arXiv:quant-ph/0407061v2 9 Aug 2006 rjc EQudrcnrc o IST-2001-37559. supp no. acknowledges contract furthermore under he visi RESQ done; hos project a was the during work for Information, present the grateful Quantum for is th AW Institute acknowledges Foundation. Caltech’s PH Fairchild EIA-0086038. Sherman no. the grant under Foundation a.j.winter@br (email: Kingdom United 1UB, BS8 tol,Bristol U 91125, CA graeme@cal Pasadena, [email protected], 107–81, [email protected], Caltech Information, Quantum nteaon fcmuiainrqie.Rcl htevery reductio that a Recall into required. directly communication translates of shared amount be the to in entangle state partial even the that in finding effect, this at of advantage sender the from communication no with all. prepared st be entangled maximally can Indeed, states. non-entangled than oig ihyetnldsae r much superdense are about states observations to entangled basic failed highly consumption most however, the coding: the result, with of That along one randomness. rate, exploit (Bob shared ebit receiver some and a qubit of with state same asymptot- entangled the can qubit [2] at (Alice) two In sender a [3]. a share [2], that ically entanglement shown one of furthermore was bit transmitting it tw one physically consuming asymptotically, by and that, communicated qubit be result can the qubits with possible, becomes osmto foeei.I h edrkosteiett of of identity coding the and superdense knows sent, bit sender be quantum the to bits If one state classical the ebit. of one two transmission of of the consumption super- by communication perhaps information the is of [1], phenomenon The coding entanglement. this dense share of parties example two best-known the if enhanced quently th showed paper earlier case minimal an the a states, In a at communication. unentangled receiver quantum preparing a and of and entanglement sender of a cost between states entangled ypyial rnmtigonly transmitting physically by icto,rmt tt rprto,spres coding superdense preparation, state remote tification, nei stoqubits. capacit two identification is sour shared quantum ebit memoryless the any an a that by demonstration of produced a result: and superdense states use our optimal entangled the of the for of protocol achievability applications without the two of res be proof all present direct two-for-one to also case, earlier need We unentangled that randomness. that the the find qubits and al of we of states at number entangled, entangled communication the no are maximally in between prepared interpolating reduction be transmitted, a to is states there the When consuming 2 A H n Sakoldetespoto h SNtoa Scie National US the of support the acknowledge University GS Mathematics, and of PH, Institute Department AA, the the with with is are Winter Smith A. 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2 1 1 dB von Neumann entropy of a state ρ as S(ρ)= Tr ρ log ρ. where we may choose C = (8π ln 2)− , and β = . − ln 2 rdA It generalizes the following lemma for rank-one ϕ, which II. THE UNIVERSAL PROTOCOL was proved in [3]. To begin, suppose that Alice would like to share a maxi- mally entangled state with Bob. Clearly, this can be accom- Lemma II.2 Let ϕ be chosen according to the Haar mea- | i plished without any communication – Alice need only perform sure on A B. Then, if 3 dB dA, operations on her half of a fixed maximally entangled state ⊗ ≤ ≤ shared between them. In particular, if ψ is an arbitrary maxi- Pr S(ϕB) < log dB α β | i d − − mally entangled state and we denote by Φ = 1 i i α2C d √d i=1
(5) | i | i| i exp (dBdA 1) 2 , a fixed maximally entangled state, then ψ can be expressed ≤ − − (log dB) | i P   as 2 1 1 dB where C = (8π ln 2)− as before and β = ln 2 d . ψ = Vψ IB Φd , (1) A ⊓⊔ | i ⊗ | i Proof (of Lemma II.1) If we let R be a space of dimension r where Vψ is a unitary transformation of Alice’s system which and τABR be a uniformly distributed state on A B R, then depends on ψ. This identity is equivalent to the following 1 | i ⊗ ⊗ Π is equal in distribution to U ϕU † , where Π circuit diagram, in which time runs from left to right: r τAB AB AB τAB is the projector onto the support of τAB. Let σ denote the r 1 I A unitary transformation − e(j+1 mod r) ej + ΠτAB that j=0 | ih | − Vψ implements a cyclic permutation on the eigenvectors ej of τ corresponding to non-zeroP eigenvalues. (There are{| r i}such Φd  ψ AB | i B | i eigenvalues except on a set of measure zero, which we will  (2) ignore.) We then have  r 1 Of course, in general, we would like to prepare an arbitrary 1 1 − k k ΠτAB = σ τAB σ− . (6) state ψAB that may not be maximally entangled, and to do r r k=0 so by| usingi as few resources as possible. Our general method X is as follows. Alice and Bob initially share a fixed maximally Eq. (6), together with the concavity of entropy, implies r entangled state Φd , to which Alice applies an isometry Vψ. | B i 1 1 k k She then sends a subsystem of dimension to Bob, who S TrA ΠτAB S(TrA σ τAB σ− ), (7) A2 dA2 r ≥ r applies a fixed unitary † . Alice’s goal is to make as   k=1 UA2B dA2 X small as possible while still reliably preparing ψAB . The which in turn gives procedure can again be summarized with a circuit| diagram,i Pr S(TrA UABϕU † ) < log dB α β although this time it is much less clear whether there exist AB − − choices of the operations Vψ and UA2B that will do the job:  r  1 k k Pr S(TrA σ τABσ− ) < log dB α β A ≤ r − − 1 k ! X=1 Vψ k k r Pr S(TrA σ τABσ− ) < log dB α β A2 ≤ − −  ψ = r Pr S(τ ) < log d α β , (8)  B B
ΦdB | i − − | i ❯  B  where the final step is a result of the
unitary invariance of UA† 2B (3)  τABR. Applying Lemma II.2 to Eq. (8) with A AR and  B B reveals that → Circuit (3) does provide a method for preparing the state → I ψ as long as UA2B ψ is maximally entangled across Pr S(TrA UABϕUAB† ) < log dB α β |thei A A B cut.⊗ (All such| statesi are related by an operation − − 1 2|  α2C  on Alice’s system alone.) We will now use this observa- (9) 12r exp rdB dA 2 . tion, together with the fact that high-dimensional states are ≤ − 4(log dB)   generically highly entangled [15], [16], [17], [18], [19], [20], ⊓⊔ to construct a protocol that prepares an arbitrary state with The idea behind the protocol is then simple: we will show I high fidelity. The precise statement about the entanglement of that there exists a single unitary UA2B such that A1 ⊗ generic states that we will need is the following lemma. UA2B ψA1A2B is almost maximally entangled across the | i A A B cut for all states ψA1A2B satisfying a bound on 1 2| | i Lemma II.1 Let ϕ be a state on A B proportional to their Schmidt coefficients and whose support on A2 B ⊗ ⊗ a projector of rank r and let UAB U(dAdB) be chosen lies in a large subspace S A2 B. Since any such ∈ I ⊂ ⊗ according to the Haar measure. Then, if 3 dB dA, A1 UA2B ψA1A2B is almost maximally entangled, we can ≤ ≤ then⊗ find an|exactly maximallyi entangled state which closely Pr S(TrA UABϕU † ) < log dB α β AB − − approximates it. This state, in turn, can be prepared by the α2C method of Circuit (3). More formally, the following general 12r exp rdAdB
, (4) ≤ − (log d )2 prescription can be made to succeed:  B  3

Protocol: To send an arbitrary pure state with maximal have Schmidt coefficient √λmax and reduction of Bob’s system ≤ Pr inf S(TrA2 UA2BϕS UA† 2B) to dimension dS. ϕ ′ | A1S i  < log dB α β δ η(γ) (12) 1) Alice and Bob share a maximally entangled state of − − − − 1 log dB = (log dS log λmax)+o(log dS) ebits on their γ  2 ′ † − A S Pr S(TrA2 UA2BϕS UA2B)

dS K F a , which implies that K log dS + log λmax + The main idea behind the proof, combining an exponential⊓⊔ log≥F ⌊2. ⌋ ≥ concentration bound with discretization, has been used a num- The− entanglement lower bound follows easily from conser- ber of times recently in quantum information theory [21], [12], vation of entanglement under local operations and classical [2]. (It is, of course, much older; see [27].) If there is a twist communication (LOCC): let Alice and Bob prepare a maxi- in the present application, it is illustrated in Eq. (13). Since mally entangled state Φ of Schmidt rank dS. If they were | 0i dS is comparable in size to dA2 dB , any prefactor significantly able to do this exactly, by the non-increase of entanglement ′ 2dA dS larger than (5/γ) 1 would have caused the probability under LOCC, they would need to start with at least log dS bound to fail. Therefore, it was crucial to first restrict to ebits. However, the protocol only succeeds in creating a state states maximally entangled between A and S, giving the ρ of fidelity F with Φ0 . By a result of Nielsen [28], this 1′ ≥ | i manageable prefactor, and then extend to general states and implies that for the entanglement of formation, larger A1 using majorization. EF (ρ) log dS 18√1 F log dS 2η 2√1 F . ≥ − − − −   III. OPTIMALITYOFTHEPROTOCOL Since EF cannot increase under LOCC, the right hand side The communication and entanglement resources of Propo- is also a lower bound on the number of ebits Alice and Bob sition II.3 are optimal up to terms of lower order than log d started with. S ⊓⊔ or log λmax: the amount of quantum communication cannot be reduced, neither can the sum of the entanglement and Corollary III.2 A superdense coding protocol of fidelity F ≥ quantum communication. (Entanglement alone can be reduced 1/2 for all dS -dimensional states with maximum Schmidt 1 at the cost of increasing the quantum communication.) We will coefficient √λmax must make use of at least 2 log dS + 1 ≤ 1 demonstrate the result in two steps. First we prove an optimal- log λmax + log F 1 qubits of communication. The 2 2 − ity result for the task of remotely preparing entangled quantum sum of qubit and ebit resources must be at least log dS − states using entanglement and classical communication. We 18√1 F log dS 2η 2√1 F . − − − then show that by teleporting the quantum communication of Proof Suppose there exists an
superdense coding protocol our superdense coding protocol for entangled states, we gener- which can prepare all d dimensional states with maximum ate the optimal remote state preparation protocol, meaning the S Schmidt coefficient √λ and which uses only Q qubits original superdense coding protocol must have been optimal. max and E ebits. Use teleportation≤ to transmit the qubits, turning it into a remote state preparation protocol. Proposition III.1 A remote state preparation protocol of fi- The qubit cost translates directly to a cbit cost of 2Q. From delity F 1/2 for all d -dimensional states with max- S Proposition III.1 we infer the lower bound on Q. The protocol imum Schmidt≥ coefficient √λ must make use of at ≤ max including teleportation requires Q + E ebits, thus the lower least log dS + log λ + log F 2 cbits and log dS max − − bound on Q + E follows from Proposition III.1 as well. 18√1 F log dS 2η 2√1 F ebits, where η(t) = ⊓⊔ − − − Thus, when F 1 and ignoring terms of order o(log dS), t log t. → −
the upper resource bounds from our protocol, and the above Proof Consider a remote state preparation protocol involving lower bound coincide. the transmission of exactly log K cbits which can, with fidelity IV. PROTOCOLFORAMEMORYLESSSOURCE F , prepare all dS dimensional states having maximum Schmidt coefficient √λmax. We will show that causality essentially The universal protocol of Proposition II.3 is easily adapted implies that K must be roughly as large as dSλmaxF . to the task of sending states produced by a memoryless source. In particular, suppose Alice wants to send Bob a message A standard application of typical subspace techniques gives dS 1 i 1,..., a , with a = λmax . One way she can control of the value of λmax and the effective size of the accomplish∈ this⌊ is⌋ by preparing (a⌈ purification⌉ of) the state states received by Bob, the two parameters determining the  1 ai σi = a k=1+a(i 1) k k on Bob’s system, with some resources consumed by the universal protocol. We model the − | ih | A1S m fixed basis k . The remote state preparation protocol will source A1S = pi, ϕ as a sequence of independent, P{| i} E { | i i}i=1 produce a state ρi for Bob which will have a fidelity F with identically distributed states with the intended state, σi. In order to decode the message, Bob A1S A1S A1S n (18) ai ϕi = ϕi1 ϕin simply measures Πi = k=1+a(i 1) k k . His probability of | i | i ⊗ · · · ⊗ | i − | ih | n decoding the message Alice intended is Tr(ρiΠi) F . occuring with probability pin = pi1 pi2 ...pi , where i = P ≥ n Now, imagine that Alice and Bob use the same protocol, i1i2 in. If we define S( S)= S ( i pi TrA1 ϕi ϕi ) and ¯ ··· E | ih | with the modification that rather than Alice sending cbits, Bob S( S ) = i piS(TrA1 ϕi ϕi ), Harrow combined coherent simply guesses which j 1,...,K Alice would have sent. classicalE communication| andih a| remoteP state preparation pro- ∈{ } P 1 ¯ The probability of Bob correctly identifying i in this case is tocol to demonstrate that a qubit rate of 2 (S( S) S( S)) F 1 1 ¯ E − E thus at least K — he has a probability K of correctly guessing and ebit rate of 2 (S( S )+ S( S)) are simultaneously achiev- j and, given a correct guess, a conditional probability F of able [11], an optimalE result [8]E which hinted at the existence correctly identifying i. However, since this protocol involves of the universal protocol. Here we show how the universal no forward communication from Alice to Bob, it can succeed protocol provides an alternate, perhaps more direct, route to dS 1 with probability no greater than − (by causality), hence Harrow’s rate pair. ⌊ a ⌋ 5

A1S Proposition IV.1 There exist protocols for superdense coding Applying the universal superdense coding protocol to σin , of entangled states with mean fidelity approaching one and we find that the number of qubits that must be sent is 1 ¯ asymptotically achieving the rate pair of 2 (S( S) S( S )) 1 n 1 E − E [log Rank Πρ,δ + log λmax]+ o(n) (27) qubits and (S( S)+ S¯( S)) ebits. 2 2 E E n ¯ 2 [S( S) S( S )] + cδ√n log(1 ǫ)+ o(n), Proof With probability pin , Alice needs to prepare the state ≤ E − E − − A1S n A1S while the number of ebits used is ϕ n . Instead, for typical i , she prepares a state σ n | i i | i i obtained by applying a typical projector and a conditional 1 [log Rank Πn log λ ]+ o(n) (28) S n 2 ρ,δ max typical projector to ϕin . When i is atypical, the protocol n − [S( S)+ S¯( S)] + log(1 ǫ)+ o(n), fails. ≤ 2 E E − Given a probability distribution q on a finite set χ, define matching the rates of the proposition. the set of typical sequences, with δ > 0, as In Section III we used the fact that teleporting the qubits⊓⊔ n n n of a superdense coding protocol leads to a remote state = x : x N(x x ) nqx δ√n qx(1 qx) , Tq,δ ∀ | | − |≤ − preparation protocol. When applied to Proposition IV.1, we (19) n p o get an alternative proof of Proposition 15 of [12]: where N(x xn) counts the numbers of occurrences of x in the n | S string x = x1x2 xn. If ρ = i piϕi has spectral decom- d ··· Corollary IV.2 There exist protocols for remote state prepa- position S R(j)Π , we then define the typical projector j=1 j P ration of entangled states with mean fidelity approaching one to be and asymptotically achieving the rate pair of S( ) S¯( ) P n S S Π = Πj1 Πj (20) E − E ρ,δ ⊗···⊗ n cbits and S( S) ebits. jn n E ⊓⊔ X∈TR,δ and the conditional typical projector to be V. IDENTIFICATION m Quantum message identification, a generalization of hy- n n Ii S (21) Πϕ ,δ(i )= Πϕi,δ, pothesis testing to the quantum setting, has been explored i=1 recently in a series of papers [30], [31], [32]. As opposed O Ii to transmission, where the goal is to reliably communicate a where Ii = j [n] : ij = i and Π refers to the typical { ∈ } ϕi,δ message over a channel, identification only allows the receiver projector in the tensor product of the systems j Ii. In terms A1S ∈ to answer a single binary question: is the message x or is of these definitions, σin , the state Alice prepares instead of A1S it not? A surprising aspect of the theory of identification is ϕ n , is proportional to i that the number of questions that can be answered grows I n n n A1 S I n n n ( A1 Π Π S (i ))ϕ n ( A1 Π S (i )Π ), (22) as a doubly exponential function of the number of uses of ⊗ ρ,δ ϕ ,δ i ⊗ ϕ ,δ ρ,δ With respect to approximation, the relevant property of these the channel, as opposed to the well-known singly exponential operators is that, defining behavior for transmission [13], [14]. In the quantum setting, a number of versions of the identification (ID) capacity have n n S n n ξin = ΠϕS,δ(i )ϕin ΠϕS,δ(i ), (23) been defined; these divide broadly into the capacities for quantum resources to identify classical messages and the we have capacities for those quantum resources to identify quantum n n n S n n n Tr[Πρ,δΠϕS,δ(i )ϕin ΠϕS,δ(i )Πρ,δ] messages. In the former case, doubly exponential growth of n the number of messages was found, with the most important = Tr[ξin ] Tr[(I Π )ξin ] − − ρ,δ n S result to date that the ID capacity of an ebit, supplemented Tr[ξin ] Tr[(I Π )ϕ n ] 1 ǫ, (24) ≥ − − ρ,δ i ≥ − with negligible rate of forward classical communication, is if δ = m 2dS/ǫ (by Lemmas 3 and 6 in [29]). The two [32]. It follows, of course, that the ID capacity of a qubit Gentle Measurement Lemma, referred to as the tender operator is also two [31]. p inequality in [29], together with a simple application of the In this section, we will instead be focusing on the capacity A1S A1S of an ebit to identify quantum messages, that is, quantum triangle inequality implies that ϕin σin 1 √8ǫ +2ǫ. For a more detailed proof of thesek facts− and furtherk ≤ information states. We will consider the model with a visible encoder and about typical projectors, see [29]. If in is typical, meaning ID-visible decoder, according to the terminology introduced it is in the set n (which occurs with probability at least in [31]. Tp,δ 1 m/δ2), then it is also true that Specifically, we say that we have a quantum-ID code on − Cd ¯ ( ) of error 0 <λ< 1 and dimension dC if there exists n n S n n n n nS( S)+cδ√n B d d ΠϕS ,δ(i )ϕin ΠϕS ,δ(i ) ΠϕS ,δ(i )2− E (25), an encoding map ε : (C C ) (C ) and a decoding map ≤ d d B → B n nS( S)+cδ√n D : C C (C ) such that for all pure states ϕ and ψ Rank Πρ,δ 2 E , (26) → B | i | i ≤ on CdC where c> 0 is independent of n and δ. Equation (25) implies λ Tr(ϕψ) Tr(ε(ϕ)Dψ) . (29) S nS¯( S)+cδ√n n that (1 ǫ)σin 2− E Πρ,δ, which in turn leads − ≤ 2 − ≤ ¯ S 1 nS( S)+cδ√n I to the conclusion that λmax(σin ) 1 ǫ 2− E =: This condition ensures that the measurement (Dψ, Dψ) ≤ − − λmax; Eq. (26) provides a bound on the effective dimension can be used on the states ε(ϕ) to simulate the test (ψ, I ψ) S n n − of the system S since σ n Π for all i . applied to the states ϕ. In the blind encoder, ID-visible decoder i ≤ ρ,δ 6

case, ε must be quantum channel and D can be an arbitrary and λmax = 1/a to conclude that for sufficiently large d, I assignment to operators 0 Dψ . It was shown in [31] that states Φϕ′ approximating Φϕ to within fidelity √2α ln 2 can for all 0 <λ< 1 there exists≤ a≤ constant c(λ) > 0 such that be prepared| i using log d +| o(logi d) ebits and o(log d) qubits d 2 on C a quantum-ID code of error λ and dC = c(λ)d of communication. By an appropriate choice of α, we can ⌊ ⌋ 2 exists. Since, for fixed λ, log dC = 2 log d const, this therefore ensure that Φϕ V ϕ λ/4. Using the triangle shows that, asymptotically, one qubit of communication− can inequality, we then find|h that| | i| ≥ identify two qubits. We claim that, again asymptotically, but now using a visible encoding map, one ebit plus a negligible Tr(ϕψ) Tr(Φ′ Dψ) λ/2 (33) | − ϕ |≤ (rate of) quantum communication can be used to identify two qubits. Rather than providing a detailed argument, we for all pure states ϕ CdC . | i ∈ ⊓⊔ simply state the method: the states ε(ϕ) that are output by There is a little subtlety in the proof that is worth consider- the blind encoding can be prepared visibly using superdense ing briefly. The states to be prepared, Φϕ , are maximally coding. Because they are extremely mixed, their purifications entangled, so one might think that they| i can be prepared are highly entangled and Proposition II.3 demonstrates that without any communication at all. The party holding Cd can, negligible communication is sufficient. indeed, create them without communication. The party holding The negligible communication cost is encountered fre- the smaller Ca, however, cannot; local unitary transformations quently in the theory of identification: the classical identifi- on Ca will not change the support of the reduction to Cd, for cation capacity of a bit of shared randomness supplemented example. Nonetheless, by appealing to Proposition II.3, we by negligible communication is a bit. In [32], it was found that see that the asymmetry disappears in the asymptotic limit if the classical identification capacity of an ebit supplemented by negligible communication is allowed. negligible communication is two bits. Our finding here that the quantum identification capacity of an ebit and negligible communication is two qubits provides an alternative proof of VI. DISCUSSION this result. We have proved the existence of protocols which allow a 2 4 sender to share entangled states with a receiver while using as Proposition V.1 If dC = c(λ)d /(log d) , then for all dC little quantum communication as is possible. These protocols states ϕ C , approximations Φ′ of the purifications | i ∈  | ϕi  interpolate between requiring no communication at all for of the states ε(ϕ) can be prepared on Ca Cd using log d + maximally entangled states and a rate of two remote qubits o(log d) ebits and o(log d) qubits of communication,⊗ in such a way that per sent qubit for product states. An immediate application of the result was a proof that the identification capacity of an ebit λ is two qubits when visible encoding is permitted. Tr[ϕψ] Tr[(TrCa Φϕ′ )Dψ] . (30) − ≤ 2 The question of efficient constructions remains – we would

Proof From the proof of Proposition 17 in [31], if we choose like to have protocols with the same ebit and qubit rates a = ǫd/2 , ǫ = (λ/48)2, and a λ/16-net in CdC , we may which are implementable in polynomial time (as has been ⌊ ⌋ dC d a let ε(ϕ) = TrCa (V ϕV †) with V : C C C a Haar demonstrated for state randomization [21] by Ambainis and → ⊗ distributed isometry, and Dψ = supp ε(ψ˜) for the state ψ˜ Smith [33]). It would also be interesting to know whether closest to ψ in the net. Then, stronger success criteria can be satisfied while still achiev- ing the same rates. Specifically, the universal remote state λ Pr ψ, ϕ such that Tr(ϕψ) Tr(ε(ϕ)Dψ) > preparation protocol of [12] produces an exact copy of the ∃ | − | 4 ! desired state when the protocol succeeds, not just a high c 4dC fidelity copy. Is such a probabilistic-exact protocol possible in 0 exp( c d2ǫ2), (31) ≤ λ − 1 the superdense coding setting? (One could even ask questions   about perfectly faithful superdense coding, in analogy to what with absolute constants c0 and c1. (Note that this statement is trivial for a = 0 or a = 1.) To be precise, in [31] the above has been done for remote state preparation in [34], [35], [36].) probability bound is derived for states in the net, but it is also Another natural question is the quantum identification capacity explained how to use triangle inequality to lift this to all states. of an ebit in the blind scenario. We have shown that it is Therefore, the states ε(ϕ) form a good quantum-ID code. possible to achieve the identification rate of two qubits per We will demonstrate how to make them using superdense ebit in the case when the identity of the encoded qubits is coding. Arguing along the lines of Eq. (13), we find that for known, but it is not at all clear whether this rate is achievable 4 when the identity of the qubits is unknown. all α> 0, dC = ad/(log a) and sufficiently large d, ⌊ ⌋ 1 Pr inf S ε(ϕ) < log a α < . (32) ϕ − 2   Acknowledgments (This is also a special case of
Theorem IV.1 from [3].) By the same reasoning given after Eq. (17), there exists a maximally It is a pleasure to thank Dave Bacon and Debbie Leung for 2 entangled state Φϕ such that Φϕ V ϕ 1 √2α ln 2. many enjoyable discussions on subjects directly and peripher- | i |h | | i| ≥ − We can, therefore, invoke Proposition II.3 with dS = d ally related to the contents of this paper. 7

REFERENCES [31] A. Winter, “Quantum and classical message identification via quantum channels,” Quantum Inf. Comuput., vol. 4, pp. 563–578, 2004, arXiv [1] C. Bennett and S. J. Wiesner, “Communication via one- and two-particle quant-ph/0401060. operators on Einstein-Podolsky-Rosen states,” Phys. Rev. Lett., vol. 69, [32] ——, “Identification via quantum channels in the presence of prior p. 2881, 1992. correlation and feedback,” arXiv quant-ph/0403203. [2] A. Harrow, P. Hayden, and D. Leung, “Superdense coding of quan- [33] A. Ambainis and A. Smith, “Small pseudo-random families of matrices: tum states,” Phys. Rev. Lett., vol. 92, p. 187901, 2004, arXiv:quant- Derandomizing approximate quantum encryption,” in RANDOM 2004, ph/0307221. 2004, arXiv quant-ph/0404075. [3] P. Hayden, D. Leung, and A. Winter, “Aspects of generic quantum [34] A. K. Pati, “Minimum classical bit for remote preparation and measure- entanglement,” Comm. Math. Phys., 2006, arXiv quant-ph/0407049. ment of a qubit,” Phys. Rev. A, vol. 63, p. 14302, 2001, arXiv:quant- [4] M. A. Nielsen and I. L. Chuang, Quantum computation and quantum ph/9907022. information. Cambridge University Press, 2000. [35] M.-Y. Ye, Y.-S. Zhang, and G.-C. Guo, “Faithful remote state preparation [5] I. Devetak, A. W. Harrow, and A. Winter, “A family of quantum using finite classical bits and a nonmaximally entangled state,” Phys. protocols,” Phys. Rev. Lett., vol. 93, p. 230504, 2004, arXiv quant- Rev. A, vol. 69, p. 022310, 2004, quant-ph/0307027. ph/0308044. [36] D. W. Berry, “Resources required for exact remote state preparation,” [6] H.-K. Lo, “Classical communication cost in distributed quantum in- Phys. Rev. A, vol. 70, p. 062306, 2004, arXiv:quant-ph/0404004. formation processing – A generalization of quantum communication complexity,” Phys. Rev. A, vol. 62, p. 012313, 2000, arXiv:quant- ph/9912009. [7] C. H. Bennett, D. P. DiVincenzo, P. W. Shor, J. A. Smolin, B. M. Terhal, and W. K. Wootters, “Remote state preparation,” Phys. Rev. Lett., vol. 87, p. 077902, 2001, arXiv:quant-ph/0006044. [8] A. Abeyesinghe and P. Hayden, “Generalized remote state preparation: Trading cbits, qubits, and ebits in quantum communication,” Phys. Rev. A, vol. 68, no. 6, p. 062319, 2003, arXiv quant-ph/0308143. [9] I. Devetak and T. Berger, “Low-entanglement remote state prepara- tion,” Phys. Rev. Lett., vol. 87, no. 9, p. 197901, 2001, arXiv quant- ph/0102123. [10] P. Hayden, R. Jozsa, and A. Winter, “Trading quantum for classical resources in quantum data compression,” J. Math. Phys., vol. 43, no. 9, pp. 4404–4444, 2002, arXiv:quant-ph/0204038. [11] A. Harrow, “Coherent communication of classical messages,” Phys. Rev. Lett., vol. 92, no. 9, p. 097902, 2004, arXiv quant-ph/0307091. [12] C. H. Bennett, P. Hayden, D. Leung, P. W. Shor, and A. Winter, “Remote preparation of quantum states,” IEEE Trans. Inf. Theory, vol. 51, pp. 56– 74, 2005, arXiv quant-ph/0307100. [13] M. O. Rabin and A. C.-C. Yao, “Unpublished manuscript,” 1979, cited in A. C.-C. Yao, “Some questions on the complexity of distributive com- puting”, in : Proc. 11th ACM Symposium on the Theory of Computing, 209-213. [14] R. Ahlswede and G. Dueck, “Identification via channels,” IEEE Trans. Inf. Theory, vol. 35, no. 1, pp. 15–29, 1989. [15] E. Lubkin, “Entropy of an n-system from its correlation with a k- reservoir.” J. Math. Phys., vol. 19, pp. 1028–1031, 1978. [16] S. Lloyd and H. Pagels, “Complexity as thermodynamic depth,” Ann. Phys., vol. 188, no. 1, pp. 186–213, 1988. [17] D. Page, “Average entropy of a subsystem,” Phys. Rev. Lett., vol. 71, pp. 1291–1294, 1993. [18] S. K. Foong and S. Kanno, “Proof of page’s conjecture on the average entropy of a subsystem,” Phys. Rev. Lett., vol. 72, pp. 1148–1151, 1994. [19] J. Sanchez-Ruiz, “Simple proof of page’s conjecture on the average entropy of a subsystem,” Phys. Rev. E, vol. 52, pp. 5653–5655, 1995. [20] S. Sen, “Average entropy of a quantum subsystem,” Phys. Rev. Lett., vol. 77, no. 1, pp. 1–3, 1996. [21] P. Hayden, D. Leung, P. Shor, and A. Winter, “Randomizing quantum states: Constructions and applications,” Comm. Math. Phys., vol. 250, no. 2, pp. 371–391, 2004, arXiv:quant-ph/0307104. [22] M. Fannes, “A continuity property of the entropy density sor spin lattice systems,” Comm. Math. Phys., vol. 31, pp. 291–294, 1973. [23] P. M. Alberti and A. Uhlmann, Stochasticity and partial order. Boston: Dordrecht, 1982. [24] M. Ohya and D. Petz, Quantum entropy and its use, ser. Texts and monographs in physics. Berlin: Springer-Verlag, 1993. [25] A. Uhlmann, “The ‘transition probability’ in the state space of a ∗- algebra,” Rep. Math. Phys., vol. 9, p. 273, 1976. [26] R. Jozsa, “Fidelity for mixed quantum states,” J. Mod. Opt., vol. 41, pp. 2315–2323, 1994. [27] V. D. Milman and G. Schechtman, Asymptotic theory of finite dimen- sional normed spaces, ser. Lecture Notes in Mathematics. Springer- Verlag, 1986. [28] M. Nielsen, “Continuity bounds for entanglement.” Phys. Rev. A, vol. 61, p. 064301, 2000. [29] A. Winter, “Coding theorem and strong converse for quantum channels,” IEEE Trans. Inf. Theory, vol. 45, pp. 2481–2485, 1999. [30] R. Ahlswede and A. Winter, “Strong converse for identification via quantum channels,” IEEE Trans. Inf. Theory, vol. 48, no. 3, pp. 569–579, 2002, arXiv quant-ph/0012127.