A Domain Theoretic Model of Qubit Channels

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A Domain Theoretic Model of Qubit Channels

Keye Martin

Naval Research LaboratoryCenter for High Assurance Computer Systems

Washington, DC [email protected]

Abstract. We prove that the spectral order provides a domain theoreticmodel of qubit channels. Specifically, we show that the spectral orderis the unique partial order on quantum states whose least element isthe completely mixed state, which satisfies the mixing law and has theproperty that a qubit channel is unital iff it is Scott continuous and has aScott closed set of fixed points. This result is used to show that the Holevocapacity of a unital qubit channel is determined by the largest value ofits informatic derivative. In particular, these channels always have aninformatic derivative that is necessarily not a classical derivative.

1 Introduction

The study of measurement was initiated within the context of computation [3].In[5], it is shown that measurement can be used to prove fixed point theorems formappings that are not monotone and unique fixed point theorems for mappingsthat are monotone. Results like these can be used to provide a unified viewof numerical algorithms, for instance. In such applications, we are primarilyconcerned with operators f whose iterates fn(x) converge to a fixed point p.The informatic derivative df(p) then measures the rate at which f convergesto p.

The view of computation taken in the study of measurement, that a compu-tation is a process that evolves on a space of informatic objects, and that as itevolves we can measure the amount of information lost or gained, in retrospectlends itself very naturally to considerations in other areas, such as physics or thestudy of communication. In[2], it was discovered that natural domain theoreticstructure existed in quantum mechanics. And developments such as[7] and[6]establish the importance of domains and measurements in classical informationtheory.

In this paper, we establish the significance of domain theory and measurement

in quantuminformation theory. We first show that a classical binary channel isScott continuous and has a Scott closed set of fixed points iff it is a binary sym-metric channel, while a qubit channel is Scott continuous and has a Scott closedset of fixed points iff it is unital. The binary symmetric channels are exactly theentropy increasing binary channels; the unital qubit channels are exactly the en-tropy increasing qubit channels. One reason such channels are important is that

L. Aceto et al. (Eds.): ICALP 2008, Part II, LNCS 5126, pp. 283297, 2008.c Springer-Verlag Berlin Heidelberg 2008


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284 K. Martin

they provide effective ways of interrupting communication. For instance, assumingall inputs are equally likely, the best way to interrupt communication for a fixedprobability of error is to use a binary symmetric channel. The class of unital qubitchannels includes most of the models used to describe noise: bit flipping, phase

flipping, bit-phase flipping, phase damping (decoherence), depolarization, uni-tary channels and projective measurements.

In fact, the connection between entropy increasing channels and Scott contin-uous channels with Scott closed sets of fixed points also turns out to uniquelydetermine the spectral order on quantum states. We have known since [2] thatthe unique partial order on 2 = {(x, y) [0, 1] : x+y = 1} that satisfies themixing law and has a least element of= (1/2, 1/2) is the Bayesian order. Whatwe prove in this paper is the quantum analogue of this result: the spectral orderis the unique partial order on two dimensional mixed quantum states 2 that

satisfies the mixing law, has least element =I/2 and the additional propertythat every unital channel is Scott continuous and has a Scott closed set of fixedpoints. This additional property is trivially satisfied in the classical case 2.

Finally, we use these results to give a method for calculating the Holevo ca-pacity of a unital qubit channel. Surprisingly, each unital qubit channel hasan informatic derivative defined everywhere except . The largest value of achannels informatic derivative determines its Holevo capacity. This informaticderivative is not a classical derivative. This demonstrates a completely new usefor informatic rates of change.

2 The Domains of Classical and Quantum States

We review the basic ideas in the study of domains and measurements, and thenthe two examples of domains that are of interest in this paper.

2.1 Domain Theory and Measurement

A domainis a partially ordered set with intrinsic notions of completeness andapproximation defined by the order. A measurement is a function that toeach informative object x assigns a number xwhich measures the informationcontent of the object x. We now define each of these terms precisely beforediscussing them further.

The intrinsic notion of completeness that a domain has is that it forms a dcpo:

Definition 1. Let (P, ) be a partially ordered set orposet. A nonempty subsetS P is directed if (x, yS)(zS) x, yz . The supremum

S ofS P

is the least of its upper bounds when it exists. A dcpo is a poset in which everydirected set has a supremum.

The intrinsic notion of approximation possessed by a domain is formalized bycontinuity:


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A Domain Theoretic Model of Qubit Channels 285

Definition 2. Let (D, ) be a dcpo. For elements x, y D, we write x y ifffor every directed subsetSwithy

S, we havex s, for somes S. We set

x:= {y D :y x} and x:= {yD :x y} x:= {y D :y x} and x:= {yD : x y}

and say D is continuousifxis directed with supremum xfor each x D.

Definition 3. A domainis a continuous dcpo. A Scott domainis a continuousdcpo in which any pair of elements with an upper bound has a supremum.

Definition 4. TheScott topologyon a continuous dcpo D has as a basis all setsof the form x for x D. A set SD is Scott closed if it is a lower set that isclosed under directed suprema.

A function f :D Ebetween domains is Scott continuousif the inverse imageof a Scott open set in Eis Scott open in D. This is equivalent[1] to saying thatf is monotone,

(x, y D) x y f(x) f(y),

and that it preserves directed suprema:

f(

S) =

f(S),

for all directedSD. In particular, for the domain [0, )

of nonnegative realsin their opposite order, a Scott continuous function : D [0, ) will satisfy

1. For allx, yD, x y x y, and2. If (xn) is an increasing sequence in D, then

n1

xn

= lim

nxn.

This is the case of Scott continuity that pertains to measurements:

Definition 5. A Scott continuous : D [0, ) is said to measure the con-tentofx D if for all Scott open sets UD,

x U( >0) x (x) U

where(x) :={yD : y x & |x y|< }

are called the -approximationsofx.We often refer to as measuring x D or as measuring X D when itmeasures each element ofX.

Definition 6. A measurement: D [0, ) is a Scott continuous map thatmeasures the content of ker() :={x D :x = 0}.


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In this paper, all measurementswe work with measure all ofD. This implies[5]that they are strictly monotone:

x y & x= y x = y

This property enables definition of the informatic derivative:

Definition 7. Let (D, ) be a domain with a measurement that measures allofD. Iff :D D is a function and p D is not compact, then

df(p) = limxp

f(x) f(p)

x p

is called the informatic derivativeoff, provided that this limit exists.

2.2 The Bayesian Order on Classical States

The set of classical states

2 :={(x, y) [0, 1]2 :x + y= 1}

has a natural domain theoretic structure introduced in [2]:

Definition 8. For x, y2,

x y (y1x11/2) or (1/2 x1y1) .

The relation on 2 is called the Bayesian order.

This order is derived from the graph of entropy H(x) = x log2(x) (1x)log2(1 x) as follows:

H

x flip

(1, 0) (0, 1)

= ( 12, 12

)

Theorem 1 ([2]). (2, ) is a Scott domain with maximal elements

max(2) ={(0, 1), (1, 0)}

and least element = (1/2, 1/2). The Shannon entropyH :2 [0, ), givenby

H(x) =x1log2(x1) x2log2(x2)

is a measurement.


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2.3 The Spectral Order on Quantum States

Let H2 denote an two dimensional complex Hilbert space with specified innerproduct|.

Definition 9. A quantum state is a density operator : H2 H2, i.e., a self-adjoint, positive, linear operator with tr() = 1. The quantum states on H2 aredenoted2.

Quantum states are also sometimes call density operators or mixed states. Theset of eigenvalues of an operator , called the spectrumof, is denoted spec().

Definition 10. A quantum state on H2 is pure if

spec() {0, 1}.

The set of pure states is denoted 2. They are in bijective correspondence withthe one dimensional subspaces ofH2.

Classical states are distributions on the set of pure states max(2).An analogousresult holds for quantum states: density operators encode distributions on theset of pure states 2.

Definition 11. Aquantum observableis a self-adjoint linear operator e : H

2

H2.

Now, if we have the operator e representing the energy observable of a system(for instance), then its spectrum spec(e) consists of the actual energy values asystem may assume. If our knowledge about the state of the system is representedby density operator , then quantum mechanics predicts the probability that ameasurement of observablee yields the value spec(e). It is

pr( e) := tr(pe ),

wherepe is the projection corresponding to eigenvalue ande is its associatedeigenspace in the spectral representationofe.

Definition 12. Let e be an observable on H2 with spec(e) = {1, 2}. For aquantum state on 2,

spec(|e) := (pr( e1), pr( e2)) 2.

We assume that all observables e have |spec(e)| = 2. Intuitively, then, e is anexperiment on a system which yields one of 2 different outcomes; if our a prioriknowledge about the state of the system is , then our knowledge about whatthe result of experiment e will be is spec(|e). Thus, spec(|e) determines ourability to predictthe result of the experiment e.

Let [a, b] =ab ba denote the commutator of operators.


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Definition 13. For quantum states , 2, we have iff there is anobservablee : H2 H2 such that [, e] = [, e] = 0 and spec(|e) spec(|e)in 2.

This is called the spectral orderon quantum states.

Theorem 2 ([2]). (2, ) is a Scott domain with maximal elements

max(2) =2

and least element = I/2, where I is the identity matrix. The von NeumannentropyS:2 [0, ) given byS() =tr( log2()) is a measurement.

The Hilbert space formalism makes things seem much more complicated thanthey really are in this case: the spectral order on2 has a much simpler descrip-

tion which we now consider.There is a 1-1 correspondence between density operators on a two dimensionalstate space and points on the unit ball B3 = {x R3 : |x| 1}: each densityoperator: H2 H2 can be written uniquely as

= 1

2

1 + rz rx iryrx+ iry 1 rz

where r = (rx, ry, rz) R3 satisfies |r| =

r2x+ r

2y+ r

2z 1. The vector r

B3

is called the Bloch vector associated to . Bloch vectors have a number ofaesthetically pleasing properties.If and are density operators with respective Bloch vectors r and s, then

(i) the eigenvalues of are (1 |r|)/2, (ii) the von Neumann entropy of isS = H((1 +|r|)/2) = H((1 |r|)/2), where H : [0, 1][0, 1] is the base twoShannon entropy, (iii) if and are pure states and r+ s = 0, then and are orthogonal, and thus form a basis for the state space; conversely, the Blochvectors associated to a pair of orthogonal pure states form antipodal points onthe sphere, (iv) the Bloch vector for a convex sum of mixed states is the convex

sum of the Bloch vectors, (v) the Bloch vector for the completely mixed stateI/2 is 0 = (0, 0, 0).Because of the correspondence between2 and B3, we regard the two as equal

for the rest of the paper.

Example 1. From[2], using the Bloch representation of density operators, thespectral order on 2 is given by x y iff the line from the origin to y passesthrough x. That is,

x y (p [0, 1]) x= py

for allx, y2.

3 Classical and Quantum Channels

We review classical binary channels, qubit channels and then a special subclassof each of them: the entropy increasing channels.


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3.1 Classical Channels

A binary channel has two inputs (0 and 1) and two outputs (0 and 1).An input is sent through the channel to a receiver. Because of noise in thechannel, what arrives may not necessarily be what the sender intended. The effect

of noise on input data is modelled by a noise matrix u. If data is sent throughthe channel according to the distribution x, then the output is distributed asy =x u. The noise matrix u is given by

u=

aab b

wherea = P(0|0) is the probability of receiving 0 when 0 is sent and b= P(0|1)is the probability of receiving 0 when 1 is sent and x:= 1 xfor x [0, 1]. Thus,the noise matrix of a binary channel can be represented by a point (a, b) in theunit square [0, 1]2 and all points in the unit square represent the noise matrix ofsome binary channel.

The noise matrix u of a binary channel defines a function f : 2 2,given by f(x) = x u, which maps an input distribution x 2 to an outputdistribution f(x) 2.

3.2 Quantum Channels

A classical binary channel f : 2 2 takes an input distribution to an

output distribution. In a similar way, a qubit channel is a function of the form: 2 2. Specifically,

Definition 14. A qubit channelis a function : 2 2 that is convex linearand completely positive.

To say that is convex linear means that preserves convex sums i.e. sums ofthe form x + (1 x) . Complete positivity, defined in [8], is a conditionwhich ensures that the definition of a qubit channel is compatible with naturalintuitions about joint systems. For our purposes, there is no need to get lost

in too many details of the Hilbert space formulation: thankfully, qubit channelsalso have a Bloch representation.

Definition 15. For a qubit channel : 2 2, the mapping it induces onthe Bloch sphere f: B

3 B3 is called the Bloch representationof.

The set of qubit channels is closed under convex sum and composition. If is aqubit channel and fis its Bloch representation, then (i) the functionfis convexlinear, (ii) composition of quantum channels corresponds to composition of Blochrepresentations: for channels 1, 2, we have f12 = f1 f2 , (iii) convex sum

of quantum channels corresponds to convex sum of Bloch representations: forchannels1, 2 and x [0, 1], we have fx1+x2 =xf1+ xf2 .

To illustrate how these properties make it simple to calculate the Bloch rep-resentation of a qubit channel, consider the bit flipping channel,

() = (1 p)I() +p x()


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where I is the identity channel and x() = xx, with x being the spin

operatorx=

0 11 0

.

The Bloch representation ofI is fI (r) = r. Using the correspondence be-

tween density operators and Bloch vectors, we calculate directly that the Blochrepresentation ofx is fx(rx, ry, rz) = (rx, ry, rz). Thus, by property (iii) ofBloch representations,

f(rx, ry, rz) = (1 p)(rx, ry, rz) +p(rx, ry, rz) = (rx, (1 2p)ry, (1 2p)rz)

Notice that states of the form (rx, 0, 0) are unchanged by this form of noise, theyare all fixed pointsoff.

3.3 Entropy Increasing Channels

The classical channels f : 2 2 which increase entropy (H(f(x)) H(x))are exactly thosefwithf() =. They are the strictmappings of domain the-ory, which are also known as binary symmetric channels in information theory.

Similarly, the entropy increasing qubit channels are exactly those for which() =. These are called unitalin quantum information theory.

Definition 16. A qubit channel: 2 2 is unital if() =.

A qubit channel is unital iff its Bloch representation f satisfiesf(0) = 0. Let

us consider a few important examples of unital channels.

Example 2. Unitary channels. If U is a unitary operator on H2, then () =U U is unital since U U = I. The Bloch representation f is given by f(r) =M rwhereMis a 3 3 orthogonal matrix with positive determinant, a rotation.

Example 3. Projective measurements. If{P0, P1}are projections withP0 + P1=I, then

() =P0P0+ P1P1

is a unital channel since P2

0 =P0 andP2

1 =P1. In this case, the Bloch represen-tation f satisfies f2 =f.

Just as with qubit channels, unital channels are also closed under convex sumand composition: if1and2are unital channels, then1 2and p1+(1p)2are unital for p [0, 1].

Example 4. Let x, y and z denote the spin operators

x = 0 11 0 y = 0i

i 0 z = 1 0

01

Each is unitary and self-adjoint.

(i) Each spin operatori defines a unital channel i() = ii. For a Blochvector r = (rx, ry, rz), the respective Bloch representations sx, sy, sz aresx(r) = (rx, ry, rz),sy(r) = (rx, ry, rz) and sz(r) = (rx, ry, rz).


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(ii) The Bit flipping channel= (1 p)I+p x is unital.(iii) The phase flipping channel= (1 p)I+p z is unital.(iv) The bit-phase flip channel = (1 p)I+p y is unital.(v) The depolarization channel

d(x) =p + (1 p)x

is unital, for a fixed p [0, 1].

Not all qubit channels are unital of course. Amplitude damping provides a well-known example of a qubit channel that is not.

4 Scott Continuity of Unital Channels

Our first result establishes that from the domain theoretic perspective, unitalqubit channels are the quantum analogue of binary symmetric channels in theclassical case.

Theorem 3

A classical channelf :2 2 is binary symmetric iff it is Scott continu-ous and its set of fixed points is Scott closed.

A quantum channelf :2 2 is unital if and only if it is Scott continuousand its set of fixed points is Scott closed.

Proof.First consider the classical case. If a classical channel f is Scott continu-ous, then it has a least fixed point, and since the set of fixed points is Scott closed, = (1/2, 1/2) must be a fixed point. This implies that f is binary symmetricsince

(1/2, 1/2)

aab b

= ((a+ b)/2, (a + b)/2) = (1/2, 1/2)

Conversely, suppose that f is binary symmetric. Then it can be written as

f(a, b) = (1 p) (a, b) +p (b, a)

for somep [0, 1]. First we show thatfis Scott continuous. For the monotonicityoff, let x, y 2 with x y. Then we want to show f(x) f(y). Writingx= (x1, x2) and y= (y1, y2), we have

(y1x11/2) or (1/2 x1 y1)

by the definition of on 2; we seek to establish

(f1(y) f1(x) 1/2) or (1/2 f1(x) f1(y))

where we have written f(x) = (f1(x), f2(x)) and f(y) = (f1(y), f2(y)). Noticethat

f1(x) = (1 2p)x1+p and f1(y) = (1 2p)y1+p.

We consider the cases y1x11/2 and 1/2 x1 y1 separately.


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In the first case,f1(y) f1(x) 1/2 holds whenp 1/2, and 1/2 f1(x)f1(y) holds when p1/2. In the second case, 1/2 f1(x)f1(y) holds when

p 1/2, and f1(y) f1(x) 1/2 holds for p 1/2. Thus, f(x) f(y),which provesfis monotone. Its Scott continuity now follows from its Euclidean

continuity and the fact that suprema in the Bayesian order coincide with limitsin the Euclidean topology.

Now we show that the fixed points offform a Scott closed set. Ifp = 0, thenf is the identity mapping, in which case its set of fixed points is Scott closed.Otherwise, its only fixed point is , since for p >0,

(a, b) =f(a, b) = (a, b) = (b, a) = (a, b) = (1/2, 1/2) = .

Either way, the fixed points offform a Scott closed subset of2.In the quantum case, any channelfthat is Scott continuous and has a Scott

closed set of fixed points must have as a fixed point, and so must be unital. Forthe converse, we first show that any unital fis Scott continuous. Recall that fcan be written in Bloch form as f(r) =M rfor some 3 3 real matrixM. Thenfis Euclidean continuous, and since suprema in the spectral order are limits inthe Euclidean topology, fis Scott continuous in the spectral order provided itis monotone.

For the monotonicity off, let r s in the spectral order on 2. Then thestraight line segment s : [0, 1]

2 from to s, given by s(t) = t s

for t [0, 1], must pass through r. To show that f(r) f(s), we must showthat the line from to f(s) passes through f(r). But this much is clear sincef(s(t)) = M(t s) = t f(s) = f(s)(t). Thus, all unital channels are Scottcontinuous.

To see that the set of fixed points fix(f) is Scott closed, we first show that it isa lower set. Ifs fix(f) andr s, thenr lies on the line segment that joins tos. But any point on this line is a fixed point off since f(s(t)) =f(s)(t) =s(t). In particular,r fix(f). The set fix(f) is closed under directed supremaby the Scott continuity off. Thus, fix(f) is Scott closed.

Selfmaps on Hausdorff spaces have closed sets of fixed points. But the Scotttopology is not Hausdorff, so the result above is meaningful. The fact that the setof fixed points is Scott closed also has experimental significance: in attemptingto prepare |0 during QKD, Alice actually prepares (1 )|00|+ |11| forsome small >0. Then this too is a fixed point of the noise operator, provided|0 is, so the only reduction in capacity is due solely to error in preparation Alice does not suffer more noise simply because she cannot prepare a qubitexactly.

5 Uniqueness of the Spectral Order

The order on2 is canonical as follows:


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Theorem 4 ([2]). There is a unique partial order on2 that satisfies the mixinglaw

x y andp [0, 1] x (1 p)x +pyy

and has:= (1/2, 1/2) as a least element. It is the Bayesian order on classical

two states.

Because of the simplicity of2, it then follows that the binary symmetric chan-nels are exactly the classical channels that are Scott continuous and have a Scottclosed set of fixed points. In this section, we prove the analogous result for thespectral order.

The special orthogonal group SO(3) is the set of 33 orthogonal real matricesMwith a positive determinant i.e. those matrices Msuch that M1 =Mt anddet(M) = +1. Such matrices are called rotations.

Lemma 1

(i) Every rotationfSO(3) is the Bloch representation of a unital channel,(ii) For any x max(2), there is a rotation f SO(3) such that f(x) =

(0, 0, 1).

Proof.(i) This is a folklore result, see [8]for instance.(ii) This follows from the fact that SO(3) is a transitive group action on S2.

However, we want to write a self-contained paper, so let us give a simpler proof.

Every unit vector x appears as the third column M(0, 0, 1) of some orthogonalmatrixMsince by the Gram-Schmidt process we can always find an orthonormalbasis {v1, v2, v3} whose first vector is v1=x. Given such an orthonormal basis,we construct an orthogonal matrix fwhose column vectors are the vectors inthe orthonormal basis with the third column being x.

So let us take an orthogonal matrix Msuch that M(0, 0, 1) =x. Then M1

is an orthgonal matrix with M1(x) = (0, 0, 1). If det(M1) =1, we set

f= 1 0 00 1 0

0 0 1 M1

then f is a rotation with f(x) = (0, 0, 1). Otherwise, det(M1) = +1, in whichcase we set f=M1.

Theorem 5. There is a unique partial order on 2 with the following threeproperties:

(i) It has least element= I /2,(ii) It satisfies the mixing law: if r s, then r tr+ (1 t)s s, for all

t [0, 1],(iii) Every unital channelf :2 2 is monotone and has a lower set of fixed

points.

It is the spectral order, and gives2 the structure of a Scott domain on which allunital channels are Scott continuous and have a Scott closed set of fixed points.


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Proof. In this proof, we work with Bloch representations. By the mixing law,the depolarization channel dt(x) =tx + (1 t)= tx is deflationary, so tx xfor eacht [0, 1]. Thus, contains the spectral order.

Now supposer s. We want to show that r precedess on the line that travels

from tos and on to a pure state. Draw the line a from to r until it hitsthe boundary of the Bloch sphere at a point a. Similarly, letb denote the linefrom to s and on to a pure state b. Since r s and s b, we have r b bytransitivity and thus r a, b.

Let f be a rotation such that f(b) = (0, 0, 1). Then f(b) = (0, 0, 1). Letp be the Bloch representation of a projective measurement in the basis whoseBloch vectors are {f(b), f(b)}. Then

Im(p) ={(0, 0, t) :t [1, 1]}

Since r b, f(r) f(b) and thus p(f(r)) p(f(b)). But, p(f(b)) = f(b), sof(r) is also a fixed point, since the fixed points of p are Scott closed. Thenf(r), f(b) Im(p). This meansf(r) andf(b) lie on a line that joins a pure stateto its antipode. Becausefis a rotation, the same is true ofr andb. However, bythe mixing law, the line fromr tob, which increases with respect to, does notpass through since is the least element (otherwise, r = and the proof isfinished). Then a= b, which means r and s lie on a line that joins to a purestate a.

So let us write r=xa and s= ya forx, y[0, 1]. Ifx y, the proof is done.Ifx > y, then s= (y/x)rr using the depolarization operator dy/x. But sincers, we have r= s by antisymmetry of.

Notice that the discrete order on 2 \ {0} with 0 adjoined as the least elementgives a domain that makes all unital channels Scott continuous with a Scottclosed set of fixed points, so requiring the mixing law is essential in uniquelycharacterizing the spectral order.

6 Holevo Capacity from the Informatic Derivative

A standard way of measuring the capacity of a quantum channel in quantuminformation is the Holevo capacity; it is sometimes called the product statecapacity since input states are not allowed to be entangled across two or moreuses of the channel.

Definition 17. For a trace preserving quantum operation f, theHolevo capacityis given by

C(f) = sup{xi,i}

S

f

i

xii

i

xi S(f(i))

where the supremum is taken over all ensembles {xi, i}of possible input statesi to the channel.


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The possible input states i to the channel are in general mixed and the xi areprobabilities with

i xi = 1. Iffis the Bloch representation of a qubit channel,

the Holevo capacity off is given by

C(f) = sup{xi,ri}

H

1 + |f(

i xiri) |2

i

xi H

1 + |f(ri)|2

where ri are Bloch vectors for density operators in an ensemble, and we recallthat eigenvalues of a density operator with Bloch vector r are (1 |r|)/2.

Theorem 6. Let(x) = 1 |x| denote the standard measurement on2. Forany unital channelf and anyp 2 different from,

df(p) = |f(p)|

|p|

Thus, the Holevo capacity offis determined by the largest value of its informaticderivative. Explicitly,

C(f) = 1 H

1

2+

1

2 supxker()

df(x)

Proof. Since x p iffx = tp for some t [0, 1], x p in the topology ifft 1, so

df(p) = limxp

f(x) f(p)x p

= limt1

f(tp) f(p)(tp) p

= limt1

|f(p)| |f(tp)|

|p| |tp|

= limt1

|f(p)|(1 |t|)

|p|(1 |t|) (Linearity off)

=

|f(p)|

|p|

Now we show that the Holevo capacity is determined by the largest value of itsinformatic derivative. By the Euclidean continuity of |f|, there is a pure stater2 for which

|f(r)|= max|x|=1

|f(x)|= m+

Setting r1 = r, r2 = r and x1 = x2 = 1/2 defines an ensemble for which theexpression maximized in the definition of C(f) reduces to 1 H((1 +m+)/2).

Notice that in this step we explicitly make use of the fact that f is unital:f(0) = 0. This proves 1 H((1 + m+)/2) C(f).For the other inequality, any term in the supremum is bounded from above

by

1 i

xi H

1 + |f(ri)|

2


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296 K. Martin

sinceH(x) 1. For each ri, there is a pure state pimax(2) withripi. Bythe Scott continuity off,

|f(ri)| |f(pi)| sup|x|=1

|f(x)|= m+,

so we have

H

1 + |f(ri)|

2

H

1 + m+

2

which then gives C(f) 1 H((1 + m+)/2).

Thus, we see that C(f) = 1 for anyrotationf sincedf= 1. Notice that df1ifffis a rotation. For eachp [0, 1], the unique channel f1 withdf =pis thedepolarization channelf=dp, so that C(dp) = 1H((1+p)/2). In fact, the map(p, 1 p) d12p defines an isomorphism from the nonnegative classical binarysymmetric channels onto the depolarization channels that preserves capacity.The only unital qubit channel with capacity zero is 0 itself.

Example 5. The two Pauli channelin Bloch form is

(r) =p r+

1 p

2

sx(r) +

1 p

2

sy(r)

where sx and sy are the Bloch representations of the unitary channels x andy. This simplifies to

(rx, ry, rz) = (prx, pry, (1 p)rz)

The matrix associated to is diagonal, so the diagonal element (eigenvalue) thathas largest magnitude also yields the largest value of its informatic derivative.The capacity of the two Pauli channel is then

1 H1 + max{p, 1 p}2

where p [0, 1].

The set of unital channels Uis compact hence closed and thus forms a dcpo asa subset of the domain [2 2].

Corollary 1. The Holevo capacityC :U [0, 1] is Scott continuous.

7 Closing

The set of unital qubit channels Uis a convex monoid and a dcpo with respectto which the Holevo capacity is monotone. In a similar way, the interval domainI[0, 1], which models classical binary channels, is a convex monoid and a dcpowith respect to which the Shannon capacity is monotone [7].


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References

1. Abramsky, S., Jung, A.: Domain theory. In: Abramsky, S., Gabbay, D.M., Maibaum,T.S.E. (eds.) Handbook of Logic in Computer Science, vol. III, Oxford University

Press, Oxford (1994)2. Coecke, B., Martin, K.: A partial order on classical and quantum states. OxfordUniversity Computing Laboratory, Research Report PRG-RR-02-07 (August 2002),http://web.comlab.ox.ac.uk/oucl/publications/tr/rr-02-07.html

3. Martin, K.: A foundation for computation. Ph.D. Thesis, Tulane University, De-partment of Mathematics (2000)

4. Martin, K.: Entropy as a fixed point. In: Daz, J., Karhumaki, J., Lepisto, A.,Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142. Springer, Heidelberg (2004)

5. Martin, K.: The measurement process in domain theory. In: Welzl, E., Montanari,U., Rolim, J.D.P. (eds.) ICALP 2000. LNCS, vol. 1853. Springer, Heidelberg (2000)

6. Martin, K.: Topology in information theory in topology. Theoretical Computer Sci-ence (to appear)

7. Martin, K., Moskowitz, I.S., Allwein, G.: Algebraic information theory for binarychannels. In: Proceedings of MFPS 2006. Electronic Notes in Theoretical ComputerScience, vol. 158, pp. 289306 (2006)

8. Nielsen, M., Chuang, I.: Quantum computation and quantum information. Cam-bridge University Press, Cambridge (2000)

9. Shannon, C.E.: A mathematical theory of communication. Bell Systems TechnicalJournal 27, 379423, 623656 (1948)
http://web.comlab.ox.ac.uk/oucl/publications/tr/rr-02-07.htmlhttp://web.comlab.ox.ac.uk/oucl/publications/tr/rr-02-07.html

A Domain Theoretic Model of Qubit Channels

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