Quantum broadcasting problem in classical low-power signal processing

Quantum broadcasting problem in classical low-power signal processing

Dominik Janzing and Bastian SteudelInstitut für Algorithmen und Kognitive Systeme, Universität Karlsruhe, Am Fasanengarten 5, 76 131 Karlsruhe, Germany

�Received 11 October 2006; revised manuscript received 15 November 2006; published 8 February 2007�

We prove a no-broadcasting theorem for the Holevo information of a noncommuting ensemble stating thatno operation can generate a bipartite ensemble such that both copies have the same information as the original.We argue that upper bounds on the average information over both copies imply lower bounds on the quantumcapacity required to send the ensemble without information loss. This is because a channel with zero quantumcapacity has a unitary extension transferring at least as much information to its environment as it transfers tothe output. For an ensemble being the time orbit of a pure state under a Hamiltonian evolution, we derive sucha bound on the required quantum capacity in terms of properties of the input and output energy distribution.Moreover, we discuss relations between the broadcasting problem and entropy power inequalities. The broad-casting problem arises when a signal should be transmitted by a time-invariant device such that the outgoingsignal has the same timing information as the incoming signal had. Based on previous results we argue that thisestablishes a link between quantum information theory and the theory of low power computing because theloss of timing information implies loss of free energy.

DOI: 10.1103/PhysRevA.75.022309 PACS number�s�: 03.67.Hk, 05.30.�d, 03.65.Yz, 06.30.Ft

I. INTRODUCTION

Quantum information theory and the theory of low-powerprocessing are currently quite different scientific disciplines.Even though future low power computers will operate moreand more on the nanoscale and therefore in the quantumregime �e.g., single electron transistors, spintronic networks�1��, superpositions of logically different states being crucialfor quantum computing �2�, are not intended to occur in low-power computing devices.

On the other hand, quantum computing research is littleinterested in issues of low power processing. The control ofquantum systems involves large laboratory equipment andeven power consumption rates for logical operations that arein the magnitude of usual classical chips seem currently to beout of reach.

To understand limitations of low-power information pro-cessing it is useful to construct theoretical models of com-puters which process information without consuming energy,i.e., the process is implemented in an energetically closedphysical system. In our opinion, discussions on fundamentalissues like bounds on power consumption require a quantumtheoretical description. Interesting quantum models of com-puters being closed physical systems can be found in Refs.�3–7�. Remarkably, it is common to all these models that thesynchronization is based upon some propagating wave orparticle and that the quantum uncertainty of its position leadsto an ill-defined logical state of the computer. In other words,the clock is entangled with the data register. It seems as if theclocking issue brings some aspects of quantum informationtheory into the field of low-power computing. This is notsurprising for the following reason: the states of a quantummechanical system have a consistent classical descriptiononly if the attention is restricted to a set of mutually com-muting density matrices. But the Hamiltonian dynamics au-tomatically generates noncommuting density matrices from agiven one. Hence the dynamical aspect makes it necessary toinclude quantum superpositions into the description. Note

that this is also in the spirit of Hardy’s paper “Quantumtheory from five reasonable axioms” �8�, saying that everystatistical theory that satisfies some very natural axioms isquantum, as soon as it makes continuous reversible dynami-cal evolution possible.

If signal propagation in future low-power devices takesplace in a system being �approximately� thermodynamicallyclosed it must be described by a quantum Hamiltonian evo-lution. The idea of this paper is that processing such signalsleads to quantum broadcasting problems for two reasons.

First, it is a natural problem to distribute signals �likeclock signals� to several devices. The timing information car-ried by a signal whose quantum state is a density operatorwithin a family of noncommuting states cannot be consid-ered as classical information, its distribution is thereforesome kind of broadcasting problem. The results in �9,10�indicate that no-broadcasting theorems are expected to getrelevant for the distribution when the signal energy is re-duced to a scale where quantum energy-time uncertainty be-comes the limiting factor for the accuracy of clocking.

The second reason why broadcasting problems come intoplay is more subtle. If such a clocking signal enters a deviceand triggers the transmission of an output signal we maydesire that the output should have as much timing informa-tion as the input �in a sense that will be further specifiedlater�. Whether channels with zero quantum capacity are ableto satisfy this requirement is a question that is linked toquantum broadcasting.

The intention of this paper is to describe a special kind ofbroadcasting problem. In contrast to the usual setting �11�,the task is not to obtain output states that are close to theinputs. The problem is to broadcast the Holevo informationof an ensemble of noncommuting quantum density matricessuch that each party gets almost the same amount of Holevoinformation as the original ensemble possessed. The use ofentropylike information measures makes it possible to drawconnections to thermodynamics.

In this paper, the ensemble of noncommuting states willalways be given by the Hamiltonian time evolution of a

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given state. Even though the problem of broadcasting Holevoinformation makes also sense for general ensembles, timeevolution is the most obvious way how noncommuting en-sembles occur in devices that are not designed to do quantuminformation processing.

It seems to be hard to derive general bounds on the infor-mation loss of each copy when the Holevo information isbroadcast �after all, we will show that it is not possible fornoncommuting ensembles to get full Holevo information forboth copies�. The intention of this paper is therefore rather topose the broadcasting problem and show its relevance than tosolve it. However, for pure input states we will give onelower bound on the loss that depends on the energy distribu-tion of input and output signals.

The paper is organized as follows. In Sec. II we introducetime-invariant signal processing devices and argue that inthis setting timing information is a resource that can never beincreased. In Sec. III we formally state the problem of broad-casting Holevo information in the general case and prove ano-broadcasting theorem. Moreover, we describe the broad-casting problem in the case of timing information. In Sec. IVwe argue that the broadcasting problem leads to the questionhow the Holevo information of an ensemble of bipartitestates is related to the information of the ensembles of thecorresponding reduced states. We discuss this informationdeficit for the special case of pure product states where theproblem is related to the entropy power inequalities of clas-sical information theory. In Sec. V we derive a bound on theinformation deficit that depends on the energy spectral mea-sure of input and output signal. In Sec. VI we show that theresults imply lower bounds on the quantum capacity requiredfor lossless transmission of signals having small energy un-certainty in a time-covariant way. Section VII derives lowerbounds on the loss of free energy implied by the loss oftiming information caused by a channel with too little quan-tum capacity. This describes an even tighter link betweenquantum information theory and the theory of low-powersignal processing.

II. QUANTUM MODEL OF TIME-INVARIANT SIGNALPROCESSING DEVICES

As already stated, the problem of transmitting noncom-muting ensembles of quantum states arises most naturally forensembles that are given by the Hamiltonian time evolutionof a given state. Such an ensemble may, for instance, de-scribe the density matrix of a propagating signal before orafter it is processed by the device. If all clocking signals thatenter a given device are included into the formal description,the quantum operation mapping the input onto the output istime invariant. As we will describe below, such a devicecannot increase the timing information. The latter is there-fore considered as a resource. The idea that devices withnonzero quantum capacity seem to deal with this resourcemore carefully than classical channels is essential for thispaper.

Now we introduce the abstract description of time-invariant devices. Here a device may be a transistor, an op-tical element or some other system with input and output

signals. The signal may, for instance, be an electric pulse, alight pulse, or an acoustical signal. We consider it as a physi-cal system with some Hilbert space H and the state is adensity operator � acting on H. For the examples mentionedabove, the space H will typically be infinite dimensionalsince one may, e.g., think of position degrees of freedom thatare encoded into �. Before and after the signal is processedin the device its free time evolution is generated by a Hamil-tonian H �i.e., a densely defined self-adjoint operator on H�and reads

�t�� ª e−iHt�eiHt. �1�

We assume that input state � and its output G�� are relatedby some completely positive trace-preserving map G satisfy-ing the covariance condition

G„�t��… = �t„G��… ∀ � . �2�

There are rather different situations where the covariancecondition is satisfied. One example would be if the interac-tions between signal and device are weak. A more interestingjustification is the following. Consider a signal propagatingtowards the device by its own autonomous Hamiltonian timeevolution until it begins to interact with the latter. Then itleaves the device �as a possibly modified signal� and as soonas the interaction with the device is negligible it is againsubjected to its Hamiltonian only. Such a process should beconsidered as a quantum stochastic analogue of a scatteringprocess �see �12� for details� and the time covariance condi-tion �2� is then a generalization of the statement that theS-“matrix” of a scattering process commutes with the freeHamiltonian evolution of the incoming and outgoing particle�13�. Note that the existence of a unitary S operator wouldrequire devices which preserve the purity of the input.

In �12� we have given a quite explicit description of theset of CP maps satisfying this covariance condition. Here it ismore interesting to discuss the implications of covariance.We first rephrase the definition of timing information used in�14� �see also �15� for a more general context�.

Recall that the Holevo information of an ensemble ofquantum states �x with probability measure p �denoted by�p�x� ,�x�x�X� is defined by �2�

I ª S�� xdp�x� −� S��x�dp�x� ,

where the measure-theoretic integral reduces to sums when pis supported by a countable set of points. Here

S�� = − tr�� log �� 3�

is the von Neumann entropy and the base of the logarithmremains unspecified. In the sequel we will measure entropyin bits or nats since sometimes one unit is more natural andsometimes the other.

Timing information refers to a specific ensemble, namelythe orbit with respect to a unitary one-parameter group.

Definition 1 (timing information). Let � be the state of aquantum system whose Hamiltonian H has discrete spec-trum. Then its timing information is defined as

DOMINIK JANZING AND BASTIAN STEUDEL PHYSICAL REVIEW A 75, 022309 �2007�

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I ª S�� − S�� , �4�

where � denotes the time average

� ª limT→�

1

T�

0

T

�t��dt = x

Rx�Rx,

�t is defined as in Eq. �1� and �Rx� is the family of spectralprojections �with eigenvalues x corresponding to the systemHamiltonian�. For pure states �= �� we have S��=0.Thus, I is the entropy of � which is then exactly the entropyof a classical random variable X describing the distributionof energy values with P�X=x�ª ��Rx��.

Note that it is a well-known question to what extent in-formation on reference frames in time or space requiresquantum communication or profits from it and to which de-gree shared reference frames are resources that are compa-rable to shared quantum states �15–20�. In this paper wewant to understand to what extent timing information shouldbe considered as quantum information by exploring the in-formation loss occurring when it is copied. In �9� we havederived lower bounds on the loss of timing information interms of Fisher information for the broadcasting problem. Toour knowledge, no results in terms of Holevo informationcan be found in the literature.

III. BROADCASTING TIMING INFORMATION

Before we pose the problem of broadcasting timing infor-mation �which we have motivated from the time-covarianttransmission of signals� we first state the more general prob-lem of “broadcasting Holevo information.” It is defined asfollows.

Definition 2 (broadcasting Holevo information). Given anensemble �p�x� ,�x�x�X of quantum states acting on someHilbert space H. Let I be its Holevo information. Find anoptimal broadcasting map in the following sense.

Let HA and HB be some arbitrary additional Hilbertspaces and G be a completely positive trace-preserving op-eration from the density operators on H on the density op-erators acting on HA � HB. Let IA ,IB denote the Holevoinformation of the ensembles given by the reduced statestrB(G��x�) and trA(G��x�), respectively.

Maximize the average information

1

2�IA + IB�

over all HA � HB and G such that it gets as close to I aspossible.

We call

� ª I −1

2�IA + IB� �5�

the broadcasting loss of a given broadcasting operation. Let�min be the minimal loss over all broadcasting operations fora given ensemble. In the context of timing information wewill also use the terminology “�min of a state �” when actu-ally referring to the information loss of the ensemble

��t��t��0,�� with uniform probability distribution over thewhole time period.

Due to the monotonicity of the Holevo information of anarbitrary ensemble with respect to CP maps �14� we certainlyhave I1�I and I2�I. The following theorem shows thatI=I1=I2 can only be achieved for ensembles of mutuallycommuting states.1

Theorem 1. (no perfect broadcasting of Holevo informa-tion). Given the terminology and assumptions of Definition2, we have �min=0 if and only if there is a basis that simul-taneously diagonalizes p-almost all �x.

Proof. If the density operators mutually commute there isa broadcasting operation with H=HA=HB that reproducesthe original states in each copy �11,21�. Then we have clearlyI=IA=IB.

To prove the converse direction we start by showing thata map with �=0 can only exist for commuting ensembles.Below, we will complete the proof by showing that otherwisethere can also be no sequence of broadcasting maps with �converging to zero. First we rephrase Example 4 in �22�which provides a proof for countable ensembles. The idea isthat if the operations

FA:� � trB„G��… and FB:� � trA„G��… �6�

both preserve the Holevo information of the ensemble there

exist operations FA and FB that restore the states �x on eachsystem. But then we have a perfect broadcasting operationfor the states which is only possible if the �x mutually com-mute. To show this formally, define

� ª x

p�x��x� x� � �x,

where ��x��x�X denotes a countable family of mutually or-thogonal states of a �possibly infinite dimensional� ancillasystem. Then the mutual information of this bipartite state isexactly the Holevo information of the ensemble. Let be thestate given by the product of the restrictions of � to the leftand the right component, respectively. Then we rewrite themutual information as

I = D�� ,

where D�� denotes the relative entropy distance betweentwo quantum states �23�. Similarly, we have

IA = D��id � FA�� id � FA�� 7�

and

IB = D��id � FB�� id � FB�� . �8�

Based upon results in �24� Theorem 3 in Ref. �22� shows thatI=IA=IB implies that there exists a map of the form

id � FA such that

�id � �FA � FA�� = �

and

1The idea of the first part of the following proof has been providedby Andreas Winter.

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�id � �FA � FA�� = ,

and, correspondingly, such a map id � FB for system B. Thisshows that

FA„FA��x�… = �x and FB„FB��x�… = �x

for all x in the support of p. Hence we can restore all thestates �x perfectly just by operating on the copies and wehave thus a perfect quantum broadcasting operation in theusual sense. This is only possible if ��x ,�x��=0 for all x ,x� inthe support of p �21�.

To generalize the argument to uncountable ensembles weintroduce the Hilbert space

H ª l2�X,p� � H � l2�X,H,p� ,

where l2�X , p� and l2�X ,H , p� denote the space of squareintegrable functions from X to C, respective H, with respectto the measure p. Then we can define the density operator �as the direct integral �25�

� ª ��

�xdp�x� .

The states �id � FA�� and �id � FB�� act on the Hilbert spaces

HAª l2�X ,HA , p� and HBª l2�X ,HB , p�, respectively. Then

the maps FA and FB restoring the original states can be con-structed like the map in Example 4 in Ref. �22� since theassumption of finite dimension is not needed there. Hence

we have FA(FA��x�)=�x for p-almost all x and similarly forsystem B. This is certainly only possible if there is a basisthat diagonalizes p-almost all �x �21�.

It remains to show that for every given noncommutingensemble the loss � has a nonzero lower bound, i.e., �min�0. We assume that there exists a sequence of broadcastingoperations Gn and corresponding operations FA

�n� and FB�n�

�see Eq. �6�� with

limn→�

D��id � Fj�n�� id � Fj

�n�� = I for j = A,B . �9�

Since we have allowed infinite dimensional Hilbert spaceswe may assume without loss of generality that the targetdensity operators act on a common Hilbert space HA � HB. Itfollows from Lemma 7 in the Appendix that there exists aCP-map G such that �id � G�� and �id � G� are clusterpoints of �id � Gn�� and �id � Gn�, respectively �with re-spect to the weak* topology�. Due to the continuity of thepartial trace operation �id � FA�� is a cluster point of �id� FA

�n�� if we define FA as in Eq. �6�. We have analoguestatements for and for FB. Relative entropy is weaklylower semicontinuous for states of an arbitrary C*-algebra�Proposition 5.23 in �26��. By interpreting the considereddensity operators as functionals on the correspondingC*-algebras of compact operators �see the Appendix� it fol-lows from �9� that the values in Eqs. �7� and �8� are bothequal to I. In analogy to the arguments above, we may thus

construct a perfect broadcasting operation �FA � FB� �G forthe states �x which is only possible if the �x mutually com-

mute. �To obtain an upper bound on �min we observe that there

are maps that provide both parties with the accessible infor-mation Iacc �2� by applying a measurement to the input stateand sending mutually orthogonal quantum states represent-ing the results to both parties. We have thus �min�I−Iacc.

Now we apply the definition of broadcasting to an en-semble given by the time orbit ��t�t��0,�� of a dynamical evo-lution with period � with uniform distribution over the wholeinterval. Then the task is to optimally broadcast the timinginformation in the sense of Definition 1. Note that informa-tion differences like that one in Eq. �5� may be well definedfor systems with continuous spectrum where the timing in-formation itself is infinite. By appropriate limits, one couldtherefore define the question on the information loss inbroadcasting operations also for systems possessing no timeaverage state.

To give an impression on the problem of broadcastingtiming information we consider the phase-covariant cloningof an equatorial qubit state, i.e., a state

��t� ª1�2

��0� + eit�1��

with unknown t� �0,2��. In the usual quantum cloningproblem one tries, for instance, to obtain two copies whosestates get as close to the original as possible with respect tothe fidelity. As shown in �27� one can generate two copies asmixed states whose Bloch vectors point in the same directionas that of the original, but are shorter than the original by thefactor 1 /�2. Thus, the density matrices of the copies have theeigenvalues 1/2±1/ �2�2�. The Holevo information of eachcopy is then given by the entropy of the time average �whichis still one bit� minus the above binary entropy when insert-ing the above eigenvalues,

I = 1 +1

2�1 +1�2

log21

2�1 +1�2

+1

2�1 −1�2

�log2

1

2�1 −1�2

� 0.399 bit.

The information of the original was 1. Here, even the sum ofthe amount of information over both copies is less than theoriginal amount. In other words, the average informationover both copies is even smaller than it was if we had givenone party the original and the other an arbitrary state that isindependent from the input.

IV. INFORMATION DEFICIT IN PURE PRODUCT STATESAND ENTROPY POWER INEQUALITIES

In the following we will not explicitly consider the broad-casting operation that generates a bipartite state from theoriginal. Since this operation can never increase the informa-tion we focus to the following problem: Given an ensembleof bipartite states, compare the Holevo information of thetwo ensembles IA and IB defined by the restrictions to thesubsystems to the information I of the joint system. Call I− �IA+IB� /2 the information deficit. In other words, the in-


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formation deficit is the broadcasting loss if the broadcastingmap is the identity and the original is already a bipartitestate.

Remarkably, the determination of the deficit is nontrivialeven when the bipartite state is a product state. Given thestate

��A� � ��B� � HA � HB,

where each subsystem is subjected to its own HamiltonianHA and HB, respectively. We may assume without loss ofgenerality that both Hamiltonians are diagonal and nonde-generate �since we restrict the attention to the time orbits ofeach state�. The distribution of energy values in the state��A� � ��B� defines a joint distribution of two stochasticallyindependent classical random variables X ,Y by

P�X = x,Y = y� ª �A�Rx��A� �B�Qy��B� , �10�

where Rx is defined as in Definition 1 and Qy similarly. SinceHA � 1+1 � HB is the Hamiltonian of the joint system, itstiming information is given by

I = S�X + Y� , �11�

where we have decided to use the same symbol for the en-tropy of classical random variables as for the von Neumannentropy of quantum states. The subsystem timing informa-tion is given by

IA = S�X� and IB = S�Y� . �12�

Note that it is a well-known problem in classical informationtheory to relate the entropy of the distributions of two inde-pendent random variables to the entropy of their sum since itaddresses the question how the entropy of a real-valued sig-nal changes when subjected to an additive noise. We re-phrase the following result that applies to continuous distri-butions. For probability densities P�X� the continuousentropy is defined by

S�X� = −� P�x�ln P�x�dx + c ,

with an unspecified constant c. For two independent randomvariables, i.e., when their density satisfies P�x ,y�= P�x�P�y�,we have the entropy power inequality �28�

e2S�X+Y� e2S�X� + e2S�Y�,

and hence

2S�X + Y� ln�1

2�2e2S�X� + 2e2S�Y��

1

2�ln 2e2S�X� + ln 2e2S�Y��

= ln 2 + S�X� + S�Y� ,

where the second inequality follows from the concavity ofthe logarithm. Assuming that the spectral measures of HAand HB are sufficiently distributed over many energy eigen-values we can approximate the discrete entropy with the con-

tinuous expression for appropriate densities. After using Eqs.�12� and �13� we obtain

I 1

2�ln 2 + IA + IB� .

Note that ln 2 corresponds exactly to one bit of informationsince the entropy power inequality refers to entropy mea-sured in natural units. We conclude that for continuous spec-trum and product states the timing information of the jointsystem is at least one-half a bit more than the average timinginformation over both systems.

V. INFORMATION DEFICIT FOR PUREENTANGLED STATES

To estimate the information deficit for entangled states wewill also use the joint distribution of X and Y on R2 given by

P�X = x,Y = y� ª tr„��Rx � Qy�… , �13�

with the spectral projections Rx and Qy. If the bipartite sys-tem is in an entangled state, Eq. �12� is no longer true. More-over, we cannot assume that both Hamiltonians are “withoutloss of generality” nondegenerate since the reduced statesmay be mixed even within a specific degenerate energyeigenspace. However, Eq. �11� still holds for pure states. Wereplace Eq. �12� by

IA = S��A� − S��A� ,

where �A denotes the reduced state on system A and obtainIB in a similar way. To derive upper bounds on the timinginformation of the subsystems we need the following lemma.

Lemma 1 (average entropy of post-measurement states).Let �Rj� j be a complete family of orthogonal projections de-fining a measurement and � be an arbitrary quantum state.Let S�p� be the Shannon entropy of the outcome probabilitiespjª tr�Rj��. Then we have

S�j

Rj�Rj � S�� + S�p� .

Proof. The statement is equivalent to

j

pjS� 1

pjRj�Rj � S�� . �14�

Let �=iqi�i be a decomposition of � into pure states. Wecan consider S�� as the Holevo information of the ensemble�qi ,�i�i. Then the left-hand side of Eq. �14� is equal to theHolevo information of the ensemble after the measurementhas been applied. It can certainly be not greater than theHolevo information of the original ensemble �see �22�, Ex-ample 4�. �

For our derivation of an upper bound on the informationof the subsystems the following lemma will be crucial.

Lemma 2 (timing information is less than conditional en-tropy). Let � be a �possibly mixed� state on a bipartite sys-tem. Then the timing information of A and B satisfies

IA � S�X�Y�, IB � S�Y�X� ,

respectively, where the joint distribution of X and Y is de-fined by Eq. �13�.


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Proof. By Definition 1 the timing information of system Ais given by

IA = S��A� − S��A� .

We decompose �A into

�A = y

p�y��A,y ,

where �A,y denotes the conditional state given that we hadmeasured the energy value y on system B. Since IA is theKullback-Leibler distance between �A and �A �see �29�� it isconvex and we get

IA � y

p�y��S��A,y� − S��A,y�� .

For each specific value y of Y,

S��A,y� − S��A,y� � S�X�y�

holds due to Lemma 1. Taking the convex sum of this in-equality over all y with weights p�y� completes the proof.�

Note that there are conditions known �30�, where the jointprobability density of two dependent random variables satis-fies the entropy power inequality,

e2S�X+Y� e2S�X�Y� + e2S�Y�X�.

Under such conditions we obtain the same lower bound onthe information deficit as in Sec. IV.

In the general case we must use other methods to derivemore explicit bounds from the bounds of Lemma 2. For do-ing so, we will need the following lemma.

Lemma 3 (information deficit and classical mutual infor-mation). The information deficit of a bipartite system beingin a pure state satisfies

� 1

2�I�X:X + Y� + I�Y:X + Y��

= S�X + Y� −1

2�S�X�Y� + S�Y�X�� ,

where I� : � denotes the mutual information between classi-cal random variables �31�.

Proof. Note that the equation I=S�X+Y� holds also forpure entangled states. Using Lemma 2 we obtain

2I − IA − IB 2S�X + Y� − S�X�Y� − S�Y�X�

= 2S�X + Y� − S�X + Y�Y� − S�X + Y�X�

= I�X + Y:Y� + I�X + Y:X� . �

It is possible to derive bounds on the information loss basedon Lemma 3, since the term on the right-hand vanishes onlyin the trivial case S�X+Y�=0 in which the joint system con-tains no timing information at all. To show this we observethat there is no joint measure where X and Y are both uncor-related to X+Y. This is seen from

C�X,X + Y� + C�Y,X + Y� = V�X + Y� , �15�

where C� , � denotes the covariance and V� � the variance.However, to derive lower bounds on the mutual information

based on these covariance terms requires additional assump-tions on the distribution. We will deal with this point later.

In order to apply the bounds of Lemma 3 it can be con-venient to relate them to other information-theoretic quanti-ties.

Lemma 4 (mutual information and relative entropy). Let Xand Y be two real-valued random variables and P the corre-sponding joint distribution on R2 with discrete support. LetP−X and PX+Y denote the marginal distribution for −X andX+Y, respectively. Denote the convolution of both byPX* PX+Y. Then we have

I�X:X + Y� D�PY � P−X * PX+Y� �16�

and

I�Y:X + Y� D�PX � P−Y * PX+Y� , �17�

where D� � � denotes here the relative entropy distance�“Kullback-Leibler distance”� between two probability distri-butions. Moreover, we have the symmetrized statement

I�X:X + Y� + I�Y:X + Y� D�1

2�PX + PY� *

1

2

��P−X + P−Y� � PX+Y� . �18�

Proof. We define measures on R2 by

Q�X = a,Y = b� ª P�X + Y = a + b�P�Y = b�

and

R�X = a,Y = b� ª P�X + Y = a + b�P�X = a� .

Then we can rewrite the mutual information on the left-handside as Kullback-Leibler distances,

I�X:X + Y� = D�P � R�

and

I�Y:X + Y� = D�P � Q� .

Due to the monotonicity of relative entropy distance undermarginalization �26� we have

D�P � Q� D�PX � QX� ,

where PX and QX denote the marginal distribution of X ac-cording to P and Q, respectively, i.e., QX�X=a�ªQ�X=a�.Similarly

D�P � R� D�PY � RY� .

We have

Q�X = a� = b

Q�X = a,Y = b� = b

P�X + Y = a + b�P�Y = b�

= c

P�X + Y = c�P�Y = c − a�

= c

P�X + Y = c�P�X = c − a� .

Hence the marginal distribution QX of Q is the convolutionproduct PX+Y � P−X and the marginal distribution RY of R is


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the product PX+Y * P−Y. This proves inequalities �16� and�17�.

We obtain the symmetrized statement from the convexityof relative entropy distance �31�. �

After applying Lemma 4 and Lemma 3 we conclude thefollowing.

Theorem 2 (information deficit for pure states). Given apure state of a bipartite system A�B. Let PX, PY, and PX+Ydenote the probability distributions for the energy of A, Band A�B, respectively. Then the difference between thejoint timing information and the average information of thesubsystems satisfies

� D�PX * P−Y � PX+Y� + D�PY � P−X * PX+Y�

D�1

2�PX + PY� �

1

2�P−X + P−Y� * PX+Y� .

The intuitive content of Theorem 2 is the following. If theenergy uncertainty of A and B are both on the same scale asthe uncertainty of X+Y, the convolution with PX+Y adds anon-negligible amount of uncertainty to �PX+ PY� /2, whichimplies that the new distribution obtained by adding addi-tional noise cannot be close to the original distribution of X.

It is often helpful to consider measures that are symmetricwith respect to exchanging X and Y, i.e., P�X=x ,Y =y�= P�X=y ,Y =x�. The following lemma shows that lowerbounds on I�X :X+Y�+ I�Y :X+Y� for symmetric joint mea-sures automatically provide bounds for asymmetric mea-sures.

Lemma 5 (symmetrization). Let P be a joint distribution of

X and Y and P its symmetrization Pª �P+ P�� /2, where P�with

P��X = x,Y = y� ª P�X = y,Y = x�

denotes the mirror image of P. Then we have

IP�X:X + Y� + IP�Y:X + Y� IP�X:X + Y� + IP�Y:X + Y� ,

where IP�:� refers to the mutual information induced by themeasure P.

Proof. We write

P�X = x,Y = y� = P�X = x�X + Y = x + y�P�X + Y = x + y� .

We obtain such a representation also for P� by replacing onlythe conditional P�X �X+Y� since the marginal distribution onX+Y coincides for P and P�. Then the lemma follows al-ready from the convexity of mutual information with respectto convex sums of conditionals with fixed marginals �Theo-rem 2.7.3 in �31��. �

A simple bound on the information deficit can be pro-vided in terms of the fourth moments of the signal energies.

Theorem 3 (information deficit in terms of energy). Givena pure bipartite state on A�B. Let ��E�2 denote the varianceof the total energy and Ej

4� denote the fourth moment of theenergy of system j=A ,B and E4� be the fourth moment ofthe total energy. Then the information deficit �measured innatural units� satisfies

� ��E�8

64� EA4� + EB

4�� E4�.

Proof. Let P, as above, be the discrete probability mea-sure on R2 describing the energy distribution of the bipartitesystem. We begin by assuming that P is symmetric �seeLemma 5�. Then we have C�X ,X+Y�=V�X+Y� /2 �see Eq.�15��. We define a measure R as in the proof of Lemma 4 andwe can rewrite the covariance as

C�X,X + Y� = xy

x�x + y�„P�x,y� − R�x,y�… .

With ZªX+Y we have

1

4V2�X + Y� = C�X,Z�2 = �

xz

xz�„P�x,z − x� − R�x,z − x�…

� �„P�x,z − x� − R�x,z − x�…�2

� xz

x2z2�P�x,z − x� − R�x,z − x��

�xz

�P�x,z − x� − R�x,z − x��

� xz

x2z2„P�x,z − x� + R�x,z − x�…�P − R�1

= � X2Z2� + X2� Z2��P − R�1

� 2� X4� Z4��P − R�1.

From the first line to the second line we have used theCauchy-Schwarz inequality which shows also X2Z2�� X4� Z4� as well as X2�� X4�.

We recall the bound

D�P � R� 1

2�P − R�2

�see Lemma 12.6.1 in �31�� for the relative entropy measuredin natural units. Then we obtain

1

2�P − R�1

2 V�X + Y�4

128 X4� �X + Y�4�.

This implies

I�X:X + Y� V�X + Y�4

128 X4� �X + Y�4�,

since the mutual information between X and X+Y is the rela-tive entropy distance between the joint measure P and themeasure R that has been defined by the product of marginaldistributions on X and X+Y. If we consider an asymmetricmeasure P we must symmetrize it first. Then we replace X4�with � X4�+ Y4�� /2 since the fourth moment of Y with re-spect to the original measure P coincides with the fourthmoment of X when calculated with respect to the reflectedmeasure P��X=x ,Y =y�ªP�X=y ,Y =x�. Using Lemma 3this proves the statement when replacing the statistical mo-ments of X ,Y, and X+Y with the more physical terms EA

4�, EB

4�, and E4�. �


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VI. QUANTUM CAPACITY REQUIRED FORLOSSLESS TRANSMISSION

In this section we will derive lower bounds on the quan-tum capacity required to transmit an ensemble with somefixed minimal information loss. The idea of the argument isthe following.2 Assume that the timing information of G��is exactly the same as that of �. Assume furthermore that Ghas zero quantum capacity. This implies, roughly speaking,that G can be modeled by a unitary that copies as muchinformation to the environment as the amount of informationthat passes the channel. But if this would be the case we hadperfect broadcast of Holevo information, an operation that isimpossible for noncommuting ensembles like time orbits. Toput this argument on a solid basis, we rephrase the followingresult of Devetak �35�. Recall that the private informationcapacity �see �35� for a formal definition� is the maximalnumber of encoded qubits per transmitted qubits, that twoparties, the sender Alice and the receiver Bob, can asymp-totically achieve in a protocol where a potential eavesdrop-per Eve, having access to the full environment of the chan-nel, gets a vanishing amount of information. The followingtheorem relates the private information capacity to the infor-mation the environment obtains when the channel is repre-sented by a unitary acting on the system and an abstractenvironment being in a pure state.3

Theorem 4 (private channel capacity). Let G be a quan-tum channel mapping density operators acting on H to den-sity operators acting on the same space. Let HE be an addi-tional Hilbert space thought of as the space of theenvironment. Moreover, let U be a unitary acting on H� HE and ��HE be a state such that

G�� = tr2�U�� U†� .

Let �x with x�X be some finite family of input states �sentby Alice with probability p�x�� and

�x ª U��x � �� U†

be the corresponding joint states of the environment and thereceiver’s �i.e., Bob’s� system. Denote the restrictions tothese subsystems by �x

B and �xE, respectively. Set

I�X:B� ª S�x

p�x��xB − p�x�

x

S��xB�

and I�X :E� similarly. Define the single copy private channelcapacity by

C1�G� ª sup�I�X:B� − I�X:E�� ,

where the supremum is taken over all ensembles (p�X� ,�x).Let G� l be the l-fold copy of G. Then the private channelcapacity is given by

Cp�G� = liml→�

1

lC1�G� l� .

Certainly, we have Cp�G� C1�G�. This is seen by transmit-ting independently distributed product states through thecopies of channels. We observe the following.

Theorem 5 (information loss in classical channels). Let�p�x� ,�x�x be an ensemble of quantum states with Holevoinformation

I�X:A� = S�x

p�x��x − x

p�x�S��x� ,

with minimal broadcasting loss �min. Let G be some channelwith

I�X:B� = S�x

p�x�G��x� − x

p�x�S„G��x�… .

Then the private channel capacity of G can be bounded frombelow by

Cp�G� 2��min − �I�X:A� − I�X:B�� .

Note that I�X :A�− I�X :B� is the information loss caused bythe channel because it is the difference between input andoutput Holevo information. Given a bound for broadcastingthe Holevo information of the considered ensemble, we havea lower bound on the quantum capacity to transmit themwithout loss.

Proof (of Theorem 5). Given some unitary operation Uextending the channel G. We have

� ª I�X:A� −1

2�I�X:B� + I�X:E�� min

by definition of �min and

Cp�G� I�X:B� − I�X:E� ,

by Theorem 4. Then simple calculations yield the stated in-equality. �

The theorem shows that for states with nonzero �min�which is every nonstationary state � due to Theorem 1� thecovariant lossless transmission requires a channel with non-zero quantum capacity. Instead of deriving lower bounds on�min, i.e., the minimum over all �, we will use the boundfrom Theorem 3 and only obtain bounds in terms of thefourth moments of the energy distribution. However, usingthis theorem is not straightforward for the following reason:Given some assumptions on the energy distribution of theinput and output signals of a device we want to derive lowerbounds on the quantum capacity required to transmit the sig-nal without information loss. To this end, we use the unitaryextension of the CP map formalizing the device because wehave only derived bounds for pure bipartite states. However,the usual construction of the unitary extension uses an ab-stract environment Hilbert space where no “environment

2This can be seen as an alternative way of quantifying the “quan-tumness” of an ensemble. In contrast, Refs. �32–34� quantify quan-tumness by the maximal achievable fidelity between input and out-put of a channel that consists in measuring the state and preparingthe estimated state afterwards.

3One should emphasize that the unitary extension gives only up-per bounds on the information transferred to the environment. Realenvironments are usually in mixed states and can therefore destroyquantum superpositions without receiving information from the sys-tem �see �36� for details�.


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Hamiltonian” is specified. And, even worse, given that wehad specified an arbitrary “environment Hamiltonian,” theunitary that models the channel could have lead to arbitraryenergy distributions for system plus environment and we ob-tained no useful statements on the fourth moments.

The following lemma shows that we can construct theunitary extension such that it is energy conserving in theconstructed joint system. We have here considered a finitedimensional system for technical reasons.

Lemma 6 (unitary extension of covariant operations). LetG be a completely positive trace-preserving map on the setof d�d density matrices that satisfies the covariance condi-tion �2� with respect to the time evolution generated by aHamiltonian H acting on Cd.

Then there is a �not necessarily finite dimensional� Hilbertspace HE, a densely defined Hamiltonian HE on HE withpurely discrete spectrum and an eigenstate �� of HE witheigenvalue 0 such that the following condition holds.

There exists a unitary U on Cn � HE commuting with theextended Hamiltonian H � 1+1 � HE which satisfies

G�� = tr2�U�� U†�

for all density matrices �.Proof. We assume without loss of generality that H is

diagonal with respect to the canonical basis. Let

G�� = j=1

k

Aj�Aj† �19�

be the Kraus representation of G �see �37��. Define �ª �x−y �x ,y�spec�H�� where spec�H� denotes the spectrumof H. As shown in Eq. �14� in �12� we can choose the Krausoperators such that for every Aj there is some real number� j �� with

�H,Aj� = � jAj . �20�

In other words, the operator Aj implements a shift of energyvalues by � j in the sense that it maps eigenstates of H witheigenvalue � onto states with energy �+� j. The idea is tochoose a unitary extension such that the energy shift causedby Aj is compensated by the opposite shift in the environ-ment. Thus, we define the Hamiltonian HE of the environ-ment such that all values in � occur as spectral gaps in HE.Set HEª l2�Z��k and

HE = j=1

k

� jMj ,

where Mj is the multiplication operator acting on the jthcomponent

Mj ª 1� j−1� diag�. . . ,− 1,0,1, . . . � � 1�k−j .

Let

Sj ª 1� j−1� S � 1�k−j

be the unitary left shift on l2�Z� acting on the jth tensorcomponent via S�n�ª �n−1� for each n�Z. Define

U ª j=1

k

Aj � Sj .

To see that U is indeed unitary we consider basis states

�l� � �z� , �21�

where l=0, . . . ,d−1 and z is in the kth-fold Cartesian prod-uct Z�k. They are all mapped onto unit vectors because j l�AjAj

†�l�=1. The images of different basis states areclearly mutually orthogonal whenever they correspond todifferent k-tuples z. If they have z in common, they are alsoorthogonal since we obtain then the inner product

j

l�AjAj†�l� z1, . . . ,zj + 1, . . . ,zk�z1, . . . ,zj + 1, . . . ,zk�

= j

l�AjAj†�l� = l�l� = 0.

To see that U commutes with the total Hamiltonian HTªH� 1+1 � HE we observe that for every eigenstate �l� of Hwith eigenvalue �l we have

Aj�l� = ��l,j� ,

where ��l,j� is some state with

H��l,j� = ��l + � j��l,j� .

We have

�Aj � Sj��l� � �z�� = ��l,j� � �z1, . . . ,zj − 1, . . . ,zk� ,

which is also an eigenstate of HT for the eigenvalue �+ j� jzj as �l� � �z� is. That is, U maps energy basis statesonto energy basis states with the same eigenvalues, i.e., itcommutes with HT. We can now choose ��ª �0� as the stateof the environment. �

Note that the state U�� U† appearing in the ex-tension of Theorem 6 has the same energy distribution withrespect to the extended Hamiltonian as � has with respect tothe original system Hamiltonian. This implies that the distri-bution of energy values in the joint state of system plus en-vironment is given in terms of the distribution of input andoutput energies. Hence we may now apply our bounds on theinformation deficit to the problem of transmitting the stateswhen only limited quantum capacity is available.

Theorem 6 (information loss and quantum capacity).Given a covariant completely positive map G with privatechannel capacity Cp�G�. Then the difference between thetiming information of input and output satisfies

Iin − Iout ��Ein�8

64�9 Eout4 � + 8 Ein

4 �� Ein4 �

−1

2Cp�G� ,

where ��Ein�2 and Ein4 � refer to the variance and the fourth

moment of the incoming signal and similarly, Eout4 � denotes

the fourth moment of the outgoing signal.Proof. Construct a unitary energy conserving extension of

G according to Lemma 6. Let EoutªEA denote the energy ofthe output signal and EB the energy of the environment. Thisimplies that EinªE=EA+EB is the initial energy. To get abound for EB

4�= �Ein−Eout�4� we use �Ein−Eout�� Ein�


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+ �Eout� and hence �Ein−Eout�4�8�Ein4 +Eout

4 �. Then we obtainthe statement using Theorem 3. �

VII. IMPLICATIONS FOR THE ENERGY LOSS

In this section we want to explain why we expect thebroadcasting problem to be specific to low-power devices.One reason is, certainly, that in current technology, informa-tion processing devices are not Hamiltonian systems. Sincethe system is not closed, a unitary description of the signalpropagation is not justified. Furthermore, quantum broad-casting gets only relevant when the time inaccuracy of aclock signal is not dominated by classical noise of highlymixed density operators. In the latter case, the energy-timeuncertainty is irrelevant. This is in agreement with the resultsin Ref. �9� showing �in terms of Fisher information� thatquantum bounds on broadcasting timing information get rel-evant when the signal energy times the considered timingaccuracy is on the scale of �. However, there is also anotherlink between energy consumption of information processingdevices and broadcasting problems that we have not men-tioned before. The idea is that loss of timing informationinevitably leads to loss of free energy in covariant devices.This is shown in �14�. We describe the relevant results.

First, we need the notion of passive devices, i.e., deviceshaving no additional energy source apart from the consideredincoming signal. In other words, all energy resources areexplicitly included into the description.

Definition 3 (passive device). A device with quantum in-put state � and output G�� is called passive if G is imple-mented without energy supply, i.e.,

F„G��… � F�� ∀ � ,

where F��ª tr��H�−kTS�� is the free energy of the systemin the state � with reference temperature T and Boltzmannconstant k.

We have shown in �14� that covariant passive channelsthat decrease the timing information decrease also the freeenergy. We rephrase this result formally.

Theorem 7 (loss of timing information implies free energyloss�. Let G be a completely positive trace-preserving mapdescribing a covariant passive device. The free energy losscaused by G can be bounded from below by the loss oftiming information,

F�� − F„G��… kT�I�� − I„G��…� .

This shows that the channel can only be thermodynamicallyreversible if it does not subject the signal to a stochasticallyfluctuating time delay, i.e., it must conserve the timing infor-mation. The result is less trivial than it may seem at firstsight. The increase of signal entropy caused by the additionaltime delay could in principle be compensated by an increaseof its inner energy such that the free energy of the system isconserved. The covariance condition is indeed required toshow �14� that the free energy splits up into the followingtwo components:

F�� = kTI�� + F�� ,

which cannot be converted into each other.

Together with Theorem 7 we even obtain statements ofthe thermodynamical irreversibility of the signal transmis-sion.

Theorem 8 (free energy loss in classical channels�. Let �be a quantum state whose timing information has the broad-casting loss �min. Then every channel G satisfies

Cp�G� 2��min −1

kT�F�� − F„G��…� .

In particular, for every channel with capacity Cp�G�=0 wehave

F�� − F„G��… 2

kT�min.

We may combine Theorem 8 and Theorem 6 and obtainthe following result.

Theorem 9 (free energy conservation and quantum capac-ity). Given a passive covariant device G with private channelcapacity Cp�G�. Let G be applied to a pure input state �.Then the free energy loss caused by applying G to � satisfies

F�� − F„G��… kT� ��Ein�8

64�9 Eout4 � + 8 Ein

4 �� Ein4 �

−1

2Cp�G� ,

with Ein and Eout as in Theorem 6.It would be desirable to find similar results for mixed

states. However, it seems to be hard to provide generalbounds. Nevertheless, Theorem 9 shows why time covari-ance brings aspects of quantum information theory into thetheory of low-power signal processing. In the context of syn-chronization protocols we have already described in �38�why covariance gives rise to additional limitations of ther-modynamically reversible information transfer with classicalchannels.

VIII. CONCLUSIONS

We have described a quantum broadcasting problem thatarises naturally in classical low-power signal processing. If atime-invariant device transmits a signal such that the outputsignal contains the same amount of Holevo informationabout an absolute time frame as the input the following twoalternatives are possible: Either the channel has nonzeroquantum capacity or it has internally solved a quantumbroadcasting problem and copied the same amount of infor-mation to its environment. But this is not possible since theHolevo information of noncommuting ensembles cannot bebroadcast without loss. Thus, the time-covariant transmissionof signals in a way that causes no stochastic time delay of thesignal requires devices with nonzero quantum capacity. Butavoiding stochastic time delays is, as we have argued, a nec-essary requirement in order to avoid loss of free energy.Thus, we have described a link between quantum informa-tion theory and the theory of classical low-power processing.

ACKNOWLEDGMENT

The authors would like to thank Rolf Gohm for bringingthe BW topology to their attention.


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APPENDIX: BOUNDED-WEAK COMPACTNESS OF THESET OF QUANTUM OPERATIONS

To state our lemma �needed in the proof of Theorem 1�we must introduce the so-called bounded-weak �BW� topol-ogy �39�. Let V ,W be two Banach spaces and L�V ,W*� bethe space of bounded linear maps from V to the dual of W. Ifwe construct a Banach space B by the closed linear span of�v � w �v�V ,w�W� with respect to some Banach crossnorm �40� we can identify L�V ,W*� with the dual space B*

by l �v � w�ª l�v��w� for every l�L�V ,W*�. The weak* to-pology induced by B on L�V ,W*�=B* is called the bounded-weak topology. In the following it will be essential that theunit sphere of B* is weak* compact due to the Banach-Alaoglu theorem �41�.

On the space L1�H� of trace-class operators acting onsome Hilbert space H we have a natural weak* topologydefined by interpreting the elements of L1�H� as boundedlinear functionals on the C*-algebra K�H�, the set of compactoperators �42�. We have then the following.

Lemma 7 (compactness of set of quantum operations). For

two Hilbert spaces H, and H let Gn :L1�H�→L1�H� be a

sequence of trace-preserving CP-maps. Let H be a third Hil-bert space. Then there is a trace-preserving CP-map G such

that for every density operator � acting on H � H the se-quence �id � Gn�� has �id � G�� as a weak* cluster point.

Proof. By setting VªL1�H � H� and W*ªL1�H � H�

=K�H � H�* we interpret the considered CP-maps as ele-ments in the unit sphere of L�V ,W*�. By the above compact-ness arguments the sequence �id � Gn� has a BW-clusterpoint. It is straightforward to check that it is a completelypositive and trace-preserving map that acts trivially on theleft tensor component. It is therefore of the form id � G withsome quantum operation G. By construction of the BW to-pology we conclude that �id � G�� is a cluster point of �id� Gn�� with respect to the weak* topology induced by the

compact operators on H � H. �

�1� Semiconductor Spintronics and Quantum Computation, editedby D. Awschalom, D. Loss, and N. Samarth �Springer, Berlin,2002�.

�2� M. Nielsen and I. Chuang, Quantum Computation and Quan-tum Information �Cambridge University Press, Cambridge,2000�.

�3� P. Benioff, J. Stat. Phys. 22, 562 �1980�.�4� R. Feynman, Optics News 11, 11 �1985�.�5� N. Margolus, “Parallel quantum computation,” in Complexity,

Entropy, and the Physics of Information, edited by W. Zurek�Addison-Wesley Longman, New York, 1990�.

�6� D. Janzing and P. Wocjan, Quantum Inf. Process. 4, 129�2005�.

�7� D. Janzing, Phys. Rev. A 72, 042324 �2005�.�8� L. Hardy, e-print quant-ph/0101012.�9� D. Janzing and T. Beth, IEEE Trans. Inf. Theory 49, 230

�2003�.�10� D. Janzing and Th. Beth, “Are there quantum bounds on the

recyclability of clock signals in low power computers?,” Pro-ceedings of the DFG-Kolloquium �VIVA, Chemnitz, 2002�.See also e-print quant-ph/0202059.

�11� H. Barnum, C. Caves, C. Fuchs, R. Jozsa, and B. Schumacher,Phys. Rev. Lett. 76, 2818 �1996�.

�12� D. Janzing, J. Math. Phys. 46, 122107 �2005�.�13� P. Lax and R. Phillips, Scattering Theory �Academic, New

York, 1967�.�14� D. Janzing, J. Stat. Phys. 125, 757 �2006�.�15� J. Vaccaro, F. Anselmi, H. Wiseman, and K. Jacobs, e-print

quant-ph/0501121.�16� G. Chiribella, G. M. D’Ariano, P. Perinotti, and M. F. Sacchi,

Phys. Rev. Lett. 93, 180503 �2004�.�17� M. de Burgh and S. D. Bartlett, Phys. Rev. A 72, 042301

�2005�.�18� T. Rudolph and L. Grover, Phys. Rev. Lett. 91, 217905 �2003�.�19� S. van Enk, Phys. Rev. A 71, 032339 �2005�.�20� S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Phys. Rev. A

70, 032307 �2004�.

�21� G. Lindblad, Lett. Math. Phys. 47, 189 �1999�.�22� P. Hayden, R. Josza, D. Petz, and A. Winter, Commun. Math.

Phys. 246, 359 �2004�.�23� H. Umegaki, Kodai Math. Semin. Rep. 14, 59 �1962�.�24� D. Petz, Commun. Math. Phys. 105, 123 �1986�.�25� O. Bratteli and D. Robinson, Operator Algebras and Quantum

Statistical Mechanics �Springer, New York, 1987�, Vol. 1.�26� M. Ohya and D. Petz, Quantum Entropy and its Use �Springer-

Verlag, Berlin, 1993�.�27� D. Bruß, M. Cinchetti, G. M. D’Ariano, and C. Macchiavello,

Phys. Rev. A 62, 012302 �2000�.�28� J. Stam, Inf. Control. 2, 101 �1959�.�29� D. Janzing and T. Beth, e-print quant-ph/0301125.�30� O. Johnson, IEEE Trans. Inf. Theory 50, 1581 �2004�.�31� T. Cover and J. Thomas, Elements of Information Theory

�Wiley, New York, 1991�.�32� C. Fuchs, Quantum Inf. Comput. 4, 467 �2004�.�33� C. Fuchs and M. Sasaki, Quantum Inf. Comput. 3, 377 �2003�.�34� K. Audenaert, C. Fuchs, C. King, and A. Winter, Quantum Inf.

Comput. 4, 1 �2004�.�35� I. Devetak, IEEE Trans. Inf. Theory 51, 44 �2005�.�36� D. Janzing, Quantum Computing Models as a Tool Box for

Controlling and Understanding the Nanoscopic World, Infor-matik in Forschung und Entwicklung, special issue Quanten-technologie �Springer-Verlag, Berlin, 2006�, pp. 83–90.

�37� K. Kraus, States, Effects, and Operations: Fundamental No-tions in Quantum Theory �Springer-Verlag, Berlin, 1983�.

�38� D. Janzing and T. Beth, e-print quant-ph/0306023.�39� V. Paulsen, Completely Bounded Maps and Dilations �Long-

man Scientific Technical, New York, 1986�.�40� A. Grothendieck, “Produits tensoriels topologiques et espace

nuclieare,” Memoirs of the American Mathematical Society�American Mathematical Society, Providence, RI, 1955�.

�41� J. Conway, A Course in Functional Analysis �Springer, NewYork, 1990�.

�42� G. Murphy, C*-Algebras and Operator Theory �Academic,Boston, 1990�.


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Quantum broadcasting problem in classical low-power signal processing

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