THE most fundamental situation in communication theory

Transcription

1 Optimal quantum source coding with quantum side information at the encoder and decoder Jon Yard, Igor Deveta arxiv: v [quant-ph] 0 Jun 007 bstract onsider many instances of an arbitrary quadripartite pure state of four quantum systems R. lice holds the part of each state, ob holds, while R represents all other parties correlated with. lice is required to redistribute the systems to ob while asymptotically retaining the purity of the global states. We prove that this is possible using Q qubits of communication and E ebits of shared entanglement between lice and ob provided that Q I(R; ) and Q + E H( ). This matches the outer bound for this problem given in []. The optimal qubit rate provides the first nown operational interpretation of quantum conditional mutual information. We also show how our protocol leads to a fully operational proof of strong subaddivity and uncover a general organizing principle, in analogy to thermodynamics, which underlies the optimal rates. I. INTRODTION THE most fundamental situation in communication theory consists of a two-terminal source coding problem. Here one user, say lice, attempts to describe a source of information to another user, who we call ob. If the information source is modeled by a sequence of independent and identically distributed (i.i.d.) random variables X, one can as for the ultimate rate at which the source can be described, in units of bits per sample. It is required that lice s description allows ob to perfectly recreate the source sequence with high probability, although decreasing the error probability generally requires bloc coding on longer source sequences. ccording to Shannon s noiseless channel coding theorem [], this ultimate rate is given by the Shannon entropy H(X) = x p(x) log p(x). Intuitively, Shannon entropy can be understood as a measure of the information contained in the random variable X. ecause Shannon entropy answers the question regarding the optimal rate for data compression, one says that the corresponding protocol for data compression provides an operational interpretation of Shannon entropy. Suppose now that ob had some a priori information regarding the random variable X, in the form of a correlated random variable Y. In this case, Slepian and Wolf demonstrated [3] that lice would only need to send to ob at a rate given by the conditional entropy H(X Y ) = H(XY ) H(Y ) jtyard@caltech.edu, Institute for Quantum Information, alifornia Institute of Technology, Pasadena, alifornia, S deveta@usc.edu, Electrical Engineering Department, niversity of Southern alifornia, S and that surprisingly, lice would not need to now ob s side information to accomplish this tas. The so-called Slepian- Wolf protocol for data compression with side information provides an operational interpretation of conditional entropy. Intuititively, one thins of H(X Y ) as a measure of the information that is to be gained by learning X for one who already nows Y. Note that there is no advantage if lice has additional side information regarding X, and that shared common randomness between lice and ob is also of no help. In this paper, we provide a complete solution to a general quantum counterpart of the above scenario. We find that, in contrast to the classical case, additional lice side information changes the problem, while quantum mechanical entanglement between lice and ob, the quantum analog of shared common randomness, is a useful resource. Our problem is fully quantum in a sense introduced by Schumacher [4], where lice is ased to transfer part of a pure quantum state to ob, while preserving the purity of the global state. For this, we consider a pure state of four quantum systems ψ R. Initially, the and systems are held by lice, while is in the possession of ob. We refer to R as the reference system and suppose that it is held by a third party, say Robert, and is inaccessible both to lice and ob. We determine the cost for lice and ob to redistribute the state, so that it is ob who holds instead of lice, thereby transferring the quantum information in to ob. More specifically, we analyze the corresponding asymptotic scenario, asing that many copies of the same state be redistributed as above, while requiring that the redistributed states have arbitrarily high fidelity with the originals in the asymptotic limit. To achieve this tas, we allow the use of two fundamental quantum mechanical resources. First, lice may send qubits (two-level quantum systems) to ob over a noiseless quantum channel. Second, we allow lice and ob to use pre-existing entanglement, shared between themselves in the form of ell states Φ + = ( 00 + ). In the sequel, we refer to such a state as an ebit (entangled bit). We therefore phrase the asymptotic cost of redistributing to ob in terms of the number Q of qubits sent and the number E of ebits consumed, per copy of the state. We allow the entanglement cost E to be negative, in which case the corresponding protocol generates entanglement rather than consuming it. Our main result (Theorem ) states that it is possible to redistribute the state ψ R as above if and

2 Q + E = H( ) I(; R ) Q I(; ) I(; ) E (FQRS) [0], [], fully quantum Reverse Shannon (FQRS) [0], [], and state merging [7], [] protocols. The paper is organized as follows. In the next subsection we fix our notational conventions. The following section gives an introduction to the resource calculus, and a formal statement of the main result, Theorem, which is proved in Section III. In Section IV, we show how our results yield a fully operational proof of strong subaddivity which, unlie previous operational proofs, is logically independent even from the subadditivity of entropy. We conclude with a discussion in Section V where we reflect on the main result and provide a novel thermodynamic interpretation of the optimal rates. Fig.. The shaded region represents contains the cost pairs from () at which it is possible to redistribute the part of ψ R from lice to ob. The figure corresponds to the case I(; ) > I(; ); otherwise the corner point, which corresponds to the optimal cost pair in (), would be in the upper-left quadrant. only if Q I(R; ), Q + E H( ). () This region is depicted in Figure. The quantities in these bounds, conditional mutual information and conditional entropy, are defined in Section I-. Simultaneously minimizing the qubit rate Q and the total sum rate Q+E gives the optimal cost pair Q = I(R; ) E = I(; ) I(; ). () The optimal qubit cost gives the first nown operational interpretation of quantum conditional mutual information. In Section IV, we show that Q cannot be negative, which leads to an operational proof of the celebrated strong subadditivity inequality [5]. This proof differs from other such operational proofs [6], [7] in that it follows solely from a direct coding theorem and not from a converse proof. In [8], where our main result was first announced, we showed that Q is symmetric under time-reversal, where now ob redistributes bac to lice, while E is anti-symmetric. The former gives an intuitive understanding to the curious identity I(; R ) = I(; R ), which holds on any quadripartite pure state. We comment further on this feature in Section V. We also demonstrated there that the corresponding protocol is perfectly composable. This constitutes an exact solution to the quantum analog of result of over and Equitz [9] on the successive refinement of classical information, although the classical problem is only nown to be exactly soluble in the presence of a Marov condition. In our protocol, the systems and which respectively remain with lice and ob can be regarded as quantum side information that is used to reduce the cost of redistribution. y disregarding that side information in various ways, our main result incorporates various quantum Shannon-theoretic protocols as special cases. s we discuss in Section V, these include the Schumacher compression [4], fully quantum Slepian Wolf. Notational conventions Throughout this paper, we assume familiarity with standard bacground material in quantum information theory; for a general reference, the reader is referred to [3]. We use capital Roman letters such as,, to denote Hilbert spaces. We write for the dimension of and use a superscripted label to associate a state to a Hilbert space, by writing ρ or ϕ. omputational basis states of are denoted with lower case Roman letters as in { i }. Tensor products of Hilbert spaces are written =. Given a pure state ϕ, we abbreviate ϕ = ϕ ϕ, while writing its partial traces as ϕ = Tr ϕ. We write π for the maximally mixed state on, and given two isomorphic Hilbert spaces and, we write Φ = i i for the unique maximally entangled state associated with the isomorphism i i. quantum channel is a completely positive, trace-preserving linear map N from density matrices on to those on. Given an isometry V, we will abbreviate its adjoint action on density matrices as V(ρ) VρV. partial isometry is an isometry when restricted to its support subspace. For the von Neumann entropy of a density matrix ϕ we write H() Tr ϕ log ϕ. i= When the underlying state could be ambiguous we write H() ϕ. Given a multipartite state ϕ, various entropic quantities can be defined in exact analogy to the classical case (see e.g. [4]). Quantum conditional entropy is defined [5] as H( ) = H() H(), quantum mutual information [5] is I(; ) = H() + H() H() and quantum conditional mutual information is given by I(; ) = H( ) + H( ) H( ). Observe that the conditional quantities above cannot generally be interepreted as averages, unless the conditioning system is purely classical. Furthermore, notice that conditional entropy

3 3 can in fact be negative, as is evidenced by any pure entangled state on. On the other hand, I(; ) is never negative, a fact which is nown as strong subaddivity [5]. In Section IV, we show how our main result leads to a self-contained proof of strong subadditivity. simulated version of this state, obtained by using the resources a i. We require that Ω and Ω are ɛ-close in either trace distance or fidelity. The particular measure is not important, as we are ultimately concerned with asymptotics, where ɛ can be arbitrarily small. II. RESORE INEQLITIES It will be convenient for us to use the high-level notation of resource inequalities [6], [7] to express our main result, as well as to describe various intermediate protocols introduced during the proof. We tae a somewhat more elementary path than that given in [7] which is sufficient for our purposes.. Finite resource inequalities single ebit shared between lice and ob is denoted [qq]. The notation [q q] represents a noiseless qubit channel from lice to ob, while a noiseless classical bit channel is written [c c]. finite resource inequality is an expression such as [q q] [c c], [q q] [qq] meaning that the resource on the left can simulate the one on the right. The above two examples respectively signify that a qubit channel can be used to send classical bits (by signaling with orthogonal pure states), or otherwise can be used to distribute entanglement (by transmitting halves of locally prepared ebits). ddition of two resources may be regarded as having each of them available. In this way, for instance, the existence of the quantum teleportation and superdense coding protocols are proofs of the respective finite resource inequalities [qq] + [c c] [q q], [qq] + [q q] [c c]. (3). pproximate resource inequalities Given two quantum states ρ and σ of the same quantum system, we may judge their closeness using either the trace distance ρ σ or the fidelity F (ρ, σ) = ρ σ. Note that when one of the states is pure, F ( ϕ, σ) = ϕ σ ϕ. useful characterization of fidelity hlmann s theorem says that if ψ is a purification of ρ, then F (ρ, σ) is the maximum of ψ φ over all purifications φ of σ. Fidelity and trace distance related by the inequalities F (ρ, σ) ρ σ (4) ρ σ F (ρ, σ). (5) Therefore, fidelity and trace distance are equivalent distance measures when one is interested in arbitrarily good approximations of states are we are here. n approximate resource inequality a i ɛ i is a finite resource inequality which holds with an error of ɛ in the following sense. onsider acting on half of a maximally entangled state with each target resource b j that is a channel, and call the resulting global state Ω. Note that Ω should also contain the b j that are quantum states. Now, let Ω be the j b j. symptotic resource inequalities The notion of finite resource inequalities can be generalized to that of an asymptotic resource inequality, which is a formal expression of the form i R (i) in a i j R (j) outb j. (6) Here the a i and b j are resources and the rates R (i) in and R(j) out are nonnegative real numbers. We shall consider the inequality (6) to be shorthand for the following formal statement: for every ɛ > 0, every set of rates R (i) in > R(i) in, R (i) out < R (i) out and all sufficiently large n, the approximate resource inequality a i ɛ nr (j) i nr (i) in j out b j holds. elow, we use Gree letters to denote linear combinations of finite resources which appear in asymptotic resource inequalities. In some asymptotic resource inequalities, we may only require a sublinear amount o(n) of a particular input resource. In such cases, we write oa + β γ if we have Ra + β γ for every R > 0. It will also be convenient for us extend the definition of asymptotic resource inequalities to have negative rates on the left. Such rates are interpreted as meaning that the corresponding resources are generated rather than consumed. Formally, these resources should be negated and moved to the right. Let us introduce two powerful lemmas which are the raison d être for the entire formalism of asymptotic resource inequalities and which play important roles in our proofs. Lemma (omposition lemma [7]): α β and β γ α γ. Lemma (ancellation lemma [7]): Given rates which satisfy R in > R out 0, R in a + β R out a + γ (R in R out )a + β γ. Otherwise, if R out R in 0, then R in a + β R out a + γ oa + β (R out R in )a + γ D. Distributed states In Schumacher data compression, lice wishes to transmit the parts of many instances of the state ψ R to ob while asymptotically preserving the entanglement with R. We introduce the following notation to describe the corresponding coding theorem: ψ + H()[q q] ψ.

4 4 The notation ψ indicates that lice holds the parts of many i.i.d. instances of some fixed purification ψ R of the density matrix ψ, while ob holds nothing. On the right, the expression ψ refers to the same purifications as on the left, only it is ob who is holding the systems. In other words, lice attempts to simulate identity channels from the systems in her lab to identical systems located in ob s lab. This channel is only required to wor well when the input is equal to ψ. In [7], the formalism of relative resources was introduced for these purposes, though our alternate notation is sufficient for our needs. State redistribution involves a purification ψ R of a tripartite density matrix ψ. We denote the distributed states before and after the protocol as ψ and ψ since lice begins by holding and ends by only holding. The rates in an asymptotic resource inequality involving such distributed states will in general be entropic expressions evaluated on the implicit but arbitrary purification into a reference system R. sing this notation, we again state the main result: Theorem : ψ + Q[q q] + E[qq] ψ (7) if and only if Q and E satisfy (), i.e. are contained in the region depicted in Figure. The converse part of the proof of Theorem, i.e. that Q and E must satisfy (), is proved in []. We thus focus on proving a coding theorem showing that (7) is satisfied whenever Q and E satisfy (). ecause [q q] [qq], it suffices for us to demonstrate (7) for the corner point (Q, E ) defined in (). III. PROOF OF THEOREM To prove Theorem we will demonstrate the existence of the following auxilliary protocol which transfers n to ob, which has the desired net communication and entanglement cost. Theorem : ψ + I(; R)[q q] + I(; )[qq] ψ + I(; )[q q] + I(; )[qq]. Together with the cancellation lemma (Lemma ), Theorem yields a proof of Theorem. However, observe that if I(; ) I(; ), the cancellation lemma still requires a sublinear amount of entanglement on the left. Similarly, if we have I(; R) = I(; R) (i.e. if strong subaddivity is saturated), a sublinear amount of communication will also be required. However, because [q q] [qq], the additional entanglement cost can be absorbed into the communication rate and is therefore only relevant if the state ψ R saturates strong subaddivity. We discuss this point further in Section V. We prove Theorem by means of another protocol which simulates coherent channels [8]. coherent channel [q qq] is a type of quantum feedbac channel which is an isometry from lice to lice and ob: sing a coherent version of teleportation, where lice and ob apply only local unitaries, it is nown that [8] [qq] + [q qq] [qq] + [q q]. Repeated concatenation yields the following asymptotic resource inequality [8]: [q qq] [q q] + [qq]. (8) In fact, the opposite direction holds as a finite resource inequality, but it will not be useful for us here. In this paper, we devote most of our efforts toward proving the following theorem which, when combined with (8) and the composition lemma (Lemma ), provides a proof of Theorem. Theorem 3: ψ + I(; R)[q q] + I(; )[qq] ψ + I(; )[q qq].. Proof of Theorem 3 Our proof of Theorem 3 relies on the following robust oneshot version, whose proof we delay until Section III-. Theorem 4 (Robust one-shot redistribution protocol): Let a pure state ψ R and a maximally entangled state Φ b b be given, where  = divides. Suppose that ϕ R and φ R are states satsifying max { ψ R ϕ R, ψ R φ R } ɛ. Then there exist a quantum system S with S = /, S encoding isometries V b and a decoding isometry W S K b under which K ψ R WV ψ R Φ b b η (9) = where η is equal to 6 ( ϕ R ɛ ϕ R S ) /4 + 4 φ 0 φ. (0) Now we show how to apply Theorem 4 to pure states of the form ( ψ R) n to obtain a proof of Theorem 3. This is accomplished via the following theorem. The direct part is proved in [], while the converse part follows from standard arguments in classical information theory (see e.g. [4]). Theorem 5 (Method of types): Let a tripartite state ψ be given. For every ɛ, δ > 0 and all sufficiently large n, there are projections Π n δ T {,, },, Π n δ, and Π n δ Tr Π T n δ (ψ T ) n ɛ nh(t ) nδ Tr Π T n δ nh(t )+nδ. such that for lso, the normalized version ϕ n n n of the subnormalized state ( )( Π n δ Π n δ Π n δ ψ R ) n

5 5 satisfies ϕ n n n ( ψ ) n ɛ a controlled isometry V = Kin V b n δ ns. and for each T {,,,,, }, nh(t ) nδ ϕ T n nh(t )+nδ 0 nh(t ) nδ ϕ T n nh(t ) nδ ϕ T n nh(t )+nδ. nh(t )+nδ The entropies in these bounds are evaluated on ψ. dditionally, the normalized version Ψ n n n of the subnormalized state ( )( n n Π n δ ψ ) n satisfies Ψ n n n ( ψ ) n ɛ. Finally, there is a constant c > 0 for which we may tae ɛ = ncδ in all of the above bounds. Proof of Theorem 3: We apply Theorem 5 two separate times to the state ( ψ R) n, obtaining two auxilliary states which we will use to control the main quantities appearing in the error bound (0) of Theorem 4. For the first, we consider ψ R to be a tripartite state of the systems,, R. We thus obtain, for every δ > 0 and all sufficiently large n, a state ϕ n n n R n which is ɛ-close to ψ n in trace distance for ɛ = ncδ, where all of the operator norms in the second term in (0) have the appropriate exponential bounds. With respect to the partition R,,, we similarly obtain another state φ n n n R n for which the operator norms in the last term of (0) are bounded accordingly. lice initiates the protocol by Schumacher compressing the system n. For this, she performs the projective measurement {Π n δ, n Π n δ } on n. ccording to Theorem 5, the first outcome occurs with probability at least ɛ. In this case, the global state is replaced by the normalized version Ψ n n n R n of the ( projected state Π ) n δ ψ R n. In case the other outcome occurs, lice declares an error and the protocol is aborted. We condition on the first case. In what follows, we identify Ψ n n n R n with its restriction Ψ n δ n R n to the support δ of the typical projection Π n δ. y the triangle inequality, each of ϕ n n n R n and φ n n n R n is ɛ-close to Ψ n δ n R n in trace distance because all three states are ɛ-close to ( ψ R) n. Therefore, the one-shot theorem (Theorem 4) implies that there exist a quantum system S and a maximally entangled state Φ b b with  S = δ, together with encoding isometries and a decoding isometry W Sn δ b n K out satisfying (9) and (0) with ɛ replaced by ɛ. If lice applies one of the isometries V uniformly at random and sends S to ob, after which he applies W, the system δ will be transferred with high global fidelity. y measuring K out, ob can, on the average, identify lice s encoding. Rather than send ob classical information, lice can instead simulate a coherent channel from a system K in to K in K out by applying V b n δ n S If she tries to send half of a maximally entangled state Φ R K in, the global pure state Ω on n δ n R n R K in K out that results from the protocol is Ω = W V Ψ n δ n R n Φ R K in Φ b b. It is then immediate from (9) that where Γ R K ink out Ψ n δ n R n Ω η. Γ R K ink out = R Kin Kout = is a GHZ state. Squaring this estimate and using monotonicity of fidelity, we obtain F ( Φ R K ink out, Ω R K ink out ) 4η. () y similar reasoning, together with (5), we find that Ω n δ n R n Ψ n δ n R n η. ecause Ψ n δ n R n is ɛ-close to ( ψ R) n in trace distance, the triangle inequality implies that Ω n δ n R n ( ψ R) n η + ɛ. () We may combine the estimates () and () using a lemma from [9], yielding F ( Φ R K ink out ( ψ R ) ) n, Ω Ω n δ n R n ( ψ R) n 3 ( F ( Φ R K ink out, Ω R K ink out ) ) ɛ η η ɛ 3 η (3) provided η is sufficiently small. t this point, all that remains is to bound the two main terms in the expression (0) for η. Taing S = nq and = nr, the first main quantity in (0) satisfies δ ϕ n R n 0 ϕ n n R n S n[h()+h(r) H(R) Q]+3nδ = n[i(;r) Q]+3nδ. (4) Therefore, this term goes to zero exponentially fast provided that For the second term, φ n n 0 φ n δ Q I(; R) + δ. (5) n[r+h() H() H()]+3nδ = n[r I(;)]+3nδ

6 6 so that if R I(; ) 4δ, this term also goes to zero exponentially with n. Since each of these terms will eventually be less than ɛ = ncδ, we find that η 6 ɛ + 4ɛ /4 + 4ɛ 5ɛ /4 if ɛ is sufficiently small. Therefore, the final bound on the fidelity in (3) can be expressed as 4ɛ /8 for sufficiently small ɛ. ombining (5) with the bound on δ = Tr Πδ n obtained in Theorem 5, we find that the rate at which this protocol uses entanglement satisfies E in = n log δ S I(; ) δ + n. ecause δ > 0 can be taen arbitrarily small, it follows that whenever Q > I(; R), E in > I(; ), and R < I(; ), we have, for all sufficiently large n, ψ + nq [q q] + ne in [qq] 5ɛ /8 ψ + nr [q qq]. Since this holds for arbitrarily small ɛ > 0, the asymptotic resource inequality of Theorem 3 follows.. Proof of Theorem 4 Our proof of Theorem 4 maes essential use of the following robust one-shot decoupling lemma, which is proved in the appendix. fter stating the lemma, we briefly recall how it is used in two previously studied special cases of our redistribution result, to help the reader understand the context into which it fits with our proof. Lemma 3 (Robust one-shot decoupling): Let a density matrix ψ E be given, fix ɛ > 0 and let ϕ E be any state satisfying ψ E ϕ E ɛ. Fix a unitary decomposition W S b of into subsystems and define, for each, Then the average state satisfies ψ S b E = W ψ E W. ψ S b E = ψ b E π b ψ E ɛ + () ψ S E b d. ϕ R 0 ϕ R S. (6) The fully quantum Slepian-Wolf theorem says that, given a tripartite pure state ψ R, ψ + I(; R)[q q] ψ + I(; )[qq]. Together with the method of types (Theorem 5), the above decoupling lemma provides an immediate proof of this resource inequality. Indeed, if lice encodes with a random unitary, Lemma 3 ensures that a system  (identified with  W I(; R) S V I(; ) V I(; R) I(; ) S W Fig.. ircuits for fully quantum Slepian-Wolf (left) and fully quantum reverse Shannon (right), related by time-reversal and swapping. We have included the rates one gets by applying the method of types (Theorem 5) to the one-shot decoupling lemma (Lemma 3). Note that for fully quantum Slepian-Wolf, the random encoding determines the decoding, while for fully quantum reverse Shannon, the random decoding determines the encoding. in the lemma), which will hold lice s half of the generated entanglement, is approximately maximally mixed and decoupled from R. This can easily be shown to imply that  is maximally entangled with S (see [], or compare with the proof of Theorem 4 below). ecause all transformations are unitary and the global state is pure, this ensures that ob can apply a local isometry to reconstruct, while at the same time obtaining the other half of the generated entanglement. This scenario is illustrated on the left of Figure. y simply reversing time (and also the labels ), a circuit for fully quantum Slepian-Wolf provides one for the fully quantum reverse Shannon theorem, which given a tripartite state ψ R, states that ψ + I(; R)[q q] + I(; )[qq]+ ψ. circuit for this is pictured on the right of Figure. We prove Theorem 3 as follows. If ob s side information is considered as part of the reference (i.e. it is disregarded as side information), the fully quantum reverse Shannon protocol can be used to transfer from lice to ob, at least maing use of lice s side information. y a modification of that protocol provided below, ob s side information can be utilized to simulate the required coherent channels [q qq] as follows. Rather than choosing a single random unitary for the decoding, we choose exponentially many (roughly ni(;) when we apply the method of types to the oneshot result). We further guarantee that if lice chooses one of the corresponding encodings uniformly at random, ob can, on average, correctly distinguish that encoding in order to apply the correct decoding. Thus, it is possible for lice to piggybac classical information on the transmitted qubits, which ob can access by means of his side information (cf. [0], []). We further ensure that this can all be done coherently, where lice instead applies a superposition of encodings by using a controlled isometry which is controlled by an arbitrary quantum state. The circuit we construct for performing this tas noncoherently is illustrated in Figure 3. Our proof of Theorem 4 relies on two other lemmas. First, we require the operator inequality []:

7 7 V I(; R) I(; ) W measure Fig. 3. ircuit for using ob s side information to piggybac extra classical (or coherent) information through the fully quantum reverse Shannon circuit on the right of Figure. Lemma 4: If 0 Π and Π Λ, then Λ / ΠΛ / ( Π) + 4(Λ Π). We also will use the following coherification lemma, which allows us to convert protocols which transmit classical information to ones which simulate coherent channels. We give a short proof in the appendix. Lemma 5: Given a pure state ψ DE and unitaries D D, let ψ DE = ψ DE. Given any other set of pure states ψ DE and a POVM {Λ D } on D, there are complex phases α such that the isometry L D DK = (α Λ ) K satisfies ψ L ψ (P + F ). = where P = Tr ψ Λ, F = ψ ψ. Proof of Theorem 4: s in the statement of the theorem, we fix nearby states ϕ and φ and let W S b be any unitary decomposition of into subsystems. Independently choose unitaries {,..., } according to the Haar measure on (). For each, define the states ψ R = ψ R ψ S b R = W ψ R = W ψ R. We define the decoupling fidelity for ψ b R R F = F (ψ b, π b ψ R ). Since Φ b b ψ R is a purification of π b ψ R, hlmann s theorem implies that there is an isometry under which V S b F = ψ S b R V Φ b b ψ R. To send the message, lice will apply the isometry V = V. We now define as ψ R = W V Φ b b ψ R, which is the state that is created after lice performs V and gives S to ob, who then applies W S b. We may therefore equivalently write F = ψ R ψ R. The average decoupling fidelity is a random variable F ave = F = which depends on the random choice of unitaries. We lower bound its expectation as follows. Define the average states with respect to Haar measure d as ψ R = ψ R d ( ) ψ S b R = W ψ R W. We now use the robust one-shot decoupling lemma (Lemma 3) to bound the expectation of F ave over the random choice of unitaries: E F ave = E F = R = F (ψ b b, π ψ R ) ϕ ɛ + R 0 ϕ R S. related estimate to be used later is E F ave E F ave ( ) ϕ R ɛ + 0 ϕ R /4 S, (7) which follows by concavity and the inequality x + y x + y, valid for x, y 0. Next, we consider ob s ability to distinguish the states. For this, we design a measurement which distinguishes the nearby states φ = φ. Let Π be the projection onto the support of φ and define ψ Π = Π, while defining the pretty good measurement Λ = Π, Λ = Λ / Π Λ /. =

8 8 The probability that this measurement fails to identify the state ψ is P = Tr( Λ )ψ. Observe that P Tr( Λ )φ ψ ψ φ ψ F + ɛ D. + ψ ecause x x is concave, we have D ave D ɛ + F ave. = φ Therefore, the average of the P can be bounded using Lemma 4, obtaining a random variable satisfying P ave P = D ave + Tr( Λ )φ = D ave + ( ( Tr Π φ ) + 4 = D ave + 4 = = Tr Π φ ) Tr Π φ The last line holds because for each, Π projects onto the support of φ. y taing the expectation over the random choice of unitaries, this yields E P ave E D ave + 4 E Tr Π φ = E D ave + 4 Tr [ E Π E φ ] = E D ave + 4 Tr [ E Π (π φ ) ] E D ave + 4 φ 0 φ E F ave + ɛ + 4 φ 0 φ. (8) We now apply Lemma 5, identifying R E and D. We can thus conclude that there is an isometry L K under which ψ L ψ (P ave + F ave ). = Taing expectations, we find that E ψ L ψ = E F ave + E P ave 4 E F ave + ɛ + 4 φ 0 φ 6 ( ) ϕ R ɛ ϕ R /4 S + 4 φ 0 φ. The second inequality is by (8), while the third is due to (7) and holds for sufficiently small ɛ, i.e. sufficiently large n. We may then conclude that for a particular value of the randomness, the same bound holds without the expectations. Finally, we define ob s decoding isometry to be W = LW, completing the proof. IV. N OPERTIONL PROOF OF STRONG SDDITIVITY Let ψ R be an arbitrary pure state. In this section, we show how our results lead to an operational proof of strong subaddivity, i.e. that I(; R ) 0. y discarding some resources on the right in Theorem, we obtain: ψ + I(; R)[q q] + I(; )[qq] I(; )[q q]. Intuitively, it maes sense that we should have I(; R) I(; ) = I(; R ) 0 since otherwise, a noiseless qubit channel could be used to faithfully transmit more than one qubit in the presence of entanglement between the sender and receiver. Of course this inequality is guaranteed by strong subadditivity. However, our aim is to provide an alternative proof of this fundamental inequality. The above asymptotic resource inequality implies that for every ɛ, δ > 0 and all sufficiently large n, we have Ψ L L + n I(;R)+nδ [q q] ɛ n I(;) [q q]. (9) Ψ L L represents the prior entanglement between lice and ob, although its precise form is irrelevant for our argument. Now consider the following lemma, whose proof we delay until the end of this section. Lemma 6: Let K and Q be quantum systems and let Ψ L L be arbitrary. onsider any potential simulation N K K( ρ K) = D QL K ( E KL P L )( ρ K Ψ L L ) of the identity quantum channel id K K by the possibly smaller one id Q Q, assisted by the bipartite state Ψ L L. If Φ K K is maximally entangled, then the entanglement fidelity [3] satisfies F ( Φ K K, ( K N )(Φ K K ) ) Q K. (0) Plugging in to the right side of (0), we find that the entanglement fidelity is upper bounded by n I(;R )+nδ. Suppose now that strong subaddivity was not satisfied. Then, for some sufficiently small δ > 0 the entanglement fidelity would tend to zero exponentially fast. However, (9) implies that the entanglement fidelity can be made arbitrarily close to. Therefore, we must have the inequality I(; R ) 0. Proof of Lemma 6: Let {E i } and {D j } be Kraus matrices for the encoding E KL Q and decoding D QL K. Fixing orthonormal bases of L and L which Schmidt-decompose the assistance state as Ψ L L = λl l L l L, e

9 9 the above Kraus matrices can be written in bloc form ] ] E i = [E i E i L, D j = [D j D j L. ecause these maps are trace-preserving, we have E i E i = KL, D j D j = QL i which in turn implies that E il E il = δ ll K, i j D jl D jl = δ ll Q. () The overall map N K K has Kraus matrices {N ij } given by N ij = λl D jl E il. l The entanglement fidelity (0) can be written in the following form [3]: F ( Φ K K, ( K N )(Φ K K ) ) = Tr N ij π K ij = Tr Nij. K On the other hand, Tr Nij = λ ltr Djl E il ij lij lij = Q l j λ l Q Tr E il D jl D jle il () λ l Tr i E il ij ( ) D jl D jl j E il = Q λ l Tr K (3) l = Q K. bove, () is by the auchy-schwartz inquality because ran ( D jl E il ) Q, while (3) follows from the identities (). The last line holds because the squares of the Schmidt coefficients sum to unity. ombining this estimate with the previous expression for the entanglement fidelity proves the lemma. V. DISSSION We have identified the cost, in terms of entanglement and transmitted qubits, of moving a subsystem of a multipartite quantum state between two spatially separated parties when the sender and receiver each have quantum side information. Our strategy was to prove a new resource inequality which, when combined with other nown results, implies the existence of the optimal protocol. The minimal rate at which qubits must be sent provides the first nown operational interpretation of quantum conditional mutual information. While operational interpretations of quantum mutual information are nown [6], [4], these do not simply lead to one for the conditional quantity by naively subtracting mutual informations. n operational interpretation for a quantity is a proof that it is an optimal rate for performing some information processing tas. Such a proof must consist of a protocol which achieves that rate, together with a converse demonstrating optimality, as was given in [] for our situation. Our interpretation provides an explaination of the quadripartite pure state identity I(; R ) = I(; R ) because the inherent reversibility of our protocol implies that the communication cost is the same in both directions. Indeed, with the exception of the Schumacher compression step, which is essentially reversible because it succeeds with high probability, the protocol constructed to prove Theorem 3 consists entirely of isometries. Moreover, the additional steps required to prove Theorem do not introduce further nonunitarity. Throughout this paper, we have adhered to the convention of always conditioning on ob s side information, although this was an arbitrary notational choice. We thus interpret quantum conditional mutual information as a measure of the quantum correlations between and R, from the perspective of either party. Technically, our result provides an interpretation for one half of the conditional mutual information; nonetheless, we observed in [8] that by teleportation, we obtain a bona fide interpretation of conditional mutual information (i.e. without the /) as the optimal classical communication rate for state merging [7]. Our protocol includes various others as special cases. When neither lice nor ob has any side information, or if they simply disregard it so that is considered part of the reference, our protocol reduces to Schumacher compression the technical components for this are contained in Theorem 5. When it is only lice who has side information, our protocol reduces to the fully quantum reverse Shannon protocol, which itself can be proved by trivially combining Theorem 5 and the robust decoupling lemma (Lemma 3). The same theorems can be shown to imply the fully quantum Slepian-Wolf theorem, in which it is only ob who uses side information. s pointed out in [8], the formal time-reversal duality between fully quantum Slepian-Wolf and fully quantum reverse Shannon observed in [0] is embodied in a more natural way by our new protocol, which is in fact self-dual with respect to time reversal. In [8], we also observed the intuitively satisfying but nonetheless surprising fact that successive redistribution can be performed optimally using the optimal redistribution protocol. State redistribution solves the most general two-terminal source coding problem that can be considered for the class of fully quantum coding problems for i.i.d. quantum states. ecause the main technical part of our proof is proved in a one-shot fashion, it is liely that it could be applied to more general quantum sources that do not satisfy the i.i.d. property but which do have some internal regularity structure; for instance, to ground states of many-body Hamiltonians in statistical physics. However, it would perhaps be most useful for such applications to have a more direct proof of Theorem which does not use coherent channels or the cancellation lemma. While it would be most desirable to have a one-shot version of Theorem, it would perhaps be most natural to

10 0 enc I(; ) it is possible to identify an underlying heuristic organizing principle governing our optimal rates which perhaps could lend itself to further generalizations of redistribution. The main tas of state redistribution is to transform between two configurations of the subsystems as follows: I(; R ) dec R R. Let initial/final (resp. ) denote the systems lice (resp. ob) holds at the beginning/end of the protocol. onsider the following dynamic potentials relative to lice ob communication: I(; ) Fig. 4. Potential one-shot redistribution circuit maing time-reversal symmetry apparent. find a one-shot version of the related resource inequality ψ + I(; R )[q q] + I(; )[qq] ψ + I(; )[qq]. The corresponding circuit for this case maes the time-reversal symmetry most apparent, as illustrated in Figure 4. We expect state redistribution to be a useful primitive for studying more complicated state transfer problems. Most generally, one can imagine n spatially separated parties all holding various parts of a global multipartite state, wishing to shuffle their subsystems around in some arbitrary but predetermined way. There is a multitude of ways in which redistribution could be applied to give achievable rate regions for such problems, where each round of communcation would fit our general setting, although they would most liely be suboptimal in general. simple example along these lines, for which the optimal solution is not yet nown, was considered in [5], where lice and ob wish to swap two systems. We expect that judicious use of state redistribution should lead to new achievable rates for this problem, by optimizing over ways of splitting the systems to be swapped into subsystems. During the proof of our main result, we found that in some cases where strong subaddivity is satisfied I(; R ) = 0, our protocol nonetheless requires a nonzero amount of quantum communication. One cannot hope for much better, as the following example illustrates. Suppose that lice wishes to redistribute to ob the parts of many copies of an arbitrary product state ψ ψ R. lice can simply throw away the states ψ and then dilute [6] her entanglement with ob to create those states nonlocally. Despite the fact that that strong subaddivity is saturated on the global state, it is nonetheless nown [7], [8] that in general, entanglement dilution requires a sublinear amount of communication. It has been apparent that one half of the mutual information plays a central role in characterizing the optimal rates in this paper. In the following, somewhat mysterious fashion, this quantity can be considered as a measure of the correlations between two subsystems. y analogy with thermodynamics, Dinitial Dfinal I(R; initial) = I(R; ) I(R; final) = I(R; ). We interpret these as indicating the correlations between lice s systems and the reference, both before and after redistribution. The optimal qubit rate for redistribution is easily shown to equal the difference between the dynamic potentials D final D initial = I(; R ) = I(; R ). We are therefore operationally justified in interpreting this difference as measuring the correlations with the reference that lice must transfer to ob to redistribute the state. nalogously, we may also define static potentials Sinitial I( initial; initial ) = I(; ) Sfinal I( final; final ) = I(; ) which indicate the correlations between lice s and ob s systems at each state of redistribution. Similarly, the optimal ebit rate can be shown to equal the difference of the static potentials S final S initial = I(; ) I(; ). It is operationally justifiable to consider this difference as the amount of excess correlation between lice and ob that is involved in going between the two configurations. Relative to the ob lice direction, the dynamic potentials are subtracted from a constant D initial/final = H(R) D initial/final while the static potentials obey S initial/final = S final/initial. Subtracting these potentials as above, we find that while D final S final D initial S initial = D final = ( S final D initial, Sinitial ), providing another explaination of the symmetry properties of the optimal rates. One could easily imagine generalizations of the above in which more complicated potentials are defined for redistribution problems involving many more parties. However, we expect it would quite challenging to find operational justifications for such a theory.

11 KNOWLEDGMENTS We would lie to than harlie ennett for suggesting the circuit pictured in Figure 4 and Toby erger for encouraging us to find a quantum counterpart to the classical result on successive refinement of information. Igor Deveta was supported in part by the NSF grants F-0548 and F (REER). Jon Yard is grateful for support from the NSF under the grant PHY PPENDIX Here we collect the proofs of some auxilliary results used in the proof of Theorem 4. Our proof of the robust decoupling lemma (Lemma 3) relies on the following non-robust version. Lemma 7 (One-shot decoupling): Let a density matrix ϕ E be given and fix a unitary decomposition W S b of into subsystems. For each unitary, define Then () ϕ S b E = W ϕ E W. E ϕ b π b ϕ E d ϕ E ϕ E 0 S. (4) Proof of Lemma 3: y convexity of the trace norm E b b ψ π ψ E b ψ E π b ψ E d, () where d is Haar measure on (). We use the triangle inequality to bound the integrand: E ψ b π b ψ E E ϕ b π b ϕ E (5) + ψ b E ϕ b E (6) + π b ψ E π b ϕ E. (7) The second term is bounded using monotonicity, unitary invariance of the trace norm, and the assumed ɛ-closeness of ψ E and ϕ E : ψ b E E ϕ b ψ SE b ψ S E b = ψ E ϕ E ɛ. (8) Similarly, the last term satisfies π b ψ E π b ϕ E = ψ E ϕ E ψ E ϕ E ɛ. ecause x x is convex, the integral of the first term satisfies ( b ϕ E π b ϕ E ) d () () ϕ b E π b ϕ E d. The theorem follows by applying Lemma 7 to this integral. Proof of Lemma 5: To begin, note that we may choose the complex phases so that ψ L ψ = ψ L ψ. Now ψ L ψ ( ψ L ψ ) ecause 0 Λ, we have ψ L ψ L(ψ ) L(ψ ). ψ L ψ = ψ Λ ψ ψ Λ ψ = ( Tr ψ Λ ). Furthermore, unitary invariance of the trace norm and (5) imply that L(ψ ) L(ψ ) = ψ ψ ψ ψ. Therefore, ψ L ψ ( Tr ψ Λ ) ψ ψ. Finally, because the functions x x and x x are convex, we find that ψ L ψ ( P ) F as required. = (P + F ) REFERENES [] Z. Luo and I. Deveta, hannel simulation with quantum side information, 006, arxiv:quant-ph/ []. E. Shannon, mathematical theory of communication, ell System Technical Journal, vol. 7, pp and , July and October 948. [3] D. Slepian and J. K. Wolf, Noiseless coding of correlated information sources, IEEE Trans. Inform. Theory, vol. 9, pp , 97. [4]. Schumacher, Quantum coding, Phys. Rev., vol. 5, no. 4, pp , pr 995. [5] E. Lieb and M.. Rusai, Proof of the strong subadditivity of quantummechanical entropy, J. Math. Phys., vol. 4, no., pp , 973. [6]. Groisman, S. Popescu, and. Winter, On the quantum, classical and total amount of correlations in a quantum state, Phys. Rev., vol. 7, p. 0337, 005, arxiv:quant-ph/ [7] M. Horodeci, J. Oppenheim, and. Winter, Partial quantum information, Nature, vol. 436, pp , 005, arxiv:quant-ph/ [8] I. Deveta and J. Yard, n operational interpretation of quantum conditional information, To appear in Phys. Rev. Lett., 007, arxiv:quantph/ [9] T. M. over and W. H. R. Equitz, Successive refinement of information, IEEE Trans. Inform. Theory, vol. 37, no., pp , 99. [0] I. Deveta, Triangle of dualities between quantum communication protocols, Phys. Rev. Lett., vol. 97, p , 006, arxiv:quantph/ []. beyesinghe, I. Deveta, P. Hayden, and. Winter, The mother of all protocols: Restructuring quantum information s family tree, 006, arxiv:quant-ph/ [] M. Horodeci, J. Oppenheim, and. Winter, Quantum state merging and negative information, 005, arxiv:quant-ph/0547. [3] M.. Nielsen and I. L. huang, Quantum omputation and Quantum Information. ambridge, K: ambridge niversity Press, 000. [4] T. M. over and J.. Thomas, Elements of Information Theory, ser. Series in Telecommunication. New Yor: John Wiley and Sons, 99. [5] N. J. erf and. dami, Negative entropy and information in quantum mechanics, Phys. Rev. Lett., vol. 79, pp , 997. [6] I. Deveta,. W. Harrow, and. Winter, family of quantum protocols, Phys. Rev. Lett., vol. 93, p , 004, arxiv:quantph/

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