some more general types of channels. Thechastic map that defines the squeezed vacuum channexplicitly calculated in Sec. VII, and the restrictions imposby the condition of complete positivity are presented. Theresults are used to determine a Kraus decomposition exitly. The geometrical picture of the channel is given in SeVIII. Finally, Sec. IX deals with the channel’s ability to senclassically encoded quantum states and its ability to sendresource of entanglement.

II. NOISY QUANTUM CHANNELS

A. Stochastic maps

The concept of a noisy quantum channel arose fromfield of quantum communication. Information is encodedquantum states and transmitted across some channel wthe receiver decodes the information to retreive the origimessage. The ability to send messages reliably dependthe noise present in the channel. The effect of the noise itake an initial quantum state and transform it to anotquantum state. The noisy quantum channel is then definea map

F:r→F~r! ~1!

which takes a quantum state described by a density oper into a quantum state described by a density operatorF(r).There are certain restrictions on the class of maps whgenerate legitimate density operators. We require tTr@F(r)#51 so that unit trace of the density operator

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DAFFER, WODKIEWICZ, AND McIVER PHYSICAL REVIEW A 67, 062312 ~2003!

conserved for all time. In addition, the image of the maF(r), must be a positive operator. A map that takes posioperators into positive operators is called a positive map.if one considers the noise to come from a larger Hilbspace of a reservoir, then the stronger condition of comppositivity is required for the process to be physical@13,14#.Therefore, we restrict our attention to completely posititrace-preserving maps, which are called stochastic maps

B. The Kraus decomposition

Noise in the channel may be considered as a reservowhich the quantum state of interest is coupled. The statethe reservoir interact unitarily for some time and they bcome correlated. If we are now interested only in the systwe trace over the environment degrees of freedom. Onethink of the reservoir as extracting information from the sytem as it will typically map pure states into mixed stateThis noise process can be described by a quantum operinvolving only operators on the system of interest. Thiscalled a Kraus decomposition and has the form

F~r!5(k

Ak†rAk, ~2!

where the condition

(k

AkAk†5I ~3!

ensures that unit trace is preserved for all time@15#. If anoperation has a Kraus decomposition, then it is complepositive. The converse is also true so that all stochastic mhave a Kraus decomposition.

C. The Lindblad form

The formalism we have outlined so far is general. Forimportant type of noise, Markov noise, we have a speclass of completely positive maps. We call a Markov quatum channel one in which the noise in the channel arifrom a coupling of the system with a reservoir under tMarkov and Born approximations. This is a commonly usapproximation in quantum optics and leads to the wknown Lindblad form of a master equation. For this typechannel, one can always write a stochastic map as

F~r!5eLtr~0!. ~4!

The equation describes the evolution of a system couplea reservoir in terms of the system of interest alone. All Linblad superoperators are stochastic maps and have a Kdecomposition. The converse is not true in general.

In this paper, we derive an equivalent equation forF ofthe form

F~r!5(i

Tr$Rir~0!%L iL i ~5!

to obtain the image of the stochastic map for Markov noiWe use a special basis of left,Li , and right,Ri , eigenopera-

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tors with damping eigenvaluesL i , which allow for an ex-plicit calculation of the stochastic map and the Kraus opetors. This method works for any Markov channel.

III. THE GENERAL LINDBLAD EQUATION

The Schro¨dinger evolution of a system coupled to a reervoir can be described in terms of a master equation ofform

r5Lr5LCr1LDr, ~6!

wherer is the density operator of the system alone, obtainby tracing over the reservoir degrees of freedom. The fiterm describes the coherent or unitary evolution and is sply given by the commutator

Lcr52i

\$H,r% , ~7!

whereH is the Hamiltonian of the undamped system.The general form of the nonunitary part which describ

the dissipation of the density operator is

LDr51

2 (i , j

N221

ci , j$@Fi ,rF j†#1@Fir,F j

†#% ~8!

valid for a finiteN-dimensional Hilbert space. The$Fi% aresystem operators which satisfy the conditions Tr(Fi

† F j )5d i , j and Tr(Fi)50. The set of complex elements$ci j %form a positive matrix.

It has been proven@16# that a linear operator on a finitN-dimensional Hilbert spaceL:M (N)→M (N) is the genera-tor of a completely positive dynamical semigroup in tSchrodinger picture if and only if it can be written in thform of L:r→Lr whereLr takes the form of Eqs.~7! and~8!. We will call the generator of the semigroup, which goerns the dissipation, the Lindbladian and denote it byLD .Equation~8! may be recognized as the master equationscribing irreversible evolution of an open quantum systunder the Markov and Born approximations. This Lindblequation is widely used in many branches of statistical mchanics and quantum optics. This form, the Lindblad forhas been shown to guarantee positivity and trace presetion of the density operator@17,18#.

IV. THE DAMPING BASIS

There are many methods for solving master equations,use of Fokker-Planck equations built on methods in stochtic processes and Monte Carlo wave functions, to name@19,20#. In this paper, we make use of the damping basisorder to solve a master equation which has the form of~6! containing both the coherent and damping dynamThis amounts to solving an eigenvalue equation. In socases, this problem can seem formidable and finding a daing basis first is useful. To solve the master equation in tfashion involves first solving the eigenvalue equation

LDr5lr ~9!

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QUANTUM MARKOV CHANNELS FOR QUBITS PHYSICAL REVIEW A67, 062312 ~2003!

for the nonunitary part of the density operator evolution dscribing an open system. This provides one with a complorthogonal basis with which to expand the density operaat any time. Such a basis is called the damping basis@21#.This basis is obtained by finding the eigenoperators ofeigenvalue equation. Likewise, the dual eigenoperatorsfound by solving the dual eigenvalue equation. The origibasis and the dual basis are orthogonal.

If the eigenoperators of Eq.~9! areRi with correspondingeigenvaluesl i , then once the initial state is known

r~0!5(i

Tr$Lir~0!%Ri , ~10!

the state of the system at any later time can be found thro

r~ t !5eLtr~0!5(i

Tr$Lir~0!%L iRi5(i

Tr$Rir~0!%L iL i ,

~11!

where L i5el i t are the damping eigenvalues andLi is thestate dual toRi . These are called the left and right eigenoerators, respectively, and satisfy the following duality retion:

Tr$LiRj%5d i j . ~12!

It is easy to show thatL and R have the same eigenvalueThe left eigenoperators satisfy the eigenvalue equation

LLD5lL ~13!

while the right eigenoperators satisfy

LDR5Rl. ~14!

This method is a simple way of finding the density operafor a givenL for all times. The solution of the left and righeigenvalue equations yields a set of eigenvalues and eisolutions:$l,L,R%. Once the damping basis is obtained,can be used to expand the density operator. Then the deoperator in this basis can be substituted back into theLiouville equation~6!. By doing this, one obtains a set ocoupled differential equations for the coefficients of the dsity operator in the damping basis. Solution of this setcoupled differential equations yields the solution to the toLiouville dynamics. The important point is that once all egenvalues and all left and right eigenoperators of the suoperator are found, the master equation can be solved ansystem observables can be computed.

V. BLOCH STOCHASTIC MAP

When studying two-level systems there is the addedvantage of a geometrical picture offered by the vector moof the density matrix. For instance, decoherence of a tlevel atom is described by the dynamics of a Bloch vecwith three components,bW 5(u,v,w), inside a unit three-sphere, governed by a set of Bloch equations@22#. Theseconstitute a set of differential equations, one for each coponent of the Bloch vector, of the form

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w521

Tw~w2weq!2Vv, ~15!

whereV is the Rabi frequency, the constantsTu andTv aredecay rates of the atomic dipole, andTw is the decay rate ofthe atomic inversion into an equilibrium stateweq . One typi-cally finds that the phenomenological decay rates inBloch equations appear as

1

Tu5

1

T2,

1

Tv5

1

T2,

1

Tw5

1

T1, ~16!

so that the parts of the atomic dipole that are in phase andof phase with the driving field are affected in the same wby the damping. This description is in terms of a two-levatom coupled to an external field as well as a reservoir. Tcoupling to the external field causes the Bloch vector totate. The coupling to the reservoir, which might be a cotinuum of vacuum field modes, causes the Bloch vectordecrease in magnitude. The combination of these two behiors leads to a spiraling in of the Bloch vector. Although wdescribe these dynamics in terms of the two-level atom,Bloch picture can describe any two-level system. Hereshall consider the more general form of Eq.~15! where allthree damping constants may be unequal. As we shallthis describes the physical situation where the two-leatom is coupled to a squeezed vacuum reservoir rather thregular vacuum field. The damping parameters lead to deherence of the system of interest.

The decoherence is caused by the presence of noisemay be viewed as a stochastic map acting on the Bloch vtor in the form of a mapping@23#

F:bW→bW 8. ~17!

Because there is a correspondence between the Bloch vbW and the density operatorr, we see that the stochastic mais a superoperator which maps density operators into denoperators:

F:r→r8. ~18!

We can expand the density operator in the Pauli ba$I ,sx ,sy ,sz% and consider how the components ofr trans-form under the map. This latter transformation is characized by a 434 matrix representation ofF. It has been foundthat the general form of any stochastic map on the secomplex 232 matrices may be represented by such a 434matrix containing 12 parameters@24#:

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T5S 1 0 0 0

t10 t11 t12 t13

t20 t21 t22 t23

t30 t31 t32 t33

D . ~19!

The 333 block of the matrixT can be diagonalized usintwo rotations. This amounts to a change of basis. Withloss of generality, we can consider the matrix

TD5S 1 0 0 0

t10 L1 0 0

t20 0 L2 0

t30 0 0 L3

D ~20!

which uniquely determines the map. To preserve HermiticT must be real. The first row must be$1,0,0,0% to preservethe trace of the density operator. We call the 333 part of thematrix TD , consisting of the damping eigenvaluesL i whichare contractions, the damping matrixL. Explicitly,

L5S L1 0 0

0 L2 0

0 0 L3

D . ~21!

In terms of the Bloch vector, a general stochastic map mbe written in the form

F:bW→bW 85LbW 1bW o, ~22!

wherebW o5(t10,t20,t30) is a translation. The overall operation consists of a damping part and translations. Due topresence of translations, the transformation is affine.

To see the properties required by the stochastic materms of the Bloch vector consider the matrix representaof the Bloch vector as an expansion in terms of the Pamatrices:

B5bW •sW 5S w u2 iv

u1 iv 2w D . ~23!

In the absence of noise, the Bloch vector remains onBloch sphere so that

detB52~u21v21w2! ~24!

has magnitude unity. The presence of stochastic noise trforms the matrixB according to

F:B→B8. ~25!

To guarantee that the mapF transforms the density operatointo another density operator, the Bloch vector can be traformed only into a vector contained in the interior of thBloch sphere, or the Bloch ball. Equivalently, we require

udetB8u<udetBu, ~26!

so that the qubit density operator

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2~ I 1B! ~27!

under the map becomes

F~r!:r→F~r!51

2~ I 1B8!. ~28!

This is possible only if the eigenvaluesL i are contractions.In the following sections we provide an explicit constructioof the stochastic mapF from the Bloch equations~15!.

VI. TWO-LEVEL ATOM IN A SQUEEZED VACUUM

The master equation~6! yields a plethora of possible completely positive dynamical maps. In the remainder of thpaper, we wish to examine a particular form of Eq.~6! whichretains a relation to the Bloch equations~15! and leads to thecase where the three components of the Bloch vector hdifferent decay rates and where the Bloch vector is shiffrom the origin of the Bloch sphere. As will be seen, thleads to a contraction of the set of states that lie on the Blsphere surface. Some states in the set become very mwhile some remain almost pure.

We will consider a special case of the Lindbladian in E~8! for a two-dimensional Hilbert space by choosing the flowing set of system operators$Fi%:

F15s, F25s†, F35s3

A2, ~29!

wheres ands† are the qubit lowering and raising operatoands3 is thez-component Pauli spin operator. If, along withis set of system operators, we choose the matrix elemci j such that

c5S 1

2T1~12weq! 2

1

T30

21

T3

1

2T1~11weq! 0

0 01

T22

1

2T1

D ,

~30!

the resulting Lindbladian is

LDr521

4T1~12weq!@s†sr1rs†s22srs†#

21

4T1~11weq!@ss†r1rss†22s†rs#

2S 1

2T22

1

4T1D @r2s3rs3#2

1

T3@s†rs†1srs#.

~31!

This part of the master equation is known to describedissipative evolution of a two-level atom coupled to a ba

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@19#. The raising and lowering operators describe transitibetween the ground and excited states, and theci j describethe losses caused by the reservoir and depend on pheenological decay constants.

The coherent part of the dynamics,LC , is described bythe Hamiltonian

H5\V

2~s†1s!, ~32!

where V is the Rabi frequency of oscillation between tground and excited states. The full dynamics describe a tlevel atom driven by a laser field subjected to irreversidecoherence by its environment. This corresponds to a linmapping from a two-dimensional Hilbert space into a twdimensional Hilbert space. More generally, this Liouvilliagenerates a completely positive dynamical map of a gentwo-level system or qubit.

The Bloch equations for a two-level system describedthis Lindblad operator~31! are given by Eqs.~15! where

1

Tu5

1

T21

1

T3,

1

Tv5

1

T22

1

T3,

1

Tw5

1

T1. ~33!

The presence of the parameterT3 is the source of the damping asymmetry between theu andv components of the Blochvector. Equation~31! describes many well-known physicaprocesses in quantum optics. We wish now to briefly discthe physics behind this Lindblad form of the master eqtion.

A. The amplitude damping channel

The amplitude damping channel is identical to the casespontaneous emission for a two-level atom. This correspoto the following choice for the parameters:

1

T15A,

1

T25

A

2,

1

T350, weq521. ~34!

The parameterA is the Einstein coefficient of spontanteoemission, which depends on the density of vacuum fimodes and how strongly the atom couples to the modes.describes the exponential decay of an atom, from exciteground state, due to vacuum fluctuations. The equilibristate isweq521 indicating that, given enough time, thatom will be in the ground state.

B. The depolarizing channel

Another type of noisy quantum channel is the depolaing channel. This channel is identical to the quantum optmodel for pure phase decay. This channel has paramete

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T150,

1

T25G,

1

T350, weq50. ~35!

This describes the process of phase randomization ofatomic dipole caused by atomic collisions. This leadsequal damping for theu and v components of the Blochvector with contractions that depend on the parameterG dueto collisions.

C. The thermal field channel

In the case of spontaneous emission, the atom is couto a vacuum reservoir. But one can consider an atom inacting with a thermal field, so that now the field has a nozero photon number. The reservoir may be consideredlarge number of harmonic oscillators such as modes offree electromagnetic field or a heat bath in equilibrium.this case, one finds that the decay constants are related tphoton number in the following way:

1

T152AS N1

1

2D ,1

T25AS N1

1

2D , ~36!

1

T350, weq52

1

2N11.

Note that settingN50 reduces this to the case of spontanous emission. In this thermal field case, the value ofweqindicates that the atomic inversion approaches a steady swhich is the ground state in the limit of zero photon numbbut asN becomes large it approaches zero. Therefore,equilibrium state for the inversion is bounded:21,weq,0. One can see from Eqs.~33! that this leads to equadamping for theu andv components of the Bloch vector.

D. The squeezed vacuum channel

A more general case occurs when all parameters arezero and explicitly are

1

T152AS N1

1

2D ,1

T25AS N1

1

2D , ~37!

1

T35AM, weq52

1

2N11.

This describes an atom in a squeezed vacuum whereN is thephoton number and M is the squeezing parameter. TherameterN is related to the two-time correlation function fothe noise operators of the reservoir^a†(t)a(t8)&5Nd(t2t8) wherea(t) is the field amplitude for a reservoir modeThe squeezing parameterM arises from the two-time correlation function involving the square of the field amplitud^a(t)a(t8)&5M !d(t2t8). These are the familiar relationobeyed by squeezed white noise, which lead to squeezina vacuum reservoir@20#. A squeezed vacuum has fluctuatioin one quadrature smaller than allowed by the uncertaprinciple at the expense of larger fluctuations in the otquadrature. This type of reservoir leads to two dipole de

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constants in the Bloch equations, the one in the squeequadrature being correspondingly smaller than that forstretched quadrature.

The steady state for the inversion is the same here athe previous case and depends only on the photon numWhen the field is not squeezed (M50), the master equationreduces to the previous case of an atom in a thermal fiThe important difference now is that the squeezing paramhas introduced a new parameterT3, which leads to unequadamping for theu andv components of the Bloch vector. Iwhat follows, we deal with the most general case, corsponding to the Lindbladian~31!, of a qubit coupled to asqueezed vacuum reservoir. This defines a noisy quanchannel that we call the squeezed vacuum channel~SVC!,which has different properties from the depolarizing and aplitude damping channels.

VII. THE SQUEEZED VACUUM CHANNEL

A. The image of the map

So far, we have introduced a general set of Bloch eqtions ~15! which correspond to the Liouvillian with Eq.~31!in addition to the coherent dynamics described byH in Eq.~32!. We are now in a position to solve this master equatiWe proceed with the method described in Sec. IV. This tus how the noise affects all states of the two-level systUsing the damping basis, the solution to the Liouville eqution ~9! with Lindblad operator of the form of Eq.~31! fol-lows. For a two-level system in a squeezed vacuum reserthe left eigenoperators are

L051

A2I , L15

1

A2~s†1s!, ~38!

L251

A2~s†2s!, L35

1

A2~2weqI 1s3!

found from solving Eq.~13! while the right eigenoperatorare

R051

A2~ I 1weqs3!, R15

1

A2~s†1s!, ~39!

R251

A2~s2s†!, R35

1

A2s3

found by solving Eq.~14!. They correspond to the followingfour eigenvalues:

l050, l152S 1

T21

1

T3D , ~40!

l252S 1

T22

1

T3D , l352

1

T1.

Of the four eigenvalues, three of them are precisely theagonal elements of the damping matrix~21! through the

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equationL5elt. We call the set$L% the damping eigenval-ues and the set$l% the eigenvalues~of the damping basis!.This indicates a relation between the decay constants inBloch equations and the damping basis. The damping eigvalues of the damping matrixL contain the decay constanfor the three components of the Bloch vector. The fourththe damping eigenvalues is unity, which is a necessary cdition for the density operator to have trace unity. The desity operator may be expanded in any complete basis. Ching the right eigenoperators as a basis we can writedensity operator as

r5 l 0R01 l 1R11 l 2R21 l 3R3 , ~41!

where the coefficients are obtained by projecting on toleft eigenbasisl i5Tr$Lir%. Next we substitute the expansioon the right eigenoperators into the equation for the toLiouville operator equation and use the fact thatLDRi5l iRi . After the substitution ofr into the total Liouvilleequation, one obtains a set of differential equations forcoefficientsl i in the right eigenbasis and the generator of tMarkovian time evolution has the following matrix represetation in the same basis:

L5S l0 0 0 0

0 l1 0 0

2 iweqV 0 l2 2 iV

0 0 2 iV l3

D . ~42!

The solution tor5Lr is

r~ t !5eLtr~0!. ~43!

One can perform a rotation of the right eigenbasis to obtthe following diagonalized form of the above matrix:

L5S l0 0 0 0

0 l1 0 0

0 0 l231x 0

0 0 0 l232x

D , ~44!

wherel235(l21l3)/2, x5 12 A(l22l3)22(2V)2, and the

l i are given in Eq.~40!. It follows that the superoperatoreLt

which maps the density operator forward in time is

eLt5S 1 0 0 0

0 L1 0 0

0 0 L23ext 0

0 0 0 L23e2xt

D . ~45!

The matrices above are represented in two different baEquation~42! is in the damping basis, while Eq.~44! is in arotated damping basis. Geometrically, one may considerfirst case to be dynamics as viewed from a shifting centethe Bloch sphere. In this case, there is one affine shift towthe south pole of the Bloch sphere. This is the viewpointthe damping basis. This can be seen by noting that in E

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~39! for the damping basis there is one shift present inR0.From the right eigenoperators of the damping basis,finds that they are almost the same as the Pauli operaFrom the viewpoint of the Pauli basis, the dynamics wotake place as seen from the stationary center of the Bsphere. In the rotated damping basis, the viewpoint is froframe that is rotating with the driving field as well as shiftinfrom the center of the Bloch sphere, and, consequently,eigenvalues lead to contractions or pure damping in thisagonal basis. Although the most general dynamics contboth coherent and incoherent parts, in certain situationspart may dominate the dynamics. We consider the cwhere the system is not isolated from its environmentunaffected by coherent dynamics. In this case, rotationscur on a time scale much longer than the dissipation soeffectivelyV→0. We will see that this case is advantageo

With these explicit formulas for the damping basis anddamping eigenvalues we can calculate the density operfor all times. This gives us the image of the map,F(r).Assuming that the initial density matrix is of the form

r5S a d

d! cD , ~46!

we find that the stochastic map generates a new densitytrix

F~r!5S A D

D! CD ~47!

in accordance with Eq.~18!. This is obtained using the set oeigenoperators and eigenvalues for the damping basis.the stochastic map that characterizes the squeezed vachannel, we have that the elements ofF(r) are given by

A51

2~a1c!~11weq!1

1

2L3@a2c2weq~a1c!#,

C51

2~a1c!~12weq!2

1

2L3@a2c2weq~a1c!#,

D51

2@d!~L12L2!1d~L11L2!#,

D!51

2@d~L12L2!1d!~L11L2!#. ~48!

These elements are in terms of the initial density matrixements as well as the damping eigenvalues which containparametersTi , and weq . The image gives us the densimatrix after the noise operatorF has acted on it. The channel capacity of a noisy quantum channel is characterizedthe types of errors that result after the input is transmittThe noise operation defines the channel.

B. Complete positivity

The restriction that this map be completely positivesomewhat stringent. We have mentioned so far that the

06231

ers.dha

ei-nsnesetc-at.etor

a-

orum

l-he

y.

p-

eration ofF on the Bloch vector must transform the vectinto another vector inside the Bloch sphere. This is just orather obvious condition. What is not as obvious is thatall states inside the Bloch sphere are allowable for the stem. In other words, the Bloch vector cannot accesspoints in the interior of the Bloch sphere. This is becausethe condition of complete positivity, related to the existenof a Kraus representation which we discuss in the followsection.

It has been shown@25# that the damping eigenvalues muobey the four inequalities

L11L22L3<1,

L12L21L3<1,

2L11L21L3<1,

2L12L22L3<1 ~49!

to guarantee complete positivity of the map. This is a necsary condition. These inequalities are the most general ci.e., they apply to the damping matrix no matter which cais considered. These inequalities are a consequence of thof equations given in the Appendix which involve the daming matrix elementsL i . There are five such equationwhich, taken with the first equation, are inner productsvectors. The fact that the inner product must be positsemidefinite leads to the inequalities above. For the speexample we present, they reduce to a more familiar foBecause for spontaneous emission we have 2T15T2, thefour inequalities become:

6coshS t

T3D<coshS t

T2D , ~50!

6sinhS t

T3D<sinhS t

T2D ,

which are satisfied if and only if

U 1

T3U<U 1

T2U. ~51!

In terms of the parametersM andN, this condition for com-plete positivity becomes

uM u<N11

2, ~52!

as expected. It is well known thatM andN are not indepen-dent and the amount of squeezing is limited by the numbephotons. A stricter inequality can be derived directly frothe matrix elementsci j of the Lindbladian form in Eq.~8!.This must be a non-negative matrix. Using the matrix ements of Eq.~30! one finds that thec matrix is non-negativeif M2<N(N11) where N is the photon number andM is thesqueezing parameter. In the case of equality, we havesqueezing. While the general inequalities of Eq.~49! are a

2-7

ind

tll

at tp-steic

, w

forro

rsht

umin

eaat

th

eom

bthh

t ofini-tedr totionre-

ult ofentve-tate.us

fiesse-er,, inisve

is-ch

ian

ex-


necessary condition for complete positivity, the stricterequality M2<N(N11) for the SVC is both necessary ansufficient.

It is worth pointing out that while contractions, withoushifts, will always map the Bloch sphere into the Bloch banot all contractions satisfy the inequalities~49!. Those thatdo not satisfy them do not have a Kraus representationhence are not completely positive maps. We shall see thacondition of complete positivity restricts the allowable ellisoids of the image. Because the Lindblad form of the maequation guarantees complete positivity, by Kraus’ theorthere must exist a decomposition with Kraus operators whacts on the system of interest alone. In the next sectiongive an explicit form of the Kraus operators.

C. The Kraus decomposition

For the special case of a qubit there can be at mostKraus operators. To see why this is true, the reader is refeto a lemma where it is shown that the minimum numberKraus operators is equal to the rank of a certain matrix@25#.It follows that the minimum number of Kraus operatoneeded to represent the map for the squeezed vacuum cnel is four. We do not show this, but rather use this factfind an operator-sum representation using the minimnumber of Kraus operators possible. We start by assumthat the four Kraus operators exist and that they are ralthough in general they are represented by complex mces. Next we expand all 232 matrices in Eq.~2! in the Paulibasis. If we let

Ak5mk0I 1mW k•sW 5Ak~mW !, ~53!

Ak†5mk0

! I 1mW k!•sW 5Ak

†~mW !!,

then Eq. ~2! becomes 12 @11si(a,c,d,d!)s i #5 1

2 @11t i(mW ,mW !,a,c,d,d!)s i #. Equating each coefficient insi andt i leads to a set of linear equations which underdetermine16 coefficients of the Kraus operators in Eq.~53! in the Paulibasis. Thus, the Kraus representation is not unique. Thements of the Kraus operators can be written as vector cponents

mj5S m1 j

m2 j

m3 j

m4 j

D , ~54!

where j 50,1,2,3. The set of linear equations that need tosatisfied is given in the Appendix. The stochastic map forsqueezed vacuum channel~47! has Kraus operators whiccan be realized in the following way:

A15m10I 1m13s3 ,

A25~m211m22!s†1~m212m22!s,

A35m31~s†1s!,

06231

-

,

ndhe

ermhe

uredf

an-o

gl,

ri-

e

le--

ee

A45m40I , ~55!

with constants given by

m1051

2

weq~12L3!

A12L12L21L3

,

m2151

2

weq~12L3!

A12L11L22L3

,

m2252i

2A12L11L22L3,

m1351

2A12L12L21L3,

m4051

2A11L11L21L32

weq2 ~12L3!2

12L12L21L3,

m3151

2A11L12L22L32

weq2 ~12L3!2

12L11L22L3. ~56!

The Kraus decomposition is not unique, so a different sefour Kraus matrices can represent the same map. The mmum number of Kraus operators, for the map construchere, is four, but one could just as well use more than fouproduce the map. This freedom in the Kraus decomposiis related to the many possible ways of performing measuments on the system. The system is decohered as a resbeing measured by the environment. The many differKraus operators correspond to the many different positioperator-valued measures which lead to the decohered sThis is why there are many different sets of possible Kraoperators which would result in the same mapF.

One can verify that this choice of Kraus operators satisthe condition in Eq.~3! so that the final density operator hatrace unity. This is a necessary condition for a tracpreserving map, to which we restrict ourselves. Howevone could consider maps that are not trace-preservingwhich case Eq.~3! becomes an inequality. Since the mapnonunital, i.e., it does not map identity into identity, we ha

F~ I !5S 11weq~12L3! 0

0 12weq~12L3!D . ~57!

This is due to the presence of the affine shift. Next, we dcuss what all this means geometrically in terms of the Blosphere.

VIII. THE GEOMETRICAL PICTURE

For a set of general Bloch equations and a Lindbladthere exists a 434 matrix of the form~20!. For the Lindbladequation we have considered here, this map can also bepressed, in terms of the 434 matrix, as

2-8

sd


FIG. 1. The effect of noise onthe set of qubit density operatorfor an atom in a squeezevacuum. Parameters areN51, M5A2, t50,0.5,1.

ane

sio

id

tot

Th

the

n

ed

stheuldchulde it

meochip-oid.

heen-andesetheex-h aheor

eseset

uumtwo

xi-ec-onlip-. As-

TD5S 1 0 0 0

0 L1 0 0

0 0 L2 0

~12L3!weq 0 0 L3

D . ~58!

This is a special case of Eq.~20! with

0,L3,L1,L2,1 ~59!

and affine shifts

t0150, t0250, t035weq~12L3!, 21,weq,0.~60!

From this we see that the components of the Bloch vectorcontracted. To guarantee that the Bloch vector is contaiwithin the Bloch ball the following condition must hold:

~L1u!21~L2v !21@L3w1~12L3!weq#2<1. ~61!

Furthermore, we see that the matrix~58! transforms theBloch sphereu21v21w251 into an ellipsoid inside theBloch ball. That is, the image of the set of pure state denmatrices under the stochastic map is given by the familyellipsoids

S u

L1D 2

1S vL2

D 2

1S w2weq~12L3!

L3D 2

51. ~62!

The shiftweq(12L3) determines the center of the ellipsowhile the eigenvalues$L1 ,L2 ,L3% define the lengths of theaxes. If we start with the set of pure state density operathat lie on the Bloch sphere, then as time progressesstates move onto the surface of a contracting ellipsoid.pure states have become mixed states. By Eq.~59!, each axisis unequally contracted, as shown in Fig. 1. Because ofsqueezed vacuum, thev component experiences very littl

06231

red

tyf

rshee

e

damping while theu component is rapidly damped, as showin Fig. 2. The explicit expression for the shift in terms ofweq

indicates that the ellipsoid is translated in the negativewdirection over time and settles into an equilibrium or fixpoint, as shown in Fig. 3.

The matrix of Eq.~20! is in the Pauli basis and describethe dissipative dynamics from the stationary viewpoint atcenter of the Bloch sphere. From this viewpoint, one woobserve a contracting ellipsoid moving away from the Blosphere center to some equilibrium point. The matrix coalso be represented in the damping basis, in which caswould be diagonal. This amounts to transforming to a frawhere, instead of being a stationary observer at the Blsphere center, one is moving along with the shifting ellsoid. In this case, one observes only a contracting ellipsThis is the geometrical picture of the damping basis.

If the effect of noise on all input states is known, then tleast noisy output states can be identified. The minimaltropy states are the states least affected by the noiseconsequently the least mixed states. Geometrically, thwould all be states whose distance from the surface ofBloch vector is a minimum. These states form a set oftreme points of the convex set of density operators. Sucset of points on the ellipsoid having minimal distance to tBloch sphere may consist of one or two points, a circle,the entire surface of the ellipsoid. For a given noise, thnearest points represent states of maximum purity in theof states affected by the noise. For the squeezed vacchannel, the set of minimal entropy states consists ofstates along the major axis of the ellipsoid.

The purest minimal entropy states are obtained by mamal squeezing of the reservoir. This results in the mostcentric ellipsoid which is stretched in one particular directiand places the two points along the major axis of the elsoid close to the Bloch sphere surface, as seen in Fig. 1the squeezing parameterM goes to zero, the ellipsoid be

-

FIG. 2. From a top view onecan see that theu component israpidly damped while thev com-ponent is slowly damped. Parameters are N51, M5A2, t50,0.5,1.

2-9

--


FIG. 3. A side view makes apparent the affine shift of the nonunital map. Parameters areN51, M5A2, t50,0.5,1.

loyees

they

gi

ordeisre

un

-alp

anitd

uE

pe

ra

oe

icut

sicfo

liath

gb

isy

onles

ates

or-ithion.ghgo-

ingor

oure-in-oneor-te is-on-

facts aanas

ter aram-

by-

finehecan

comes less elongated and the minimal entropy statestheir purity. WhenM is identically zero, the minimal entroppoints lie in a circle. Equivalently, asM approaches zero, thdamping for theu component of the Bloch vector approachthat for thev component.

If we had included the coherent dynamics as well asdissipative dynamics, the minimal entropy points on thelipsoid would be more mixed than without the coherent dnamics. The reason for this is that the ellipsoid would beto rotate and the minimal entropy points along thev compo-nent would rotate away from this axis where there is mnoise. Consequently, the minimal entropy points getgraded by being moved away from the axis with least noso that for this channel it is advantageous to keep cohedynamics suppressed.

Other noisy quantum channels have been introduced sas the depolarizing channel, the amplitude-damping chanand the phase-damping channel@12#. The depolarizing channel has all three components of the Bloch sphere equdamped and it is a unital channel. The set of minimal entropoints would lie on a sphere. The amplitude-damping chnel has two of the three components equally damped anda nonunital channel. It arises from the master equationscribing spontaneous emission as in Eq.~34!. The set of allstates moves on an ellipsoid which shifts toward the sopole. The phase-damping channel is the case described in~35!. It has two equally damped components, one undamcomponent, and is unital. It has two extreme points, onethe north pole and one at the south pole. Given any typenoise, the geometrical picture is a useful aid and allows fosimple analysis of the channel capacity. To transmit informtion over the channel it is ideal to use as input states thstates which result in the minimum amount of measuremerror.

IX. CHANNEL CAPACITY

A. Encoding classical information in qubits

One way to use a quantum channel is to send classinformation encoded in quantum states. Although quantinformation is being sent because qubits are sent throughchannel, these qubits are being used in an entirely clasway. A natural extension of the classical channel capacitya quantum channel was proposed by Holevo@10#. In thiscase, quantum states are used to transmit messages reacross some noisy quantum channel. Also known asproduct state capacity, because messages are sent usinsor products of the input states which comprise the alpha

06231

se

el--n

e-ent

chel,

lyy-ise-

thq.d

atofa-

sent

almhealr

blyeten-et,

the channel capacity for classical information over a noquantum channel has been defined as

C~F!5 max(pi ,r i )

H SFFS (i

pir i D G2(i

piS@F~r i !#J .

~63!

S is the von Neumann entropy defined asS(r)52Tr(r logr) and is the quantum analog of the Shannentropy. The maximum is taken over all possible ensembof input states. The input alphabet consists of a set of str i which are transmitted with probabilitypi . For the case ofthe SVC, the maximization is achieved using the twothogonal input states for which the damping is smallest wa uniform distribution over these states. With the conventalready chosen, this is thev component of the Bloch sphereThus, the optimal way to send classical information throuthe SVC is to prepare product states using the two orthonal input statesr05u0&^0uv and r15u1&^1uv , which lie inthe equatorial plane of the Bloch sphere, with correspondprobabilitiesp051/2 andp151/2 to encode messages. Fexample, r0

(1)^ r1

(2)^ •••^ r1

(N) , requiring N uses of thechannel, would represent theN-length classical bit string01•••1.

The noise operationF causes the output of these twstates to be nonorthogonal, mixed states. No single measment at the receiving end can perfectly determine whichput state was sent. There are two errors that can occur;error arises because of the nondistinguishability of nonthogonal states while the other arises because the stamixed. The first term in Eq.~63! takes into account the information loss due to the fact that the output states are northogonal while the second term takes into account thethe the output states are mixed. The Holevo capacity hanice geometrical interpretation. To see this we considerexample. For simplicity, let us assume that the SVC hmaximum squeezing and the input states are received af1 s pass through the channel. Then, using the same paeters as in Figs. 1–3, namely,N51, M5A2, andt51, wefind explicitly that the Holevo capacity isC(SVC)50.9320.1150.82 qubits per transmission. This is calculatedusingr05u0&^0uv andr15u1&^1uv as input states with corresponding probabilitiesp051/2 andp151/2 and calculat-ing the quantity inside the braces of Eq.~63!.

Note that the first term is present because of the afshift while the second term is due to a contraction of tBloch sphere. To see the various errors that can occur wewrite C(SVC)512(120.93)20.1150.82 qubits per trans-

2-10

atencthr oioomsr

otatno

tte

the

tinexs

hatu

a

t ifen-by

mto ay

d

ty

obe


mission. Now the first term is the number of qubits thcould be sent if there was no noise present. The secondis the error caused by the affine shift. This is the distafrom the center of the Bloch sphere to the center ofellipsoid. This error occurs because the states lose theithogonality. The last term is the error due to the contractof the ellipsoid. The contractions cause the states to becmixed and consequently there will be measurement errorthis kind. This term is a distance from the Bloch sphesurface to the surface of the ellipsoid. It is interesting to nthat the affine shift actually takes the maximally mixed stinto a state with less entropy. This can occur because gealized measurements can decrease entropy. Also, if no nwere present the capacity would beC(I )51 qubit per trans-mission, meaning that one error-free qubit can be transmiin one use of the channel.

B. Entanglement transmission

Another way in which a quantum channel can be useddoes not have a classical counterpart is for entanglemtransmission. In this case, one is interested in distribuparts of an entangled state to different locations. Forample, a source may generate Einstein-Podolsky-Ro~EPR! pairs, and one may be interested in sending one-of this EPR pair through some channel to a receiver. Na

ndngbnin-

th

bi

06231

trmeer-

ne

ofeeeer-ise

d

atntg-

enlf-

rally, noise will corrupt the transmitted state and lead todecrease in the entanglement of the joint state.

A channel has the capacity to transmit entanglemenafter passing through the channel there is any nonzerotanglement present in the joint state. It has been shownHorodecki et al. that any nonseparable bipartite systewhich has entanglement, however small, can be distilledsinglet form @26#. Therefore, if a channel can transmit anentanglement, it is a useful channel.N copies can be sent ana singlet state can be distilled.

Assuming the initial Bell state has the following densioperator representation:

rAB51

2 S 1 0 0 1

0 0 0 0

0 0 0 0

1 0 0 1

D , ~64!

the output state will be determined by the operation

rAB8 51A^ FB@rAB#, ~65!

which describes the process of sending one qubit to Bwhile Alice keeps her qubit intact. In matrix notation, thoutput state for the joint system is

rAB8 51

4 S 11weq1L3~12weq! 0 0 L11L2

0 12weq2L3~12weq! L12L2 0

0 L12L2 11weq2L3~11weq! 0

L11L2 0 0 12weq1L3~11weq!

D . ~66!

ring

oint

At the initial time the joint state is maximally entangled athe noise process should result in a decrease in the entament until some critical time when the joint state is separaand remains separable thereafter. Using the Peres criteriothe positivity of the partial transpose, one can determwhen the state becomes separable@27#. A necessary and sufficient condition for the output state to be nonseparablethat the partial transpose map be negative@28#. To check thepositivity of the partial transpose it suffices to examineeigenvalues of the operator given by

r8ABTB 51A^ TB@rAB8 #, ~67!

whereTB denotes the transpose of the state of Bob’s quThe four eigenvalues of the partial transpose matrix are

e151

4$11L32A~L12L2!21@weq~12L3!#2%,

e251

4$11L31A~L12L2!21@weq~12L3!#2%,

le-leof

e

is

e

t.

FIG. 4. The eigenvaluee3 is shown as a function of time fophoton numberN51. The three curves correspond to squeezparameterM50 ~dash-dotted!, M50.8Mmax ~dashed!, and M5Mmax. The larger the squeezing parameter, the longer the jstate is nonseparable.

2-11

e

id

ancaphder

heanfie

ai

ndthe

too

totuquont

nW

tica

dnt

noqueaatc

re

es

re,thetionini-

ns-o—

singan-ingome

toto thean-umdon-thectmayon

c-byis-

sauset ofec.torsre


e351

4$12L32A~L11L2!21@weq~12L3!#2%,

e451

4$12L31A~L11L2!21@weq~12L3!#2%. ~68!

Initially, the eigenvalues have valuese151/2, e251/2, e3521/2, ande451/2, while in the steady state they become151/6, e252/6, e351/6, ande452/6. We find that thenonseparability is determined solely by the eigenvaluee3.The transmitted EPR state remains nonseparable provthat

12L3,A~L11L2!21@weq~12L3!#2. ~69!

There is some critical time when the eigenvalue is zerothen remains positive thereafter. The value of this crititime depends on the parameters of the reservoir. As theton number of the reservoir increases the critical timecreases. One would expect this to be the case since theervoir is more noisy. It is not as obvious what effect tsqueezing of the reservoir has on the entanglement becthe squeezing results in a trade-off of more noise in ocomponent and a decrease in another component. Wethat a squeezed reservoir results in a longer entanglemtime for the inital maximally entangled state. The critictime is longer for a maximally squeezed vacuum. Thisshown by the solid curve in Fig. 4. In both the case of seing product states and for distributing EPR states,squeezing parameterM is able to enhance the capacity of thchannel.

X. CONCLUSION

In this paper we provided a method for calculating a schastic map for any quantum Markov channel. Our methuses a special damping basis of left and right eigenoperaThis basis provides a natural way to generate noisy quanchannels from a quantum optical approach. This techniallows one to calculate the stochastic map and from thiscan then determine a set of Kraus operators which definenoisy quantum channel.

We used this method to calculate explicitly a noisy quatum channel we called the squeezed vacuum channel.showed the relationship between a set of quantum opBloch equations and the damping basis. Some known qutum channels such as the amplitude-damping channel andepolarizing channel arise from this set of Bloch equatioThese quantum Markov channels are special cases ofsqueezed vacuum channel. The channel we derived is aunital stochastic map which is characterized by three unedamping eigenvalues. By using the damping basis, we wable to find a set of Kraus operators for the stochastic mFrom this, the effect of noise on the set of input states winterpreted geometrically. The Bloch picture was usedstudy the effect of noise present in the channel and theherent dynamics was considered in addition to the incohedynamics.

The procedure to calculate the channel capacity requir

06231

ed

dlo--es-

useendnt

ls-e

-drs.mee

he

-e

aln-thes.hen-alrep.s

oo-nt

a

maximization over all input states. Using the Bloch pictuwe were able to determine the channel capacity forsqueezed vacuum channel to transmit classical informain quantum states. We found that the channel has two mmal entropy points which should be used to optimally tramit information. A geometrical interpretation of the Holevcapacity was given and used to identify two types of errorsthose arising from nonorthogonal states and those arifrom mixed states. We also discussed the ability of this chnel to distribute EPR states. We found that, after sendone-half of the EPR state through the channel, there is scritical time after which the state becomes separable.

This paper shows that with thea priori knowledge of theeffect of noise given by a Lindblad form, one can chooseencode messages using the pure states that are closestfinal states of minimum entropy. The squeezed vacuum chnel is a more general noisy channel derived from quantoptical two-level systems. Unlike previously introducechannels, it has unequal damping eigenvalues and it is nunital. We found that channel capacity is enhanced bysqueezing parameterM, whether it is used to send produstates or used to transmit EPR states. This channelprove useful as a testing ground for future conjecturesquantum channel capacities.

ACKNOWLEDGMENTS

We thank Dr. G. H. Herling for discussions and corretions to the final draft. This work was partially supportedKBN Grant No. 2PO3B 02123 and the European Commsion through the Research Training Network QUEST.

APPENDIX

For the Hilbert space of 232 matrices, one may alwayrepresent the quantum channel using four or fewer Kroperators. To find a representation, one can construct a ssimultaneous equations via the prescription given in SVI B. The squeezed vacuum channel has Kraus operawhich must satisfy the following set of equations. There afour for the shifts:

m0!m01m1

!m11m2!m21m3

!m351, ~A1!

m0!m11m0m1

!1 i ~m2!m32m2m3

!!5t01,

m0!m21m0m2

!2 i ~m1!m32m1m3

!!5t02,

m0!m31m0m3

!1 i ~m1!m22m1m2

!!5t03.

For the coefficients ofa, c, d, andd!:

i ~m0!m22m0m2

!!2~m1!m31m1m3

!!50,

m0!m01m1

!m12m2!m22m3

!m31 i ~m1!m21m1m2

!!

2~m0!m32m0m3

!!5L1 ,

2-12

int

reg

thol-ily

iven


m0!m01m1

!m12m2!m22m3

!m32 i ~m1!m21m1m2

!!

1~m0!m32m0m3

!!5L1 ,

m0!m12m0m1

!2 i ~m2!m31m2m3

!!50,

m0!m02m1

!m11m2!m22m3

!m32 i ~m1!m21m1m2

!!

2~m0!m32m0m3

!!5L2 ,

m0!m02m1

!m11m2!m22m3

!m31 i ~m1!m21m1m2

!!

1~m0!m32m0m3

!!5L2 ,

m0!m02m1

!m12m2!m21m3

!m35L3 ,

m0!m12m0m1

!1m1!m31m1m3

!

1 i ~m0!m22m0m2

!1m2!m31m2m3

!!50,

m0!m12m0m1

!2m1!m32m1m3

!

1 i ~2m0!m21m0m2

!1m2!m31m2m3

!!50.

~A2!

And to satisfy the conditionAkAk†5I :

m0!m31m0m3

!2 i ~m1!m22m1m2

!!50. ~A3!

The set of equations leads to restrictions on the dampeigenvalues as in Eq.~49!. As an example of the explici

y

, a

ev

. A

-

o

06231

g

construction of Kraus matrices, we will consider the puphase decay case of Eq.~35!, which is the phase-dampinchannel. For this case,L15L2 , L350, and t015t025t0350. Any solution of the set of linear equations along withese conditions will give a Kraus representation. The flowing solution for elements of the Kraus matrices is easfound to bem250W , m350W , and

m05S m10

0

0

0

D , m35S 0

0

0

m43

D ~A4!

with

m105A11L

2, m435A12L

2. ~A5!

The phase-damping channel has a Kraus representation gby

A15A11L

2I , ~A6!

A25A12L

2s3 . ~A7!

th.

d

el

pl.

v.

A

@1# C. H. Bennett and P. W. Shor, IEEE Trans. Inf. Theory44,2724 ~1998!.

@2# W. H. Zurek, Phys. Today44~10!, 36 ~1991!.@3# C. E. Shannon, Bell Syst. Tech. J.27, 379 ~1948!.@4# T. Cover and J. Thomas,Elements of Information Theor

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