One-shot multi-sender decoupling and simultaneous decoding for the quantum MAC
aa r X i v : . [ qu a n t - ph ] F e b One-shot multi-sender decoupling and simultaneousdecoding for the quantum MAC
Sayantan Chakraborty ∗ Aditya Nema ∗ Pranab Sen * Abstract
In this work, we prove a novel one-shot ‘multi-sender’ decoupling theorem generalisingDupuis’ result. We start off with a multipartite quantum state, say on A A R , where A , A are treated as the two ‘sender’ systems and R is the reference system. We apply independentHaar random unitaries in tensor product on A and A and then send the resulting systemsthrough a quantum channel. We want the channel output B to be almost in tensor with theuntouched reference R . Our main result shows that this is indeed the case if suitable entropicconditions are met. An immediate application of our main result is to obtain a one-shot simul-taneous decoder for sending quantum information over a k -sender entanglement unassistedquantum multiple access channel (QMAC). The rate region achieved by this decoder is thenatural one-shot quantum analogue of the pentagonal classical rate region. Assuming a simul-taneous smoothing conjecture, this one-shot rate region approaches the optimal rate region ofYard et al. [YDH05] in the asymptotic iid limit. Our work is the first one to obtain a non-trivialsimultaneous decoder for the QMAC with limited entanglement assistance in both one-shotand asymptotic iid settings; previous works used unlimited entanglement assistance. The paradigm of decoupling , that is the process of removing correlations between systems, hasturned out to be a powerful and general technique for obtaining inner bounds for transmissionof quantum information in quantum Shannon theory. Its importance can be seen by its role inobtaining coding strategies for sending quantum information over a quantum channel, one of themost basic tasks in quantum Shannon theory. Let | ψ i RA be a pure state, where R is the so-called reference system that will be untouched by all operations of our protocol. We want to isometricallyencode the message system A into a system A ′ G and send A ′ through a noisy quantum channel N A ′ → B so that the receiver can decode the output B to obtain a state close to | ψ i RA . Considerthe Stinespring dilation of N , namely U A ′ → BE N , where the system E is treated as the purifying environment . Consider the global pure state | ψ i RBEG . Suppose the following decoupling conditionholds: ψ REG ≈ ψ R ⊗ σ EG for some state σ on EG . Let I denote the identity superoperator. Then byUhlmann’s theorem one can immediately conclude that there exists a decoding isometry D B → AF such that ( D B → AF ⊗ I REG | ψ i RBEG ≈ | ψ i RA | σ i FEG , * School of Technology and System Science, Tata Institute of Fundamental Research, Mumbai, India. Email: {kings-bandz, aditya.nema30, pranab.sen.73}@gmail.com | σ i FEG is a purification of σ EG . Thus if a suitable isometric encoder of A into A ′ G canbe found which satisfies the above decoupling condition, we can achieve quantum informationtransmission over a quantum channel without even constructing an explicit decoder. In otherwords, decoupling has allowed us to solve a quantum coding problem doing only half the workas compared to the classical setting!The decoupling paradigm was used to obtain a protocol for the Fully Quantum Slepian Wolf(FQSW) problem [ADHW09] aka the mother protocol of quantum Shannon theory as it can inturn be used in a black-box fashion to obtain many other protocols for useful quantum informationtheoretic tasks in the asymptotic iid setting. An useful and powerful one-shot decoupling theorem,generalising many earlier decoupling constructions including that of [ADHW09], was obtained byDupuis [Dup10]. We shall refer to this result henceforth as the single sender decoupling theorem . Ahigh level description follows. Suppose Alice holds the A register of a bipartite mixed state ρ AR ,where R is the reference system. Alice applies a Haar random unitary U on A followed by acompletely positive (CP) map T A → E . Then if certain entropic conditions are met, the resultingstate typically is close to the decoupled state ω E ⊗ ρ R where ω E is a state depending only on thechannel T and not on the state ρ AR nor on the unitary U A .The intuition described in the first paragraph of the introduction above can be made preciseand indeed Dupuis’ used his single sender decoupling theorem to obtain nearly optimal one shotinner bounds for sending quantum information over a point to point quantum channel with lim-ited entanglement assistance [Dup10], generalising an earlier result by Buscemi and Datta [BD10]for the same problem without entanglement assistance. In the asymptotic iid setting, Dupuis’ re-sult recovers the well known regularised coherent information inner bound when there is no en-tanglement assistance [Llo97,Sho02,Dev05], and the well known mutual information inner boundwhen there is unlimited entanglment assistance [BSST02].The main contribution of this work is the generalisation of the single sender decoupling theo-rem to the case of independent multiple senders . To be precise, we prove a theorem of the followingkind. Consider the multipartite state ρ A A ... A k R where the users Alice , Alice and so on only haveaccess to their respective registers A , A , . . . A k . Let Alice i apply a Haar random unitary U i to herregister A i independently of the other Alices. After the individual unitaries are applied, a CP map T A A ... A k → E is also applied. We show, if certain entropic conditions are met, that the resultingstate typically is close to the decoupled state ω E ⊗ ρ R where ω E is a state depending only on thechannel T and not on the state ρ AR nor on the unitaries U A i i .We prove our multi-sender decoupling theorem by suitably extending Dupuis’ proof of hissingle sender decoupling theorem. As will become clear during the course of its proof, one of themain bottlenecks in proving such a theorem turns out to be defining and using the correct one-shot entropic quantities in order to bound the error in the protocol. We show that a modificationof the conditional Rényi 2-entropy defined recently in [NS20] turns out to be the right quantity forour purpose. Our simultaneous decoding inner bound is thus stated in terms of a modified Rényi -coherent information and its smoothed version derived from the above quantity. A similar multisender decoupling theorem was earlier proved by Dutil [Dut11]. However his formulation did notuse the correct entropic quantity required to get strong bounds in applications e.g. inner boundsfor quantum multiple access channels.As an important application of our multi sender decoupling theorem, we consider the prob-lem of proving inner bounds for the Quantum Multiple Access Channel (QMAC) with limitedentanglement assistance in the one-shot setting i.e. when the channel is used only once. Previous2orks only considered the QMAC in the asymptotic iid setting, either with no entanglement assis-tance [YDH05] or with unlimited entanglement assistance [HDW08]. In a very recent companionpaper [CNS21], one shot inner bounds were shown for the QMAC with limited entanglement as-sistance which approach the previously known bounds in the asymptotic iid setting both withoutentanglement assistance as well as with unlimited entanglement assistance. However all theseworks use successive cancellation decoding to obtain their inner bounds. Successive cancellationtends to have faster decoding strategies than simultaneous decoding . However it also has somedrawbacks like the difficulty of clock synchronisation that arises when used together with timesharing in the asymptotic iid setting. This drawback can be eliminated by a technique called ratesplitting first developed for the classical asymptotic iid setting by [GRUW01] and later extendedto the one-shot quantum setting by [CNS21]. However using rate splitting in the one shot settingbrings a new feature which is aesthetically unappealing viz. the obtained inner bound is a subsetof the familiar ‘pentagonal’ inner bound for the MAC. Note that the pentagonal inner bound holdsboth in the classical asymptotic iid setting [Ahl71,Lia72], as well as for transmitting classical infor-mation over a quantum MAC both with and without entanglement assistance [Sen18b]. Provinga (super) pentagonal inner bound in the one shot setting requires simultaneous decoding.Our multi sender decoupling theorem allows us, for the first time, to get a simultaneous de-coder for sending quantum information over a QMAC with limited entanglement assistance. Thisallows us to obtain the (super) pentagonal rate region as shown in Figure 1.Bob Alice (
0, ˜ I δ ( B > C ))(
0, ˜ I δ ( B > AC )) ( ˜ I δ ( A > C ) , 0 )( ˜ I δ ( A > BC ) , 0 ) Q A + Q B = ˜ I ǫ δ ( AB > C ) Figure 1: Inner bound for the unassisted QMAC obtained by simultaneous decoding. The quantity˜ I δ ( · > · ) is the modified Rényi 2-coherent information and ˜ I ǫ δ ( · > · ) its smoothed versiondefined in Section 2. O ( log ǫ ) additive factors have been ignored in the figure.Simultaneous decoders are an essential building block for obtaining the best inner bounds forseveral multiterminal channels in classical network information theory e.g. Marton’s inner boundwith common message for the broadcast channel. It is expected that our simultaneous decoder forthe QMAC will pave the way for similar results in quantum network information theory too.A shortcoming of our results is that we are unable to show that our one shot inner bound forthe unassisted QMAC recovers the optimal asymptotic iid result of [YDH05]. To do so, one wouldrequire the existence of a single state which simultaneously smooths and (nearly) maximises allthe three entropic quantities that arise in our simultaneous decoding inner bound picturised inFigure 1. The existence of such a state is a major open problem in quantum information theoryand is known as the simultaneous smoothing conjecture . The interested reader is referred to [Sen18a]3or more details.In this paper we will be using a variant of the 2 entropy, defined by Nema and Sen in [NS20].The interested reader is referred to [Tom12] for an extremely comprehensive survey of these quan-tities and their properties. A näive choice of the smoothed conditional Rényi 2-entropy does notwork, for technical reasons that we will describe in Section 4. The rest of the paper is organised as follows:• In Section 2 we define the one-shot entropic quantities that we require to prove our theo-rems, along with the statements of useful facts about these quantities and other identities ingeneral.• In Section 4 we state and prove the multi sender decoupling theorem.• In Section 5 we use the 2 sender version of the multi sender decoupling theorem to deriveinner bounds for sending quantum information via the QMAC.• We conclude with Section 6 by mentioning an immediate open problem which might beuseful for other problems in quantum Shannon theory.
All vector spaces considered the paper are finite dimensional inner product spaces, also calledfinite dimensional Hilbert spaces, over the complex field and denoted by H . We use |H| to denotethe dimension of a Hilbert space H . Logarithms are all taken in base two. We tacitly assume thatthe ceiling is taken of any formula that provides dimension or value of t in unitary t -design. Thesymbols E , P denote expectation and probability respectively. The abbreviation "iid" is used tomean identically and independently distributed, which just means taking the tensor power of theidentical copies of the underlying state. The notation ":=" is used to denote the definitions of theunderlying mathematical quantities.The notation L ( A , A ) denotes the Hilbert space of all linear operators from Hilbert space A to Hilbert space A with the inner product being the Hilbert-Schmidt inner product h X , Y i : = Tr [ X † Y ] . For the special case when A = A we use the phrase operator on A and the symbol L ( A ) . The symbol I A denotes the identity operator on vector space A . The matrix π A denotes theso-called completely mixed state on system A , i.e., π A : = I A | A | . We use the notation U ◦ A as a shorthand to denote the conjugation of the operator U on the operator M , that is, U · M : = U MU † .The symbol ρ usually denotes a quantum state, also called as a density matrix which is a Her-mitian positive semidefinite matrix with unit trace, and D ( C d ) denotes the set of all d × d densitymatrices. The symbol Pos ( C d ) denotes the set of all d × d positive semidefinite matrices, and thesymbol U ( d ) denotes the set of all d × d unitary matrices with complex entries. For a positivesemidefinite matrix σ , we use σ − to denote the operator which is the orthogonal direct sum of theinverse of σ on its support and the zero operator on the orthogonal complement of the support.This definition of σ − is also known as the Moore-Penrose pseudoinverse . The symbol | ψ i denotes a4ector ψ of unit Schatten 2-norm or the Frobenius norm, and h ψ | denotes the corresponding linearfunctional. A pure quantum state is rank one density matrix. For brevity, a pure quantum state | ψ i h ψ | is denoted by ψ to emphasise that it is a density matrix. For two Hermitian matrices A , B of the same dimension, we use A ≥ B as a shorthand to imply that the matrix A − B is positivesemidefinite.Let X ∈ L ( A ) . The symbol Tr X denotes the trace of operator X . Trace is a linear map from L ( A ) to C . Let A , B be two vector spaces. The partial trace Tr A [ · ] obtained by tracing out A isdefined to be the unique linear map from L ( A ⊗ R ) to L ( R ) satisfying Tr A [ X ⊗ Y ] = ( Tr X ) Y forall operators X ∈ L ( A ) , Y ∈ L ( R ) .A linear map T : M m → M d , that maps a linear operator to another linear operator is calleda superoperator. A superoperator T is said to be positive if it maps positive semidefinite operatorsto positive semidefinite operators, and completely positive if T ⊗ I is a positive superoperator for allidentity superoperators I . A superoperator T is said to be trace preserving if Tr [ T ( X )] = Tr [ X ] forall X ∈ M m . Completely positive and Trace Preserving (abbreviated as CPTP) superoperators arecalled quantum operations or quantum channels . In this paper all the superoperators considered arecompletely positive and trace non-increasing superoperators, unless stated otherwise. The symbol I denotes the identity or the noiseless channel which does not alter the input at all or I ( X ) = X ,for all operators X .The adjoint of a superoperator is defined with respect to the Hilbert-Schmidt inner product onmatrices. If T : M m → M d is a superoperator, then its adjoint T † : M d → M m is a superoperatoruniquely defined by the property that hT † ( A ) , B i = h A , T ( B ) i for all A ∈ M d , B ∈ M m . In this section we define the relevant entropic quantities used in the proof of our general multi-user decoupling theorem. We start with the conditional min entropy, followed by conditional2-entropy, a variant of which will be used in most of our proofs.
Definition 2.1
Let ≤ ǫ < . The ǫ -smooth conditional min-entropy of ρ AB is defined as:H ǫ min ( A | B ) ρ : = min σ AB ∈ Pos ( B ) : k σ AB − ρ AB k ≤ ǫ { Tr ( γ B ) : γ B ∈ Pos ( B ) , σ AB ≤ I A ⊗ γ B } . When ǫ = , this is just H min ( A | B ) with σ AB replaced by ρ AB . Definition 2.2
Let ≤ ǫ < . The ǫ -smooth conditional Rényi 2-entropy for a bipartite positive semidef-inite operator ρ AR on systems A and R is defined as:H ǫ ( A | R ) ρ : = − σ AR ∈ Pos ( AR ) : k ρ AR − σ AR k ≤ ǫω R ∈D ( R ) : ω R > R {k ( ω R ⊗ I A ) − σ AR ( ω R ⊗ I A ) − k } . When ǫ = , we simply refer to the above quantity as conditional Rényi -entropy and denote it byH ( A | R ) ρ and define ˜ ρ AR : = ( ω R ⊗ I A ) − ρ AR ( ω R ⊗ I A ) − .The advantage of working with smoothed conditional Rényi 2-entropy is that in the asymptotic iidlimit it is appropriately bounded by conditional Shannon entropy, as mentioned in the followingFact 2.3: 5 act 2.3 Let ǫ > . Then, H ǫ ( A | B ) ω ≤ H ( A | B ) ω + ǫ log | A | + + ǫ − . and H ǫ min ( A | B ) ρ ≥ H ( A | B ) − | A | × q log ǫ The proof of the bounds on H min can be found in [TCR09, Theorem 9] and for H can be deducedby combining [TBH14, Equation 8] with [TCR09, Theorem 7, Lemma 2, Equation 33] and thenapplying the Alicki-Fannes inequality [AF04], respectively .For our proofs we will be using a slightly modified version of the 2-entropy, where we fix the σ B to a special state instead of optimising over it. The justification for this definition is as follows:1. This quantity is much more tractable than the optimised 2-entropy.2. The smoothed version of this new quantity indeed approaches the conditional Shannon en-tropy in the asymptotic iid limit, as proved in [NS20]. Definition 2.4 δ -Tilde Conditional -Entropy Given a state ρ AB on the registers AB and δ ∈ (
0, 1 ) the δ -Tilde conditional -entropy of A given B is defined as ˜ H δ ( A | B ) ρ : = − log Tr [ (cid:0) ( I A ⊗ ρ B δ ) − ρ AB ) (cid:1) ] where ρ B δ is that positive semidefinite matrix that is obtained by zeroing out the smallest eigenvalues of ρ B that sum to less than or equal to δ . The smoothed variant of this quantity, as defined by Nema and Sen was shown in [NS20] to ap-proach the Shannon conditional entropy in the asymptotic iid limit. We call this the ǫ -smooth δ -tilde conditional 2-entropy. In the interest of brevity, we will refer to this quantity simply as thesmooth tilde 2-entropy from now on. We will require some additional definitions before introduc-ing this quantity: Definition 2.5 ǫ -smooth Max Entropy Given a state ρ A and positive ǫ , the ǫ -smooth Max Entropy maxentropy is given by H ǫ max ( A ) ρ : = ρ ′ ≥ k ρ ′ − ρ k ≤ ǫ Tr [ p ρ ′ ] Definition 2.6 δ -Tilde Max Entropy Given the state ρ A , consider the state ρ δ which is obtained byzeroing out the smallest eigenvalues of ρ A which sum to less than or equal to δ . Then the δ -Tilde MaxEntropy is given by ˜ H δ max ( A ) ρ : = log (cid:13)(cid:13)(cid:13)(cid:0) ρ B δ (cid:1) − (cid:13)(cid:13)(cid:13) ∞ We are now ready to define the smooth tilde 2-entropy:
Definition 2.7 ǫ -smooth δ -Tilde Conditional -Entropy Given a state ρ AB , consider the state ρ ′ B ǫ , δ thatis obtained by zeroing out those eigenvalues of ρ B which are smaller than − ( + δ ) ˜ H ǫ max ( B ) ρ . Then, we define ˜ H ǫ δ ( A | B ) ρ : = − log min ≤ η AB ≤ ρ AB k η − ρ k ≤ ǫ Tr [ (cid:0) ( I A ⊗ ρ ′ B ǫ , δ ) − ρ AB ) (cid:1) ] Fact 2.8
For n ∈ N and ǫ , δ > , given a quantum state ρ AB and its iid extension ρ AB ⊗ n the followingholds lim ǫ , δ → lim n → ∞ ˜ H ǫ δ ( A n | B n ) ρ n n ≥ H ( A | B ) ρ .3 Useful Facts Fact 2.9 [Wat18, NS20] Any superoperator T A → B can be represented as: T A → B ( M A ) = Tr Z { V AC → BZ T ( M A ⊗ ( | i h | ) C )( W AC → BZ T ) † } where V T , W T are operators that map vectors from A ⊗ C to vectors in B ⊗ Z. Systems C and Z areconsidered as the input and output ancillary systems respectively, such that | A || C | = | B || Z | . Without lossof generality, | C | ≤ | B | and | Z | ≤ | A | . Furthermore, in the following special cases V T , W T have additionalproperties.1. T is completely positive if and only if V T = W T .2. T is trace preserving if and only if V − T = W † T . Thus, T is completely positive and trace preservingif and only if V T = W T and are unitary operators.3. T is completely positive and trace non-decreasing if and only if V T = W T and k V T k ∞ ≤ . Fact 2.10 [Dup10]
Swap Trick
Given two linear operators M and N on the system A and the swapoperator F AA ′ , where we denote by A ′ an isomorphic copy of A, the following holds: Tr [ MN ] = Tr [ (cid:0) M ⊗ N (cid:1) F AA ′ ] Fact 2.11 [Wat18]
Uhlmann’s Theorem
For quantum states M and N with purifications | φ i XY and | ψ i XZ respectively (referring to the systems Y , Z as purifying systems and systems Y , Z need not beisomorphic). Then, F ( M , N ) = max V Y → Z | h φ | V † | ψ i | where the maximization is over all partial isometries V ( ≡ V † V = I Y ) from Y to Z with dim ( Z ) ≥ dim ( Y ) . Fact 2.12
Given a linear operator M on A ⊗ it holds that E ( M ) = Z U ∈ U ( A ) (cid:0) U ⊗ · M (cid:1) dU = α I AA ′ + β F AA ′ where α and β are the solutions of the equations Tr [ M ] = α | A | + β | A | and Tr [ FM ] = α | A | + β | A | andthe integration is over the Haar measure over the Unitary group. Fact 2.13
Given a positive semidefinite operator ρ AB on the system AB | A | ≤ Tr [ ρ AB ] Tr [ ρ B ] ≤ | A | Fact 2.14
Let M be any linear operator and σ be a positive semidefinite operator on system A. Then k M k ≤ q Tr [ σ ] Tr [ σ − M σ − M † σ − ] and in particular, when M is Hermitian k M k ≤ q Tr [ σ ] Tr [ (cid:0) σ − M σ − (cid:1) ] Inner Bounds for the QMAC using the Multi Sender DecouplingTheorem
We will first consider the task of entanglement transmission. As before, we first consider theseemingly more general problem: We are given a QMAC N A ′ B ′ → C and two states pure states ψ AC R and ϕ BC R , with Alice holding the system A , Bob the system B and Charlie the systems C C . R R are the reference registers. Alice and Bob wish to send the registers A and B to Charliethrough one use of the channel N , such that at the end of the protocol, the state that Charlie holdsis close to ψ AC R ϕ BC R . To do this, we must show the existence of encoders E and E and adecoder D such that (cid:13)(cid:13) D ◦ N ◦ (cid:0)
E ⊗ E (cid:1)(cid:0) ψ ⊗ ϕ (cid:1) − ψ ⊗ ϕ (cid:13)(cid:13) ≤ ǫ We consider the complementary channel ¯ N A ′ B ′ → E and the randomized encoders E and E such that¯ N A ′ B ′ → E ◦ (cid:0) E A → A ′ ⊗ E B → B ′ (cid:1)(cid:0) ψ AR ⊗ ϕ BR (cid:1) ≈ ¯ N A ′ B ′ → E ◦ (cid:0) E A → A ′ ⊗ E B → B ′ (cid:1)(cid:0) ψ A ⊗ ϕ B (cid:1) ⊗ (cid:0) ψ R ⊗ ϕ R (cid:1) (1)We will first fix a pure control state | ω i A ” B ” CE : = U A ′ B ′ → CE N | Ω i A ” A ′ | ∆ i B ” B ′ . We consider ran-domized encoders E and E , where the randomness is derived from independently picked uni-taries U and U , each of which is identically distributed with respect to the Haar measure. Thesingle user decoupling theorem will clearly not work here, and hence we use our multisenderdecoupling theorem instead. Using that theorem, we show that there exist decoders which obeyEq. (1), as long as the following entropic inequalities are satisfied:1. − ˜ H δ ( A ” B ” | E ) − ˜ H δ ( A | R ) ψ − ˜ H δ ( B | R ) ϕ ≤ log ǫ − ˜ H δ ( B ” | E ) ω − ˜ H δ ( B | R ) ϕ ≤ log ǫ − ˜ H δ ( A ” | E ) ω − ˜ H δ ( A | R ) ϕ ≤ log ǫ One should note that to finish the argument, one still has to show the existence of two fixed isometric encoders V and V which perform almost as well as the randomized decoders. Thisargument however does not require the power of the multi sender decoupling theorem, a twoseparate applications of the single sender decoupling theorem provide a proof, with error esti-mates in terms of H max ( A ) ψ − ˜ H δ ( A ” ) ω and H max ( B ) ϕ − ˜ H δ ( B ” ) ω . The reader is referred toTheorem 5.1 for a detailed proof of the claims made above.It is now easy to show inner bounds for the entanglement transmission for the QMAC fromthese bounds. We simply set ψ AC R to be Φ R M ⊗ Φ ˜ AC and ϕ BC R to be Φ R M ⊗ Φ ˜ BC . Recallthat Φ ˜ AC and Φ ˜ BC represent pre-shared entanglement. So the inner bounds we derive are for partial entanglement assistance. For the unassisted inner bounds we simply set the systems ˜ A and˜ B to be trivial. Please refer to Theorem 5.2 for details.We note that it is possible to design an entanglement generation protocol for the QMAC aswell using the multi sender decoupling theorem. The idea is as follows: we consider the control8tate | ω i ABCE : = U A ′ B ′ → CE N | Ω i AA ′ | ∆ i BB ′ . Consider the projectors Π A → R and Π B → R , of rank | R | and | R | , where R and R are subspaces of A and B respectively. The idea is we hit the systems A and B with the operators | A || R | Π A → R U A and | B || R | Π A → R U B respectively, where U A and U B areHaar random unitaries. We want to show that, on average, the systems R and R are decoupledfrom the system E .We note that, in contrast to the case of entanglement transmission, where the random unitaryis a part of the encoder and acts on the system to be transmitted, in the case of entanglementgeneration, the averaging is actually done of the action of the random unitaries on the purifyingsystem . Because of this, the case for entanglement generation seems potentially more challenging,as, instead on acting on two tensored subsystems ( ψ AR ⊗ ϕ BR ), the random unitaries act on astate which are entangled via the system E viz. ω ABE .Nonetheless, our multisender decoupling theorem is general enough to handle this case aswell (see Theorem 4.2) and we are indeed able to show that on average the systems R and R aredecoupled with E .The proofs of the claims made above are included in the Appendix. Before we move on to the general multi sender decoupling theorem it will be instructive to see theproof for the case of only 2 senders. The proof for more than 2 senders requires heavy notation.Hence we defer its exposition to a later section.
Notation :
We will sometimes abbreviate the symbol for the unitary U A i as U i to ease the notation.As before we will use the ′ accent in conjunction with the name of a register to denote isomorphiccopies of the original system, for e.g. A and A ′ .We will require the following lemma: Lemma 4.1
Given a linear operator M on the space A A ′ A A ′ the following holds = Z (cid:0) U ⊗ ⊗ U ⊗ (cid:1) · M dU dU = α I A A ′ ⊗ I A A ′ + α F A A ′ ⊗ I A A ′ + α I A A ′ ⊗ F A A ′ + α F A A ′ ⊗ F A A ′ where the coefficients α ij are given by | A A | (cid:20) | A | | A | (cid:21) O (cid:20) | A | | A | (cid:21) α α α α = Tr [ M ] Tr [ F A A ′ ⊗ I A A ′ M ] Tr [ I A A ′ ⊗ F A A ′ M ] Tr [ F A A ′ ⊗ F A A ′ M ] roof: The proof is an easy extension of Theorem 2.12 and the linearity of integration. We give itfor completeness.We expand M A A ′ A A ′ in a Schmidt decomposition as : M = Σ i (cid:16) X A A ′ i ⊗ ( X ′ i ) A A ′ (cid:17) . Then: Z (cid:0)(cid:2) U ⊗ ⊗ U ⊗ (cid:3) · M (cid:1) dU dU = Z (cid:16) Σ i (cid:2) U ⊗ ⊗ U ⊗ (cid:3) · { X A A ′ i ⊗ ( X ′ i ) A A ′ } (cid:17) dU dU = Σ i (cid:20) Z (cid:16)(cid:2) U ⊗ · X A A ′ i (cid:3) ⊗ (cid:2) U ⊗ · ⊗ ( X ′ i ) A A ′ (cid:3)(cid:17) dU dU (cid:21) = Σ i (cid:20) Z (cid:16) U ⊗ · X A A ′ i (cid:17) dU ⊗ Z (cid:16) U ⊗ · ⊗ ( X ′ i ) A A ′ (cid:17) dU (cid:21) a = Σ i h(cid:16) α i I A A ′ + β i F A A ′ (cid:17) ⊗ (cid:16) α ′ i I A A ′ + β ′ i F A A ′ (cid:17)i b = α I A A ′ ⊗ I A A ′ + α F A A ′ ⊗ I A A ′ + α I A A ′ ⊗ F A A ′ + α F A A ′ ⊗ F A A ′ where,• (a) follows by Theorem 2.12.• (b) holds by the identification that α : = Σ i α i , α : = Σ i β i , α : = Σ i α ′ i and α : = Σ i β ′ i .In order to obtain the values of the coefficients { α kl } k , l = , we use the identities Tr (cid:2) R U ⊗ · X i dU (cid:3) = Tr [ X i ] = α i | A | + β i | A | and Tr (cid:2) F R U ⊗ · X i dU (cid:3) = Tr [ FX i ] = α i | A | + β i | A | in equality (b) andsolving the system of equations simultaneously, completes the proof. (cid:3) We now state and prove the main technical tool of this work, the 2 sender decoupling theorem:
Theorem 4.2
Let ρ A A R be a density operator, T A A → E be a CP map, and define ω A ′ A ′ E : = ( I A A ⊗T )( Φ A A ′ ⊗ Φ A A ′ ) . For a given δ > , we define σ E : = ω E δ as the positive semidefinite matrix obtainedby zeroing out the smallest eigenvalues of ω E that sum to at most δ . We define ζ : = ρ R δ similarly.Define1. ˜ T : = σ E − · T ˜ ρ A A R : = ζ R − · ρ A A AR ˜ ω A ′ A ′ E : = ˜ T ( Φ A A ′ ⊗ Φ A A ′ ) Then, Z (cid:13)(cid:13)(cid:13) T { ( U A ⊗ U A ) · ρ A A R ) } − ω E ⊗ ρ R (cid:13)(cid:13)(cid:13) dU A dU A ≤ (cid:16) δ + | A | | A | − n (cid:13)(cid:13)(cid:13) ˜ ρ A R (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ ω A ′ E (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ˜ ρ A R (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ ω A ′ E (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ˜ ρ A A R (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ ω A ′ A ′ E (cid:13)(cid:13)(cid:13) o(cid:17) where | A | > | A | . Proof:
We will start by using Theorem 2.14 with the weighting matrix ( σ E ⊗ ζ R ) . First observethat by definition Tr [ σ E ⊗ ζ R ] ≤
1. Then, Z (cid:13)(cid:13)(cid:13) T { ( U A ⊗ U A ) · ρ A A R } − ω E ⊗ ρ R (cid:13)(cid:13)(cid:13) ≤ s Z Tr (cid:20)(cid:0) ˜ T { ( U A ⊗ U A ) · ˜ ρ A A R } − ˜ ω E ⊗ ˜ ρ R (cid:1) (cid:21) T commute we see that, Z Tr (cid:20)(cid:0) ˜ T { ( U A ⊗ U A ) · ˜ ρ A A R } − ˜ ω E ⊗ ˜ ρ R (cid:1) (cid:21) = Z Tr (cid:20)(cid:0) ˜ T { ( U A ⊗ U A ) · ˜ ρ A A R } (cid:1) (cid:21) − " ˜ T n Z ( U A ⊗ U A ) · ˜ ρ A A R dU dU o ( ˜ ω E ⊗ ˜ ρ R ) (cid:21) + Tr (cid:20) ( ˜ ω E ⊗ ˜ ρ R ) (cid:21) = Z Tr (cid:20)(cid:8) ˜ T ( U A ⊗ U A ) · ˜ ρ A A R ) (cid:9) (cid:21) − Tr (cid:20) ( ˜ ω E ⊗ ˜ ρ R ) (cid:21) By standard manipulations, using Theorem 2.10 and the definition of adjoint of an operator, itfollows that Z Tr (cid:20)(cid:8) ˜ T ( U A ⊗ U A ) · ˜ ρ A A R ) (cid:9) (cid:21) dU dU = Z Tr (cid:20)(cid:0) ˜ ρ A A R (cid:1) ⊗ n(cid:16)(cid:0) U †1 ⊗ ⊗ U †2 ⊗ (cid:1) · ˜ T † ⊗ (cid:0) F EE ′ (cid:1)(cid:17) O F RR ′ o(cid:21) dU dU = Tr (cid:20)(cid:0) ˜ ρ A A R (cid:1) ⊗ n Z (cid:16)(cid:0) U †1 ⊗ ⊗ U †2 ⊗ (cid:1) · ˜ T † ⊗ (cid:0) F EE ′ (cid:1)(cid:17) dU dU O F RR ′ o(cid:21) (2)We will now use Theorem 4.1 by plugging in the matrix ˜ T † ⊗ (cid:0) F EE ′ (cid:1) into M . The first step is tocompute the entries of the vector on the R.H.S. in the matrix equation in Theorem 4.1. We willdemonstrate one such computation, the rest follow along similar lines.11 omputing Tr (cid:20) ( F A A ′ ⊗ F A A ′ ) ˜ T † ⊗ (cid:0) F EE ′ (cid:1)(cid:21) : Tr (cid:20) ( F A A ′ ⊗ F A A ′ ) ˜ T † ⊗ (cid:0) F EE ′ (cid:1)(cid:21) = Tr (cid:20) ˜ T ⊗ (cid:0) F A A ′ ⊗ F A A ′ (cid:1) F EE ′ (cid:21) = Tr (cid:20) ˜ T ⊗ (cid:0) F A A ′ (cid:0) I A ⊗ I A ′ (cid:1) ⊗ F A A ′ (cid:0) I A ⊗ I A ′ (cid:1)(cid:1) F EE ′ (cid:21) a = | A | | A | Tr (cid:20) ˜ T ⊗ (cid:0) F A A ′ (cid:0) Tr ˆ A ( Φ A ˆ A ) ⊗ Tr ˆ A ′ ( Φ A ′ ˆ A ′ ) (cid:1) ⊗ F A A ′ (cid:0) Tr ˆ A ( Φ A ˆ A ) ⊗ Tr ˆ A ′ ( Φ A ′ ˆ A ′ ) (cid:1)(cid:1) F EE ′ (cid:21) = | A | | A | Tr (cid:20) ˜ T ⊗ (cid:0) Tr ˆ A , ˆ A ′ (cid:0) ( F A A ′ ⊗ I ˆ A ˆ A ′ )( Φ A ˆ A ⊗ Φ A ′ ˆ A ′ ) (cid:1) ⊗ Tr ˆ A , ˆ A ′ (cid:0) ( F A A ′ ⊗ I ˆ A ˆ A ′ )( Φ A ˆ A ⊗ Φ A ′ ˆ A ′ ) (cid:1) F EE ′ (cid:21) = | A | | A | Tr (cid:20) Tr ˆ A , ˆ A ′ , ˆ A , ˆ A ′ (cid:8)(cid:0) ˜ T ⊗ ⊗ I ˆ A , ˆ A ′ , ˆ A , ˆ A ′ (cid:1)(cid:0)(cid:0) ( F A A ′ ⊗ I ˆ A ˆ A ′ )( Φ A ˆ A A ′ ˆ A ′ ) (cid:1) ⊗ (cid:0) ( F A A ′ ⊗ I ˆ A ˆ A ′ )( Φ A ˆ A A ′ ˆ A ′ ) (cid:1)(cid:9) F EE ′ (cid:21) = | A | | A | Tr (cid:20) (cid:8)(cid:0) ˜ T ⊗ ⊗ I ˆ A , ˆ A ′ , ˆ A , ˆ A ′ (cid:1)(cid:0)(cid:0) ( F A A ′ ⊗ I ˆ A ˆ A ′ )( Φ A ˆ A A ′ ˆ A ′ ) (cid:1) ⊗ (cid:0) ( F A A ′ ⊗ I ˆ A ˆ A ′ )( Φ A ˆ A A ′ ˆ A ′ ) (cid:1)(cid:9) (cid:0) F EE ′ ⊗ I ˆ A ˆ A ′ ⊗ I ˆ A , ˆ A ′ (cid:1)(cid:21) b = | A | | A | Tr (cid:20) (cid:8)(cid:0) ˜ T ⊗ ⊗ I ˆ A , ˆ A ′ , ˆ A , ˆ A ′ (cid:1)(cid:0)(cid:0) ( I A A ′ ⊗ ( F T ) ˆ A ˆ A ′ )( Φ A ˆ A A ′ ˆ A ′ ) (cid:1) ⊗ (cid:0) ( I A A ′ ⊗ ( F T ) ˆ A ˆ A ′ )( Φ A ˆ A A ′ ˆ A ′ ) (cid:1)(cid:9) (cid:0) F EE ′ ⊗ I ˆ A ˆ A ′ ⊗ I ˆ A , ˆ A ′ (cid:1)(cid:21) = | A | | A | Tr (cid:20) (cid:8)(cid:0) ˜ T ⊗ ⊗ I ˆ A , ˆ A ′ , ˆ A , ˆ A ′ (cid:1)(cid:0) Φ A A ˆ A ˆ A ⊗ Φ A ′ A ′ ˆ A ′ ˆ A ′ (cid:1) × (cid:0) F EE ′ ⊗ ( F T ) ˆ A ˆ A ′ ⊗ ( F T ) ˆ A ˆ A ′ (cid:1)(cid:21) = | A | | A | Tr (cid:20) (cid:16)(cid:0) ˜ ω ˆ A ˆ A ′ E ⊗ ˜ ω ˆ A ′ ˆ A ′ E ′ (cid:1)(cid:0) F EE ′ ⊗ ( F T ) ˆ A ˆ A ′ ⊗ ( F T ) ˆ A ˆ A ′ (cid:1)(cid:21) c = | A | | A | (cid:13)(cid:13)(cid:13) ˜ ω ˆ A ˆ A E (cid:13)(cid:13)(cid:13) where,• (a) follows by defining systems A ∼ = ˆ A ′ , A ∼ = ˆ A ′ , Φ A ˆ A , Φ A ′ ˆ A ′ , Φ A ˆ A , Φ A ′ ˆ A ′ are themaximally entangled states and the fact that I A m = | A | Tr ˆ A m ( Φ A m ˆ A m ) for m = {
1, 2 } ;12 (b) follows from the fact that for maximally entangled states, say Φ AA ′ and any operator M A it holds that ( M A ⊗ I A ′ ) Φ AA ′ = ( I A ⊗ ( M T ) A ′ ) Φ AA ′ with the identification of the systems as A = A A ′ , A ′ = ˆ A ˆ A ′ and the operator M as the swap operator F ; and• (c) follows from the Theorem 2.10 and the observation that F T is equivalent to F .Using similar arguments it can be shown that1. Tr (cid:20) ˜ T † ⊗ (cid:0) F EE ′ (cid:1)(cid:21) = | A | | A | (cid:13)(cid:13) ˜ ω E (cid:13)(cid:13)
2. Tr (cid:20) F A A ′ ⊗ I A A ′ ˜ T † ⊗ (cid:0) F EE ′ (cid:1)(cid:21) = | A | | A | (cid:13)(cid:13)(cid:13) ˜ ω ˆ A E (cid:13)(cid:13)(cid:13)
3. Tr (cid:20) I A A ′ ⊗ F A A ′ ˜ T † ⊗ (cid:0) F EE ′ (cid:1)(cid:21) = | A | | A | (cid:13)(cid:13)(cid:13) ˜ ω ˆ A E (cid:13)(cid:13)(cid:13) Finally, to get meaningful bounds we need to bound the values of α , α , α and α . To do thiswe first invert the matrix in Theorem 4.1 and observe the following: α α α α = | A A | (cid:0) | A | − (cid:1)(cid:0) | A | − (cid:1) (cid:20) | A | − − | A | (cid:21) O (cid:20) | A | − − | A | (cid:21) (cid:13)(cid:13) ˜ ω E (cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ ω ˆ A E (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ ω ˆ A E (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ ω ˆ A ˆ A E (cid:13)(cid:13)(cid:13) (3) = | A A | (cid:0) | A | − (cid:1)(cid:0) | A | − (cid:1) | A || A | −| A | −| A | −| A | | A || A | −| A |−| A | | A || A | −| A | −| A | −| A | | A || A | (cid:13)(cid:13) ˜ ω E (cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ ω ˆ A E (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ ω ˆ A E (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ ω ˆ A ˆ A E (cid:13)(cid:13)(cid:13) (4) Bounding α Consider the quantity I : = | A || A | (cid:13)(cid:13)(cid:13) ˜ ω E (cid:13)(cid:13)(cid:13) − | A | (cid:13)(cid:13)(cid:13) ˜ ω ˆ A E (cid:13)(cid:13)(cid:13) − | A | (cid:13)(cid:13)(cid:13) ˜ ω ˆ A E (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ˜ ω ˆ A ˆ A E (cid:13)(cid:13)(cid:13) By Theorem 2.13 we have that1. (cid:13)(cid:13)(cid:13) ˜ ω ˆ A ˆ A E (cid:13)(cid:13)(cid:13) ≤ | A | (cid:13)(cid:13)(cid:13) ˜ ω ˆ A E (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ ω ˆ A E (cid:13)(cid:13)(cid:13) ≥ | A | (cid:13)(cid:13) ˜ ω E (cid:13)(cid:13) I ≤ | A | · (cid:0) | A | − (cid:1) | A | (cid:13)(cid:13)(cid:13) ˜ ω E (cid:13)(cid:13)(cid:13) Thus on solving the system of Equations 3 and the bound on I further implies that α = | A || A | (cid:0) | A | − (cid:1)(cid:0) | A | − (cid:1) I ≤ | A | | A | − (cid:13)(cid:13)(cid:13) ˜ ω E (cid:13)(cid:13)(cid:13) Bounding α Define II : = (cid:2) −| A | | A || A | −| A | (cid:3) (cid:13)(cid:13) ˜ ω E (cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ ω ˆ A E (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ ω ˆ A E (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ ω ˆ A ˆ A E (cid:13)(cid:13)(cid:13) By Theorem 2.13 we have (cid:13)(cid:13)(cid:13) ˜ ω ˆ A E (cid:13)(cid:13)(cid:13) ≤ | A | (cid:13)(cid:13) ˜ ω E (cid:13)(cid:13) and (cid:13)(cid:13)(cid:13) ˜ ω ˆ A ˆ A E (cid:13)(cid:13)(cid:13) ≥ | A | − (cid:13)(cid:13)(cid:13) ˜ ω ˆ A E (cid:13)(cid:13)(cid:13) , which im-plies that α = | A || A | (cid:0) | A | − (cid:1)(cid:0) | A | − (cid:1) II ≤ | A | | A | − (cid:13)(cid:13)(cid:13) ˜ ω ˆ A E (cid:13)(cid:13)(cid:13) Bounding α Define
III : = (cid:2) −| A | | A || A | −| A | (cid:3) (cid:13)(cid:13) ˜ ω E (cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ ω ˆ A E (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ ω ˆ A E (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ ω ˆ A ˆ A E (cid:13)(cid:13)(cid:13) By Theorem 2.13 we have (cid:13)(cid:13)(cid:13) ˜ ω ˆ A E (cid:13)(cid:13)(cid:13) ≤ | A | (cid:13)(cid:13)(cid:13) ˜ ω ˆ A ˆ A E (cid:13)(cid:13)(cid:13) and (cid:13)(cid:13) ˜ ω E (cid:13)(cid:13) ≥ | A | − (cid:13)(cid:13)(cid:13) ˜ ω ˆ A E (cid:13)(cid:13)(cid:13) , which im-14lies that α = | A || A | (cid:0) | A | − (cid:1)(cid:0) | A | − (cid:1) III ≤ | A | | A | − (cid:13)(cid:13)(cid:13) ˜ ω ˆ A E (cid:13)(cid:13)(cid:13) Bounding α Define IV = (cid:2) −| A | −| A | | A || A | (cid:3) (cid:13)(cid:13) ˜ ω E (cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ ω ˆ A E (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ ω ˆ A E (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ ω ˆ A ˆ A E (cid:13)(cid:13)(cid:13) By Theorem 2.13 we have (cid:13)(cid:13)(cid:13) ˜ ω ˆ A E (cid:13)(cid:13)(cid:13) ≥ | A | − (cid:13)(cid:13)(cid:13) ˜ ω ˆ A ˆ A E (cid:13)(cid:13)(cid:13) and (cid:13)(cid:13) ˜ ω E (cid:13)(cid:13) ≤ | A | (cid:13)(cid:13)(cid:13) ˜ ω ˆ A E (cid:13)(cid:13)(cid:13) , which im-plies that α = | A || A | (cid:0) | A | − (cid:1)(cid:0) | A | − (cid:1) IV ≤ | A | | A | − (cid:13)(cid:13)(cid:13) ˜ ω ˆ A ˆ A E (cid:13)(cid:13)(cid:13) Collating all these bounds and the fact that systems ˆ A i can be relabelled as A ′ i for i = {
1, 2 } wesee that Z Tr (cid:20)(cid:0) ˜ T { ( U A ⊗ U A ) · ˜ ρ A A R } (cid:1) (cid:21) dU dU ≤ | A | | A | − (cid:20) (cid:13)(cid:13)(cid:13) ˜ ρ R (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ ω E (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ˜ ρ A R (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ ω A ′ E (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ˜ ρ A R (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ ω A ′ E (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ˜ ρ A A R (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ ω A ′ A ′ E (cid:13)(cid:13)(cid:13) (cid:21) Plugging this bound into the main expression of the theorem we find that Z (cid:13)(cid:13)(cid:13) T ( U A ⊗ U A · ρ A A R ) − ω E ⊗ ρ R (cid:13)(cid:13)(cid:13) ≤ | A | − (cid:13)(cid:13)(cid:13) ˜ ρ R (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ ω E (cid:13)(cid:13)(cid:13) + | A | | A | − (cid:20) (cid:13)(cid:13)(cid:13) ˜ ρ A R (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ ω A ′ E (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ˜ ρ A R (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ ω A ′ E (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ˜ ρ A A R (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ˜ ω A ′ A ′ E (cid:13)(cid:13)(cid:13) (cid:21) By the choice of the weighting matrices σ E and ζ R in the statement of the theorem, (cid:13)(cid:13) ˜ ρ R (cid:13)(cid:13) (cid:13)(cid:13) ˜ ω E (cid:13)(cid:13) ≤
1, for instance σ E is the matrix obtained by zeroing out the smallest eigen values that sum up to δ and hence (cid:13)(cid:13) ˜ ω E (cid:13)(cid:13) ≤ ζ R by curtailing the marginal density operator ρ R and finally choosing | A | large such that the first term is less that δ . This concludes the proof. (cid:3) We now state a corollary that will be useful in stating our coding theorems:15 orollary 4.3
Given the same conditions as in Theorem 4.2 the following holds Z (cid:13)(cid:13)(cid:13) T { ( U A ⊗ U A ) · ρ A A R } − ω E ⊗ ρ R (cid:13)(cid:13)(cid:13) dU A dU A ≤ (cid:16) δ + · − ˜ H δ ( A | R ) ρ − ˜ H δ ( A | E ) ω + − ˜ H δ ( A | R ) ρ − ˜ H δ ( A | E ) ω + − ˜ H δ ( A A | R ) ρ − ˜ H δ ( A A | E ) ω (cid:17) Proof:
The proof is easy, since the term | A | | A | − ≤ | A | . The rest follows trivially bythe definition of ˜ H δ ( ·|· ) . (cid:3) We now generalise the tensor product decoupling theorem for k > Theorem 4.4 (Generalised k sender Tensor Product Decoupling Theorem) Let ρ A ... A k − R be a den-sity operator that can be thought of as an entangled state between k senders with each sender denoted by { A i } k − i = and a reference R. Let T A ... A k − → E be a CP map, and define ω ˆ A ... ˆ A k − E : = ( I ⊗ T ) · ( N i Φ A i ˆ A i ) ,the Choi state for the superoperator T . Then, Z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) T (cid:8)(cid:0) O i U A i RAND (cid:1) · ρ A ... A k − R (cid:9) − ω E ⊗ ρ R (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) dU . . . dU k − ≤ (cid:16) | A . . . A k − | ( | A k − | − )( | A | − ) − (cid:17) · (cid:13)(cid:13)(cid:13) ˜ ω E (cid:13)(cid:13)(cid:13) · (cid:13)(cid:13)(cid:13) ˜ ρ R (cid:13)(cid:13)(cid:13) + | A . . . A k − | ( | A k − | − )( | A | − ) ∑ b = k (cid:13)(cid:13)(cid:13) ˜ ω A b E (cid:13)(cid:13)(cid:13) (cid:0) (cid:13)(cid:13)(cid:13) ˜ ρ A b R (cid:13)(cid:13)(cid:13) · k + (cid:13)(cid:13)(cid:13) ˜ ρ R (cid:13)(cid:13)(cid:13) (cid:1) where we assume that | A | is the smallest among the dimensions of the registers and the indices b ∈{
0, 1, . . . , k − } are represented as bit strings of length k. Proof:
For brevity of notation, let A [ k ] : = A k − k − denote the system representing the jointstate of all the k senders with dimension | A [ k ] | : = Π k − i = | A i | . We will use the same definitionsof ˜ T , ˜ ω and ˜ ρ as in Theorem 4.2, that is, to represent that ˜ · denotes conjugation of the underlyingoperator by appropriate weighting matrices arising due to Fact 2.14. We begin with the applicationof Fact 2.14 as follows: Z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) T (cid:8)(cid:0) O i U A i RAND (cid:1) · ρ A ... A k − R (cid:9) − ω E ⊗ ρ R (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) dU . . . dU k − ≤ s Z Tr h(cid:16) ˜ T (cid:8)(cid:0) O i U A i RAND (cid:1) · ˜ ρ A ... A k − R (cid:9) − ˜ ω E ⊗ ˜ ρ R (cid:17) i dU . . . dU k − Recall that Z Tr h(cid:16) ˜ T (cid:8)(cid:0) O i U A i RAND (cid:1) · ˜ ρ A ... A k − R (cid:9) − ˜ ω E ⊗ ˜ ρ R (cid:17) i dU . . . dU k − = Z Tr h(cid:16) ˜ T (cid:8)(cid:0) O i U A i RAND (cid:1) · ˜ ρ A ... A k − R (cid:9)(cid:17) i d (cid:0) O i U i (cid:1) − Tr [( ˜ ω E ) ] · Tr [( ˜ ρ R ) ] Z Tr h(cid:16) ˜ T (cid:8)(cid:0) O i U A i RAND (cid:1) · ˜ ρ A ... A k − R (cid:9)(cid:17) i d (cid:0) O i U i (cid:1) = Tr h ( ˜ ρ A ... A k − R ) ⊗ (cid:16) E U † A ,..., U † Ak − k − (cid:2) O i (cid:0) U † A i i ⊗ U † A ′ i i (cid:1) · M A [ k ] A ′ [ k ] (cid:3) ⊗ F RR ′ (cid:17)i (5)where M A [ k ] A ′ [ k ] : = (cid:0) ˜ T † (cid:1) ⊗ (cid:0) F EE ′ (cid:1) and the expectation is taken over independent choice of Haarrandom unitaries { U i } k − i = .From Theorem 4.1 we have: E U † A ,..., U † Ak − k − [ O i (cid:0) U † A i i ⊗ U † A ′ i i (cid:1) · M A [ k ] A ′ [ k ] ]= E U † A ,..., U † Ak − k − [ O i (cid:0) U A i i ⊗ U A ′ i i (cid:1) · M A [ k ] A ′ [ k ] ] (6) = k ∑ a : = a k − ... a = k α a O i (cid:16) F A i A ′ i (cid:17) a i (7)where in the last equality we represent the indices { a } k − a = in binary as a k − , . . . , a for each a i ∈
0, 1. To evaluate the coefficients α a we again apply Lemma 4.1 with the following equalities:Tr ( M A [ k ] A ′ [ k ] ) = Tr (cid:18) E ⊗ k − i = U Aii h ( ⊗ k − i = ( U † A i i ⊗ U † A ′ i i )) ◦ ( M A [ k ] A ′ [ k ] ) i(cid:19) (8)and Tr (cid:16) ( ⊗ k − i = ( F A i A ′ i ) a i )( M A [ k ] A ′ [ k ] ) (cid:17) (9) = Tr (cid:18) ( ⊗ k − i = ( F A i A ′ i ) a i ) E ⊗ k − i = U Aii h ( ⊗ k − i = ( U † A i i ⊗ U † A ′ i i )) ◦ ( M A [ k ] A ′ [ k ] ) i(cid:19) . (10)This gives the matrix equation K · α k ... α a ... α k = | A [ k ] | ... (cid:13)(cid:13)(cid:13) ˜ ω ˆ A b E (cid:13)(cid:13)(cid:13) ... (11)where, for the bit string b : = b k − . . . b we define:˜ ω ˆ A b E : = ˜ ω ˆ A bk − k − ... ˆ A b E K is a 2 k × k matrix with rows indexed by bit vector a ∈ {
0, 1 } k and columns indexedby the bit vector b ∈ {
0, 1 } k and is obtained from Eq. (8) and Eq. (9) with entries-1 ( K ) b , a = | A [ k ] | k − ∏ i = A ( b i ⊕ a i ) i , (12)where ⊕ denotes the bit-wise XOR. This is not hard to see as the i -th term in the product is the ( b i , a i ) term of the i -th 2 × | A i | b i ⊕ a i .Also note that The RHS of Eq. (11) comes from the fact thatTr [ O i (cid:0) F A i A ′ i (cid:1) b i M ] = (cid:0) ∏ i ∈{ k − } | A i | (cid:17) (cid:13)(cid:13)(cid:13) ˜ ω ˆ A b E (cid:13)(cid:13)(cid:13) This leads to the following representation of K : K = | A k − || A k − | . . . | A | (cid:18) | A k − | | A k − | (cid:19) ⊗ (cid:18) | A k − | | A k − | (cid:19) ⊗ . . . ⊗ (cid:18) | A | | A | (cid:19) = | A [ k ] | O i ∈{ k − k − } (cid:18) | A i | | A i | (cid:19) (13)From Eq. (13) we note that: K − = | A [ k ] | (cid:0) | A k − | − (cid:1) . . . (cid:0) | A | − (cid:1) O i ∈{ k − k − } (cid:18) | A i | − − | A i | (cid:19) (14)Then coupled with Eq. (12) and Eq. (11), Eq. (14) implies that ( K − ) a , b = | A [ k ] | (cid:0) | A k − | − (cid:1) . . . (cid:0) | A | − (cid:1) ∏ i ∈{ k − k − } | A i | b i ⊕ a i ( − ) a i ⊕ b i (15)This directly implies that α a = | A [ k ] | (cid:0) | A k − | − (cid:1) . . . (cid:0) | A | − (cid:1) k ∑ b = k (cid:16) ∏ i ∈{ k − k − } | A i | b i ⊕ a i ( − ) a i ⊕ b i (cid:17) (cid:13)(cid:13)(cid:13) ˜ ω A b [ k ] E (cid:13)(cid:13)(cid:13) We will differentiate between the following two cases. Define c : = a ⊕ b . Case 1: a = k and let without loss of generality A be the register with the smallest dimension. Then, α a = | A k − . . . A | (cid:0) | A k − | − (cid:1) . . . (cid:0) | A | − (cid:1) (cid:16) (cid:13)(cid:13)(cid:13) ˜ ω E (cid:13)(cid:13)(cid:13) − (cid:13)(cid:13) ˜ ω A E (cid:13)(cid:13) | A | (cid:17) + | A k − . . . A | (cid:0) | A k − | − (cid:1) . . . (cid:0) | A | − (cid:1) ∑ c ′ : = c k − ... c = k − h(cid:16) ∏ i = | A ¯ c i i | ( − ) c i (cid:17) | A | (cid:13)(cid:13)(cid:13) ˜ ω A c ′ E (cid:13)(cid:13)(cid:13) − (cid:16) ∏ i = | A ¯ c i i | ( − ) c i (cid:17) (cid:13)(cid:13)(cid:13) ˜ ω A c ′ A E (cid:13)(cid:13)(cid:13) i (16)18or c ′ with odd parity and Theorem 2.13 the term inside the summation is: (cid:16) ∏ i = | A ¯ c i i | (cid:17)h (cid:13)(cid:13)(cid:13) ˜ ω A c ′ A E (cid:13)(cid:13)(cid:13) − | A | (cid:13)(cid:13)(cid:13) ˜ ω A c ′ E (cid:13)(cid:13)(cid:13) i ≤ c ′ with even parity and again Theorem 2.13 we have that: (cid:16) ∏ i = | A ¯ c i i | (cid:17)h | A | (cid:13)(cid:13)(cid:13) ˜ ω A c ′ E (cid:13)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:13) ˜ ω A c ′ A E (cid:13)(cid:13)(cid:13) i ≤ (cid:16) ∏ i = | A ¯ c i i | (cid:17) · (cid:16) | A | − | A | (cid:17) · (cid:13)(cid:13)(cid:13) ˜ ω A c ′ E (cid:13)(cid:13)(cid:13) (18)Substituting Eq. (17) and Eq. (18) in equation Eq. (16) and using the bound Theorem 2.13 for (cid:13)(cid:13) ˜ ω A E (cid:13)(cid:13) ≥ (cid:13)(cid:13) ˜ ω E (cid:13)(cid:13) / | A | , we get: α k ≤ (cid:13)(cid:13)(cid:13) ˜ ω E (cid:13)(cid:13)(cid:13) · | A [ k ] | ( | A k − | − )( | A | − ) h − | A | i + ∑ c ′ = k − (cid:13)(cid:13)(cid:13) ˜ ω A c ′ E (cid:13)(cid:13)(cid:13) · | A | · ∏ i = | A ¯ c i + i | ( | A k − | − )( | A | − ) h | A | − | A | i = (cid:13)(cid:13)(cid:13) ˜ ω E (cid:13)(cid:13)(cid:13) · | A . . . A k − | ( | A k − | − )( | A | − )+ ∑ c = k , c = (cid:13)(cid:13)(cid:13) ˜ ω A c E (cid:13)(cid:13)(cid:13) · ∏ i = | A ¯ c i + i | ( | A k − | − )( | A | − ) (19) Case 2: a = k Firstly observe that, given a fixed a ∈ {
0, 1 } k , and c = a ⊕ b for some b ∈ {
0, 1 } k , (cid:16) ∏ i | A ¯ c i i | (cid:17) · (cid:13)(cid:13)(cid:13) ˜ ω A a ⊕ c E (cid:13)(cid:13)(cid:13) ≤ (cid:16) ∏ i | A i | (cid:17) · (cid:13)(cid:13)(cid:13) ˜ ω A a E (cid:13)(cid:13)(cid:13) (20)This is easy to verify on a case by case basis, by considering any fixed index i ∈ [ k ] and iteratingthrough all possible values of the tuple ( a i , c i ) . The above identity holds in each of these fourpossible cases, which is seen either directly or by invoking Theorem 2.13, as the case demands.Then we simply bound the value of α a as follows: α a ≤ | A [ k ] | ( | A k − | − )( | A | − ) (cid:13)(cid:13)(cid:13) ˜ ω A a E (cid:13)(cid:13)(cid:13) + | A [ k ] | ( | A k − | − )( | A | − ) ∑ b = (cid:16) ∏ i | A ¯ c i i | (cid:17) · (cid:13)(cid:13)(cid:13) ˜ ω A a ⊕ c E (cid:13)(cid:13)(cid:13) ≤ | A [ k ] | ( | A k − | − )( | A | − ) (cid:13)(cid:13)(cid:13) ˜ ω A a E (cid:13)(cid:13)(cid:13) · k (21)19here in the first inequality we upper bound every term by its absolute values and use Eq. (20).Finally we collate the estimates for α a for the two different cases from Eq. (19) and Eq. (21) andsubstitute these values in Eq. (6) to get: E ⊗ k − i = U Aii h ( ⊗ k − i = ( U † A i i ⊗ U † A ′ i i )) · M A [ k ] A ′ k i ≤ | ( A k − . . . A ) | ( A k − − ) . . . ( A − ) k ˜ ω E k ! ( I A [ k ] A ′ [ k ] )+ ∑ c = k , c = (cid:13)(cid:13)(cid:13) ˜ ω A c E (cid:13)(cid:13)(cid:13) · ∏ i = | A ¯ c i + i | ( | A k − | − )( | A | − ) ( I A [ k ] A ′ [ k ] )+ k ∑ b = k | A [ k ] | ( | A k − | − )( | A | − ) (cid:13)(cid:13)(cid:13) ˜ ω A a E (cid:13)(cid:13)(cid:13) · k ! O i (cid:16) F A i A ′ i (cid:17) a i (22)By substituting Eq. (22) in Eq. (5) we get: E U ,..., U k − Tr h(cid:16) ˜ T (cid:8)(cid:0) O i U A i RAND (cid:1) · ˜ ρ A ... A k − R (cid:9) − ˜ ω E ⊗ ˜ ρ R (cid:17) i ≤ (cid:16) | A . . . A k − | ( | A k − | − )( | A | − ) − (cid:17) · (cid:13)(cid:13)(cid:13) ˜ ω E (cid:13)(cid:13)(cid:13) · (cid:13)(cid:13)(cid:13) ˜ ρ R (cid:13)(cid:13)(cid:13) + | A . . . A k − | ( | A k − | − )( | A | − ) ∑ b = k (cid:13)(cid:13)(cid:13) ˜ ω A b E (cid:13)(cid:13)(cid:13) (cid:0) (cid:13)(cid:13)(cid:13) ˜ ρ A b R (cid:13)(cid:13)(cid:13) · k + (cid:13)(cid:13)(cid:13) ˜ ρ R (cid:13)(cid:13)(cid:13) (cid:1) This concludes the proof. (cid:3)
Remark 4.5
Just as in Theorem 4.2 we can justify that k ˜ ω E k k ˜ ρ R k is much smaller than the second termin Theorem 4.4 above. For this, recall ˜ ω E : = ( ω ′′ δ ) − E ω E ( ω ′′ ǫ ) − E , with ( ω ′′ ǫ ) E as the operator defined byzeroing out the smallest eigen values of ω E that sum up to δ . Thus, k ˜ ω E k ≤ .Similarly, define ζ R as the operator obtained by zeroing out those eigen values of ρ R that sum to δ . Thus, k ˜ ρ R k ≤ . Hence, ⇒ k ˜ ω E k k ˜ ρ R k ≤ Thus the term with k ˜ ω E k k ˜ ρ R k can be neglected in the multi-user tensor product decoupling theorem above. The second term involving (cid:13)(cid:13)(cid:13) ˜ ω A b E (cid:13)(cid:13)(cid:13) × (cid:13)(cid:13)(cid:13) ˜ ρ A b E (cid:13)(cid:13)(cid:13) serves as the entropic quantity that gives the rate region for reliable communicationfor QMAC. In this section we will use the results of the previous section to derive coding theorems, in the oneshot regime for the Quantum Multiple Access Channel. The task we will consider is the following: Given a quantum multiple access channel N A ′ B ′ → C with Alice and Bob as senders and Charlieas receiver, consider two pure states ψ AC R and ϕ BC R where A and B are registers that belongto Alice and Bob respectively, C and C belong to Charlie and R and R are purifying systems.The task is for Alice and Bob to send their shares of these states to Charlie by a single use of20he channel. To do this we will show the existence of encoding isometries U A LICE and V BOB and adecodimg CPTP map D such that (cid:13)(cid:13)(cid:13) D ◦ N ( U ALICE ⊗ V
BOB · ψ AC R ⊗ ϕ BC R ) − ψ AC R ⊗ ϕ BC R (cid:13)(cid:13)(cid:13) ≤ ǫ for some small ǫ . The strategy we follow closely resembles the one for the point to point channel. Theorem 5.1
Let ψ AC R and ϕ BC R be pure states and the registers C and C are held by Charlie. N A ′ B ′ → C is a CPTP map. ω A ” B ” CE : = U N ( Ω A ′ A ” ⊗ ∆ B ′ B ” ) where Ω A ′ A ” and ∆ B ′ B ” are pure statesand | A ” | = | A ′ | and | B ” | = | B ′ | . Then there exist encoding isometries U ALICE and V BOB and a decodingCPTP D CC C → ˆ A ˆ BC C such that (cid:13)(cid:13)(cid:13) D ◦ N (( U ALICE ⊗ V
BOB ) · ψ AC R ⊗ ϕ BC R ) − ψ AC R ⊗ ϕ BC R (cid:13)(cid:13)(cid:13) ≤ p δ where, for some δ , δ , δ , δ , δ > , we have: δ + − ˜ H δ ( A ” B ” | E ) ω − ˜ H δ ( A | R ) ψ − ˜ H δ ( B | R ) ϕ + − ˜ H δ ( B ” | E ) ω − ˜ H δ ( B | R ) ϕ + − ˜ H δ ( A ” | E ) ω − ˜ H δ ( A | R ) ψ : = δ H max ( A ) ψ − ˜ H δ ( A ” ) ω : = δ H max ( B ) ϕ − ˜ H δ ( B ” ) ω : = δ δ + p δ + p δ + p δ δ : = δ Proof:
Let the states | Ω i ◦ AA ′ and | ∆ i ◦ BB ′ to be copies of the original | Ω i A ” A ′ and | ∆ i A ” A ′ states,where | ◦ A | = | A ” | and | ◦ B | = | B ” | . Define T ◦ A ◦ B → E ( ρ ) : = | A ” B ” | ¯ N ( op ◦ A → A ′ ( | Ω i ) ⊗ op ◦ B → B ′ ( | ∆ i ) · ρ ) Firstly, observe that : ω A ” B ” E = Tr C [ ω A ” B ” CE ] (23) = ¯ N A ′ B ′ → E ( Ω A ” A ′ ⊗ ∆ B ” B ′ ) (24) = T ⊗ I A ” B ” ( Φ ◦ AA ” ⊗ Φ ◦ BB ” ) (25)Let W A → ◦ A and W B → ◦ B be two isometries. Then the tensor product decoupling theorem thenimplies: Z (cid:13)(cid:13)(cid:13)(cid:13) T (( U ◦ A RAND W ⊗ V ◦ B RAND W ) · ( ψ AR ⊗ ϕ BR )) − ω E ⊗ ψ R ⊗ ϕ R (cid:13)(cid:13)(cid:13)(cid:13) dUdV ≤ q δ + − ˜ H δ ( A ” B ” | E ) ω − ˜ H δ ( A | R ) ψ − ˜ H δ ( B | R ) ϕ + − ˜ H δ ( B ” | E ) ω − ˜ H δ ( B | R ) ϕ + − ˜ H δ ( A ” | E ) ω − ˜ H δ ( A | R ) ψ = δ Now we show the existence of two isometries U ALICE and V BOB which approximately emulate theaction of the operators p | A ” | op ◦ A → A ′ ( Ω ) U ◦ A RAND W and p | B ” | op ◦ B → B ′ ( ∆ ) V ◦ B RAND W . To that enddefine the maps : 21. E ◦ A → G ( ρ ) : = | A ” | Tr [ op ◦ A → A ′ ( Ω ) · ρ ] F ◦ B → G ′ ( ρ ) : = | B ” | Tr [ op ◦ B → B ′ ( ∆ ) · ρ ] where G and G ′ are one dimensional systems. Then, using the vanilla (non smooth) decouplingtheorem twice we get1. Z (cid:13)(cid:13)(cid:13) I C R ⊗ E ( U RAND W · ψ AC R ) − ψ C R (cid:13)(cid:13)(cid:13) dU ≤ H max ( A ) ψ − ˜ H δ ( A ” ) ω = δ Z (cid:13)(cid:13)(cid:13) I C R ⊗ F ( V RAND W · ϕ BC R ) − ϕ C R (cid:13)(cid:13)(cid:13) dV ≤ H max ( B ) ϕ − ˜ H δ ( B ” ) ω = δ where we have used the facts that E ( Φ ◦ AA ” ) = ω A ” and F ( Φ ◦ BB ” ) = ω B ” .Consider the random variables defined as follows:1. X : = (cid:13)(cid:13)(cid:13) T (( U ◦ A RAND W ⊗ V ◦ B RAND W ) · ( ψ AR ⊗ ϕ BR )) − ω E ⊗ ψ R ⊗ ϕ R (cid:13)(cid:13)(cid:13) Y : = (cid:13)(cid:13) I R ⊗ E ( U RAND W · ψ AC R ) − ψ C R (cid:13)(cid:13) Z : = (cid:13)(cid:13) I R ⊗ F ( V RAND W · ϕ BC R ) − ϕ C R (cid:13)(cid:13) and the following events:1. E : = { X ≥ δ } E : = { Y ≥ δ } E : = { Z ≥ δ } Now by Markov’s inequality (for instancePr [ E ] ≤ (cid:2) R k I C R ⊗E ( U RAND W · ψ AC R ) − ψ C R k dU δ (cid:3) ≤ ) and union bound for events E , E , E we getPr [ E ∩ E ∩ E ] > U ◦ A RAND and V ◦ B RAND which satisfy the event E ∩ E ∩ E . Fix such a pair of unitaries. Then, from Uhlmann’s theorem we see that Fact 2.11,there exist isometries U A → A ′ ALICE and V B → B ′ BOB such that: (cid:13)(cid:13)(cid:13)(cid:13) | A ” | op ◦ A → A ′ ( Ω ) U ◦ A FIXED W · ψ AC R − U A → A ′ ALICE · ψ AC R (cid:13)(cid:13)(cid:13)(cid:13) ≤ p δ (26) (cid:13)(cid:13)(cid:13)(cid:13) | B ” | op ◦ B → B ′ ( ∆ ) V ◦ B FIXED W · ϕ BC R − V B → B ′ BOB · ϕ BC R (cid:13)(cid:13)(cid:13)(cid:13) ≤ p δ (27)Define Tr [ | B ” | op ◦ B → B ′ ( ∆ ) V ◦ B FIXED W · ψ BC R ] : = c . Since trace is a quantum operation, from theequations above we see that | c − | ≤ p δ (cid:13)(cid:13)(cid:13)(cid:13) | A ” B ” | ( op ◦ A → A ′ ( Ω ) U ◦ A FIXED W ⊗ op ◦ B → B ′ ( ∆ ) V ◦ B FIXED W ) · ( ψ AC R ⊗ ϕ BC R ) − U ALICE ⊗ V
BOB · ψ AC R ⊗ ϕ BC R (cid:13)(cid:13)(cid:13)(cid:13) (28) ≤ (cid:13)(cid:13)(cid:13)(cid:13) | B ” | op ◦ B → B ′ ( ∆ ) V ◦ B FIXED W · ψ BC R (cid:13)(cid:13)(cid:13)(cid:13) × (cid:13)(cid:13)(cid:13)(cid:13) | A ” | op ◦ A → A ′ ( Ω ) U ◦ A FIXED W · ψ AC R − U A → A ′ ALICE · ψ AC R (cid:13)(cid:13)(cid:13)(cid:13) (29) + (cid:13)(cid:13)(cid:13) U A → A ′ ALICE · ψ AC R (cid:13)(cid:13)(cid:13) × (cid:13)(cid:13)(cid:13)(cid:13) | B ” | op ◦ B → B ′ ( ∆ ) V ◦ B FIXED W · ϕ BC R − V B → B ′ BOB · ϕ BC R (cid:13)(cid:13)(cid:13)(cid:13) (30) ≤ ( + p δ ) × ( p δ ) + p δ (31)where we bound c by ( + √ δ ) . Finally, we use the triangle inequality and the monotonicity of1-norm under a quantum operation (which is partial trace over C C followed by ¯ N ) to obtain: (cid:13)(cid:13)(cid:13) ¯ N ( U ALICE ⊗ V
BOB · ψ AR ⊗ ϕ BR ) − ω E ⊗ ψ R ⊗ ϕ R (cid:13)(cid:13)(cid:13) ≤ δ + p δ + p δ + p δ δ (32) = δ (33)We conclude by invoking Uhlmann’s theorem Fact 2.11 again for the last inequality to prove theexists a decoder D CC C → F ˆ A ˆ BC C such that: (cid:13)(cid:13)(cid:13) D U ¯ N ( U ALICE ⊗ V
BOB · ψ AC R ⊗ ϕ BC R ) − λ FE ⊗ ψ AC R ⊗ ϕ BC R (cid:13)(cid:13)(cid:13) ≤ p δ where λ FE is some purification of ω E . (cid:3) We are now ready to state our coding theorem.The task is to use the QMAC to send arbitrary states, tensored across the registers belonging toAlice and Bob, with high fidelity to Charlie. It was shown in [KW03] that this task is equivalentto sending one half of two maximally entangled states (one belonging to Alice and one to Bob)across the channel. For a more general setting where there may be entanglement assistance, wereformulate the problem in the language of Theorem 5.1:We set the states ψ AC R and ϕ BC R as Φ R M ⊗ Φ ˜ AC and Φ R M ⊗ Φ ˜ BC respectively. Here theregisters M ˜ A and M ˜ B play the roles of A and B . For a given ǫ > ( Q A , E A , Q B , E B ) is ǫ -achievable if there exist encoding isometries U A LICE , V B OB and decodingCPTP D , with | M | = Q A , | ˜ A | = E A , | M | = Q B and | ˜ B | = E B , such that (cid:13)(cid:13)(cid:13) D ◦ N ( U ALICE ⊗ V
BOB · ψ AC R ⊗ ϕ BC R ) − ψ AC R ⊗ ϕ BC R (cid:13)(cid:13)(cid:13) ≤ ǫ The rate pair ( Q A , Q B ) is achievable for entangled assisted transmission if there exist E A , E B ≥ ( Q A , Q B , E A , E B ) is ǫ -achievable. The pair ( Q A , Q B ) is achievable for unassisted trans-mission of ( Q A , Q B , 0, 0 ) is ǫ -achievable. The one shot capacity region is the union of all achievablepoints ( Q A , Q B ) for a fixed ǫ , over all controlling states ω as defined in Theorem 5.1. Theorem 5.2
Given a quantum multiple access channel N A ′ B ′ → C and fixed δ > , define ǫ : = δ where δ is as defined in Theorem 5.1. Let Ω A ′ A ” and ∆ B ′ B ” be pure states and ω A ” B ” CE : = U N ( Ω ⊗ ∆ ) . Then the ate quadruple ( Q A , E A , Q B , E B ) is ǫ -achievable for quantum transmission with rate limited entanglementassistance through N if Q A − E A + Q B − E B < ˜ H δ ( A ” B ” | E ) ω + ( − δ ) Q A − E A < ˜ H δ ( A ” | E ) ω + log ( − δ ) Q B − E B < ˜ H δ ( B ” | E ) ω + log ( − δ ) Q A + E A < ˜ H δ ( A ” ) ω Q B + E B < ˜ H δ ( B ” ) ω Proof:
The proof is essentially an application of Theorem 5.1. First, set the states ψ ˜ AM C R = Φ R M ⊗ Φ ˜ AC and ϕ ˜ BM C R = Φ R M ⊗ Φ ˜ BC where the registers ˜ AM and ˜ BM are placeholdersfor the registers A and B in Theorem 5.1. Then, invoking Theorem 5.1 for the channel N withcontrolling state ω , we see that there exist encoding isometries U A LICE , V BOB and decoding CPTP D such that (cid:13)(cid:13)(cid:13) D ◦ N ( U ALICE ⊗ V
BOB · ψ ˜ AM C R ⊗ ϕ ˜ BM C R ) − ψ ˜ AM C R ⊗ ϕ ˜ BM C R (cid:13)(cid:13)(cid:13) ≤ ǫ , where ǫ = δ and δ + − ˜ H δ ( A ” B ” | E ) ω − ˜ H δ ( ˜ AM | R ) ψ − ˜ H δ ( ˜ BM | R ) ϕ + − ˜ H δ ( B ” | E ) ω − ˜ H δ ( ˜ BM | R ) ϕ + − ˜ H δ ( A ” | E ) ω − ˜ H δ ( ˜ AM | R ) ψ = δ H max ( ˜ AM ) ψ − ˜ H δ ( A ” ) ω = δ H max ( B ) ϕ − ˜ H δ ( B ” ) ω = δ δ + p δ + p δ + p δ δ = δ Observe that ˜ H δ ( ˜ AM | R ) ψ & ≥ − Q A + E A + log ( − δ ) ˜ H δ ( ˜ BM | R ) ϕ ≥ − Q B + E A + log ( − δ ) H max ( ˜ AM ) ψ ≤ Q A + E A H max ( ˜ BM ) ϕ ≤ Q B + E B Plugging in these estimates in the expressions for δ , δ , δ such that δ = O ( √ δ ) we conclude thatthe statement of the theorem is true. (cid:3) In this paper, we have proven a decoupling theorem which involves multiple random unitaries, intensor product with each other, chosen independently from the Haar measure as the decouplingunitary and the QMAC channel as the superoperator that results in decoupling the reference sys-tem of the input states and the channel environment, when expectation is taken over these uni-taries in tensor product. The unitaries in tensor product can be thought of as independent en-coders and the decoupling is achieved as a decoding step. The analysis of the error rate in ourdecoupling theorem leads to the characterization of an achievable rate region. We then proceed24o evaluate the asymptotic iid limit of our rate region. However, we cannot recover the asymp-totic iid rate region of Yard et al. in [YDH05]. The reason being an immediate open problem, thatis to find an optimising state that simultaneously smoothes the three different conditional Rényi2-entropies mentioned in Theorem 5.2.
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A Appendix
We present inner bounds for the
QMAC for the task of entanglement generation.The following is an easy corollary of Theorem 4.2 :
Corollary A.1
Given orthogonal projectors Π A → E and Π A → E , define the map T A A → E E : = | A || E | · | A || E | · Π A → E ⊗ Π A → E hen given the density operator ρ A A R the following holds: Z (cid:13)(cid:13)(cid:13) T ( U A RAND ⊗ U A RAND · ρ A A R ) − π E ⊗ π E ⊗ ρ R (cid:13)(cid:13)(cid:13) dU A dU A ≤ (cid:16) δ + · | E | − ˜ H δ ( A | R ) ρ + | E | − ˜ H δ ( A | R ) ρ + | E E | − ˜ H δ ( A A | R ) ρ (cid:17) Proof:
First, observe that Tr A ′ A ′ T ( Φ A A ′ ⊗ Φ A A ′ ) = π E ⊗ π E Then, define ˜ ω A ′ A ′ E E : = σ E − ⊗ σ E − · T ( Φ A A ′ ⊗ Φ A A ′ ) where σ E : = π E and σ E : = π E . Then note that:1. ˜ ω E = I E √ | E | and ˜ ω E = I E √ | E |
2. ˜ ω A ′ E E = ˜ ω A ′ E ⊗ ˜ ω E and ˜ ω A ′ E E = ˜ ω A ′ E ⊗ ˜ ω E .It is easy to see that Tr [ ˜ ω A ′ E ] = | E | sinceTr [ ˜ ω A ′ E ] = (cid:16)q | E | · | A || E | (cid:17) · Tr [ Π | Φ i A A ′ h Φ | Π | Φ i A A ′ h Φ | Π ] (34) = | A | | E | (cid:16) h Φ | Π | Φ i (cid:17) (35) = | E | (cid:16) Tr [ Π ] (cid:17) (36) = | E | (37)Similarly one can show that Tr [ ˜ ω A ′ E ] = | E | . We conclude by noting that (cid:13)(cid:13)(cid:13) ˜ ω A ′ E E (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) ˜ ω A ′ E (cid:13)(cid:13)(cid:13) · (cid:13)(cid:13) ˜ ω E (cid:13)(cid:13) = | E | and similarly (cid:13)(cid:13)(cid:13) ˜ ω A ′ E E (cid:13)(cid:13)(cid:13) = | E | . This completes the proof. (cid:3) To get a channel coding theorem we will use Theorem A.1, but with some overloading of nota-tion. Consider the quantum multiple access channel N A ′ B ′ → C with isometric extension U A ′ B ′ → CE where we use the register E to mean the environment. Consider the controlling state | ω i ABCE : = U A ′ B ′ → CE | Ω i AA ′ | ∆ i BB ′ , where | Ω i AA ′ and | ∆ i BB ′ are arbitrary pure states. Then the followingtheorem holds: Theorem A.2
Given δ as in Theorem 4.4 and a positive ǫ , the rates ( m ALICE , n BOB ) for entanglement gen-eration over the channel N A ′ B ′ → E are achievable wheneverm ALICE < ˜ H δ ( A | E ) ω + log ǫ n BOB < ˜ H δ ( B | E ) ω + log ǫ m ALICE + n BOB < ˜ H δ ( AB | E ) ω + log ǫ with error √ δ + ǫ . Proof:
We simply relabel terms from Theorem A.1:27. Registers: A ← A , B ← B and E ← R .2. Registers: R ← E and R ← E .3. State: ω ABE ← ρ A A R .4. | R | = m ALICE and | R | = n BOB .Define T AB → R R : = | A || R | · | B || R | · Π A → R ⊗ Π B → R Then applying Theorem A.1 we see that: Z (cid:13)(cid:13)(cid:13) T ( U A RAND ⊗ U B RAND · σ ABE ) − π R ⊗ π R ⊗ ρ E (cid:13)(cid:13)(cid:13) dU A dU B ≤ (cid:16) δ + · | R | − ˜ H δ ( A | E ) + | R | − ˜ H δ ( B | E ) + | R R | δ − ˜ H ( AB | E ) (cid:17) Finally, the above equation implies that there exist fixed unitaries U A FIXED and U B FIXED such that theabove inequality still holds. To conclude, by using the usual argument of applying Uhlmann’stheorem to the purifying register C and requiring that every term inside the curly braces be < ǫ we conclude the proof. (cid:3) Remark A.3
We can emulate the T operation in the usual way by picking unitaries independently fromusing two -designs instead of two Haar random unitaries. This reduces the required amount of sharedrandomness necessary to implement the T operation from infinite to the log of product of the cardinalityof the designs. Classical communication is necessary however so that Alice and Bob can let Charlie knowwhich code they are using.operation from infinite to the log of product of the cardinalityof the designs. Classical communication is necessary however so that Alice and Bob can let Charlie knowwhich code they are using.