QQuantum Communication With Zero-CapacityChannels
Graeme Smith and Jon Yard IBM TJ Watson Research Center1101 Kitchawan Road, Yorktown Heights, NY 10598 Quantum InstituteCenter for Nonlinear Studies (CNLS)Computer, Computational and Statistical Sciences (CCS-3)Los Alamos National LaboratoryLos Alamos, NM 87545
Communication over a noisy quantum channel introduces errors inthe transmission that must be corrected. A fundamental bound onquantum error correction is the quantum capacity, which quanti-fies the amount of quantum data that can be protected. We showtheoretically that two quantum channels, each with a transmissioncapacity of zero, can have a nonzero capacity when used together.This unveils a rich structure in the theory of quantum communica-tions, implying that the quantum capacity does not uniquely specifya channel’s ability for transmitting quantum information. a r X i v : . [ qu a n t - ph ] F e b oise is the enemy of all modern communication links. Cellular, internet and satellitecommunications all depend crucially on active steps taken to mitigate and correct for noise.The study of communication in the presence of noise was formalized by Shannon ( ), whosimplified the analysis by making probabilistic assumptions about the nature of the noise.By modeling a noisy channel N as a probabilistic map from input signals to output signals,the capacity C ( N ) of N is defined as the number of bits which can be transmitted perchannel use, with vanishing errors in the limit of many transmissions. This capacity iscomputed via the formula C ( N ) = max X I ( X ; Y ) where the maximization is over randomvariables X at the input of the channel, Y is the resulting output of the channel andthe mutual information I ( X ; Y ) = H ( X ) + H ( Y ) − H ( X, Y ) quantifies the correlationbetween input and output. H ( X ) = − (cid:80) x p x log p x denotes the ‘Shannon entropy’,which quantifies the amount of randomness in X . The capacity, measured in bits perchannel use, is the fundamental bound between communication rates that are achievablein principle, and those which are not. The capacity formula guides the design of practicalerror correction techniques by providing a benchmark against which engineers can test theperformance of their systems. Practical implementations guided by the capacity resultnow come strikingly close to the Shannon limit ( ).A fundamental prediction of the capacity formula is that the only channels with zerocapacity are precisely those for which the input and output are completely uncorrelated.Furthermore, suppose one is given simultaneous access to two noisy channels N and N .The capacity of the product channel N ×N , where the channels are used in parallel, takesthe simple form C ( N × N ) = C ( N ) + C ( N ), i.e., the capacity is additive. Additivityshows that capacity is an intrinsic measure of the information conveying properties of achannel.Quantum data is an especially delicate form of information and is particularly suscepti-2le to the deleterious effects of noise. Because quantum communication promises to allowunconditionally secure communication ( ), and a quantum computer could dramaticallyspeed up some computations ( ), there is tremendous interest in techniques to protectquantum data from noise. A quantum channel N models a physical process which addsnoise to a quantum system via an interaction with an unobservable environment (Fig. 1),generalizing Shannon’s model and enabling a more accurate depiction of the underlyingphysics. In this setting, it is natural to ask for the capacity of a quantum channel fortransmitting quantum mechanical information ( ), and whether it has a simple formulain analogy with Shannon’s.Just as any classical message can be reversibly expressed as a sequence of bits, aquantum message, i.e. an arbitrary state of a given quantum system, can be reversiblytransferred to a collection of two-level quantum systems, or ‘qubits’, giving a measure ofthe size of the system. The goal of quantum communication is to transfer the joint stateof a collection of qubits from one location to another (Fig. 2). The quantum capacity Q ( N ) of a quantum channel N is the number of qubits per channel use that can bereliably transmitted via many noisy transmissions, where each transmission is modeledby N . Although noiseless quantum communication with a noisy quantum channel isone of the simplest and most natural communication tasks one can imagine for quantuminformation, it is not nearly as well understood as its classical counterpart.An analogue for mutual information in the quantum capacity has been proposed ( )and called the ‘coherent information’: Q (1) ( N ) = max ρ A ( H ( B ) − H ( E )) . (1)The entropies are measured on the states induced at the output and environment of thechannel (Fig. 2) by the input state ρ A , where H ( B ) is the ‘von Neumann entropy’ of the3tate ρ B at the output. Coherent information is rather different from mutual information.This difference is closely related to the no-cloning theorem ( ), which states that quantuminformation cannot be copied, as the coherent information roughly measures how muchmore information B holds than E . The no-cloning theorem itself is deeply tied to thefundamentally quantum concept of entanglement, in which the whole of a quantum systemcan be in a definite state while the states of its parts are uncertain.The best known expression for the quantum capacity Q is given ( ) by the‘regularization’ of Q (1) : Q ( N ) = lim n →∞ n Q (1) ( N × n ) . Here N × n represents the parallel use of n copies of N . The asymptotic nature of thisexpression prevents one from determining the quantum capacity of a given channel inany effective way, while also making it difficult to reason about its general properties.In contrast to Shannon’s capacity, where regularization is unnecessary, here it cannot beremoved in general (
11 , 12 ). Consequently, even apparently simple questions, such asdetermining from a channel’s description whether it can be used to send any quantuminformation, are currently unresolved. We find that the answer to this question dependson context; there are pairs of zero-capacity channels which, used together, have a positivequantum capacity (Fig. 3). This shows the quantum capacity is not additive, and thus thequantum capacity of a channel does not completely specify its capability for transmittingquantum information.While a complete characterization of zero-capacity channels is unknown, certain classesof zero-capacity channels are known. One class consists of channels for which the jointquantum state of the output and environment is symmetric under interchange. These‘symmetric channels’ are quite different from Shannon’s zero-capacity channels, as theydisplay correlations between the input and output. However, they are useless by them-4elves for quantum communication because their symmetry implies that any capacitywould lead to a violation of the no cloning theorem ( ). Another class of zero-capacity channels are entanglement-binding channels (
14 , 15 ), also called ‘Horodeckichannels’, which can only produce very weakly entangled states satisfying a conditioncalled positive partial transposition ( ).Even though channels from one or the other of these classes cannot be combined tofaithfully transmit quantum data, we find that when one combines a channel from eachclass, it is sometimes possible to obtain a positive quantum capacity. We do this byproving a new relationship between two further capacities of a quantum channel: theprivate capacity ( ) and the assisted capacity ( ).The private capacity P ( N ) of a quantum channel N is the rate at which it can beused to send classical data that is secure against an eavesdropper with access to theenvironment of the channel. This capacity is closely related to quantum key distributionprotocols ( ) and was shown ( ) to equal the regularization of the ‘private information’: P (1) ( N ) = max X,ρ Ax ( I ( X ; B ) − I ( X ; E )) , (2)where the maximization is over classical random variables X and quantum states ρ Ax onthe input of N depending on the value x of X .In order to find upper bounds on the quantum capacity, an ‘assisted capacity’ wasrecently introduced ( ) where one allows the free use of arbitrary symmetric channels toassist quantum communication over a given channel. Letting A denote a symmetric chan-nel of unbounded dimension (the strongest such channel), the assisted capacity Q A ( N )of a quantum channel N satisfies ( ) Q A ( N ) = Q ( N × A ) = Q (1) ( N × A ) . Because the dimension of the input to A is unbounded, we cannot evaluate the assisted5apacity in general. Nonetheless, the assisted capacity helps to reason about finite-dimensional channels.While Horodecki channels have zero quantum capacity, examples of such channels withnonzero private capacity are known (
18 , 19 ). One of the two zero-capacity channels wewill combine to give positive joint capacity is such a ‘private Horodecki channel’ N H , andthe other is the symmetric channel A . Our key tool is the following new relationshipbetween the capacities of any channel N (Fig. 4): P ( N ) ≤ Q A ( N ) . (3)A channel’s assisted capacity is at least as large as half its private capacity. It followsthat any private Horodecki channel N H has a positive assisted capacity, and thus the twozero-capacity channels N H and A satisfy Q A ( N H ) = Q ( N H × A ) > . Although our construction involves systems of unbounded dimension, one can showthat any private Horodecki channel can be combined with a finite symmetric channel togive positive quantum capacity. In particular, there is a private Horodecki channel actingon a four-level system ( ). This channel gives positive quantum capacity when combinedwith a small symmetric channel – a 50% erasure channel A e with a four-level input whichhalf of the time delivers the input state to the output, otherwise telling the receiver thatan erasure has occurred. We show ( ) that the parallel combination of these channelshas a quantum capacity greater than 0.01.We find this ‘superactivation’ to be a startling effect. One would think that the ques-tion, “can this communication link transmit any information?” would have a straight-forward answer. However, with quantum data, the answer may well be “it dependson the context”. Taken separately, private Horodecki channels and symmetric channels6re useless for transmitting quantum information, albeit for entirely different reasons.Nonetheless, each channel has the potential to “activate” the other, effectively cancelingthe other’s reason for having zero capacity. We know of no analogue of this effect in theclassical theory. Perhaps each channel transfers some different, but complementary kindof quantum information. If so, can these kinds of information be quantified in an opera-tionally meaningful way? Are there other pairs of zero-capacity channels displaying thiseffect? Are there triples? Does the private capacity also display superactivation? Can allHorodecki channels be superactivated, or just those with positive private capacity? Whatnew insights does this yield for computing the quantum capacity in general?Besides additivity, our findings resolve two open questions about the quantum capacity.First we find ( ) that the quantum capacity is not a convex function of the channel.Convexity of a capacity means that a probabilistic mixture of two channels never has ahigher capacity than the corresponding average of the capacities of the individual channels.Violation of convexity leads to a counterintuitive situation where it can be beneficial toforget which channel is being used. We also find ( ) channels with an arbitrarily largegap between Q (1) – the so-called ‘hashing rate’ ( ) – and the quantum capacity. Ithad been consistent with previous results (
11 , 12 ) to believe that Q and Q (1) would beequal up to small corrections. Our work shows this is not the case and indicates that thehashing rate is an overly pessimistic benchmark against which to measure the performanceof practical error-correction schemes. This could be good news for the analysis of faulttolerant quantum computation in the very noisy regime.Forms of this sort of superactivation are known in the multiparty setting, where severalseparated parties communicate via a quantum channel with multiple inputs or outputs(
21 , 22 , 23 , 24 ), and have been conjectured for a quantum channel assisted by classicalcommunication between the sender and receiver (
25 , 26 ). Because these settings are7ather complex, it is perhaps unsurprising to find exotic behavior. In contrast, the problemof noiseless quantum communication with a noisy quantum channel is one of the simplestand most natural communication tasks imaginable in a quantum mechanical context.Our findings uncover a level of complexity in this simple problem that had not beenanticipated and point towards several fundamentally new questions about informationand communication in the physical world. 8 eferences and Notes
1. C. E. Shannon,
Bell Syst. Tech. J. , 379 (1948).2. T. Richardson, R. Urbanke, IEEE Communications Magazine , 126 (2003).3. C. H. Bennett, G. Brassard, Proceedings of the IEEE International Conference onComputers, Systems and Signal Processing p. 175 (1984).4. P. W. Shor,
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Phys. Rev. A , 1613 (1997).9. P. W. Shor, Quantum error correction. Lecture notes, MSRIWorkshop on Quantum Computation, 2002. Available online at .10. I. Devetak, IEEE Trans. Inform. Theory , 44 (2005).11. D. DiVincenzo, P. W. Shor, J. A. Smolin, Phys. Rev. A , 830 (1998).12. G. Smith, J. A. Smolin, Phys. Rev. Lett. , 030501 (2007).13. C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, Phys. Rev. Lett. , 3217 (1997).14. M. Horodecki, P. Horodecki, R. Horodecki, Phys. Lett. A , 1 (1996).95. P. Horodecki,
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Phys. Rev. Lett. , 1413 (1996).17. G. Smith, J. Smolin, A. Winter, The quantum capacity with symmetric side channels. arXiv:quant-ph/0607039 .18. K. Horodecki, M. Horodecki, P. Horodecki, J. Oppenheim, Phys. Rev. Lett. , 160502(2005).19. K. Horodecki, L. Pankowski, M. Horodecki, P. Horodecki, IEEE Trans. Inform. The-ory , 2621 (2008).20. Further details can be found in Supporting Online Material.21. P. Shor, J. Smolin, A. Thapliyal, Phys. Rev. Lett. , 107901 (2003).22. W. Dur, J. Cirac, P. Horodecki, Phys. Rev. Lett. , 020503 (2004).23. R. Duan, Y. Shi, Phys. Rev. Lett. (2008).24. L. Czekaj, P. Horodecki, Nonadditivity effects in classical capacities of quantummultiple-access channels. arXiv:0807.3977 .25. P. Horodecki, M. Horodecki, R. Horodecki,
Phys. Rev. Lett. , 1056 (1999).26. P. Shor, J. Smolin, B. Terhal, Phys. Rev. Lett. , 2681 (2001).27. We are indebted to Charlie Bennett, Clifton Callaway, Eddy Timmermans, Ben Tonerand Andreas Winter for encouragement and comments on an earlier draft. JY issupported by the Center for Nonlinear Studies (CNLS) and the Quantum Institutethrough grants provided by the LDRD program of the U.S. Department of Energy.10igure 1: Representation of a quantum channel. A channel reversibly transfers the stateof a physical system in the laboratory of the sender to the combination of the systempossessed by the receiver and an ‘environment’ which is inaccessible to the users of thechannel. Discarding the environment results in a noisy evolution of the state. The inputand output denote separate places in space and or time, modeling for example a leakyoptical fiber or the irreversible evolution of the state of a quantum dot.11 lice Bob Figure 2: The quantum capacity of a quantum channel. Quantum data is held by a sender(traditionally called Alice), who would like to transmit it to a reciever (Bob) with manyparallel uses of a noisy quantum channel N . Alice encodes the data with a collectiveencoding operation E which results in a joint quantum state on the inputs of the channels N × n . The encoded state is sent through the noisy channels. When Bob receives thestate, he applies a decoding operation D which acts collectively on the many outputsof the channels. After decoding, Bob holds the state which Alice wished to send. Thequantum capacity is the total number of qubits in the state Alice sends divided by thenumber of channel uses. 12igure 3: (A) Alice and Bob attempt to separately use two zero-capacity channels N and N to transfer quantum states. Alice uses separate encoders E and E for each group ofchannels and Bob uses separate decoders D and D . Any attempt will fail because thecapacity of each channel is zero. (B) The same two channels being used in parallel for thesame task. Alice’s encoder E now has simultaneous access to the inputs of all channelsbeing used and Bob’s decoding D is also performed jointly. Noiseless communication isnonetheless possible because Q is not additive.13igure 4: Relating the private capacity and the assisted capacity. A straightforward proofof Eq. 3 uses the expression ( ) Q A ( N ) = max ρ XAC ( I ( X ; B | C ) − I ( X ; E | C )) . Here, I ( X ; B | C ) is the ‘conditional mutual information’: H ( XC ) + H ( BC ) − H ( XBC ) − H ( C ).It is evaluated on the state obtained by putting the A part of a state ρ XAC into thechannel N , which can be thought of as mapping A → BE as in Fig. 2. The maximizationhere is similar in form to Eq. 2, but is over a less constrained type of state. Therefore, P (1) ( N ) ≤ Q A ( N ). This bound holds for the associated regularized quantities and sinceregularization does not change Q A , Eq. 3 follows.14 uantum Communication with Zero-Capacity Channels (Supporting Online Material) Graeme Smith and Jon Yard
In this supporting material, we assume a basic knowledge of quantum informationtheory at the level of ( ). We denote the dimension of a Hilbert space A as | A | whileabbreviating tensor products as AB = A ⊗ B and the restriction of a density matrix ρ AB tothe subsystem A as ρ A . By an ensemble of states { p x , ρ Ax } , we mean that the density matrix ρ Ax on A occurs with probability p x . We freely associate such ensembles with the jointstate (cid:80) x p x | x (cid:105)(cid:104) x | X ⊗ ρ Ax , where {| x (cid:105) X } denotes an orthonormal basis for X . For a channelwith input A , output B and environment E , we abbreviate I c ( N , ρ A ) = H ( B ) − H ( E ) sothat Q (1) ( N ) = max ρ A I c ( N , ρ A ). Superactivation with finite dimensional channels
Our key tool in this work was the relationship P ( N ) ≤ Q ( N ⊗ A ) = Q A ( N ), whichis valid for any quantum channel N . A disadvantage of this result is that the input andoutput systems of the channel A are infinite. Guided by that result, we now give a weaker,more manageable bound in which a 50%-erasure channel A e with finite-dimensional inputand output systems plays the role of A .Consider a channel N with input A , output B and environment E . By Eq. 2 in themain text, observe that every ensemble { p x , ρ Ax } yields the lower bound I ( X ; B ) − I ( X ; E ) ≤ P (1) ( N )where the mutual informations are evaluated on the state ρ XBE = (cid:88) x p x | x (cid:105)(cid:104) x | X ⊗ ρ BEx ρ BEx is the joint state of the output and environment when ρ Ax is sent through thechannel. Furthermore, if the input to N is finite, this bound is achievable using a finiteensemble ( ).Currently, every known private Horodecki channel N H also satisfies P (1) ( N H ) > I ( X ; B ) − I ( X ; E ) > Theorem:
Given an ensemble { p x , ρ Ax } and a channel N with input A , output B andenvironment E , let A e be a 50%-erasure channel with input space C of dimension equalto the sum of the ranks of the states ρ Ax . Then there is a state ρ AC such that I c ( N ⊗ A e , ρ AC ) = (cid:0) I ( X, B ) − I ( X ; E ) (cid:1) . The next section describes a private Horodecki channel ( ) N (4) H with a four-dimensionalinput and an ensemble with two rank-two states such that I ( X ; B ) − I ( X ; E ) > . A (4) e with a four-dimensional input C and a state ρ AC such that Q ( N H ⊗ A e ) ≥ I c ( N ⊗ A e , ρ AC ) > . ρ Ax = | ρ x (cid:105)(cid:104) ρ x | A , we have the identity ( ) I c ( N , ρ A ) = I ( X ; B ) − I ( X ; E )where ρ A = (cid:80) x p x | ρ x (cid:105)(cid:104) ρ x | . Consequently, the coherent information is obtained by restrict-ing the maximization for the private information to pure state ensembles. Since we beginwith a mixed state ensemble, we consider a related ensemble of purified states and sendthe purifying system through a 50%-erasure channel.2 roof of Theorem: Define purifications | ρ x (cid:105) AC of the states ρ Ax such that the supportsof the ρ Cx are disjoint. Then the pure state | ρ (cid:105) XAC = (cid:88) x √ p x | x (cid:105) X | ρ x (cid:105) AC is a purification of the state (cid:80) x p x | x (cid:105)(cid:104) x | X ⊗ ρ Ax associated to the ensemble. We willevaluate the coherent information resulting from sending A through N and C through A e . Denoting the output of A e by D and the environment of A e by F , we obtain thefollowing chain of inequalities: I c ( N ⊗ A e , ρ AC ) = H ( BD ) − H ( EF ) (S1)= (cid:0) H ( B ) − H ( EC ) (cid:1) + (cid:0) H ( BC ) − H ( E ) (cid:1) (S2)= (cid:0) H ( B ) − H ( XB ) (cid:1) + (cid:0) H ( XE ) − H ( E ) (cid:1) (S3)= (cid:0) I ( X ; B ) − I ( X ; E ) (cid:1) . (S4)In Eq. S1, the entropies are evaluated on the state obtained by sending the AC parts of | ρ (cid:105) XAC through the respective channels. Eq. S2 holds because with equal probability, N e either delivers C to its output D or to its environment F , so this difference of entropiescan be rewritten in terms of quantities evaluated on the state on XBEC obtained bysending only the A part of | ρ (cid:105) XAC through N . Eq. S3 is true because bipartitions of anypure state have the same entropy and Eq. S4 uses the definition of mutual informationafter adding H ( X ) to the first term and subtracting it from the second. (cid:117)(cid:116) four-dimensional private Horodecki channel For convenience we give an explicit description of the four-dimensional private Horodeckichannel N (4) H from ( ). The action of any channel N can be written in Kraus form as N ( ρ ) = (cid:88) k N k ρN † k , where the Kraus matrices N k satisfy (cid:80) k N † k N k = I . We denote the input of N (4) H as atensor product of two qubits A = A A and denote the output as B . This channel isspecified by the following six Kraus matrices: (cid:113) q I ⊗ | (cid:105)(cid:104) | , (cid:113) q Z ⊗ | (cid:105)(cid:104) | , (cid:113) q Z ⊗ Y, (cid:113) q I ⊗ X, (cid:112) − qX ⊗ M , (cid:112) − qY ⊗ M . Here, q = √ √ , while X , Y and Z are the usual Pauli matrices and M = (cid:32) (cid:112) √ (cid:112) − √ (cid:33) , M = (cid:32) (cid:112) − √ (cid:112) √ (cid:33) . A lower bound of 0 .
02 on the private capacity P ( N (4) H ) is obtained via the ensembleconsisting of two equiprobable states ρ Ax = | x (cid:105)(cid:104) x | A ⊗ I A because the state ρ XBE resultingfrom putting A into the channel N (4) H satisfies ( ) I ( X ; B ) − I ( X ; E ) ≥ − q log q − (1 − q ) log (1 − q ) > . . onconvexity of quantum capacity We now use the results of the first section to show that Q is not convex. Fix a privateHorodecki channel N H and a 50%-erasure channel A e such that the input A to N H andthe input C to A e have the same dimension, and such that there is a state ρ AC which issymmetric under interchanging A and C satisfying I c ( N H ⊗ A e , ρ AC ) >
0. In particular,the channel N (4) H from the previous section and a four-dimensional erasure channel A (4) e satisfy these criteria with respect to the state ρ AC = 12 (cid:0) | (cid:105)(cid:104) | A ⊗ | (cid:105)(cid:104) | C + | (cid:105)(cid:104) | A ⊗ | (cid:105)(cid:104) | C (cid:1) ⊗ | φ + (cid:105)(cid:104) φ + | A C where | φ + (cid:105) = √ ( | (cid:105) + | (cid:105) ) , A = A A and C = C C . Identifying A (cid:39) C , we definethe channel M p = p N H ⊗ | (cid:105)(cid:104) | + (1 − p ) A e ⊗ | (cid:105)(cid:104) | where 0 ≤ p ≤
1. With probability p , this channel applies N H to the input and otherwiseapplies A e , while telling the receiver which channel was applied. Although M p is a convexcombination of the zero-capacity channels N H ⊗ | (cid:105)(cid:104) | and A e ⊗ | (cid:105)(cid:104) | , we will show thatfor small enough values of p , their convex combination M p has a positive capacity. Onany input state ρ , we have I c ( M p ⊗ M p , ρ ) = p I c ( N H ⊗ N H , ρ ) + p (1 − p ) I c ( N H ⊗ A e , ρ )+ p (1 − p ) I c ( A e ⊗ N H , ρ ) + (1 − p ) I c ( A e ⊗ A e , ρ ) . Since A e ⊗ A e is a symmetric channel, the last term is always zero. Choosing the state ρ = ρ AC , which was assumed to be symmetric, we find that I c ( M p ⊗ M p , ρ AC ) = 2 p (1 − p ) I c ( N H ⊗ A e , ρ AC ) + p I c ( N H ⊗ N H , ρ AC ) . For 0 < p <
1, the first term is positive by assumption. The second term can never begreater than zero because Q ( N H ) = 0, although it is lower bounded by − p c , where5 = log | E | . Here c is finite because | E | is finite when the input and output of N H havefinite dimension. Simple algebra reveals that I c ( M p ⊗ M p , ρ AC ) > p satisfying0 < p < I c ( N H ⊗ A e , ρ AC ) c + I c ( N H ⊗ A e , ρ AC ) . For the four-dimensional example of the previous section, one has c = log < p < . Arbitrarily large gap between Q (1) and Q Although it has long been known that Q can be strictly greater than Q (1) , there hasbeen speculation that deviations of Q from Q (1) may be fairly small. Thus, while theregularized nature of the capacity expression is unwieldy, we might hope that for practicalpurposes the quantum capacity is well approximated by Q (1) and analysis could proceed byconsidering the computable function Q (1) . Our work shows that this is not true, as thereexist channels M with Q (1) ( M ) = 0 for which the actual capacity can be arbitrarily large.Let N H be a private Horodecki channel and let A e be a 50%-erasure channel with the sameinput dimension for which Q (1) ( N H ⊗A e ) >
0. For example, the four-dimensional channelsdiscussed above would work. Define M to be a channel with input A = A A , where A isa qubit and A is the input space of N H and A e . The channel measures the first qubit A inthe {| (cid:105) , | (cid:105)} basis and, depending on the outcome, applies one of the channels N H or A e to A . The outcome of the measurement is revealed to the receiver. This channel can be seento have Q (1) ( M ) = 0 because Q (1) ( N H ) = Q (1) ( A e ) = 0. However, the sender has controlover which channel is applied to which input, so Q (1) ( M ⊗ M ) ≥ Q (1) ( N H ⊗ A e ) >
0. Byreplacing N H in the above discussion with n instances of N H , and similarly for A e , thisviolation can be made arbitrarily large. 6 eferences and Notes
1. M. Nielsen, I. Chuang,
Quantum Information and Computation (Cambridge UniversityPress, 2000).2. I. Devetak,
IEEE Trans. Inform. Theory , 44 (2005).3. K. Horodecki, L. Pankowski, M. Horodecki, P. Horodecki, IEEE Trans. Info. Theory54