a r X i v :q u a n t -p h /0403074v 1 9 M a r 2004
Purity and State Fidelity of Quantum Channels via Hamiltonians
Paolo Zanardi
Institute for Scienti?c Interchange (ISI),Villa Gualino,Viale Settimio Severo 65,I-10133Torino,Italy and Department of Mechanical Engineering,Massachusetts Institute of Technology,Cambridge Massachusetts 02139
Daniel A.Lidar
Chemical Physics Theory Group,Chemistry Department,University of Toronto,
80St.George Street,Toronto,Ontario M5S 3H6,Canada
We associate to every quantum channel T acting on a Hilbert space H a pair of Hamiltonian operators over the symmetric subspace of H ?2.The expectation values of these Hamiltonians over symmetric product states give either the purity or the pure state ?delity of T .This allows us to analytically compute these measures for a wide class of channels,and to identify states that are optimal with respect to these measures.
PACS numbers:
I.INTRODUCTION
The study of open quantum systems [1]is of interest in ?elds as diverse as quantum information science [2],quan-tum control [3],and foundations of quantum physics [4].Let T ∈CP(H )be a completely positive trace-preserving quan-tum map i.e.,a channel over the ?nite-dimensional quantum state-space H .The channel T has a (non-unique)Kraus oper-ator sum representation [5]
T (X )=
i
A i X A ?i ,
[X ∈End(H )](1)where the Kraus operators A i satisfy the constraint i A ?
i A i =1
1,which guarantees preservation of the trace of a state (density operator)X =ρ.A fundamental prop-erty of a state is its purity p [ρ]=Tr(ρ2).States are called pure iff p =1and mixed if p <1.In the paradigmatic scenario of open quantum systems,a state starts out as pure,ρ=|ψ ψ|,and is then mapped,e.g.,via the interaction with an environment,to a mixed state by the action of a channel T :p [T (ρ)]=Tr[T 2(|ψ ψ|)]<1.In this case we say that the state |ψ has been decohered by the channel.A typical goal of,e.g.,quantum information processing,is to maximize the purity of a state that is transmitted via some channel T .To this end a variety of decoherence-reduction techniques have been developed,such as quantum error correcting codes (QECCs)[6,7,8,9]and decoherence-free subspaces (DFSs)[10,11].In this work we are interested in the intrinsic purity of quan-tum channels:
De?nition 1The purity of the channel T over the subspace C ?H is
P (T,C ):=min |ψ ∈C
Tr[T 2(|ψ ψ|)].
(2)
Minimization is required since we must consider the worst-case scenario.We invoke subspaces in our de?nition since we know from the theory of QECCs and DFSs that it is possible to encode quantum information in a manner that maximizes purity by restricting to a subspace.In particular:
De?nition 2If P (T,C )=1we say that that C is a decoherence-free subspace with respect to T ,in short a T -DFS.
In many cases it will not be possible to ?nd a T -DFS.A
central question we shall be concerned with here is the char-acterization of those states that optimally approximate a T -DFS,i.e.,those states for which P (T,C )is as large as possi-ble.Thus:
De?nition 3The optimal purity of T is
P (T ):=max C?H
P (T,C )
(3)
Note that P (T )=1?the set of T -DFSs is non-empty.However,this situation is rather rare and generally requires that there be a symmetry in the system-environment interac-tion.Associated with this symmetry is a conserved quantity:quantum coherence.This in turn leads to the preservation of quantum information.Here we wish to depart from the no-tion of a strict symmetry and explicitly consider the situation where one can only expect optimal,as opposed to ideal pu-rity.However,the optimization problem de?ned by P (T )is a hard one,since it involves a search over all possible subspaces C ?H ;the number of such subspaces grows combinatorially in the dimension of H ,which itself may be exponential in the number of particles,in a typical quantum information process-ing application.Moreover,even if one restricts the problem to the computation of P (T,C )(for a given,?xed subspace),one is still faced with a complicated-looking functional.
In this work we focus on the the computation of P (T,C )and we associate a Hamiltonian to each channel .This “chan-nel Hamiltonian”is a mathematical trick,rather than a physi-cal Hamiltonian.But,as we shall show,this has the advantage that it allows us to cast the purity problem into the familiar framework of computing eigenvalues of Hermitian operators.In addition,we show that our channel Hamiltonian leads to an elegant physical (re-)interpretation of the channel purity in terms of the expectation value of the SWAP operator.
Our work is also related to questions about channel capac-ity;indeed recently it has been shown that multiplicativity of generalized maximal purities implies additivity of the mini-mal output entropy of the quantum channel.The latter in turn is equivalent to the additivity of the Holevo channel capacity [12].
We introduce the ?rst channel Hamiltonian in Section II.We then derive a number of properties and bounds on the pu-
2 rity based on this formalism in Section III.We then devote
Section IV to a number of examples designed to illustrate our
formalism,and derive some interesting properties for a class
of channels.In Section V we derive an alternative interpre-
tation of the expression for the channel purity,in terms of a
dual map.It turns out that the same methods we introduce
for the channel purity also apply to the pure state?delity of
the channel.In particular,we can introduce a second channel
Hamiltonian to this end.This is addressed in Section VI.We
conclude in Section VII.
II.A HAMILTONIAN OPERATOR FOR QUANTUM
CHANNELS
Associated to the channel T we de?ne an operator over
H?2:
De?nition4The channel purity-Hamiltonian is:
?(T):= ij??ij??ij,?ij=A?i A j.(4) (We shall refer to?(T)simply as the“channel Hamilto-
nian”until our discussion of the pure state?delity in Sec-tion VI.)It follows immediately from??ij=?ji that?(T)is a symmetric,Hermitian operator.Thus?(T)has the status of
a Hamiltonian over H?2.Moreover,?(T)is independent of the particular Kraus operator sum representation chosen for T:
all possible operator sum-representations of T are obtained by considering new Kraus operators of the form A′i= j U ij A j where the U ij’s are the entries of unitary matrix.By inserting this expression into the de?nition(4)one can explicitly check
that?(T)is invariant.
We now come to our key result:a representation of the pu-
rity of quantum channels as the expectation value of the chan-nel Hamiltonian:
Proposition0For every quantum channel T and subspace C one has the identity
P(T,C)=min
|ψ ∈C
ψ?2|?(T)|ψ?2 .(5) Proof.One has
Tr T2(|ψ ψ|)= ij ψ|A?j A i|ψ ψ|A?i A j|ψ
= ij| ψ|A?j A i|ψ |2=Tr[|ψ ψ|?2A?j A i?A?i A j]
=Tr[|ψ ψ|?2?(T)].(6)
Equation(5)now follows by taking the minimum over|ψ ∈C.Lemma1Let T be unital[T(11)=11].Then?|ψ ∈H we have:
?(T)|ψ ?2 ≤1.(8) Proof.Let p ij:=||A?j A i|ψ ||.One has the following nor-malization condition
ij
p2ij= ij ψ|A?i A j A?j A i|ψ = i ψ|A?i A i|ψ =1,
where in the?rst(second)equality we used the unitality(CP-
map)condition j A j A?j=11( i A?i A i=11).Now
?(T)|ψ ?2 = ij A?j A i|ψ ?A?i A j|ψ
≤ ij A?j A i|ψ A?i A j|ψ = ij p ij p ji
≤ ij p2ij=1,(9)
where in the last line we used the Cauchy-Schwartz inequality for the Hilbert-Schmidt product of matrices,
ij
p ij p ji=Tr P2= P,P? ≤ P P?
= P 2= ij p2ij.
3 i)If?|ψ ∈C and?i it holds that:A i|ψ =
αi U|ψ ,A?i|ψ =α?i U?|ψ ,where U is unitary,then
|ψ ?2∈H?.
ii)Let T be unital.Then C is a T-DFS?C?2?H?.
iii)T-DFS?the?rst inequality in Eq.(7)is an equality.
Proof.i)Notice?rst that from the CP-map condition,
i A?i A i=11,it follows that i|αi|2=1.Now for
|ψ ∈C one has that ψ?2|?(T)|ψ?2 = ij|αiαj|2=
( i|αi|2)2=1.
ii)(?)If C is a T-DFS then min|ψ ∈C ψ?2|?(T)|ψ?2 =
1.But from the Cauchy-Schwartz inequality and Lemma1
above one has that ψ?2|?(T)|ψ?2 ≤1(?|ψ )and
the equality holds iff?(T)|ψ?2 =|ψ?2 (?|ψ ∈C).
Now,if|Ψ is in the symmetric part of C?2,one has that
|Ψ =Π+(C)|Ψ =α(C) C|ψ?2 ψ?2|Ψ [whereα(C):=
dim C(dim C+1)];therefore
?(T)|Ψ =α(C) C?(T)|ψ?2 ψ?2|Ψ
=α(C) C|ψ?2 ψ?2|Ψ .(10)
It follows that|Ψ ∈H?.
(?)If?|ψ ∈C it holds that|ψ ?2∈H?1,then C is clearly
a T-DFS,i.e.,P(T,C)=1.A fortiori this holds if all the
elements of the symmetric part of C?2are in H?.
iii)We have just seen that?|Ψ =|ψ ?2such that|ψ ∈C
one has Ψ|?(T)|Ψ =1.By integrating over|ψ one obtains
that the average[leftmost part of Eq.(7)]coincides with the
minimum[middle term in(7)].
p0p iσ0σi=√
p j p i?ijkσk(i=1,2,3);?ii=p i11,(i=0,...,3).It
then follows that
?(T)=
3
i=0
αiσi?σi(11)
whereα0= 3i=0p2i,αk=2(p0p k+p i p j)(i=j= k,k=1,2,3).Note that 3i=0αi=( 3i=0p i)2=1,that α0∈[1/4,1],andαk∈[0,1/2],(k=1,2,3).The eigen-states of?(T)are the Bell states|ψ? =(|01 ?|10 )/√
2,|ψ+ =(|01 +|10 )/√2}(triplet).Their respec-tive eigenvalues are2α0?1and{1?2α1,1?2α3,1?2α2}. Further note that Spec?(T)?[?1/2,1]and that?(T)in the triplet sector is a positive operator.The triplet sector is symmetric,while the singlet is anti-symmetric.From Eq.(7) we thus know that the minimal eigenvalue in the triplet sector provides a lower bound on the purity:
P(T)≥1?2max
i=1,2,3
αi≥0(12)
(of course in this single-qubit example there are no non-trivial subspaces:C=H).In this general case we cannot directly determine the actual purity or?nd the corresponding maxi-mally robust state(s),since the Bell triplet-states are not prod-uct states.To?nd the optimal purity states in such a case one has to resort to other optimization techniques.However,in certain special cases the eigenstates of?(T)will be product states,whence our method directly yields the optimally ro-bust states.For instance,consider the case p0=p1=1/2 and p2=p3=0;one?nds?(T)=(11+σx?σx)/2. In this case|ψ? ,|φ? are degenerate,as are|ψ+ ,|φ+ . We then?nd,respectively,the symmetric product eigenstates [(|0 ?|1 )/
√2]?2,both with eigen-value1.The states|± :=(|0 ±|1 )/
√
pαpβσασβ?σασβand
?= α=0,x,y,z p2α(I?I)?2+ α=β=γpαpβ(σγ?σγ)?2.(14)
This example can be solved directly by observing that each of the Bell states has eigenvalue+1or?1under the action of σγ?σγ,whence the purity is one.Thus the Bell states are T-DFSs.Notice that this result appears to be related to the communication problem for channels with correlated noise studied in Ref.[17].
One can also?nd the minimal purity states by differenti-ating ? :=?2 ψ|?(T)|ψ ?2as a function of expansion
4
parameters of|ψ over the Bell states.This yields ? min=
α=0,x,y,z p2α,and the corresponding minimally robust set of
states are superpositions of pairs of Bell states with arbitrary
phases:
|ψ1 =e iα12(|01 ?|10 )
|ψ2 =e iα22(|01 +|10 ).
This channel thus has the interesting property that the maxi-
mally entangled Bell states are more robust than any separable
(pure)state.
Example3Amplitude damping.
Let T(ρ)=|0 0|,?ρ∈S(
C|0 .
Example4Projective measurements.
Let T(ρ)= iΠiρΠi,ΠiΠj=δijΠi, iΠi=11.Then
?ij=δijΠi,from which
?(T)= iΠi?Πi.(16)
If H i:=ImΠi then H?2
i?H?,i.e.,from Proposition2all
the eigenvectors of theΠi’s are1D T-DFSs.The maximum
eigenvalue of?(T)is1;this follows from Ψ|?(T)|Ψ ≤
1(?|Ψ ).The latter inequality results from the following ar-
gument:11?2=( iΠi)?2= ijΠi?Πj=?(T)+
i=jΠi?Πj.The last term is a non-negative operator(sum
of products of non-negative operators),whence11??(T)≥0.
Taking the expectation value of the last inequality with respect
to|Ψ proves the bound above.
In the following we use the operator norm A
∞:=
max |ψ | =1 A|ψ .We shall write A for simplicity. Example5Unitary mixture of a group representation.
This is a rather general and quite important example,which includes Examples1,2above.Let
T(ρ)= g p g U gρU?g(17)
where g→p g is a probability distribution over the group G={g}and g→U g a unitary representation of G.One?nds ?gh=√p g p h U g?1h;thus
?(T)= k∈G q k U k?U k?1,(18)where q k:= g p g p gk?1is also a probability distribution. If|ψ ∈H is a G-singlet,i.e.,U g|ψ =|ψ (?g∈G) then|ψ ?2∈H?,i.e.,all the G-singlets are1D T-DFSs. Here again,the maximum eigenvalue of?(T)is1:Indeed, it is easy to see that ?(T) ≤1: ?(T) ≤ k q k U k?U k?1 = k q k=1.
As a particular instance of this kind of channel let us con-sider an N-qubit case with the U k’s generating an Abelian subgroup of G the Pauli group(all tensor products of Pauli matrices on N qubits),as was the case in Examples1,2 above.The set of G-singlets is now given by the the stabi-lizer of G[denoted Stab(G)],i.e.,the subspace generated by the|ψ such that U k|ψ =|ψ (?k).Since U k=U?k one ?nds immediately that elements of the form|ψ ?2,where |ψ ∈Stab(G),are eigenvectors of?(T)with maximum eigenvalue k q k.These states also play
the role of code-words of stabilizer QECCs[8].We thus see that,in this ex-ample,the stabilizer-QECC codewords are maximally robust, though no active error correction is assumed.
In fact,the connection to quantum error correction can be made more general:the formalism developed so far allows us to establish an intriguing identity for the purity of states belonging to a QECC C for the CP-map T:ρ→ i A iρA?i. If|ψα ,|ψβ ∈C then the error-correction condition is:
ψα|A?i A j|ψβ =c ijδαβ,(19)
where the matrix c ij is Hermitian,non-negative,and has trace one[7].For non-degenerate codes c ij has maximal rank.Let us now consider states of the form|ψα ?2(|ψα ∈C).From Eq.(4)and the error correction condition one has that
P(T,C)= ψ?2
α|?(T)|ψ?2α =
ij
| ψα|A?i A j|ψα |2 = ij|c ij|2=Tr(c2).(20)
Viewing c as a state(density operator),we have thus found that the purity of the channel acting on the codewords of C is just the purity of the“state”c associated to the code itself. For example,for DFSs c is simply a rank one matrix with unit trace[14],whence Tr c2=1and the maximum eigenvalue condition is readily recovered.As a more interesting exam-ple,consider a CP-map T with A i=√
p i,and from the CP-condition r R?r R r=11we?nd r|λri|2=p i.As-sume for simplicity that there is a unique recovery operator per error,i.e.,λri=λiδri,λi=0?i(this is an example of a non-degenerate code).Then|λi|2=p i and c=diag(p i); it follows that the purity over such a QECC associated to T is simply given by i p2i.
5
V .THE DUAL REPRESENTATION
We now develop an alternative representation of the
channel-Hamiltonian,which is useful for the derivation of several additional results,and sheds new light on the physi-cal interpretation of the channel purity.
De?nition 6The dual T ?of a CP-map T [see Eq (1)]
is:
T
?(X )=
i A ?i XA i .
Proposition 3Let S be the SWAP operator (de?ned above).Then:
?(T )=T ?2
?(S )S.
(21)
We give two different proofs.
Proof.a)
T ?2
?(S )S = ij
(A ?
i ?A ?j )S (A i ?A j )S
= ij (A ?
i ?A ?j )(A j ?A i )
=
ij
A ?i A j
?
A ?j A i
=?(T )(22)
b)By writing the SWAP operator explicitly as S = lm |m l |?|l m |and applying T ?2
?one obtains: lm,ij A ?
i |m l |A i ?A ?j |l m |A j .Then the proof follows by explicity comparing the matrix elements of the latter operator times S with the ones of ?(T ).
The following corollary contains a general derivation,based on the dual representation of ?(T ),of a fact that was already proved for speci?c examples in Section IV.Corollary 2 ?(T ) ≤1.
Proof.One has ?(T ) = T ?2?(S )S ≤ T ?2
?(S ) S ≤ T ?2?(S ) .Since T ?2
?is the dual of a CP map elements smaller (greater)than the identity (minus the identity)are mapped onto elements smaller than the identity.Since ?11≤
S ≤11one has ?11≤T ?2
?(S )≤11.This relation implies in particular that the maximum eigenvalue of the Hermitian
operator T ?2
?(S )is smaller than one.Since this maximum
eigenvalue coincides with the · ∞norm of T ?2
?(S )the in-equality is proved.
Tr[T ?2(|ψ ψ|)]ψ
=
1
d (d +1)
=
1
In other words,the Haar-average purity of a channel T is
given by the expectation value of ?(T )over the maximally mixed state Π+(H )=1/d (d +1)(11+S )over the symmetric
subspace of H ?2
.
Corollary 3Using Eq.(24)one can get the Haar-averaged purities of the channels considered above:
(i)One qubit depolarizing channel:(1+2α0)/3.(ii)Amplitude damping channel:(1?p )2+p 2+2p (1?p )/d .(iii)Projective measurements:[d + i (TrΠ
i )2]/d (d +1).(iv)Unitary mixing:[d + g ∈G q g |Tr U g |2]/d (d +1)?From (iii)and (iv)it follows that:
(a)One-dimensional projective measurements achieve the minimal average purity,of 2/(d +1).
(b)For unitary mixing and assuming a Haar-uniform distri-bution (all q g equal,i.e.,the fully depolarizing channel),mini-mal purity is obtained for U g ’s in a G -irrep.Indeed,one has in general that (1/|G|) g ∈G |Tr U g |2
= J n 2J where n J is the multiplicity of the J -th G irrep [16].The minimum is clearly achived when just one irrep appears,i.e.,the irreducible case.Before concluding this section we would like to point out that the formula Ψ|?(T )|Ψ =Tr ST ?2
(|Ψ Ψ|) allows us to give an operational meaning to the operator ?,and in the particular case in which |Ψ =|ψ ?2,to the purity of T (|ψ ψ|).Indeed,this expectation value of ?(T )is noth-ing but the expectation value of the observable S in the state T ?2(|Ψ Ψ|).The latter state can in turn be viewed as the result of an action of the channel on a pair of,possibly entan-gled,input states from H .
VI.PURE STATE FIDELITY OF A CHANNEL
We now show how many of the techniques introduced above for the channel purity carry over to the (simpler)prob-lem of calculating the
De?nition 7Pure state ?delity:
F (T,|ψ ):= ψ|T (|ψ ψ|)|ψ .
(26)
6 Proposition5
(i)
F(T,|ψ )= ψ?2|?1(T)|ψ?2 (27)
where?1(T)=(11?T?)(S)S.
(ii)
?1(T)= i A i?A?i.(28)
(iii)
Tr[?1(T)+S(11?T(11))].(29)
d(d+1)
In particular,for a unital map one has average pure state?-
delity given by[d+ i|Tr A i|2]/d(d+1).
Proof.
(i)
F(T,|ψ )= ψ|T(|ψ ψ|)|ψ =Tr[|ψ ψ|T(|ψ ψ|)]
=Tr[S|ψ ψ|?T(|ψ ψ|)]
=Tr S(11?T)|ψ ψ|?2
=Tr (11?T?)(S)|ψ ψ|?2
= ψ?2|(11?T?)(S)S|ψ?2 .
(ii)
(11?T?)(S)S= i(11?A?i)S(11?A i)S
= i(11?A?i)(A?i?11)
= i A i?A?i.
(iii)
.
d(d+1)
Notice that the second term inside the square brackets is,
for unital maps,simply Tr S=d.
7
[2]M.A.Nielsen and I.L.Chuang,Quantum Computation and
Quantum Information(Cambridge University Press,Cam-bridge,UK,2000).
[3]P.W.Brumer and M.Shapiro,Principles of the Quantum Con-
trol of Molecular Processes(Wiley,2003).
[4]A.Peres,Quantum Theory:Concepts and Methods(Kluwer,
Dordrecht,1998).
[5]K.Kraus,States,Effects and Operations,Fundamental Notions
of Quantum Theory(Academic,Berlin,1983).
[6]P.W.Shor,Phys.Rev.A52,2493(1995);A.M Steane,Phys.
Rev.Lett.77,793(1997).
[7]E.Knill and https://www.doczj.com/doc/e017117938.html,?amme,Phys.Rev.A55,900(1997).
[8]D.Gottesman,Phys.Rev.A54,1862(1996).
[9]For a review see A.M.Steane,in Introduction to Quantum Com-
putation and Information,edited by H.K.Lo,S.Popescu and T.P.Spiller(World Scienti?c,Singapore,1999),pp.184–212.
[10]P.Zanardi and M.Rasetti,Phys.Rev.Lett.79,3306(1997);
Mod.Phys.Lett.B11,1085(1997).
[11]D.A.Lidar,I.L.Chuang,and K.B.Whaley,Phys.Rev.Lett.81,
2594(1998);For a review see D.A.Lidar and K.B.Whaley, in Irreversible Quantum Dynamics,V ol.622of Springer Lec-ture Notes in Physics,edited by F.Benatti and R.Floreanini (Springer,Berlin,2003),p.83,eprint quant-ph/0301032. [12]P.W.Shor,quant-ph/0305035,to appear in Comm.in Math.
Phys..
[13]P.Zanardi,C.Zalka,and L.Faoro,Phys.Rev.A62,030301(R)
(2000).
[14]D.A.Lidar,D.Bacon and K.B.Whaley,Phys.Rev.Lett.82,
4556(1999).
[15]Here is a proof:Tr(SA?B)= ij,lm S ij,lm A l,i B m,j= ij,lmδi,mδj,l A l,i B m,j= ij A j,i B i,j=Tr AB.
[16]J.F.Cornwell,Group Theory in Physics,V ol.I and II of Tech-
niques of Physics:7(Academic Press,London,1984). [17]C.Macchiavello and G.M.Palma,Phys.Rev.A65,050301
(2002).
[18]M.Nielsen,Phys.Lett.A303,249(2002).
【Fidelity：一个基金巨头的失败总记录】作者：裴明宪 从1977年到1990年，历史上最伟大的共同基金经理彼得-林奇(Peter Lynch)在Fidelity(富达投资)的麦哲伦(Megallen)基金完成了传奇般的旅程语出Fidelity官方网站)。对于许多人来说，彼得-林奇就意味着Fidelity，他让Fidelity赢得了聪明的选股者(Intelligent Stock Pickers)的威名。在那个时代，Fidelity几乎代表着整个基金管理业。没有一家银行或证券公司敢于藐视这个名字。没有一个投资者不对这个名字感到亲切。现在一切都变了。 麦哲伦基金早已变成了财经记者和基金分析师们嘲笑的对象。根据几年前的一个研究报告，这只基金与标准普尔500指数的相关性居然达到0.99，而它收取的管理费却是任何一只标准普尔500指数基金的10倍以上。The Barron's周刊的编辑送给麦哲伦基金一个绰号：昂贵的指数基金。现在，经历了大换血的麦哲伦基金的资产仍然高达440亿美元之巨，但已经比它在高潮时期的资产跌去一半左右(为了防止这只基金过于庞大难以管理，Fidelity的领导层已经禁止了一切新投资的进入)。 麦哲伦基金的困境只是Fidelity整体败局的一部分。毫无疑问，麦哲伦这只传奇基金的失败给Fidelity的威名蒙上了一层阴影，但它的其他股票基金表现也算不上优秀。除了America ContraFund等少数王牌基金，Fidelity的大部分积极管理股票基金都没有发挥出足够的潜力，其中许多不但长期落后于大盘，还落后于主要的竞争对手。在传统积极管理领域，T.Rowe Price，American Fund和Franklin等竞争对手已经迎头赶上，它们拥有比较优秀的综合回报率和比较低的费率，尤其是T.Rown Price 以不收费基金(No-load Fund)给Fidelity的高费率基金带来了严重冲击。在工程型积极管理领域，Fidelity几乎一事无成，眼睁睁地看着Barclays Global Investors夺走这个新兴市场的主导权。 在消极管理领域，Fidelity已经输掉了它好不容易赢得的一席之地。在2000年之前，它一直试图与强大的指数投资者Vanguard竞争，但现在它的最大敌人变成了Barclays和State
User Manual Version 1.0 Revision Date: 08/12/2013 Product name: I-5? 2X High-Fidelity Master Mix Cat #: I5HM-200, I5HM-5000, I5HM-OEM Description: I-5? High-Fidelity DNA Polymerases is an ultra-high fidelity and high processivity enzyme. It produces the most accurate copies of DNA, and performs at an ultra-high rate. I-5? is perfect for applications like cloning in vitro amplified material for protein expression, SNP analysis by sequencing, and high-specificity PCR. Convenient and Usage: I-5? 2X High-Fidelity Master Mix has a buffer system that is designed to offer maximum performance of the I-5? High-Fidelity DNA Polymerase. It offers the highest fidelity and incredibly robust performance. The convenient master mix allows reaction to be set up in room temperature and only requires the additions of primers and DNA template for the amplification. Application: - High-specificity PCR amplification - High-Throughput PCR - Various Cloning technologies - Difficult Amplification Recommended Storage Conditions: -20°C.
方梦之主编：《译学词典》，上海外语教育出版社，第29页 功能翻译理论functionalist translation theory 又称“功能目的论”（Skopos theory）。1971年，德国的莱斯（K. Reiss）首先提出“把翻译行为所要达到的特殊目的”作为翻译评价的新模式。1984年她在与费米尔（H. J. Vermeer）合写的General Foundation of Translation Theory一书中声称：译者在整个翻译过程中的参照系不应是“对等”翻译理论所注重的原文及其功能，而应是译文在译语文化环境中所预期达到的一种或若钟交际功能。20世纪90年代初，德国学者克利斯蒂安·诺德（Christiane Nord）进一步拓展了译文功能理论。她强调译文与原文的联系，但这种联系的质量与数量由译文的预期功能确定。这就是说，根据译文语境，原文中的哪些内容或成分可以保留，哪些需调整或改写，该由译文的预期功能确定。 功能目的理论的两项基本原则是：1. 翻译各方面的交互作用受翻译目的所决定；2. 目的随接受对象的不同而变化。按照这两项原则，译者可以为了达到目的而采用任何他自己认为适当的翻译策略。换句话说，目的决定方式（The end justifies the means）。 作为受文化制约的语言符号，原文语篇和译文语篇受到各自交际环境的影响，译文功能与原文功能可相似或保持一致，也可能完全不同。根据不同的语境因素和预期功能，选择最佳的处理方法，这是功能翻译理论比以对等为基础的翻译理论或极端功能主义的翻译理论更为优越之处。翻译功能理论指导下的翻译方法表现出较大的灵活性，较高的科学性和易操作性。Toury 把“功能目的论”看作是“译文文本中心论”的翻版。 Skopos theory (plural Skopos theories) 1.(translation studies) The idea that translating and interpreting should primarily take into account the function of both the source and target text. o1995, Paul Kussmaul, Training The Translator, John Benjamins Publishing Co, p. 149: The functional approach has a great affinity with Skopos theory. The function of a translation is dependent on the knowledge, expectations, values and norms of the target readers, who are again influenced by the situation they are in and by the culture. These factors determine whether the function of the source text or passages in the source text can be preserved or have to be modified or even changed. Introduction to the Skopos Theory The Skopos theory is an approach to translation which was put forward by Hans Vemeer and developed in Germany in the late 1970s and which oriented a more functionally and socioculturally concept of translation. Translation is considered not as a process of translation, but as a specific form of human action. In our mind, transla tion has a purpose, and the word “Skopos” was from Greek. It?s used as the technical term for the purpose of the translation.
WIFI全称Wireless Fidelity，又称802.11b标准，它的最大优点就是传输速度较高，可以达到11Mbps，另外它的有效距离也很长，同时也与已有的各种802.11 DSSS设备兼容。今夏最流行的笔记本电脑技术——迅驰技术就是基于该标准的。 IEEE 802.11b无线网络规范是IEEE 802.11网络规范的变种，最高带宽为11 Mbps，在信号较弱或有干扰的情况下，带宽可调整为5.5Mbps、2Mbps和1Mbps，带宽的自动调整，有效地保障了网络的稳定性和可靠性。其主要特性为：速度快，可靠性高，在开放性区域，通讯距离可达305米，在封闭性区域，通讯距离为76米到122米，方便与现有的有线以太网络整合，组网的成本更低。 Wi-Fi WirelessFidelity，无线保真 技术与蓝牙技术一样，同属于在办公室和家庭中使用的短距离无线技术。该技术使用的使2.4GHz附近的频段，该频段目前尚属没用许可的无线频段。其目前可使用的标准有两个，分别是IEEE802.11a和IEEE802.11b。该技术由于有着自身的优点，因此受到厂商的青睐。 Wi-Fi技术突出的优势在于： 其一，无线电波的覆盖范围广，基于蓝牙技术的电波覆盖范围非常小，半径大约只有50英尺左右 约合15米 ，而Wi-Fi的半径则可达300英尺左右 约合100米 ，办公室自不用说，就是在整栋大楼中也可使用。最近，由Vivato公司推出的一款新型交换机。据悉，该款产品能够把目前Wi-Fi无线网络300英尺 接近100米 的通信距离扩大到4英里 约6.5公里 。 其二，虽然由Wi-Fi技术传输的无线通信质量不是很好，数据安全性能比蓝牙差一些，传输质量也有待改进，但传输速度非常快，可以达到11mbps，符合个人和社会信息化的需求。
a r X i v :q u a n t -p h /9907068v 1 21 J u l 1999 Fidelity for multimode thermal squeezed states Gh.-S.Paraoanu ? Department of Physics,University of Illinois at Urbana–Champaign,1110W.Green St.,Urbana,IL 61801,USA;e-mail: paraoanu@https://www.doczj.com/doc/e017117938.html, Horia Scutaru Department of Theoretical Physics,Institute of Atomic Physics,PO BOX MG-6,R-76900Bucharest-Magurele,Romania; e-mail:scutaru@theor1.theory.nipne.ro In the theory of quantum transmission of information the concept of ?delity plays a fundamental role.An important class of channels,which can be experimentally realized in quantum optics,is that of Gaussian quantum channels.In this work we present a general formula for ?delity in the case of two arbitrary Gaussian states.From this formula one can get a previous result [1],for the case of a single mode;or,one can apply it to obtain a closed compact expression for mul-timode thermal states.The concept of ?delity used in this paper is the standard one [2–4].It can be de?ned by F (ρ1,ρ2)def = max |ψ1>,|ψ2> | ψ1|ψ2 |2, where |ψi >,i =1,2are puri?cations of the density matrices ρi . 03.65.Bz,42.50.Dv,89.70.+c I.INTRODUCTION Within recent years,the quantum theory of informa-tion,an extension of the classical theory of information to the quantum realm,has emerged as a fascinating re-search ?eld.A great deal of e?ort has been devoted to the issue of transmitting a state through noisy quantum channels despite the quantum-mechanical uncertainties in our knowledge about that state.This is a di?erent problem from the classical situation,where the states are mutually exclusive and the input system may remain in its initial state;in the quantum case the states are density operators on a Hilbert space and the non-cloning theo-rem [5]precludes the input system,in general,to retain its original state.Moreover,for a noisy quantum chan-nel the state is subject to a decoherence process,due to the interaction with an external environment,which fur-ther decreases the reliability of information processing.Thus,a fundamental problem is to extend the classical encoding and decoding procedures to quantum channels ρ1ρ2 √
“音乐传真”的起源 “音乐传真”（Musical Fidelity）的创办人Antony Michaelson出生于1951年，从小就热爱音乐，他本身是一位专业音乐师并且以吹奏单簧管为主要学习和演奏乐器。在他31岁的时候（1982年），由于一次偶然的机会（失业），Antony Michaelson和他的妻子毅然创立了“Musical Fidelity”公司。Antony Michaelson一开始已经确定了自己的经营方针和宗旨，就是要用最低成本制造出拥有丰富音乐感和杰出表现的音响器材。 由于在那个时候，市场上的所有高级音响器材均取价高昂，故此要购买一部具有质素的放大器，实在不是一般人所能够负担的，而Antony Michaelson在他那股热爱音乐的诚意带动之下，便决心把高级音响推进普及化市场，制造他认为拥有绝佳声音且是大部分人可以承担得起的放大器。他先把主要生产放大器的厂房设于英国的Wembley，这间厂房的设备和员工在短短的十多年间不断扩充，到今天已经拥有超过40个员工，成为英国具规模的音响制造商之一。 Antony Michaelson创业初期所生产的放大器主要仍是以他自己一手一脚所制作出来的简单制品，其中最知名的是一部名为“The Pre-Amps”的前级放大器。当时主要是卖给自己一些爱好欣赏音乐的朋友，而生产的费用，是从他的太太和岳母所借出的三百多英镑加上他自己从银行信用卡所借贷得来的。到了1984年，Antony Michaelson在一次意大利旅程中获得灵感开始设计他的成名作品——A1合并式放大器。 以A系列打头炮 从Antony Michaelson开始创业之时已经把生产音响器材建基于音乐化之上，故此他坚持的理念就是要生产一些价格定得合理和音质得到绝大部分爱好音乐的人所认同的器材，而A1的制作和往后在市场上所取得的空前成功证明他的努力没有白费。A1合并式放大器的设计和理念事实非常明确和简单直接，它采用了纯甲类输出，是音色甜美的高品质机种。 在那时由于市场上的大部分放大器只是朝着大功率和力量型的方向，而忽略了音质的细致度，因此在Antony Michaelson设计这部A1放大器之时，便决定要根据他本人所持有的古典音乐多年来的修养来“调声”，并以纯甲类输出来设计A1。 在设计和制作这部A1之时实在是困难重重，由于甲类放大器的线路比较复杂、零件的数量和质素要求较高，而最重要的是在散热的安排和设计上一定要造得妥当，才能够取得稳定而安全的效果。而A1在最后的设计上把散热器安装在整个机身的顶部。 A1合并机在1986年正式推出国际市场时简直是平地一声雷，取得了极高的声誉，直到今天仍然有人认为它是合并式放大器的经典制品。 继往开来，推陈出新 由于国际市场上的需要和渴望，Antony Michaelson在A1推出后的翌年便开始尝试利用相同的设计，制作更大输出功率的合并式放大器，并终于在1987年成功推出了输出功率比A1大1倍有多的A100（每边输出50瓦），由于这部A100在推动扬声器的宽容度比A1大得多，故此甫一推出就得到好评如潮，不论在英国
X5 High-Fidelity DNA Polymerase ( with dNTPs ）使用说明书 产品名称 单位 货号 X5 High-Fidelity DNA Polymerase 100U MF003-01 X5 High-Fidelity DNA Polymerase 5×100U MF003-05 【储存条件】 长期保存，请置于-20?C ，有效期24个月。经常使用，可置于4?C 保存至少六个月。 【产品简介】 聚合美X5超保真DNA 聚合酶具有5’ 到3’ DNA 聚合酶活性和3’ 到5’ 的外切酶活性(即校读活性)，能够纠正DNA 扩增过程中产生的碱基错配现象，在标准缓冲液中其保真度约相当于普通Taq DNA Polymerase 的50倍、Pfu DNA Polymerase 的6倍。同时该酶还具有快速的DNA 合成速度，约相当于普通Taq DNA Polymerase 的4-6倍、Pfu DNA Polymerase 的8-12倍。该酶合成能力很强，即使是复杂的模板，也能快速准确的完成反应，尤其适用于对保真性要求高的DNA 长片段的快速扩增，如基因克隆、测序、定点突变、SNP 分析等，也可用于DNA 片段的末端补平。使用聚合美X5超保真DNA 聚合酶扩增得到的PCR 产物无3＇端突出碱基，不可直接用于TA 克隆，可以直接连入聚合美特制的平末端TOPO 克隆载体（货号MF021和MF022）。 【单位定义】 用大马哈鱼精子DNA 作为模板/引物，在74?C 、30分钟内，摄入10 nmol 的全核苷酸为酸性不溶物所需的酶量。 【产品组份】 X5 High-Fidelity DNA Polymerase (1U/μl) 100 μl 10× X5 PCR Buffer 1.0 ml 25mM MgSO 4 1.0 ml 2mM dNTPs 1.0 ml 【质量控制】 以λ DNA 为模板，能有效扩增20 kb 的DNA 片段；以基因组DNA 为模板，能有效扩增单拷贝基因；无内切酶和外切酶污染。 【适用范围】 高保真扩增，长片段的快速扩增，如基因克隆、定点突变等；高GC 含量、具有二级结构的复杂模板的扩增。 【操作示例】 按下表配制PCR 反应体系: 10× X5 PCR Buffer 2.5 μl 25mM MgSO 4 2.5 μl 2mM dNTPs 2.5 μl Primer 1 (10 μM) 0.5 μl Primer 2 (10 μM) 0.5 μl Template DNA* 10-200ng X5 High-Fidelity DNA Polymerase 0.5μl ddH 2O 补足至 25 μl <*模板量：10~200 ng 基因组 DNA ，10~30 ng 质粒，或 1~2 μl RT-PCR 反应后的 cDNA 。以上举例为常规 PCR 反应系统，仅供参考。实际反应条件因模板、引物等的结构不同而各异，需根据模板、引物、目的片段的特点设定最佳反应条件，并根据比例放大或缩小反应体系>。 【备注】 本产品仅供科研使用。在确认产品质量出现问题时，本公司承诺为客户免费更换等量的质量合格产品。 X5 High-Fidelity DNA Polymerase ( with dNTPs ）User Manual Product Name Units Cat.# 建议的PCR 条件： 95°C 2 min. 32-36 cycles of: 94°C 25 sec. 53–64°C 20-35sec. 68°C 15-30sec./1kb 68°C 5 min. 4°C forever
语言学派翻译理论 奥古斯丁发展了亚里士多德的“符号”理论，提出了语言符号的“能指”、“所指”和译者“判断”的三角关系，开创了西方翻译理论的语言学传统。20世纪初，索绪尔提出普通语言学理论，标志这现代语言学的诞生，也为当代翻译研究的各种语言学方法奠定了基础。虽然出现了各种不同流派的代表人物和理论方法，却存在一个共同的特征，就是以语言为核心，从语言的结构特征出发研究翻译的对等问题。一般认为，西方语言学派开始对翻译进行”科学“研究的标志是美国著名学者尤金.奈达Eugene Nida 于1947年发表的《论<圣经>翻译的原则和程序》(Bible Translation: An Analysis of Principles and Procedures with Special Reference to Aboriginal Languages)。语言学派代表人物主要集中于英美，代表人物有奈达、卡特福德(J.C.Catford)、纽马克(Peter Newmark)、哈蒂姆(Hatim)等。 罗曼.雅科布逊Roman Jakobson 原籍俄国，后移居捷克；二战时迁至美国，加入美籍。作为学派的创始人之一，他对翻译理论的贡献主要体现在《论翻译的语言学问题》(On Linguistic Aspects of Translation)之中。文章从语言学的角度，对语言和翻译的关系、翻译的重要性、以及翻译中存在的问题做出了详尽的分析和论述。自１９５９年发表后，此文一直被西方理论界奉为翻译研究的经典之一。雅科布逊的论述主要有五点：（１）翻译分为三类：语内翻译
（intralingual translation）、语际翻译（interlingual translation）和符际翻译（intersemiotic translation）。所谓语内翻译，是指在同一语言内用一些语言符号去解释另一些语言符号，即通常的“改变说法”(rewording)。所谓语际翻译，是指在两种语言之间即用一种语言的符号去解释另一种语言的符号，即严格意义上的翻译。所谓符际翻译，是指用非语言符号系统解释语言符号，或用语言符号解释非语言符号，比如把旗语或手势变成言语表达。（２）对于词义的理解取决于翻译。他认为，在语言学习和语言理解过程中，翻译起着决定性作用。（３）准确的翻译取决于信息对称。翻译所涉及的是两种不同语符中的对等信息。（４）所有语言都具有同等表达能力。如果语言中出现词汇不足，可通过借词、造词或释义等方法对语言进行处理。（５）语法范畴是翻译中最复杂的问题。这对于存在时态、性、数等语法形式变化的语言，尤其复杂。 尤金.奈达Eugene Nida 语言学派最重要的代表人物之一，著述极丰富，其理论对西方当代翻译研究作出了很大的贡献.提出了“翻译的科学”这一概念。在语言学研究的基础上，把信息论应用于翻译研究，认为翻译即交际，创立了翻译研究的交际学派。提出了“动态对等”的翻译原则，并进而从社会语言学和语言交际功能的观点出发提出“功能对等”的翻译原则。就翻译过程提出“分析”、“转换”、“重组”和“检验”的四步模式。功能对等理论由美国人尤金·A·奈